| {"record_id":"research_section_01","section_index":1,"title_latex":"The concrete contribution and its physical scope","text_latex":"\\section{The concrete contribution and its physical scope}\nThe central proposed mechanism is a controller that \\emph{writes geometric\nmemory while using that memory to choose its next motion}. Avenyra supplies the\nexact finite rule. The extension here makes its operational timing, noise\nresolution, energy exchange, and measurable readout explicit.\n\nThe strongest structural theorem is a complete observable classification:\n\\[\n\\boxed{\\begin{gathered}\n\\Delta\\,[x^T\\phi(-A^2)x]\\ge0\\text{ for every legal finite stroke}\\\\\n\\iff\\phi(0)=0,\\ \\phi|_{(0,\\infty)}\\ge0\\text{ and nondecreasing}.\n\\end{gathered}}\n\\]\nThis is complete within the quadratic scalar spectral class, not a complete\nreachability order or a classification of all Lyapunov functions.\nSection~\\ref{sec:spectral-cone} derives the exact endpoint measure transport,\nproves necessity and sufficiency, and separates full cone order from pointwise\nridge order. The rank-continuous ridge member\n$F_\\lambda=x^T[-A^2(\\lambda^2I-A^2)^{-1}]x$ retains its gap-free error budget:\nif $\\norm{Aa}\\le\\epsilon\\norm a$, its decrease is at most\n$(\\epsilon/\\lambda)(\\norm x^2+\\norm{x+a}^2)$.\nSection~\\ref{sec:ridge} gives its thermal readout; exact samples at any $2m$\npositive resolutions recover the visible $m$-mode measure and hence every\ncone certificate, under the exact-data assumptions of\nSection~\\ref{sec:finite-tomography}.\n\nThe gate also has a precise conditional selection principle. A bilinear\nskew write equivariant under every orthogonal frame change must be\n$\\gamma x\\wedge a$. For nonzero $\\gamma$, requiring the squared memory norm\nnever to fall for any endpoint $x$, while permission depends only on $A,a$,\nallows precisely $a\\in\\ker A$ (Section~\\ref{sec:intrinsicgeometry}).\nThe isotropy, bilinearity and robust memory-norm requirement are declared\ndesign axioms; observed physical matter must justify them independently.\n\nThe strongest combined readout compares two monitoring resolutions. In a seeded\nthree-dimensional sample-held controller,\n\\begin{equation}\\label{eq:core}\n\\boxed{\\begin{aligned}\n\\dd U&=2\\nu f(c)R\\,\\dd t,\\\\\n\\dd R&=2\\nu f(c)\\frac{RQ^2}{U}\\,\\dd t,\\\\\n\\dd Q&=\\sqrt{2\\nu}\\,\\dd W-\\nu f(c)\\frac{RQ}{U}\\,\\dd t,\n\\end{aligned}\\qquad f(c)=1-\\frac{1-e^{-c}}c.}\n\\end{equation}\nAn independent passive readout has\n\\begin{equation}\\label{eq:core-survival}\n\\boxed{S_{\\rm resolved}=U_0/U,\n\\qquad S_{\\rm diffusion\\ first}=(U_0/U)^{1/f}\\quad(f>0).}\n\\end{equation}\nThe first applies complete projection after each finite stroke. The second takes\nthe controller diffusion limit at finite loss and then increases the loss rate.\nSection~\\ref{sec:continuous-dark} proves the limit distinction. This is ordinary\npassive quantum mechanics tied to a particular classical geometric controller;\nclassical waves obey the same amplitude law. Coherent transport can avoid the\nattenuation. Known noisy optical physics is explicitly credited below.\n\nVersion 3 fills the positive joint-scaling interval between these sequential\nlimits. With $\\ell_h=\\gamma_h h\\to\\ell\\in(0,\\infty]$, define\n\\[\n\\boxed{S_\\ell(U)=\\exp\\!\\left[-\\frac1{f(c)}\\int_{U_0}^{U}\nF_c(e^{-\\ell u/2})\\frac{\\dd u}{u}\\right],\\quad\nF_c(q)=1-\\frac{(1-e^{-c})(1-q)}{c(1-e^{-c}q)}.}\n\\]\nFor $\\ell=\\infty$ the result is $U_0/U$. Its positive-$\\ell$ limit as\n$\\ell\\downarrow0$ is $(U_0/U)^{1/f}$; arbitrary joint paths with\n$\\ell_h\\to0$ are not covered. Section~\\ref{sec:joint-loss} proves the\nstate-dependent averaging and a localized\n$O_{\\mathbb P}(\\sqrt{h\\log(1/h)})$ log-error bound. In particular, the\nsurvival curve can depend on microscopic lag structure that a controller's\ncoarse diffusion alone does not identify.\n\nThe physical proposal has three layers which must remain distinct:\n\\begin{enumerate}\n\\item A classical history register $(x,A)$ and a specified powered controller.\n\\item A controlled mechanical, electrical, magnetic, or optical plant whose\ncouplings are set by that register.\n\\item An independent probe whose fluctuations, heat, or survival provide an\nobservable test.\n\\end{enumerate}\nNeither the probe displacement nor an unknown quantum state is silently\nidentified with the noiseless controller coordinate $x$. Such an identification\nwould require additional dynamics and invalidate the present proofs.\n\n\\subsection{What is inherited and what is added}\nAvenyra v1.0.0 \\cite{avenyra} proves the pre-gated reachability theorem, kernel\ntransport, rank-growth schedules, exterior-volume monotonicity, and a spectral\nrank-one reduction. Those results are inputs here. The additions are the\ntiming-family classification, complete continuous isospectrality, sharp\nfull-rank impulse threshold, seeded stochastic closure and colored-noise limit,\nand the explicit thermal and quantum readout laws. Version 2 adds the resolved\nexclusion family, finite-error certificates, continuous-loss theorem, selected\nentrance, arbitrary-rank intrinsic closure and finite-rate construction.\n\nVersion 3 adds the complete quadratic spectral cone, exact finite visible\ntomography, positive joint loss-scaling theorem, intrinsic geometry with a\nshort-history spatial error, an axiomatic gate selection, a conditional Jacobi\nobstruction and a finite-resource thermal writing certificate. These extend\nthe specified controller without identifying geometric area with physical\nenergy, spacetime curvature or a universal entropy.\n\nThe prior-art search found established ancestry in signatures, nonholonomic\nintegration, stochastic feedback, irreversible metamaterials, and reservoir\nengineering. Its outcome is not a proof of first priority. The manuscript does\nnot rely on an unverified proof of another open problem or on unseen manuscripts.\n\n","labels":["eq:core","eq:core-survival"],"source_line_start":90,"source_line_end":194,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_02","section_index":2,"title_latex":"State, units, and the original gate","text_latex":"\\section{State, units, and the original gate}\nLet $V=\\R^d$ have its fixed Euclidean metric and anchor. The controller state is\n$x\\in V$ and a real skew matrix $A^T=-A$. For vectors $x,a$, define\n\\[\nx\\wedge a=xa^T-ax^T,\n\\qquad \\area=\\norm{x\\wedge a}_{\\Lambda^2}^2\n=\\norm{x}^2\\norm a^2-\\inner xa^2.\n\\]\nThe bivector norm is half the squared Frobenius norm of the corresponding skew\nmatrix. Write $E(A)=\\norm A_F^2/2$. Original Avenyra is\n\\begin{equation}\\label{eq:pre}\nAa=0,\\qquad x'=x+a,\\qquad A'=A+x\\wedge a.\n\\end{equation}\nAll controller formulas in this manuscript use dimensionless registers:\n$x=x_{\\rm physical}/L_*$ and $A=A_{\\rm physical}/L_*^2$ for a fixed calibrated\nlength $L_*$. Thus $\\nu$ has inverse-time units and the dark-mode coefficient\n$\\gamma$ has inverse-time units. A dimensional path convention is equally\npossible: then $A$ has length-squared units, $U$ has length-to-the-fourth units,\nand the coupling coefficients must be rescaled. The independent thermal probe\n$q$ retains physical length units. Neither $E$ nor $U$ is energy without an\nadditional physical Hamiltonian or hardware calibration.\n\nWith $K=\\ker A$, $H=\\im A=K^\\perp$, decompose $x=u+w$, where $u\\in H,w\\in K$.\nThe original exclusion radius is\n\\[\n\\rho(x,A)=\\norm{P_Hx}.\n\\]\nThe inherited global theorem is\n\\[\n\\min_{y\\in\\operatorname{Reach}(x,A)}\\norm y=\\rho(x,A),\n\\qquad\\operatorname*{argmin}\\norm y=\\{u\\}.\n\\]\nThe legal displacement $a=-w$ reaches $u$ in one step. The theorem concerns\nfinite legal words, not every closure point of the reachable set.\n\nThe memory is anchor dependent: translating $x$ while retaining the same $A$\nchanges the next area update. A physical implementation needs a named origin or\na relative coordinate $x=p-o$; simultaneous translation of $p,o$ is harmless.\nThe present model has no spacetime metric or relativistic covariance premise.\n\n","labels":["eq:pre"],"source_line_start":194,"source_line_end":234,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_03","section_index":3,"title_latex":"Gate timing controls spectrum, rank, and reversal","text_latex":"\\section{Gate timing controls spectrum, rank, and reversal}\nHold the algebraic update fixed and vary the gate:\n\\begin{equation}\\label{eq:theta}\nB=x\\wedge a,\\qquad (A+\\theta B)a=0,\\qquad A'=A+B,\n\\qquad x'=x+a.\n\\end{equation}\nThe values $\\theta=0,1/2,1$ test the pre-update, midpoint, and post-update tensor.\nThese are different legal transition relations. A straight stroke with\ncontinuous area interpolation has $A(s)=A+sB$, so the parameter has an\noperational interpretation as the location of a permission sample. Midpoint\nor post-gated pulses must be computed prospectively or solved implicitly; they\nare not obtained by accidentally using a delayed pre-gate sensor.\n\n\\begin{theorem}[Exact timing budget and adjoint gate]\\label{thm:budget}\nEvery $\\theta$-legal transition satisfies\n\\begin{equation}\\label{eq:timing-budget}\nE(A')-E(A)=(1-2\\theta)\\area.\n\\end{equation}\nThe inverse algebraic transition is legal under the $(1-\\theta)$-gate. For an\narea-writing transition it is legal under the same gate exactly when\n$\\theta=1/2$.\n\\end{theorem}\n\\begin{proof}\nFor arbitrary $a$, expansion gives\n\\[\nE(A+B)-E(A)=\\area+2x^TAa.\n\\]\nThe gate makes $Aa=-\\theta Ba$ and $x^TBa=\\area$. For reversal use $-a$ from\n$x+a$; its area increment is $(x+a)\\wedge(-a)=-B$. The reversed gate becomes\n$-(A+\\theta B)a=0$. Testing it with the original $\\theta$ instead gives\n$-(1-2\\theta)Ba=0$. For $a\\ne0$, $Ba=0$ implies $x\\wedge a=0$.\n\\end{proof}\n\n\\begin{theorem}[Congruence and midpoint isospectrality]\\label{thm:congruence}\nFor $a\\ne0,\\theta\\ne0$, let $P=aa^T/\\norm a^2$ and\n$T_\\theta=I-P/\\theta$. Then\n\\begin{equation}\\label{eq:congruence}\nA'=T_\\theta A T_\\theta.\n\\end{equation}\nAt $\\theta=1/2$, $T_\\theta=I-2P$ is a Householder reflection, hence the full\ncharacteristic polynomial and all singular values are preserved.\n\\end{theorem}\n\\begin{proof}\nSkewness gives $PAP=0$ and\n\\[\nPA+AP=\\frac{Aa\\wedge a}{\\norm a^2}.\n\\]\nThe gate gives $Aa=-\\theta\\norm a^2P_{a^\\perp}x$, so\n$B=-(PA+AP)/\\theta$. Expansion of the congruence proves the identity. At the\nmidpoint $T_\\theta^T=T_\\theta=T_\\theta^{-1}$, making it an orthogonal conjugation.\n\\end{proof}\n\n\\begin{corollary}[Spectrum and rank classification]\nFor $\\theta<1/2$, all ordered singular values are nondecreasing; for\n$\\theta>1/2$, all are nonincreasing. At the midpoint they are constant.\nRank is constant for $\\theta\\notin\\{0,1\\}$. The pre-gate can create two ranks;\nthe post-gate can delete two ranks. A history starting at $A=0$ cannot write\nnonzero area using only gates with $\\theta\\ne0$.\n\\end{corollary}\n\\begin{proof}\nFor nonzero $\\theta<1/2$, $T_\\theta$ is expansive, with least singular value one;\nfor $\\theta>1/2$ it is contractive, with norm one. The standard singular-value\nbounds for left and right multiplication apply to \\eqref{eq:congruence}.\nThe zero-gate case uses the inherited positive rank-one spectral reduction.\nFor $\\theta\\notin\\{0,1\\}$ the congruence is invertible. At $\\theta=1$ it is a\nprojection congruence; rank cannot increase and its decrease is at most two,\nbecause $\\rank B\\le2$ and skew ranks are even. If $A=0,\\theta\\ne0$, the gate\ngives $Ba=0$, hence $B=0$.\n\\end{proof}\n\nThe rank-birth endpoint is therefore singular: an arbitrarily small but nonzero\n$\\theta$ cannot nucleate rank from precisely zero under exact legality. This\nstatement concerns exact rank, not the robustness of a numerical threshold.\n\n\\subsection{An orientation observable beyond the singular values}\nIn dimension $d=2m$, Pfaffian congruence gives\n\\begin{equation}\\label{eq:pfaffian}\n\\Pf(A')=\\left(1-\\frac1\\theta\\right)\\Pf(A),\\qquad\\theta\\ne0.\n\\end{equation}\nThus a nonzero midpoint pulse from full-rank memory reverses the Pfaffian sign\nwhile preserving all positive skew frequencies. No continuous full-rank\nisospectral path in the same oriented frame can do this: the continuous\nnonzero Pfaffian would have to cross zero to change sign. A real plant must\ntherefore execute a finite hybrid update, use an auxiliary space, or pass through\nan internal rank singularity while the signal is held or decoupled. The\nendpoint theorem does not prescribe a continuously protected interpolation.\n\n\\subsection{The exact full-rank impulse threshold}\nOriginal pre-gating freezes invertible $A$. Other timing gates allow a jump,\nbut not an arbitrarily small one.\n\\begin{theorem}[Sharp jump threshold]\\label{thm:threshold}\nIf $A$ is invertible, $x\\ne0$, and $\\theta\\ne0$, the smallest norm of a nonzero\n$\\theta$-legal displacement is\n\\begin{equation}\\label{eq:threshold}\n\\min\\norm a=\\frac1{|\\theta|\\norm{A^{-1}x}}.\n\\end{equation}\nIt is attained by $a=-A^{-1}x/(\\theta\\norm{A^{-1}x}^2)$, which is perpendicular\nto $x$.\n\\end{theorem}\n\\begin{proof}\nFor a nonzero legal $a$, put $\\lambda=\\theta\\inner xa$. Rearranging the gate\ngives $(\\lambda I-A)a=\\theta\\norm a^2x$. Since invertible skew $A$ has no real\neigenvalue, $v=(\\lambda I-A)^{-1}x$ exists for all real $\\lambda$ and\n$a=v/(\\theta\\norm v^2)$. Conversely every such resolvent vector gives a legal\ncontrol, since $\\inner xv=\\lambda\\norm v^2$.\nIn skew normal form, with positive frequencies $\\sigma_i$ and plane projections\n$x_i$, $\\norm v^2=\\sum_i\\norm{x_i}^2/(\\lambda^2+\\sigma_i^2)$.\nIts maximum occurs at $\\lambda=0$, where $v=-A^{-1}x$. Since\n$\\norm a=1/(|\\theta|\\norm v)$, this proves the minimum. Skewness of $A^{-1}$\ngives the perpendicularity.\n\\end{proof}\nIf $x=0$ and $A$ is invertible there is no nonzero legal control, for any\n$\\theta$. The threshold theorem excludes that case explicitly.\n\n","labels":["eq:theta","thm:budget","eq:timing-budget","thm:congruence","eq:congruence","eq:pfaffian","thm:threshold","eq:threshold"],"source_line_start":234,"source_line_end":348,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_04","section_index":4,"title_latex":"Continuous motion and the origin of the operational arrow","text_latex":"\\section{Continuous motion and the origin of the operational arrow}\n\\begin{theorem}[Complete continuous isospectrality]\\label{thm:continuous}\nSuppose $x,A$ are absolutely continuous on a compact interval and almost\neverywhere $\\dot x=v$, $\\dot A=x\\wedge v$, $Av=0$. Then the entire characteristic\npolynomial of $A$ is constant. In particular its rank cannot change.\n\\end{theorem}\n\\begin{proof}\nFor every $m\\ge1$, the polynomial chain rule gives\n\\[\n\\frac{\\dd}{\\dd t}\\tr A^{2m}\n=2m\\tr[A^{2m-1}(xv^T-vx^T)]=0,\n\\]\nbecause $A^{2m-1}v=0$ and $v^TA^{2m-1}=0$. Odd traces of real skew matrices\nvanish. Newton identities determine the characteristic polynomial from\nfinitely many power traces.\n\\end{proof}\nThis extends the original zero-memory continuous-collapse result to every\ninitial spectrum. Finite legal jumps are not merely an interchangeable\nparameterization of the continuous rule.\n\n\\subsection{The screenshot's fundamental-arrow claim}\nThe supplied image suggests that reversible algebra plus feedback spontaneously\nproduces a fundamental arrow. The unrestricted area update is invertible, but\nthe \\emph{complete pre-gated transition relation} is already asymmetric under\nreversal. The inverse of a writing step is rejected at its new tensor. The\nmidpoint comparison isolates this asymmetry algebraically.\n\nMore precisely, any time-reversal involution $\\Theta$ that preserves $E$ cannot\nreverse a writing step under the same pre-gate. If it did, monotonicity would\nimply $E(s)\\ge E(s')>E(s)$. Orthogonal changes of frame and sign changes of $x,A$\npreserve $E$, so the usual candidate reversors are covered. Larger reversible\napparatuses retaining a clock and control record are not excluded: they can\nreverse the unrestricted shear while using the opposite gate phase.\n\nAn exact nonwriting stationary probability measure is possible; a stationary\nmeasure with positive probability of writing is not. To see this without any\nmoment assumption, apply stationarity to the bounded increasing function\n$E/(1+E)$. Its one-step increment is nonnegative and strictly positive on every\nwriting transition, so its stationary expected increment cannot be zero unless\nwriting has zero probability. A stationary nonzero writing regime needs reset,\nleakage, or a larger controlled state. A positive long-run writing rate forces\nunbounded memory; nonstationary summable writing is not excluded.\n\nA finite-dimensional closed unitary has a related elementary obstruction. If\n$U^\\dagger F U-F\\succeq0$ for a Hermitian observable $F$, its trace is zero, so\nthe difference is zero. Universal strict increase cannot be achieved by that\nclosed unitary. Open channels and external control avoid those hypotheses.\nThese no-go statements set the scope; they do not prohibit controlled geometric\nmemory devices or establish a universal energy lower bound.\n\n","labels":["thm:continuous"],"source_line_start":348,"source_line_end":398,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_05","section_index":5,"title_latex":"A seeded three-dimensional noisy controller","text_latex":"\\section{A seeded three-dimensional noisy controller}\nIn three dimensions use $Av=-b\\times v$. Then the update is\n\\[\nb'=b+x\\times a.\n\\]\nAssume $U=\\norm b^2>0$ and define $n=b/\\sqrt U$. A legal pulse has $a=s n$.\nSet\n\\[\nQ=x\\cdot n,\\qquad R=\\norm x^2-Q^2=\\rho(x,A)^2.\n\\]\nHere $U$ is the squared area-memory strength, not thermodynamic energy, and\n$R$ is the squared exclusion radius.\n\n\\begin{theorem}[Exact scalar pulse closure]\\label{thm:closure}\nEvery pre-gated scalar pulse obeys\n\\begin{equation}\\label{eq:closure}\n\\boxed{U'=U+s^2R,\\quad Q'=\\sqrt{\\frac U{U+s^2R}}(Q+s),\\quad\nR'=R+\\frac{s^2R(Q+s)^2}{U+s^2R}.}\n\\end{equation}\nBoth $U$ and $R$ are nondecreasing for every realized pulse, including negative\nones. The formulas depend only on $U,R,Q,s$, not on the orientation of $b$.\n\\end{theorem}\n\\begin{proof}\n$b\\perp x\\times n$ gives the first identity. Triple products give\n$x'\\cdot b'=\\sqrt U(Q+s)$, and division by $\\sqrt{U'}$ gives the second.\nFinally $\\norm{x'}^2=R+(Q+s)^2$; subtract $(Q')^2$.\n\\end{proof}\n\n\\subsection{Independent Gaussian strokes}\nLet $s_j=\\sqrt{2\\nu h}\\,\\xi_j$, with independent standard Gaussian $\\xi_j$.\nThe vector updates are Euler--Maruyama for\n\\begin{equation}\\label{eq:ito}\n\\dd x=\\sqrt{2\\nu}\\,n\\,\\dd W,\\qquad\n\\dd b=\\sqrt{2\\nu}(x\\times n)\\,\\dd W\\quad\\text{(It\\^o)}.\n\\end{equation}\nFor $U_0>0$ this has a global strong solution. The coefficients are smooth away\nfrom $b=0$ and have linear growth; the next identities keep $b$ away from zero.\nStandard localized Euler estimates and moment bounds justify the pulse limit.\n\n\\begin{theorem}[Pathwise stochastic locking]\\label{thm:locking}\nThe solution of \\eqref{eq:ito} satisfies\n\\begin{equation}\\label{eq:ito-closure}\n\\dd U=2\\nu R\\,\\dd t,\\qquad\n\\dd R=2\\nu\\frac{RQ^2}{U}\\,\\dd t,\\qquad\n\\dd Q=\\sqrt{2\\nu}\\,\\dd W-\\nu\\frac{RQ}{U}\\,\\dd t.\n\\end{equation}\nIf $R_0>0$, $U(t)\\ge U_0+2\\nu R_0t$ and $R(t)\\ge R_0$. If $R_0=0$, the\npath remains a Brownian line and writes no memory.\n\\end{theorem}\n\\begin{proof}\nIt\\^o's formula gives $\\dd U=2\\nu\\norm{x\\times n}^2\\dd t$; its martingale\nterm vanishes. Normalizing $b$ gives\n\\[\n\\dd n=\\frac{\\sqrt{2\\nu}}{\\sqrt U}(x\\times n)\\dd W-\\frac{\\nu R}{U}n\\dd t.\n\\]\nThe product rule for $Q=x\\cdot n$ has zero cross term and zero triple product,\ngiving the stated $Q$ equation. Also\n\\[\n\\dd\\norm x^2=2\\sqrt{2\\nu}Q\\dd W+2\\nu\\dd t,\n\\quad\n\\dd Q^2=2\\sqrt{2\\nu}Q\\dd W+(2\\nu-2\\nu RQ^2/U)\\dd t.\n\\]\nSubtract to obtain $\\dd R$. The inequalities follow by integration; at $R=0$\nthe cross product vanishes and $n$ is fixed.\n\\end{proof}\n\nThe controller's mean position remains $x_0$ in this independent-stroke limit,\nbecause $x$ is a martingale. Nevertheless two nonlinear geometric observables\ngrow on each path. This is an exact rectification of fluctuations by the\nspecified driven gate, not a rectification of a single equilibrium heat bath.\n\nSeveral additional identities make the prediction quantitative:\n\\begin{align}\n\\E\\norm{x(t)}^2&=\\norm{x_0}^2+2\\nu t,\\\\\n\\E U(t)&\\le U_0+2\\nu\\norm{x_0}^2t+2\\nu^2t^2,\\\\\n\\dd(x\\cdot b)&=\\sqrt{2\\nu U}\\,\\dd W,\\\\\n\\frac{\\dd^2U}{\\dd t^2}&=4\\nu^2RQ^2/U\\ge0.\n\\end{align}\nThe last derivative exists because $R$ has a continuous finite-variation\nderivative. Memory growth is pathwise convex. An explicit expression for radius\ngrowth is $R(t)=R_0\\exp(2\\nu\\int_0^t Q^2/U\\,\\dd r)$ when $R_0>0$.\n\nIn the independent-stroke Brownian limit, angular quadratic variation satisfies\n\\begin{equation}\\label{eq:angular}\n\\tr\\dd[n]=2\\nu R/U\\,\\dd t=\\dd\\log U.\n\\end{equation}\nThis exact logarithm will reappear in the ideal dark-mode readout. It must not be\ntransferred unchanged to arbitrary colored-noise limits.\n\n\\begin{figure}[htbp]\\centering\n\\includegraphics[width=\\linewidth]{results/stochastic_memory.pdf}\n\\caption{Seeded finite-pulse simulation: four sample trajectories and an\nensemble of 2,048 paths, 8,192 steps, $\\nu=0.3,T=2$. Both geometric observables\ngrow along the displayed paths. The lower-right curve averages the ideal\nfully relaxed projection protocol; it is simulation, not photon data.}\n\\end{figure}\n\n","labels":["thm:closure","eq:closure","eq:ito","thm:locking","eq:ito-closure","eq:angular"],"source_line_start":398,"source_line_end":495,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_06","section_index":6,"title_latex":"A colored-noise experiment with an exact crossover","text_latex":"\\section{A colored-noise experiment with an exact crossover}\nController timing is experimentally meaningful when noise has a correlation\ntime. Let $\\eta(t)$ be a stationary Ornstein--Uhlenbeck velocity with\n\\[\n\\E[\\eta(t)\\eta(r)]=(\\nu/\\tau)e^{-|t-r|/\\tau}.\n\\]\nDuring each interval of length $h$, hold the current direction $n_j$ fixed and\nexecute $s_j=\\int_{jh}^{(j+1)h}\\eta(t)\\dd t$. The path is straight within that\nstroke and the finite update is exactly \\eqref{eq:closure}.\n\n\\begin{proposition}[Stroke covariance budget]\\label{prop:ou}\nWith $c=h/\\tau$ fixed,\n\\begin{align}\n\\operatorname{Var}(s_j)&=2\\nu h f(c),\n&f(c)&=1-(1-e^{-c})/c,\\\\\n\\operatorname{Cov}(s_j,s_{j+k})&=\\nu\\tau(1-e^{-c})^2e^{-(k-1)c},&& k\\ge1.\n\\end{align}\nAs $h,\\tau\\to0$ at this ratio, the accumulated driver converges uniformly in\nprobability on bounded intervals to $\\sqrt{2\\nu}W$, while the accumulated\n\\emph{resolved stroke squares} converge uniformly in probability to\n$2\\nu f(c)t$.\n\\end{proposition}\n\\begin{proof}\nIntegrate the OU covariance over one or two sampling intervals. Summing the\ncovariances gives\n$\\operatorname{Var}(s_j)+2\\sum_{k\\ge1}\\operatorname{Cov}(s_j,s_{j+k})=2\\nu h$.\nThe Brownian variance is therefore $2\\nu$, rather than $2\\nu f(c)$.\nThe OU equation on a common Brownian coupling gives\n\\[\n\\int_0^t\\eta(r)\\dd r=\\sqrt{2\\nu}W(t)-\\tau[\\eta(t)-\\eta(0)].\n\\]\nThe second term tends uniformly to zero in probability; stationary Gaussian\nmaximal bounds give order $\\sqrt{\\tau\\log(1/\\tau)}$ on compact intervals.\nThe unobserved fraction of the final stroke also vanishes uniformly.\nGaussian fourth moments give, for $N h=t$,\n\\[\n\\operatorname{Var}\\!\\left(\\sum_{j<N}s_j^2\\right)\n=2\\left[Nv^2+2\\sum_{k=1}^{N-1}(N-k)c_k^2\\right]=O(h),\n\\]\nwhere $v=\\operatorname{Var}s_j$ and $c_k$ is its lag-$k$ covariance. Hence the\nsquare sum concentrates at $2\\nu f(c)t$. Monotonicity of its cumulative process\nand continuity of this deterministic limit upgrade fixed-time concentration\nto uniform convergence on compact intervals. Gaussian moments also give\n$\\E\\sum_{j<N}|s_j|^3=O(\\sqrt h)$.\n\\end{proof}\n\n\\Needspace{17\\baselineskip}\n\\begin{lemma}[Deterministic pulse-convergence lemma]\\label{lem:pulse-limit}\nFor pulse families, suppose the cumulative sums $Z_h$ converge uniformly to a\ncontinuous $Z$, the cumulative squares $V_h$ converge uniformly to $\\kappa t$,\nand $\\sum|s_j|^3\\to0$. With $U_0>0$, the exact scalar recurrences converge on\nevery bounded time interval to\n\\begin{align}\nU(t)&=U_0+\\kappa\\int_0^tR\\dd r,\\\\\nR(t)&=R_0+\\kappa\\int_0^tRQ^2/U\\dd r,\\\\\nQ(t)&=Q_0+Z(t)-(\\kappa/2)\\int_0^tRQ/U\\dd r.\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nOn any compact domain with $U\\ge U_0$, Taylor expansion of the exact recurrences\nhas uniform remainders:\n\\[\n\\Delta U=Rs^2,\\quad\n\\Delta R=(RQ^2/U)s^2+O(|s|^3),\\quad\n\\Delta Q=s-(RQ/(2U))s^2+O(|s|^3).\n\\]\nThe cumulative remainders vanish. Remove the additive input by writing\n$\\widetilde Q_h=Q_h-Z_h$. The remaining increments are Riemann--Stieltjes sums\nagainst the positive measures $\\dd V_h$. Their bounded total masses, the\nuniform convergence of $Z_h,V_h$, and compact-state Lipschitz coefficients\ngive equicontinuity and subsequential limits solving the displayed integral\nequations. Ordinary Gronwall uniqueness identifies all subsequential limits.\nFor completeness, the localization is removable deterministically. The exact\n$Q$ recursion is $Q_{j+1}=c_j(Q_j+s_j)$ with $0<c_j\\le1$. Expansion of the\nrecursion and Abel summation against the cumulative driver give\n$\\max_j|Q_j|\\le |Q_0|+2\\max_j|Z_h(jh)|$. With this bound $Q_*$ and\n$s_*=\\max_j|s_j|$, the other recurrences give\n\\[\nR_j\\le R_0\\exp[(Q_*+s_*)^2V_h(T)/U_0],\\qquad\nU_j\\le U_0+R_{\\max}V_h(T).\n\\]\nUniform convergence of $Z_h,V_h$ therefore supplies a common compact region.\nThe limiting integral equation has the same bound by variation of constants\nand integration by parts with the continuous additive input. This argument\ndoes not require a guess of stochastic integration convention.\n\\end{proof}\n\n\\begin{theorem}[Sample-held colored-noise locking]\\label{thm:colored}\nFor the stationary OU sampling above and $U_0>0$, the scalar controller converges\nto \\eqref{eq:core} on bounded time intervals. Its solution is global; $U,R$\nremain nondecreasing. For $R_0>0$,\n$U(t)\\ge U_0+2\\nu f(c)R_0t$.\n\\end{theorem}\n\\begin{proof}\nApply Proposition \\ref{prop:ou} and Lemma \\ref{lem:pulse-limit} on stopped compact\ndomains with $\\kappa=2\\nu f(c)$ and $Z=\\sqrt{2\\nu}W$. For globality set\n$X=R+Q^2$. It\\^o's formula gives\n$\\dd X=2\\sqrt{2\\nu}Q\\dd W+2\\nu\\dd t$ independently of $f$.\nStopped expectation bounds control excursions of $X\\ge0$; also\n$U\\le U_0+\\kappa\\int X\\dd t$, and $U\\ge U_0$. Localization can therefore be\nremoved. Local Lipschitz coefficients on $U>0$ give uniqueness. The inequalities\nfollow from the first two scalar equations.\n\\end{proof}\n\nFor $c\\ll1$, $f(c)=c/2+O(c^2)$: a gate updated much faster than the correlation\ntime writes little memory. For $c\\gg1$, $f(c)=1-1/c+O(e^{-c}/c)$: a held stroke\nspanning many correlation times approaches the independent-pulse write rate.\nThe same macroscopic driver variance can therefore yield different geometric\nwriting. This is a prediction for this sampling protocol and covariance model,\nnot a universal law for all noise spectra or controllers.\n\n\\begin{figure}[htbp]\\centering\n\\includegraphics[width=.85\\linewidth]{results/colored_crossover.pdf}\n\\caption{Exact normalized resolved-stroke budget and independent integrated-OU\nsimulation: 1,024 paths at each ratio, 8,192 sampling intervals. Error bars show\ntwo standard errors. The curve does not claim novelty of stochastic feedback\ncrossover in general; it tests the specific area-memory controller.}\n\\end{figure}\n\n\\subsection{Why this is not an It\\^o--Stratonovich notation choice}\nFor continuous smooth monitoring, the limiting geometric equation is\n\\[\n\\dd x=\\sqrt{2\\nu}n\\circ\\dd W,\\qquad\n\\dd b=\\sqrt{2\\nu}(x\\times n)\\circ\\dd W.\n\\]\nIts ordinary chain rule gives $\\dd U=\\dd R=0$ and $\\dd Q=\\sqrt{2\\nu}\\dd W$.\nIn It\\^o form its vector drifts are\n\\[\n\\dd x=\\frac\\nu{\\sqrt U}(x\\times n)\\dd t+\\sqrt{2\\nu}n\\dd W,\n\\quad\n\\dd b=\\frac\\nu{\\sqrt U}x\\times(x\\times n)\\dd t\n+\\sqrt{2\\nu}(x\\times n)\\dd W.\n\\]\nThey cancel the spectral writing of the held-gate It\\^o model. In the colored\nsample-held vector limit their coefficients are multiplied by $1-f(c)$.\nThe two models have different controllers and different drifts. In contrast,\nthe antisymmetric area integral itself is identical in It\\^o and Stratonovich\nfor a fixed semimartingale $x$: their correction is symmetric in the two\ncoordinates and cancels under antisymmetrization.\n\nFor $f(c)<1$, the angular quadratic variation of the limiting continuous\nsemimartingale is $2\\nu R/U\\,\\dd t$, whereas the resolved sampling-grid square\nbudget tends to $2\\nu f(c)R/U\\,\\dd t=\\dd\\log U$. This is not a contradiction.\nQuadratic variation is not continuous under smooth or correlated-noise\napproximations. Equation \\eqref{eq:angular} identifies the two only in the\nindependent-pulse limit. A readout protocol must retain which resolution its\nprojections use.\n\n\\subsection{Seed requirement for the initial diffusion theorems}\nThe theorems assume $b_0\\ne0$. At zero, the kernel changes discontinuously when\nrank is created. A first vanishing pulse can leave a random permission\norientation even as its memory magnitude tends to zero. In two dimensions a\ngeneric first writing pulse exhausts rank and freezes further motion. A formal\nprojector SDE across this singularity is not a proved diffusion model.\nA physical device can create a \\emph{finite} two-pulse seed first; that operation\nhas its own amplitude, work, and calibration. Subsequent diffusion is then\ncovered by the seeded theorems. Section~\\ref{sec:zero} now proves a selected\nscalar entrance and vector weak existence, while excluding a seed-independent\nFeller extension. An undefined projector alone is not a weak-solution no-go.\n\n","labels":["prop:ou","lem:pulse-limit","thm:colored"],"source_line_start":495,"source_line_end":655,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_07","section_index":7,"title_latex":"Finite resolution, gate error, and actuator work","text_latex":"\\section{Finite resolution, gate error, and actuator work}\nFor arbitrary $a$ let $r_\\theta=(A+\\theta B)a$ be the implemented residual.\nThe exact error identity is\n\\begin{equation}\\label{eq:residual}\n\\Delta E=(1-2\\theta)\\area+2x^Tr_\\theta.\n\\end{equation}\nIf $\\norm{r_\\theta}\\le\\epsilon\\norm a$, then $r_\\theta\\perp a$ and\n\\[\n|2x^Tr_\\theta|\\le2\\epsilon\\sqrt\\area.\n\\]\nFor a pre-gated stroke, $\\sqrt\\area\\ge2\\epsilon$ therefore guarantees\nnonnegative writing. This is a finite-resolution threshold, not a robustness\nclaim for exact rank or exact reachability. With sufficiently small intended\narea increments, a fixed residual can reverse the sign. The screenshot's\nabsolute irreversibility cannot be transferred to hardware without error bars.\n\nFor exact pre-gated histories,\n\\[\n\\Delta E=\\sum_j\\norm{x_j\\wedge a_j}^2\n\\le M^2\\sum_j\\norm{a_j}^2,\\qquad M=\\max_j\\norm{x_j}.\n\\]\nBounded variation refinement sends the square budget to zero. Brownian or\nheld-noise scaling retains it. This is the quantitative reason noise changes\nthe continuum outcome.\n\nFor a \\emph{specified drag-limited commanded actuator} with coefficient $\\zeta$\nand stroke duration $h$, Cauchy--Schwarz gives\n$W_{{\\rm drag},j}=\\zeta\\int\\norm v^2\\dd t\\ge\\zeta\\norm{a_j}^2/h$.\nThus $W_{{\\rm drag},{\\rm total}}\\ge\\zeta\\Delta E/(M^2h)$ for equal-duration\npulses. A positive fixed memory increment with bounded coordinates then costs\ndivergent drag work as $h\\to0$ in this actuator model. This is not a universal\nthermodynamic bound on Brownian matter. Microscopic inertia, bath exchanges,\nand finite bandwidth have to be modeled before using singular stochastic paths\nas physical trajectories.\n\n","labels":["eq:residual"],"source_line_start":655,"source_line_end":690,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_08","section_index":8,"title_latex":"A stable thermal probe with exact work and heat","text_latex":"\\section{A stable thermal probe with exact work and heat}\nTo make a genuine thermodynamic observable, introduce an independent physical\ndisplacement $q\\in\\R^d$, momentum $p$, mass $M_p>0$, frequency $\\omega_0>0$ and\ncoupling $g$ with frequency units. Dimensionless $A$ programs\n\\begin{equation}\\label{eq:probe}\nH_A(q,p)=\\frac{p^Tp}{2M_p}+\\frac{M_p}2q^TK(A)q,\n\\qquad K(A)=\\omega_0^2I-g^2A^2.\n\\end{equation}\n$K(A)$ is positive definite for every $A$, since $-A^2\\succeq0$. A standard\nunderdamped thermal model at fixed $A$ is\n\\[\n\\dd q=(p/M_p)\\dd t,\\quad\n\\dd p=-M_pKq\\dd t-\\gamma_p p\\dd t+\n\\sqrt{2\\gamma_pM_p\\kT}\\,\\dd W_d.\n\\]\nIt has the canonical equilibrium state. The proposed cycle equilibrates this\nprobe, quenches $A$ while $q,p$ stay fixed, then allows reequilibration. Registers\nand spring-control hardware have additional costs; the calculations below\naccount for the probe's work and heat only.\n\nSet $\\eta_g=g^2/\\omega_0^2$ and $G=I-\\eta_g A^2$. For a pre-gated pulse let\n\\[\n\\response=\\norm a^2x^TG^{-1}x-\\inner xa^2.\n\\]\nThe notation $\\response$ is a response-weighted area scalar and is distinct\nfrom the three-dimensional radius $R$.\n\n\\begin{theorem}[Exact thermal writing law]\\label{thm:thermal}\nFor a pre-gated pulse,\n\\begin{align}\n0\\le\\response&\\le\\area,\n&\\frac{\\det G'}{\\det G}&=(1+\\eta_g\\response)^2,\\\\\n\\Delta F&=\\kT\\log(1+\\eta_g\\response),\n&\\E W&=\\frac{\\kT\\eta_g}2(\\area+\\response).\n\\end{align}\nThe irreversible work of quench plus thermal relaxation is\n\\begin{equation}\\label{eq:wirr}\nW_{\\rm irr}=\\kT\\left[\\eta_g\\response-\\log(1+\\eta_g\\response)\n+\\frac{\\eta_g}2(\\area-\\response)\\right]\\ge0.\n\\end{equation}\nFor $g\\ne0$, it is strictly positive whenever the pulse writes nonzero area.\n\\end{theorem}\n\\begin{proof}\nDecompose $x=u+w$ in image and kernel and resolve $u=\\sum u_i$ into skew planes\nwith positive frequencies $\\sigma_i$. Let\n$D_0=\\norm a^2\\norm w^2-\\inner wa^2\\ge0$. The inherited spectral rank-one\nreduction and the determinant lemma give\n\\[\n\\frac{\\det G'}{\\det G}=\n\\left[1+\\eta_gD_0+\\eta_g\\norm a^2\n\\sum_i\\frac{\\norm{u_i}^2}{1+\\eta_g\\sigma_i^2}\\right]^2.\n\\]\nThe bracket is $1+\\eta_g\\response$. Its nonnegative terms prove\n$\\response\\ge0$, while $G^{-1}\\preceq I$ gives $\\response\\le\\area$.\nFor $a\\ne0$, $\\response=0$ implies $u=0$ and $w,a$ dependent, hence $B=0$.\nThe classical partition function gives\n$F(A)=\\mathrm{constant}+(\\kT/2)\\log\\det K(A)$.\n\nAn instantaneous quench has average work\n$\\E W=(\\kT/2)\\tr[(K'-K)K^{-1}]$.\nSince $K'-K=-g^2(AB+BA+B^2)$, the two linear cross traces vanish using\n$Aa=0$ and commutation of $G^{-1}$ with $A$. Finally\n\\[\n-B^2=\\norm a^2xx^T+\\norm x^2aa^T-\n\\inner xa(xa^T+ax^T),\\qquad G^{-1}a=a,\n\\]\ngives the work formula. Subtract $\\Delta F$ and use $z-\\log(1+z)\\ge0$.\n\\end{proof}\n\nThe equilibrium positional covariance is $(\\kT/M_p)K^{-1}$, so its determinant\ncontracts by $(1+\\eta_g\\response)^{-2}$. Individual coordinate variances need\nnot all contract: the axes can rotate. Ordered stiffness frequencies harden,\nbut the physical probe must not be confused with a pure gyroscopic oscillator,\nwhose frequency branches have different behavior.\n\nAt fixed temperature, the classical mean probe energy is $d\\kT$ independently\nof $K$. The full quench work $\\E W$ is therefore released as heat to the bath\nduring reequilibration. It is not the same as $W_{\\rm irr}$. The free-energy\nchange describes reduced equilibrium entropy and recoverable controlled work;\nthe excess describes irrecoverable work in this cycle.\nFor $A=0$ one has $\\response=\\area$ and\n\\[\n\\Delta F=\\kT\\log(1+\\eta_g\\area),\\quad\n\\E W=\\kT\\eta_g\\area.\n\\]\n\n\\subsection{Isospectral orientation writing can still dissipate}\nAt a nonzero midpoint pulse let $v=a/\\norm a$ and $H_v=I-2vv^T$. Then\n$K'=H_vKH_v$ and $\\Delta F=0$, but the sudden quench has\n\\begin{equation}\\label{eq:midpoint-heat}\n\\boxed{Q_{\\rm bath}=W_{\\rm irr}=\\E W\n=2\\kT[(v^TKv)(v^TK^{-1}v)-1]\\ge0.}\n\\end{equation}\nExpand $H_vKH_v$ in the trace formula to obtain the equality. Cauchy--Schwarz\nfor $K^{1/2}v,K^{-1/2}v$ proves the inequality. Equality holds exactly when\n$v$ lies in one stiffness eigenspace, in which case $K'=K$.\n\nThis separates spectral memory from orientation memory operationally. Midpoint\ngating can preserve every skew frequency while producing positive heat by\nrotating an anisotropic stiffness relative to the old equilibrium covariance.\nThe effect is supplied by the controller; neither isospectrality nor geometric\nirreversibility by itself determines entropy production.\n\n\\begin{figure}[htbp]\\centering\n\\includegraphics[width=\\linewidth]{results/timing_work.pdf}\n\\caption{Left: a matched-area timing comparison from Section\n\\ref{sec:magnetic}. Right: exact normalized free-energy and quench-work laws\nfor an initially zero-memory thermal probe. The difference is irreversible\nwork for this specified protocol.}\n\\end{figure}\n\n","labels":["eq:probe","thm:thermal","eq:wirr","eq:midpoint-heat"],"source_line_start":690,"source_line_end":801,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_09","section_index":9,"title_latex":"Thermal reversibility requires additional channels","text_latex":"\\section{Thermal reversibility requires additional channels}\nAn area-writing edge of the exact pre-gated state graph has no pre-gated inverse.\nOn a thermally resolved even-state graph with strictly positive equilibrium\nweights, ordinary detailed balance cannot give that edge a positive forward\nrate and exactly zero backward rate. Local detailed balance and its limits are\nestablished topics \\cite{maes,zeraati}; the present graph is an application.\n\nA physical completion can add the labelled inverse channel\n\\[\n(x',A')\\mapsto(x'-a,A'-x'\\wedge a),\n\\qquad [A'-x'\\wedge a]a=0.\n\\]\nThis is the post-gated adjoint, deliberately extending the original rule.\nIf a fuel reservoir supplies chemical free energy $\\Delta\\mu$ and the resolved\nstate free-energy difference is $\\Delta\\mathcal F$, rates can be assigned as\n\\[\nk_\\pm=k_*\\exp\\left[\\pm\\frac{\\Delta\\mu-\\Delta\\mathcal F}{2\\kT}\\right].\n\\]\nThey satisfy $\\log(k_+/k_-)=(\\Delta\\mu-\\Delta\\mathcal F)/\\kT$. On a countable\ngraph finite outgoing rate sums and nonexplosion must also be checked;\nfinite-edge truncations avoid that additional issue. Requiring\n$k_-/k_+\\le\\varepsilon$ gives the conditional reliability cost\n$\\Delta\\mu\\ge\\Delta\\mathcal F+\\kT\\log(1/\\varepsilon)$.\nIf a register is assigned $E_A=\\chi E(A)$ and the probe is quasistatically\nequilibrated, one possible $\\Delta\\mathcal F$ is\n$\\chi\\area+\\kT\\log(1+\\eta_g\\response)$. Sudden-quench dissipation is additional.\n\nThere is no universal infinite-energy claim. Absorbing effective models, zero\ntemperature, odd time-reversal variables, hidden controller states, and different\nreverse protocols require distinct accounting. The exact mathematical arrow\nis an ideal transition-language property. At finite temperature, a real device\nmay approximate it with small leakage and externally maintained nonequilibrium\nresources.\n\n","labels":[],"source_line_start":801,"source_line_end":835,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_10","section_index":10,"title_latex":"A completely positive dark-mode readout","text_latex":"\\section{A completely positive dark-mode readout}\nUse an independent signal Hilbert space\n$\\mathcal H=\\mathbb C|0\\rangle\\oplus\\mathbb C^d$, containing vacuum and at most\none photon. A fixed register programs\n\\begin{equation}\\label{eq:lindblad}\nL_j(A)=\\sqrt\\gamma\\sum_kA_{jk}|0\\rangle\\langle k|,\n\\qquad \\dot\\varrho=\\sum_j\\left(L_j\\varrho L_j^\\dagger-\n\\tfrac12\\{L_j^\\dagger L_j,\\varrho\\}\\right).\n\\end{equation}\nThe matrix $A^TA$ acts on the signal sector only, so\n$\\sum L_j^\\dagger L_j=\\gamma[0\\oplus A^TA]$ on the full Hilbert space.\nThe dark one-photon space is\nexactly $\\ker A$. This is an ordinary engineered loss bath within GKSL quantum\nmechanics \\cite{lindblad,gks}. No Hamiltonian mixing dark and bright modes is\nincluded. Classical randomization of the register gives a convex mixture of\ncompletely positive trace-preserving channels.\n\nFor dwell $t_d$, the surviving one-photon amplitude is multiplied by\n$M_A=\\exp(-\\gamma A^TAt_d/2)$, extended by zero on the vacuum when used below.\nFor an orthonormal signal eigenbasis $A^TAv_r=\\lambda_rv_r$, Kraus operators are\n\\[\nK_0=|0\\rangle\\langle0|+M_A,\n\\qquad K_r=\\sqrt{1-e^{-\\gamma\\lambda_rt_d}}|0\\rangle\\langle v_r|.\n\\]\n$K_0^\\dagger K_0+\\sum_rK_r^\\dagger K_r=I_{\\mathcal H}$ exactly.\nBright population enters vacuum;\nthere is no bath reexcitation. Bosonic operators\n$L_j=\\sqrt\\gamma\\sum_kA_{jk}\\hat b_k$ give the equivalent passive linear loss\nmodel restricted to the one-photon sector. Moving protected subspaces,\nZeno transport, and Lindbladian geometry have close established ancestors\n\\cite{burgarth,albert,yamamoto}. Damped auxiliary modes are a\nstandard reservoir-engineering route, but hardware realization and bandwidth\nremain engineering work.\n\n\\subsection{The endpoint-only survival telescope}\nIn three dimensions $A^TA=U(I-nn^T)$. For a pre-gated pulse,\n$b\\cdot b'=U$ and therefore\n\\[\nn\\cdot n'=\\sqrt{U/U'}>0.\n\\]\nPrepare one photon in $n_0$, quench the programmed loss to $A_1$, fully damp its\nbright subspace, and repeat. No intermediate outcome readout is necessary.\n\\begin{theorem}[Exact relaxed-stage survival]\\label{thm:survival}\nIn the full-relaxation limit for any finite seeded three-dimensional pre-gated\ncontroller path,\n\\[\nP_{n_N}\\cdots P_{n_1}n_0=\\sqrt{U_0/U_N}\\,n_N,\n\\]\nand the final quantum state is\n\\[\n\\varrho_N=(U_0/U_N)|n_N\\rangle\\langle n_N|\n+(1-U_0/U_N)|0\\rangle\\langle0|.\n\\]\nThus final photon survival equals $U_0/U_N$ exactly.\n\\end{theorem}\n\\begin{proof}\nEach projection multiplies the surviving amplitude by\n$n_j\\cdot n_{j-1}=\\sqrt{U_{j-1}/U_j}$. The product telescopes. Trace preservation\nplaces the missing population in vacuum.\n\\end{proof}\nOn random controller paths, the ensemble survival is\n$\\E[U_0/U_N]$, generally not $U_0/\\E[U_N]$. In the independent-pulse limit,\n\\eqref{eq:angular} yields $S(t)=\\exp[-\\int\\tr\\dd[n]]=U_0/U(t)$ for the ideal\nprojection protocol. The bath rate and pulse resolution must satisfy the\nconditions below; a coupling quench alone is not a projection.\n\n\\subsection{Finite dwell has a controlled error}\nFor the $j$th stage, let\n$M_j=P_j+\\epsilon_j(I-P_j)$, with\n$\\epsilon_j=e^{-\\gamma U_jt_{d,j}/2}$. The physical survival is\n$S_N=\\norm{M_N\\cdots M_1n_0}^2$.\nA telescoping product identity and contraction of suffixes give\n\\begin{equation}\\label{eq:errorbound}\n\\left|S_N-\\frac{U_0}{U_N}\\right|\\le\n2\\sum_{j=1}^N\\epsilon_j\n\\sqrt{\\frac{U_0(U_j-U_{j-1})}{U_{j-1}U_j}}\n\\le2\\epsilon_{\\max}\\sqrt{N\\log(U_N/U_0)}.\n\\end{equation}\nFor the first inequality, the $j$th error acts on the ideal prefix\n$\\sqrt{U_0/U_{j-1}}n_{j-1}$ and its bright component has the displayed norm.\nFor the second, Cauchy--Schwarz and\n$\\sum(U_j-U_{j-1})/U_j\\le\\log(U_N/U_0)$ suffice.\nWith minimum dwell $t_d$, a conservative sufficient error criterion is\n\\[\n\\gamma U_0t_d\\ge2\\log\\left(\\frac{2\\sqrt{N\\log(U_N/U_0)}}{\\delta}\\right)\n\\]\nwhen the logarithm is positive. This is sufficient, not a claimed optimal\nresource lower bound.\n\n\\subsection{A two-pulse interference falsifier}\nChoose $b_0=e_3,x_0=e_1+e_3,a_1=-e_3$. Then $b_1=e_2+e_3$, $x_1=e_1$ and\n$U_1=2$. The legal second pulse $a_2=-(e_2+e_3)$ gives $b_2=2e_2,U_2=4$.\nFor an initial $e_3$ photon,\n\\begin{equation}\\label{eq:two-pulse}\nS_2(t_1,t_2)=\\frac{(1-e^{-\\gamma t_1})^2+\n(1+e^{-\\gamma t_1})^2e^{-4\\gamma t_2}}4.\n\\end{equation}\nWith complete second relaxation, it ranges from zero at $t_1=0$ to $1/4$ at\n$t_1\\to\\infty$. The final memory is the same in both cases. Thus a finite-dwell\ndevice can have survival \\emph{below} the ideal telescope value; multiplying\nsingle-stage survival numbers is incorrect when bright amplitudes remain.\n\nAt fixed finite $\\gamma$ and vanishing pulse spacing, the signal obeys\n$\\dot\\psi=-(\\gamma/2)U(t)(I-P_{n(t)})\\psi$. Survival is the norm of its\ntime-ordered propagator. Section~\\ref{sec:continuous-dark} now solves the\nstrong-loss continuation and the colored-controller order dependence.\nController-history time and laboratory dwell time must be distinguished when\nreplaying a recorded history.\n\n\\subsection{A coherent control removes the attenuation}\nSince a nonzero $b$ retains a one-dimensional dark mode, an external unitary\nrotation can map $n_j$ to $n_{j+1}$ before damping. It then preserves photon\nsurvival. More generally, in the inherited rank-preserving kernel graph,\nthe principal rotation angle is $\\arctan(\\norm a\\norm{(A|_{\\im A})^{-1}u})$.\nCounterdiabatic or mode-transport control can implement that rotation. The\nsurvival telescope describes the \\emph{unrotated quench-and-damp} protocol,\nnot an unavoidable universal price of memory.\nNo quantum-computational advantage is established: a linear classical wave\nwith the same damping matrix has the analogous surviving intensity law.\n\n\\Needspace{10\\baselineskip}\n","labels":["eq:lindblad","thm:survival","eq:errorbound","eq:two-pulse"],"source_line_start":835,"source_line_end":956,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_11","section_index":11,"title_latex":"A magnetic feedback realization","text_latex":"\\section{A magnetic feedback realization}\\label{sec:magnetic}\nIn three dimensions a uniform magnetic field has the same skew Lorentz tensor\nup to sign and units. One candidate plant is a commanded stage or charged\nparticle with externally programmed $B$. For a calibrated coefficient\n$\\lambda_B$, update\n\\[\nB'=B+\\lambda_Bx\\times a,\n\\qquad a\\parallel B+\\theta\\lambda_Bx\\times a.\n\\]\nThe timing budget becomes\n\\begin{equation}\\label{eq:magnetic-budget}\n\\norm{B'}^2-\\norm B^2=(1-2\\theta)\\lambda_B^2\\norm{x\\times a}^2.\n\\end{equation}\nAt the midpoint, the field is rotated by $\\pi$ about the pulse direction and\nits magnitude is preserved. At the pre-gate it grows; at the post-gate it can\nshrink. These endpoints do not assert that the whole stroke is continuously\nLorentz-force free. A causal implementation holds the field and direction for\nthe stroke, stops or clamps the stage, switches the coil state, and then reads\nthe next permission. Midpoint and post-gated pulses are solved in advance.\n\nIf a matched isotropic three-axis coil system has storage\n$U_B=C\\norm B^2/2$, then its storage increment is the right side of\n\\eqref{eq:magnetic-budget} times $C/2$. Anisotropic inductance requires a\ndifferent metric and need not retain this simple identity. Real coil drivers\nalso incur resistive losses, switching transients, and mechanical costs.\nFaraday's law induces electric fields during switching; the field cannot be\nchanged for free while ignoring the charged particle's response. Holding or\ndecoupling the stage during switching is part of the protocol.\n\n\\subsection{A matched-area timing calibration}\nIn dimensionless units let $x=e_1$, $B=e_3$, $\\lambda_B=1$, and take\n\\[\na_\\theta=(0,-\\theta\\ell^2,\\ell\\sqrt{1-\\theta^2\\ell^2}),\n\\qquad |\\theta|\\ell<1.\n\\]\nThis is exactly $\\theta$-legal, has $\\norm a=\\ell$ and\n$\\norm{x\\times a}^2=\\ell^2$ for every comparison. Thus\n$\\norm{B'}^2=1+(1-2\\theta)\\ell^2$.\nPre, midpoint and post trials hold the initial state and the area magnitude\nfixed. The residual identity \\eqref{eq:residual} provides error bars. This is a\nprospective bench calibration, not a record of a built apparatus.\n\n","labels":["sec:magnetic","eq:magnetic-budget"],"source_line_start":956,"source_line_end":998,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_12","section_index":12,"title_latex":"A rank-continuous family of robust geometric certificates","text_latex":"\\section{A rank-continuous family of robust geometric certificates}\\label{sec:ridge}\nExact kernel projectors are discontinuous when an arbitrarily small skew\nfrequency crosses zero. A useful physical observable must survive that change.\nThe following family does, and retains the original one-step monotonicity.\nFor a fixed resolution $\\lambda>0$, define\n\\begin{equation}\\label{eq:ridge}\nF_\\lambda(x,A)=x^T[-A^2(\\lambda^2I-A^2)^{-1}]x,\n\\qquad m=\\norm x^2-F_\\lambda\n=\\lambda x^T(\\lambda I-A)^{-1}x.\n\\end{equation}\nAll eigenvalues of its matrix weight lie in $[0,1]$. Thus\n$0\\le F_\\lambda\\le\\rho^2$, and $F_\\lambda\\uparrow\\rho^2$ as\n$\\lambda\\downarrow0$. Unlike $\\rho$, $F_\\lambda$ is smooth at every real skew\nmatrix, including zero. It is a lower bound on squared reachable distance,\nnot a redefinition of the exact finite-word minimum.\n\n\\begin{theorem}[Resolved exclusion is exactly monotone]\\label{thm:ridge}\nFor every dimension, skew $A$, and legal pre-gated displacement $Aa=0$, put\n$c=\\norm a^2$, $d=x\\cdot a$ and $D=cm-d^2$. Then\n\\begin{equation}\\label{eq:ridge-increment}\n\\boxed{F_\\lambda(x+a,A+x\\wedge a)-F_\\lambda(x,A)\n=\\frac{D(m+2d+c)}{\\lambda^2+D}\\ge0.}\n\\end{equation}\nNo nonzero seed, constant rank, or spectral gap is required. The inequality is\nstrict exactly when $x\\wedge a\\ne0$.\n\\end{theorem}\n\\begin{proof}\nThe symmetric part of $(\\lambda I-A)^{-1}$ is\n$\\lambda(\\lambda^2I-A^2)^{-1}$, proving the expression for $m$.\nSince $a\\in\\ker A$, both $(\\lambda I-A)^{-1}a=a/\\lambda$ and its\ntranspose identity hold. The two-column Woodbury identity applied to\n$\\lambda I-A-xa^T+ax^T$ yields\n\\[\nm'=\\lambda^2\\frac{m+2d+c}{\\lambda^2+cm-d^2}.\n\\]\nThis can also be verified by multiplying the two-by-two update matrix\n$\\left(\\begin{smallmatrix}1-d/\\lambda&c/\\lambda\\\\-m/\\lambda&1+d/\\lambda\\end{smallmatrix}\\right)$,\nwhose determinant is $(\\lambda^2+D)/\\lambda^2$.\nLet $w=P_{\\ker A}x$. The spectral formula gives $m\\ge\\norm w^2$.\nThen $d=w\\cdot a$ implies $D\\ge0$, while\n$m+2d+c\\ge\\norm{w+a}^2\\ge0$. Subtract $m'$ from\n$\\norm{x+a}^2$ and use \\eqref{eq:ridge}.\n\\end{proof}\n\n\\begin{theorem}[Gate-error certificate without a spectral gap]\\label{thm:ridge-error}\nIf $a\\ne0$ and $\\norm{Aa}\\le\\epsilon\\norm a$, the same area update satisfies\n\\begin{equation}\\label{eq:ridge-error}\nF_\\lambda(x+a,A+x\\wedge a)\n\\ge F_\\lambda(x,A)-\\frac\\epsilon\\lambda\n\\bigl(\\norm x^2+\\norm{x+a}^2\\bigr).\n\\end{equation}\nConsequently a whole measured history obeys\n\\[\n\\norm{x_N}^2\\ge F_\\lambda(x_0,A_0)\n-\\frac1\\lambda\\sum_{j<N}\\epsilon_j\n(\\norm{x_j}^2+\\norm{x_{j+1}}^2).\n\\]\nThe right-hand side can be clipped at zero. Calibration uncertainty must be\nadded separately; the displayed tolerance is the actual pre-gate residual.\n\\end{theorem}\n\\begin{proof}\nFor $P=aa^T/\\norm a^2$, the nearby skew matrix\n$\\bar A=(I-P)A(I-P)$ obeys $\\bar Aa=0$ and\n$\\norm{A-\\bar A}_{\\rm op}=\\norm{Aa}/\\norm a\\le\\epsilon$.\nThe resolvent norms $\\norm{(\\lambda I\\pm A)^{-1}}_{\\rm op}$ are at most\n$1/\\lambda$. Expressing the weight in \\eqref{eq:ridge} as\n$I-(\\lambda/2)[(\\lambda I-A)^{-1}+(\\lambda I+A)^{-1}]$ and using the\nresolvent identity proves\n$\\norm{J_\\lambda(A)-J_\\lambda(B)}_{\\rm op}\\le\\norm{A-B}_{\\rm op}/\\lambda$.\nApply Theorem \\ref{thm:ridge} to $\\bar A$; compare the two initial and the two\nfinal quadratic forms. Summation gives the history bound.\n\\end{proof}\n\nTaking $\\lambda$ too small amplifies uncertainty as $\\epsilon/\\lambda$.\nA useful polynomial consequence is\n\\[\n\\norm{(A+x\\wedge a)(x+a)}^2-\\norm{Ax}^2\n=\\area\\norm{x+a}^2.\n\\]\nIt follows by taking $\\lambda^2 F_\\lambda\\to\\norm{Ax}^2$ in the exact gain\nidentity, or directly since $Ax$ is perpendicular to both $x$ and $a$.\nThe finite-error construction also bounds every ordered positive skew frequency:\n$\\sigma_i(A')\\ge\\sigma_i(A)-2\\epsilon$. Apply the inherited exact spectral\nmonotonicity to $\\bar A$, and Weyl's singular-value inequality at the two\nendpoints, each within $\\epsilon$ of the actual endpoint.\n\nResolution therefore has a measurable tradeoff: a larger $\\lambda$ provides a\nweaker exact-distance bound with better noise tolerance. This resolves the\noperational stability problem by a continuous certificate. It does not make the\nunregularized rank projector continuous.\n\n\\subsection{A thermal measurement of the certificate}\nAn independent classical oscillator probe with squared-frequency matrix\n$K_\\lambda=\\omega_0^2(I-A^2/\\lambda^2)$ and potential\n$M_p q^TK_\\lambda q/2$ is positive at every rank. At fixed held $x,A$ in canonical\nequilibrium, the measured coordinate $Y=x\\cdot q$ has variance\n\\begin{equation}\\label{eq:ridge-readout}\nV_\\lambda=\\E Y^2\n=\\frac{\\kT}{M_p\\omega_0^2}\\bigl(\\norm x^2-F_\\lambda\\bigr).\n\\end{equation}\nThus the controller supplies $x$ while the probe variance supplies\n$F_\\lambda$. For $N$ independent equilibrium Gaussian samples, an unbiased\nsample-variance estimator gives\n\\[\n\\operatorname{Var}(\\widehat F_\\lambda)\n=\\frac{2(\\norm x^2-F_\\lambda)^2}{N-1}.\n\\]\nThe independence assumption requires sampling beyond the relaxation time or\naccounting for autocorrelation. Both preparation and stiffness control are\npowered operations. This is a concrete readout of a geometric certificate,\nnot an energy source.\n\n\nThere is also an exact gain/free-energy relation for this specified probe.\nWith $C=m+2d+c>0$ and $g=\\Delta F_\\lambda$, the determinant and quench formulas\nfrom the thermal section, specialized to $\\eta=1/\\lambda^2$, give\n\\[\n\\Delta F_{\\rm probe}=\\kT\\log(1+D/\\lambda^2),\\qquad\n\\boxed{g/C=1-e^{-\\Delta F_{\\rm probe}/\\kT}.}\n\\]\nThe corresponding sudden-quench mean work is\n$\\langle W\\rangle=\\kT(\\area+D)/(2\\lambda^2)$, with $\\area\\ge D$.\nThis relates continuous geometric locking to fluctuation-compression free\nenergy in the chosen Hamiltonian; it is not a universal energy price for information.\n\n","labels":["sec:ridge","eq:ridge","thm:ridge","eq:ridge-increment","thm:ridge-error","eq:ridge-error","eq:ridge-readout"],"source_line_start":998,"source_line_end":1123,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_13","section_index":13,"title_latex":"Continuous stochastic dark-space transport","text_latex":"\\section{Continuous stochastic dark-space transport}\\label{sec:continuous-dark}\nThe finite-stage law $S_N=U_0/U_N$ does not determine a probe that continuously\nfollows a limiting noisy permission line. Here the formerly open strong-loss\nquestion is resolved for real, rank-one continuous dark spaces.\n\n\\begin{theorem}[Strong loss detects angular quadratic variation]\\label{thm:strong-loss}\nOn $[0,T]$, let $n(t)$ be a real unit-vector continuous It\\^o semimartingale,\n\\[\n\\dd n=\\sigma\\dd W+\\beta\\dd t,\n\\quad n\\cdot\\sigma=0,\n\\quad n\\cdot\\beta=-\\norm\\sigma^2/2.\n\\]\nAssume locally bounded coefficients and continuous adapted $U(t)\\ge u_0>0$.\nLet a passive signal evolve from $\\psi_\\gamma(0)=n(0)$ by\n\\[\n\\dot\\psi_\\gamma=-\\frac\\gamma2U(t)(I-nn^T)\\psi_\\gamma.\n\\]\nThen, uniformly on compact time intervals in probability,\n\\begin{equation}\\label{eq:strong-loss}\n\\boxed{\\psi_\\gamma(t)\\longrightarrow\n\\exp\\!\\left[-\\frac12\\int_0^t\\norm\\sigma(r)^2\\dd r\\right]n(t),\\quad\nS_\\infty(t)=\\exp\\!\\left[-\\int_0^t\\norm\\sigma(r)^2\\dd r\\right].}\n\\end{equation}\nThe bounded survival probabilities also converge in every finite $L^p$.\nThese are unnormalized surviving one-photon amplitudes of the passive channel;\nthe remaining probability is vacuum. The law also applies to classical waves.\n\\end{theorem}\n\\begin{proof}\nWrite $P=nn^T$, $z=(I-P)\\psi_\\gamma$, $\\alpha=n\\cdot\\psi_\\gamma$,\n$\\kappa=\\norm\\sigma^2$ and $k=\\beta+\\kappa n/2$. Exact dissipation gives\n\\[\n\\frac\\dd{\\dd t}\\norm{\\psi_\\gamma}^2=-\\gamma U\\norm z^2,\n\\qquad \\int_0^T\\norm z^2\\dd t\\le1/(\\gamma u_0).\n\\]\nSince $\\psi_\\gamma$ has finite variation, the product rule gives\n\\[\n\\dd\\alpha=-\\tfrac12\\kappa\\alpha\\dd t+k\\cdot z\\dd t+\\sigma\\cdot z\\dd W.\n\\]\nOn a localization with $\\kappa\\le C$, $\\norm k\\le K$, the cumulative forcing\n$F=\\int k\\cdot z\\dd t+\\int\\sigma\\cdot z\\dd W$ obeys\n$\\E\\sup|F|^2\\le2(TK^2+4C)/(\\gamma u_0)$ by Cauchy--Schwarz and Doob.\nThe dissipative scalar equation for\n$\\alpha-\\exp(-\\int\\kappa/2)$ gives $\\sup|\\alpha-\\alpha_*|\\le2\\sup|F|$;\nhence its squared expected supremum is at most\n$8(TK^2+4C)/(\\gamma u_0)$.\n\nFor the transverse component, freeze $P(t)$ only inside a variation-of-constants\ncalculation over the preceding interval of length $\\delta$. The positive damping\nkernel has total mass at most one. With\n$\\omega_P(\\delta)=\\sup_{|t-s|\\le\\delta}\\norm{P(t)-P(s)}_{\\rm op}$ this gives\n\\[\n\\sup_{t\\le T}\\norm{z(t)}\n\\le e^{-\\gamma u_0\\delta/2}+\\omega_P(\\delta).\n\\]\nFor $t<\\delta$, the initial dark state removes the exponential term and gives\njust the modulus bound. Choosing $\\delta\\downarrow0$ with\n$\\gamma\\delta\\to\\infty$ proves uniform pathwise transverse convergence,\nsince $P$ is continuous. Remove coefficient localization to obtain convergence\nin probability. Norms never exceed one, proving the stated $L^p$ consequence.\n\\end{proof}\n\n\\subsection{Two distinct Avenyra survival exponents}\nThe colored scalar controller has the vector lift\n\\[\n\\begin{aligned}\n\\dd x&=\\sqrt{2\\nu}n\\dd W+\\frac{(1-f)\\nu}{\\sqrt U}(x\\times n)\\dd t,\\\\\n\\dd b&=\\sqrt{2\\nu}(x\\times n)\\dd W+\\frac{(1-f)\\nu}{\\sqrt U}x\\times(x\\times n)\\dd t.\n\\end{aligned}\n\\]\nIts permission direction obeys\n\\begin{equation}\\label{eq:colored-n}\n\\dd n=\\sqrt{\\frac{2\\nu}U}(x\\times n)\\dd W+\n\\left[\\frac{(1-f)\\nu Q}U(x-Qn)-\\frac{\\nu R}U n\\right]\\dd t.\n\\end{equation}\nThis follows by normalizing the displayed vector equations; its\nnormal drift is required by $\\norm n=1$. In particular,\n$\\norm\\sigma^2=2\\nu R/U$, whereas $\\dd\\log U=2\\nu fR/U\\dd t$.\nTheorem \\ref{thm:strong-loss} therefore gives, for fixed $f>0$,\n\\begin{equation}\\label{eq:two-exponents}\n\\boxed{S_{\\rm resolved}=\\frac{U_0}{U(t)},\\qquad\nS_{\\rm continuous}=\\left(\\frac{U_0}{U(t)}\\right)^{1/f}.}\n\\end{equation}\nThe first quantity comes from fully relaxed projections at the original held\nstroke boundaries. The second takes the controller diffusion limit first,\nthen increases the continuous loss rate. At $f=1$ they agree. At $f=0$ the\ncontinuously monitored model instead has\n$S_{\\rm continuous}=\\exp(-2\\nu R_0t/U_0)$ even though $U,R$ are constant.\nA passive readout can therefore detect angular stochastic motion that a\nresolved area-memory budget does not retain. This is a proposed protocol\ncomparison, not an experimental observation or a universal thermodynamic cost.\n\n\\begin{proof}[Why the actual held-register vector limit has this drift]\nLet $Y=(x,b)$, $g(Y)=(n,x\\times n)$, and $\\Phi_z$ be the complete flow of\n$\\dd Y/\\dd z=g(Y)$. This flow preserves $U,R$ and changes $Q$ to $Q+z$.\nFor the exact finite update $Y'=Y+sg(Y)$, Taylor expansion gives\n\\[\n\\Phi_{-s}(Y+sg(Y))=Y-\\tfrac12s^2Dg(Y)g(Y)+O(|s|^3).\n\\]\nWrite $Z_j=\\sum_{i<j}s_i$ and transform $V_j=\\Phi_{-Z_j}(Y_j)$.\nThe transformed increments are Riemann--Stieltjes sums of\n$-D\\Phi_{-Z_j}(Y_j)Dg(Y_j)g(Y_j)/2$ against the resolved square budget.\nThe cumulative input converges to $Z=\\sqrt{2\\nu}W$, the square budget to\n$2\\nu ft$, and cubic remainders vanish by the OU estimates already proved.\nCompact-domain Gronwall gives convergence of $V$; the deterministic scalar\nbounds remove localization and bound the vector state. Transforming back gives\n$\\dd Y=g(Y)\\circ\\dd Z-\\nu fDg(Y)g(Y)\\dd t$.\nThe It\\^o correction $\\nu Dg\\,g$ leaves exactly $(1-f)\\nu Dg\\,g$,\nwith $Dg\\,g=((x\\times n)/\\sqrt U,x\\times(x\\times n)/\\sqrt U)$.\nThis proves the displayed vector limit without guessing a convention.\n\\end{proof}\nThis is a theorem for the ideal held-register model. A calibrated finite-rate\napparatus must scale its bandwidth, input range and losses to approximate that\nlimit; fixed component bandwidth does not do so automatically. Higher-corank\nprojector fields require the tangency conditions in Section~\\ref{sec:multi}.\n\nMore precisely, let $S_{h,\\gamma}$ use endpoint losses held during dwells $h$\non the same correlated finite controller histories. At fixed $h$, infinite\nloss gives the exact projection telescope. At fixed $\\gamma$, uniform vector\nconvergence gives the continuous loss ODE by Duhamel/Gronwall. Therefore\n\\[\n\\boxed{\\lim_{h\\to0}\\lim_{\\gamma\\to\\infty}S_{h,\\gamma}\n=\\frac{U_0}{U},\\qquad\n\\lim_{\\gamma\\to\\infty}\\lim_{h\\to0}S_{h,\\gamma}\n=\\left(\\frac{U_0}{U}\\right)^{1/f}.}\n\\]\nThese limits differ strictly for $0<f<1$, $\\nu R_0T>0$. At $f=0$ the\ncorresponding values are $1$ and $e^{-2\\nu R_0T/U_0}$; at $f=1$ these survival\nlimits agree. Ensemble survival averages each nonlinear expression over the\ncontroller histories. It does not insert the mean memory into that expression.\nA physical reservoir's finite auxiliary bandwidth also limits the accessible\n$\\gamma$; the statement is a limit of the specified GKSL model.\n\n\\subsection{A known optical benchmark and noncommuting limits}\nFor $n=(\\cos\\theta,\\sin\\theta,0)$, $\\theta=\\sqrt{k}\\,W$ and constant $U$, an\nindependent two-dimensional rotating-frame benchmark has moments\n\\[\n\\frac\\dd{\\dd t}\\begin{pmatrix}\\E y_1^2\\\\\\E y_2^2\\end{pmatrix}\n=\\begin{pmatrix}-k&k\\\\k&-(\\Gamma+k)\\end{pmatrix}\n\\begin{pmatrix}\\E y_1^2\\\\\\E y_2^2\\end{pmatrix},\n\\qquad \\Gamma=\\gamma U.\n\\]\nWith the initial dark state, $d=(\\Gamma^2+4k^2)^{1/2}$ gives\n\\[\n\\E S_\\gamma(t)=e^{-(\\Gamma+2k)t/2}\n\\left[\\cosh(dt/2)+\\frac{\\Gamma+2k}{d}\\sinh(dt/2)\\right]\n\\longrightarrow e^{-kt}.\n\\]\nThis fixture is not an Avenyra trajectory. Its moment equation matches\nKofman--Kurizki--Opatrn\\'y \\cite{kofman}; the independently implemented\nwhite-noise formula checks the theorem. Generic noisy optical attenuation\nis established ancestry, not a novelty claim.\nFor a smooth fixed record, $\\sigma=0$ and the strong-loss survival tends to one.\nConsequently smoothing a Brownian angle before taking infinite loss and taking\nthe diffusion limit before infinite loss give different answers, $1$ and\n$e^{-kt}$. The ideal order of limits must be specified in every proposed device.\nNo universal joint scaling formula for arbitrary finite bandwidth follows from\nthis result.\n\n\\begin{figure}[htbp]\\centering\n\\includegraphics[width=.85\\linewidth]{v2/results/readout_stress_check.pdf}\n\\caption{Independent continuous-loss benchmark. Exact ensemble survival for a\nBrownian angle approaches $e^{-0.8}$; a smooth angle rotated at constant rate\napproaches one. The Brownian simulation uses 2,048 paths and 16,384 time steps,\nwith two-standard-error bars. These are calculations, not optical data.}\n\\end{figure}\n\n","labels":["sec:continuous-dark","thm:strong-loss","eq:strong-loss","eq:colored-n","eq:two-exponents"],"source_line_start":1123,"source_line_end":1289,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_14","section_index":14,"title_latex":"Zero memory: a selected entrance and a universal obstruction","text_latex":"\\section{Zero memory: a selected entrance and a universal obstruction}\\label{sec:zero}\nThe discontinuous permission projector cannot be repaired by declaring a unique\norientation at $A=0$. We can nevertheless resolve both a meaningful scalar\nstartup and the obstruction to a seed-independent vector law.\n\n\\begin{theorem}[No seed-independent Feller extension]\nFor $\\nu>0$ and $0\\le f\\le1$, the off-zero seeded controller semigroup has no Feller\nextension at any state $(x_0,b_0=0)$. Here Feller means that every bounded\ncontinuous final-state observable has a continuous expectation as its initial\nstate approaches that point.\n\\end{theorem}\n\\begin{proof}\nFor $x_0\\ne0$, seeds $b_\\epsilon=\\epsilon x_0/\\norm{x_0}$ have $R_0=0$ and\n$U(t)=\\epsilon^2$ forever. Perpendicular seeds have $R_0=\\norm{x_0}^2$ and\n$U(t)\\ge\\epsilon^2+2\\nu f\\norm{x_0}^2t$.\nThe bounded continuous observable $U/(1+U)$ therefore tends to zero on the\nfirst sequence and is bounded away from zero on the second, for every $t>0$.\nBoth initial sequences approach the same state. At $f=0$ use the bounded\nobservable $e^{-\\norm x^2}$ instead: writing is absent and $U,R$ are constant,\nwhile $Q=Q_0+\\sqrt{2\\nu}W$. Parallel and perpendicular seeds give respectively\n$e^{-r_0/(1+4\\nu t)}/\\sqrt{1+4\\nu t}$ and\n$e^{-r_0}/\\sqrt{1+4\\nu t}$, which differ for $r_0,t>0$.\nSeeded finite-time radial\nmoment bounds make these laws uniformly tight; compact cutoffs preserve the\nseparation if Feller is defined on $C_0$ rather than bounded continuous tests. At $x_0=0$, instead choose\n$x_\\epsilon=\\epsilon v$, $b_\\epsilon=\\epsilon v$ with unit $v$. These are\nBrownian-line solutions $x(t)=(\\epsilon+\\sqrt{2\\nu}W_t)v$ with constant $b$.\nTwo different directions give different limits of a bounded coordinate\nobservable such as $x_1^2/(1+x_1^2)$. Continuity again fails.\n\\end{proof}\nThis excludes a universal seed-independent continuous extension. It does not\nexclude weak entrance laws, an enlarged state carrying a direction or startup\nrule, or a selected class of seeds.\n\n\\begin{theorem}[Unique scalar writing entrance]\\label{thm:entrance}\nFix $r_0>0$, $\\kappa=2\\nu f>0$, and a continuous input $Z(0)=0$ which is\nH\\\"older at zero with some exponent $\\alpha>0$. There is a unique continuous\nscalar writing solution with $R(0)=r_0$, $Q(0)=U(0)=0$, $R\\ge r_0$, and\n$U(t)=\\kappa\\int_0^tR(s)\\dd s$. For $t>0$ it obeys\n\\[\n\\dd R=\\kappa RQ^2/U\\dd t,\n\\qquad Q(t)=Z(t)-\\frac\\kappa2\\int_0^tRQ/U\\dd s.\n\\]\nFor $Z=\\sqrt{2\\nu}W$, this is an adapted, pathwise unique scalar entrance.\nIt is selected by perpendicular vanishing seeds, including a first legal\narea-creating pulse. The next theorem identifies the full immediate-writing vector law.\n\\end{theorem}\n\\begin{proof}\nWrite $I_R(t)=\\int_0^tR(s)\\dd s$. The integrating factor for $Q$ gives\n\\[\nQ_R(t)=Z(t)-\\frac1{2\\sqrt{I_R(t)}}\n\\int_0^t\\frac{R(s)Z(s)}{\\sqrt{I_R(s)}}\\dd s,\n\\qquad\n\\Phi(R)(t)=r_0\\exp\\left[\\int_0^t\\frac{Q_R(s)^2}{I_R(s)}\\dd s\\right].\n\\]\nOn $r_0\\le R\\le2r_0$, if $|Z(t)|\\le Ct^\\alpha$, positive averaging gives\n$|Q_R(t)|\\le2Ct^\\alpha$. A direct difference estimate gives\n\\[\n|Q_R(t)-Q_S(t)|\\le\n\\frac{3Ct^\\alpha}{2r_0(\\alpha+1/2)}\\norm{R-S}_\\infty.\n\\]\nThe exponent of $\\Phi$ is at most $2C^2T^{2\\alpha}/(\\alpha r_0)$.\nUsing $I_R\\ge r_0t$, the difference of the exponents has Lipschitz constant\nat most $[6/(\\alpha+1/2)+4]C^2T^{2\\alpha}/(2\\alpha r_0^2)$.\nFor sufficiently small $T$, $\\Phi$ maps the displayed interval into itself\nand is a contraction. Its causal fixed point gives the solution.\nGlobal continuation follows from $|Q(t)|\\le2\\sup_{s\\le t}|Z(s)|$ and\n\\[\n\\log(R(t)/r_0)\\le\\frac4{r_0}\n\\int_0^t\\frac{\\sup_{u\\le s}|Z(u)|^2}{s}\\dd s<\\infty.\n\\]\nNear zero the integral is finite by H\\\"older regularity; after any positive\ntime ordinary local uniqueness applies. The same contraction is uniform for\nperpendicular seeds with $U_0\\downarrow0$, $R_0\\to r_0$, $Q_0=0$, proving\nthe selected scalar limit. Brownian paths are H\\\"older with every exponent\nbelow $1/2$ on bounded intervals, so the stochastic conclusion follows.\n\\end{proof}\n\nThere is a sharp singularity in the resolution parameter: $R,Q$ in this\nzero-seed writing branch are independent of $\\kappa>0$, since it cancels from\nthe two integral equations, while $U=\\kappa I_R$ scales linearly with it.\nThis must not be extrapolated to a unique branch at $f=0$ or $r_0=0$.\nFor Brownian input the entrance has\n\\[\nU(t)=\\kappa r_0t+o(t),\\qquad\nQ(t)=\\sqrt{\\frac{2\\nu}{t}}\\int_0^t\\sqrt{s}\\,\\dd W_s+o_P(\\sqrt t),\n\\qquad Q(t)/\\sqrt t\\Rightarrow N(0,\\nu).\n\\]\nThe first statement is pathwise. The scalar leading Gaussian follows from\n$R(t)=r_0+O(t^{2\\alpha})$ and the Volterra representation, with any fixed\nBrownian H\\\"older exponent $\\alpha\\in(1/6,1/2)$.\nAngular quadratic variation between $t_0$ and $t$ diverges as\n$f^{-1}\\log(t/t_0)$ when $t_0\\downarrow0$. Thus a permission direction need\nnot have a continuous entrance value even though $b\\to0$ and the scalar\nobservables do. Vector weak existence follows from tight transverse-seed limits: on stopped\nbounded $x$ domains, $U_\\epsilon\\ge U_{0,\\epsilon}+\\kappa(r_0/2)t$ bounds\nvector drifts by $C/\\sqrt t$ and noise coefficients by constants. Brownian\nmoment bounds give tight continuous paths. Away from zero the coefficients\nare smooth, and martingale identities pass to the limit; the integrable\n$C/\\sqrt t$ bound then permits passage to the entrance. The scalar fixed\npoint identifies its observables. The next theorem upgrades this existence result to uniqueness in the immediate-writing class.\nA Borel extension assigning $n(x,0)=x/\\norm x$ also admits a nonwriting\nBrownian-line weak solution; the writing law spends zero time at $b=0$ and\nsolves that same extension. An explicit boundary policy is therefore needed.\nThe two following theorems now resolve nonzero-anchor vector uniqueness and\nselfsimilar writing from the complete origin; arbitrary boundary-policy\nuniqueness remains excluded.\n\nIdeal fully relaxed survival $U_0/U(t)$ tends to zero as the seed vanishes at\nfixed positive $t$ on this writing branch. A finite-rate loss probe is different:\n$S_\\gamma(t)\\ge\\exp[-\\gamma\\int_0^tU(s)\\dd s]>0$ for any initial unit signal,\nso zero seed does not prohibit finite-rate probing. Its generator\n$U(I-P_n)=UI-bb^T=A^TA$ extends polynomially to $b=0$; the finite-rate\nprobe ODE has no normalized-direction singularity. At zero every signal mode\nis initially dark, and its chosen preparation becomes additional data.\nFor a prescribed two-pulse seed with $x\\cdot b=0$, $R=\\norm x^2>0$ and\n$U=\\norm b^2$, all preparations are\n$p=\\alpha x+(x\\times b)/R$, $q=x-p$.\nTheir sharp minimum quadratic stroke budget is $R/2+2U/R$, and minimum total\nlength is $\\sqrt{R+4U/R}$, attained at $\\alpha=1/2$. Optimizing over $R$\ngives budgets $2\\sqrt U$ and $2U^{1/4}$. These are geometric preparation costs;\nphysical work still depends on the apparatus.\nStarting at $(x,b)=(0,0)$, two finite legal strokes $e_1,e_2$ give\n$(x,b)=(e_1+e_2,e_3)$ and provide an explicit operational seed.\n\n\\subsection{The nonzero-anchor vector entrance is unique in law}\n\\begin{theorem}[Canonical immediate-writing vector law]\\label{thm:vector-entrance}\nLet $x_0\\ne0$, $\\nu>0$, and $0<f\\le1$. Among continuous causal weak solutions\nof the off-zero vector controller with $(x,b)(0)=(x_0,0)$ and $U(t)>0$ at every\npositive time, the joint path law $(x,b,W)$ exists and is unique. No such solution\nis adapted solely to the natural filtration of its specified innovation $W$.\nThe whole family of shrinking transverse legal seeds converges weakly to this\nlaw. This statement selects immediate writing; it does not select it over\nnonwriting or delayed boundary policies.\n\\end{theorem}\n\\begin{proof}\nFirst scalar selection is automatic. The vector equations give\n$\\dd(x\\cdot b)=\\sqrt{2\\nu U}\\dd W$ and $U'=\\kappa R$, with\n$\\kappa=2\\nu f$. Both identities extend to zero by continuity and local boundedness.\nIntegration by parts, using the increasing $U$, gives\n\\[\nQ(t)=\\sqrt{2\\nu}\\left[W_t-\\frac1{2\\sqrt{U(t)}}\n\\int_0^t\\frac{U'(s)W_s}{\\sqrt{U(s)}}\\dd s\\right],\n\\qquad |Q(t)|\\le2\\sqrt{2\\nu}\\sup_{s\\le t}|W_s|.\n\\]\nThe normalized positive integral has total weight one. Hence $Q\\to0$ and\n$R\\to r_0=\\norm{x_0}^2$. Theorem~\\ref{thm:entrance} identifies the unique\nscalar functions of $W$. For each Brownian H\\\"older exponent $0<\\alpha<1/2$,\n\\[\nQ=O(t^\\alpha),\\quad R=r_0+O(t^{2\\alpha}),\\quad\nU=\\kappa r_0t[1+O(t^{2\\alpha})].\n\\]\n\nFor completeness, the singular direction has a constructive causal gauge.\nLet $e=(x-Qn)/\\sqrt R$, $m=e\\times n$, $O=[e,n,m]\\in SO(3)$, and put\n\\[\na=\\sqrt{2\\nu R/U},\\quad c=\\sqrt{2\\nu/U}\\,Q,\\quad\nd=(1+f)\\nu/\\sqrt U,\\quad k=(1-f)ac/2,\\quad h=(1+f)ac/2.\n\\]\nDirect It\\^o products yield $\\dd O=ON\\dd W+OC\\dd t$, where\n\\[\nN=\\begin{pmatrix}0&0&c\\\\0&0&-a\\\\-c&a&0\\end{pmatrix},\\qquad\nC=\\begin{pmatrix}-c^2/2&k&d\\\\h&-a^2/2&0\\\\-d&0&-(a^2+c^2)/2\\end{pmatrix}.\n\\]\nThus $N^T=-N$ and $C+C^T=N^2$, checking orientation and orthogonality.\nWrite $N_0=a_0J$, where $a_0=(ft)^{-1/2}$ and\n$J=\\left(\\begin{smallmatrix}0&0&0\\\\0&0&-1\\\\0&1&0\\end{smallmatrix}\\right)$.\nLet $N_r=N-N_0$, $C_r=C-N_0^2/2$. The displayed scalar estimates imply\n\\[\n\\norm{N_r}=O(t^{\\alpha-1/2}),\\qquad\n\\norm{C_r-N_rN_0}=O(t^{\\alpha-1}).\n\\]\nThe first is square integrable and the second integrable at zero.\n\nWe identify the singular phase without assuming it is independent randomness\nadded at time zero. Choose a reference frame $O_0=[e_0,u,v]$ with\n$e_0=x_0/\\sqrt{r_0}$ and $v=e_0\\times u$. The vector drift is $O(t^{-1/2})$\nand its diffusion bounded, so $e-e_0=O(t^\\alpha)$ after localization.\nSet $\\xi=e_0\\cdot n$, $\\eta=e_0\\cdot m$, $\\zeta=e_0\\cdot e$ and\n$\\rho^2=1-\\xi^2$. On a sufficiently small stopped interval, the circle angle\n$\\theta$ of the projection of $n$ onto $e_0^\\perp$ satisfies\n\\[\n\\dd\\theta=A_\\theta\\dd W+D_\\theta\\dd t,\\quad\nA_\\theta=\\frac{a\\zeta}{\\rho^2},\\quad\nD_\\theta=-\\frac{k\\eta}{\\rho^2}\n+\\frac{a^2\\zeta\\xi\\eta}{\\rho^4}.\n\\]\nHere $A_\\theta-a_0=O(t^{2\\alpha-1/2})$ and\n$D_\\theta=O(t^{2\\alpha-1})$. Consequently\n$r(t)=\\int_0^t(A_\\theta-a_0)\\dd W+\\int_0^tD_\\theta\\dd s$ exists and\ntends to zero. Stop when $\\norm{e-e_0}=1/4$, then extend the corrected\ncircle $z=\\exp[i(\\theta-r)]$ by\n\\[\n\\dd z=ia_0z\\dd W-a_0^2z\\dd t/2.\n\\]\nFor $j\\ne0$, its increment from $s$ to $t$ has Gaussian phase variance\n$f^{-1}\\log(t/s)$. Its covariance with any fixed Brownian cylinder is bounded,\nsince $\\int_s^t a_0\\dd u\\le2\\sqrt{t/f}$. Conditional Gaussian characteristic\nfunctions therefore give, for a cylinder characteristic $G$, a finite $C_G$,\n\\[\n|\\E[z_t^jG]|\\le\\exp[-j^2\\log(t/s)/(2f)+C_G]\\longrightarrow0.\n\\]\nFourier uniqueness makes $z_t$ Haar uniform and independent of the entire\nspecified Brownian record. The increment equation then determines the unique\njoint law of $(z,W)$ on every finite positive-time interval. This is the\nclassical eternal-circle mechanism of \\cite{emery}; only its nonlinear\ncontroller reduction is a new candidate here.\n\nSet $H(z)=\\exp(J\\arg z)$, independent of the angle representative. The regular\nmatrix $Y=OH(z)^T$ satisfies\n\\[\n\\dd Y=YB\\dd W+YG\\dd t,\\quad\nB=HN_rH^T,\\qquad G=H(C_r-N_rN_0)H^T,\\qquad Y(0)=O_0.\n\\]\nThese coefficients depend only on $(z,W)$, with finite squared-noise and\nabsolute-drift integrals. The linear equation has a unique continuous solution\nand preserves $SO(3)$. Reconstructing $O=YH$, $x=\\sqrt R e+Qn$ and\n$b=\\sqrt U n$ proves joint weak uniqueness.\nFor existence take stationary circular Brownian motion on its entire log-time\naxis $u=(\\log t)/f$ and define\n$W_t=\\int_{-\\infty}^{(\\log t)/f}\\sqrt{fe^{fu}}\\dd\\beta_u$.\nIts quadratic variation is $t$ and its circle solves the equation above.\nThe regular reconstruction then constructs the required weak entrance.\n\nIf the vectors were measurable from their specified $W$ alone, so would be\n$z_t$, contradicting its independent nondegenerate Haar law. This concerns that\ninnovation, not every possible Brownian generator of the solution filtration;\nthe latter distinction is essential in \\cite{emery}. Finally transverse-seed\ntightness was proved above, and each subsequential limit is in this uniquely\nidentified writing class. Hence the whole seed family converges.\n\\end{proof}\n\n\\subsection{A writing entrance from the complete origin}\nThe completely zero state has a different scaling. Existence is not inferred\nfrom evaluating a singular projector there. We construct a positive stationary\nlaw in logarithmic time and verify continuity of the observable vectors.\n\n\\begin{theorem}[Unique selfsimilar scalar origin writing law]\\label{thm:origin}\nFor $\\nu>0$ and $\\kappa=2\\nu f>0$, there exists a continuous immediately\nwriting scalar weak entrance from $U=R=Q=0$. It is unique in law in the\nselfsimilar positive writing class. There also exists a rotationally invariant\nselfsimilar vector weak entrance from $(x,b)=(0,0)$. Define a dimensionless\nconstant $c_*$ through the unique positive stationary law below. Then\n\\begin{equation}\\label{eq:origin-means}\n\\boxed{\\E R(t)=\\nu c_*t,\\quad\n\\E U(t)=\\frac{\\kappa\\nu c_*}{2}t^2,\\quad\n\\E Q(t)^2=\\nu(2-c_*)t,\\qquad 1<c_*<2.}\n\\end{equation}\nThe constant is independent of $\\nu,\\kappa>0$ for this model. It is not\na claimed universal constant of nature. Neither uniqueness of all origin\nvector laws nor universal selection by finite seeds is asserted.\n\\end{theorem}\n\\begin{proof}\nFor $s=\\log t$, set\n$U=\\kappa\\nu t^2X(s)$, $R=\\nu tY(s)$, $Q=\\sqrt{\\nu t}\\,z(s)$.\nIt\\^o's rule gives the parameter-free diffusion\n\\begin{equation}\\label{eq:origin-log}\n\\dd X=(Y-2X)\\dd s,\\quad\n\\dd Y=Y(z^2/X-1)\\dd s,\\quad\n\\dd z=\\sqrt2\\dd B-\\tfrac12(1+Y/X)z\\dd s.\n\\end{equation}\nIts domain is $D=\\{0<X\\le Y,\\ z\\in\\R\\}$.\nIt is invariant because $(Y-X)'=Yz^2/X-2(Y-X)$.\nAlso $X(s)\\ge X(0)e^{-2s}$ and $Y(s)\\ge Y(0)e^{-s}$ on finite forward\nintervals. With $H=Y/X$ and $S=Y+z^2$,\n\\begin{equation}\\label{eq:origin-S}\n\\dd S=(2-S)\\dd s+2\\sqrt2 z\\dd B.\n\\end{equation}\nSince $z^2\\le S$, localized expectation and BDG bounds prevent explosion.\nThus the positive-domain diffusion is globally well posed and Feller.\n\nHere is the essential persistence estimate, including its constants.\nLet $\\ell=1/40$. A nonincreasing $C^2$ cutoff $\\chi$ equals one below $\\ell$\nand zero above $2\\ell$; in between put\n$\\chi=1-10r^3+15r^4-6r^5$, $r=(Y-\\ell)/\\ell$.\nThen $|\\chi'|\\le75$. Define\n\\[\nF=\\chi(Y)Y^{-1}e^{-z^2},\\quad G=X^{-1/2}e^{-S},\\quad\n\\mathcal W=S+X/4+F+G.\n\\]\nThis function is proper on $D$: bounded $S$ prevents escape to infinity and\n$G$ diverges if $X\\downarrow0$ with bounded $S$.\nFor the generator $\\mathcal L$ of \\eqref{eq:origin-log}, direct differentiation gives\n\\[\n\\frac{\\mathcal L(Y^{-1}e^{-z^2})}{Y^{-1}e^{-z^2}}\n=-1+[5+H(1-1/Y)]z^2,\\qquad\n\\frac{\\mathcal LG}{G}=-1-H/2+Y+5z^2,\n\\]\nand $\\mathcal L(S+X/4)\\le2-3S/4-X/2$.\nFor $Y\\le\\ell$, $\\mathcal LF\\le-F-Hz^2F/(4Y)$ and\n$\\mathcal LG\\le-(1/2+H/2)G+5z^2G$.\nThe positive term is absorbed since\n$20Y^{3/2}e^{-Y}/\\sqrt H<1/2$, giving\n$\\mathcal L(F+G)\\le-F-G/2$.\nFor $Y\\ge\\ell$, $F\\le40$ and $\\mathcal LF\\le75$:\nthe $\\chi'Hz^2$ term is nonpositive and the remaining cutoff derivative is\nat most $75e^{-z^2}$. If $\\mathcal LG+G/4>0$, then $H<10S$ and\n\\[\nG\\le20\\sqrt S e^{-S},\\qquad\n(\\mathcal LG+G/4)_+\\le100S^{3/2}e^{-S}\\le41.\n\\]\nCombining both regions proves the global bound\n\\begin{equation}\\label{eq:origin-foster}\n\\boxed{\\mathcal L\\mathcal W\\le128-\\mathcal W/4.}\n\\end{equation}\nDynkin's formula, localization and Gronwall give\n$\\E\\mathcal W(s)\\le e^{-s/4}\\mathcal W(0)+512(1-e^{-s/4})$.\nProperness makes occupation measures tight inside $D$. The Feller occupation\nargument supplies an invariant probability $\\pi$, with $\\pi\\mathcal W\\le512$.\nThe face $X=Y$ immediately enters the interior under Brownian forcing, so\n$\\pi$ is supported on $0<X<Y$.\n\nWe also establish uniqueness, rather than merely choose an invariant law.\nFor drift $b=(Y-2X,Yz^2/X-Y,-(1+Y/X)z/2)$ and noise $V_1=\\partial_z$,\n\\[\n[\\partial_z,b]=(0,2Yz/X,-(1+Y/X)/2),\\quad\n[\\partial_z,[\\partial_z,b]]=(0,2Y/X,0)=V_2.\n\\]\nThe $X$ component of $[V_2,b]$ is $2Y/X\\ne0$.\nThese fields span all three directions. The localized parabolic\nH\\\"ormander theorem \\cite{hormander} gives smooth transition densities and\nthe strong Feller property in the interior.\nA common accessible point follows from explicit controls. Put $a=X/Y$;\ncontrolled paths obey\n\\[\na'=1-a-z^2/Y,\\qquad (\\log X)'=1/a-2.\n\\]\nConversely prescribe a smooth $0<a<1$ with $a'\\le1-a$, solve the second\nequation and choose $z^2=Y(1-a-a')$. Holds with $a<1/2$ grow $X$ by any\ndesired factor; holds with $a>1/2$ shrink it. Feasible ratio transitions and\nsmooth changes of $z$ connect these holds. Their durations adjust the final\nmagnitude. A final $a=1/2$ segment can reach any neighborhood of\n$(X,Y,z)=(1,2,1)$. The noise control realizes each smooth $z$ path.\nSupport follows directly here: subtract the additive noise from $z$,\nsolve the locally Lipschitz random ODE on a compact positive-domain tube,\nand use continuity in the uniform driving path and positive Wiener probability\nof each Cameron--Martin tube. This is the localized mechanism behind the\nsupport theorem \\cite{stroock}, and gives a common accessible interior point. Strong Feller plus this accessibility excludes two invariant\nprobabilities. This does not assert reachability of every state at every short time.\n\nWe record the moments needed to justify the singular inverse time change.\nEquation~\\eqref{eq:origin-S} gives $\\pi S=2$. Since\n$(-\\log X)_+\\le2G+2S$ and $\\log_+X\\le S$,\n$\\pi|\\log X|<\\infty$. Stationarity of $(\\log X)'=H-2$ gives $\\pi H=2$.\nThe square equation for $z$ then gives $\\pi[(1+H)z^2]=2$, with its\nmartingale square integrable because $\\pi z^2\\le2$. Hence\n\\[\n\\pi(H|z|)\\le2,\\quad \\pi\\sqrt H\\le\\sqrt2,\n\\quad\\pi\\sqrt{SH}\\le2.\n\\]\nTake a two-sided stationary diffusion and set\n$W_t=\\int_{-\\infty}^{\\log t}e^{s/2}\\dd B_s$; its quadratic variation is $t$.\nThe inverse time change recovers the original scalar equations on $t>0$.\nStationarity and BDG applied to \\eqref{eq:origin-S} bound expected unit-block\nsuprema of $S$ uniformly. Markov's inequality and Borel--Cantelli show\n$e^{-n}\\sup_{[-n-1,-n]}S\\to0$ almost surely. Thus $R,Q,U\\to0$ almost surely\nas $t\\downarrow0$. The improper $Q$ drift is absolutely integrable because\n\\[\n\\E\\int_0^T\\frac{\\kappa R|Q|}{2U}\\dd t\n=\\frac{\\sqrt\\nu}{2}\\int_0^Tt^{-1/2}\\pi(H|z|)\\dd t<\\infty.\n\\]\nThe other two equations have nonnegative integrable derivatives. They all\nhold in integrated form from zero. Stationarity implies selfsimilarity.\nConversely any selfsimilar positive scalar origin entrance induces an\ninvariant law in logarithmic time (and $X\\le Y$ follows from\n$U=\\kappa\\int_0^tR\\le\\kappa tR$). Uniqueness of $\\pi$ and positive-time\nwell-posedness therefore establish uniqueness in this selfsimilar class.\n\nFor a vector lift, augment the stationary base by the frame equation in the\npreceding proof, with time and scalar coefficients transformed to logarithmic\ntime. Its body coefficients depend only on $(X,Y,z)$ and it is a\nright-multiplicative orthogonal flow. Conditional on the base and Brownian\nrecord, this flow preserves Haar measure. Consequently\n$\\pi\\times\\mathrm{Haar}_{SO(3)}$ is invariant for the joint process and has a\ntwo-sided stationary realization. Define\n\\[\nx(t)=\\sqrt{\\nu t}\\,[\\sqrt{Y(\\log t)}e(\\log t)+z(\\log t)n(\\log t)],\\quad\nb(t)=\\sqrt{\\kappa\\nu}\\,t\\sqrt{X(\\log t)}n(\\log t).\n\\]\nThe It\\^o product rule recovers the colored vector controller. The preceding\nblock bound proves $x,b\\to0$ almost surely. Its coordinate drift is bounded\nin expectation by a constant times $t^{-1/2}\\pi\\sqrt H$, and memory drift\nby a constant times $\\pi\\sqrt{SH}$. Both integrate at zero. The coordinate\ndiffusion is bounded, and the memory diffusion has squared magnitude with\nfinite expectation proportional to $t\\pi Y$. Thus this is a continuous\nrotationally invariant vector weak writing entrance, including time zero.\n\nFinally let $c_*=\\pi Y$. Since $H>1$ and the additive noise makes\n$\\pi z^2>0$, $\\pi[(1+H)z^2]=2$ implies $0<\\pi z^2<1$.\nTogether with $\\pi S=2$ and stationarity of $X'=Y-2X$ this yields\n$1<c_*<2$, $\\pi X=c_*/2$ and \\eqref{eq:origin-means}.\n\\end{proof}\nAn explicit invariant density or a numerical value of $c_*$ has not been\ncertified. This proof closes selfsimilar scalar uniqueness and vector existence;\nit does not classify all nonselfsimilar origin laws. At $f=0$ the integrated\nequation instead forces $U(t)=0$ from $U(0)=0$, excluding positive writing.\nThe positive-$f$ entrance uses indefinitely many log-time frame rotations\nnear zero. A finite-bandwidth apparatus regularizes that region through an\nactual seed and schedule; its selection of this law remains to be tested.\n\n","labels":["sec:zero","thm:entrance","thm:vector-entrance","thm:origin","eq:origin-means","eq:origin-log","eq:origin-S","eq:origin-foster"],"source_line_start":1289,"source_line_end":1689,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_15","section_index":15,"title_latex":"Multiple drivers: valid limits and rank-burst collapse","text_latex":"\\section{Multiple drivers: valid limits and rank-burst collapse}\\label{sec:multi}\nA higher-dimensional noise source is not obtained simply by replacing the scalar\nBrownian motion with a vector in a kernel projector equation.\nLet $P=P_{\\ker A}$, $w=Px$, and $h=A^+x$. Every existing absolutely continuous\nlegal path is isospectral, so it remains on a fixed-rank stratum. Its tangent\nmust have zero kernel--kernel block:\n\\begin{equation}\\label{eq:tangency}\nP\\dot A P=w\\wedge v=0.\n\\end{equation}\nWhen $w\\ne0$, this confines its kernel velocity to the line through $w$.\nFor an actual legal path, differentiation gives\n$\\dot P=-hv^T-vh^T$, $\\dot w=v-h(v\\cdot w)$ and\n$\\dd(\\norm x^2-\\norm w^2)/\\dd t=0$.\n\nFor fixed input columns $g_i$, put $p_i=Pg_i$ and\n$F_i=(p_i,x\\wedge p_i)$. If two projected Brownian input directions are\nindependent, a pure continuous Stratonovich equation with these fields cannot\nhave a local solution satisfying the fixed-rank chain rules. Tangency requires\n$w\\wedge p_i=0$ for each driver; with two independent directions it forces\n$w=0$ on a neighborhood. But then $\\dd w$ has nonzero diffusion coefficient\n$PG$, contradicting $w\\equiv0$. This is a nonexistence result for that proposed\ncontinuous equation, not for every discrete controller or controlled path.\n\n\\subsection{The actual generic even-dimensional pulse limit}\nSuppose each proposed pulse, conditioned on the preceding history, has a\nnondegenerate density on the current kernel. This holds for full ambient\nGaussian forcing and nondegenerate fresh OU innovations, for example\n$NN^T\\succ0$ with input map $G$ surjective onto the ambient space. If the initial\ncorank is $k$, put $m=\\lfloor k/2\\rfloor$. For $m>0$, rank saturates almost\nsurely after exactly\n\\[\nN=\\begin{cases}2m,&w_0=0,\\\\2m-1,&w_0\\ne0.\\end{cases}\n\\]\nFor $m=0$ it is already maximally ranked; freezing then holds only in even dimension. Indeed a nonzero $w$ and a generic new\nstroke are independent, creating rank two and setting the new kernel component\nof $x$ to zero. A stroke from $w=0$ preserves rank and creates a nonzero kernel\ncomponent. These two cases alternate by the exact kernel transport theorem.\nIn even dimension the final kernel is zero, so all subsequent motion freezes.\nIf each of these finitely many strokes is $O_P(\\sqrt h)$, their entire prefix\nchanges $x,A$ by $O_P(\\sqrt h)$. On every fixed time horizon the controller\ntherefore converges uniformly in probability to its stationary initial state.\nFull rank at each finite $h$ can converge to a lower-rank matrix. This identifies\nthe actual collapse rather than assigning a spurious diffusion to the singular\nprojector. Odd dimension leaves one permission line, but vanishing newly born\nfrequencies can still produce a singular entrance.\n\n\\subsection{A matrix colored-noise law on a valid seeded sector}\nA smooth invariant sector does exist: retain a unit kernel line $n$, require\n$w=Qn$ and $PG=n\\ell^T$, and seed the nonzero image spectrum away from zero.\nFinite legal pulses along $n$ preserve this sector and rank. The remaining\nkernel directions stay orthogonal to every input column. Here the smooth-field\ncolored-noise calculus is valid.\n\nFor stationary $m$-channel OU velocity\n$\\dd\\eta=-M\\eta\\dd t/\\tau+N\\dd B/\\tau$, with all eigenvalues of $M$ having positive real parts, write\n$\\operatorname{Cov}(\\eta)=C/\\tau$ and $MC+CM^T=NN^T$.\nFor strokes integrated over $h=c\\tau$, direct covariance integration yields\n\\begin{equation}\\label{eq:matrix-ou}\n\\boxed{\\begin{aligned}\n\\Sigma&=M^{-1}C+CM^{-T},\\\\\nK(c)&=c^{-1}C(M^T)^{-2}(I-e^{-M^Tc}),\\\\\nC_0(c)&=\\Sigma-K(c)-K(c)^T,\\qquad\n\\Gamma(c)=\\tfrac12(K(c)-K(c)^T).\n\\end{aligned}}\n\\end{equation}\nHere $\\Sigma$ is the accumulated Brownian covariance, $C_0$ is the resolved\nstroke-square covariance per unit time, and $K$ is the earlier--later covariance\nsum. The limit is\n\\[\n\\dd Y=\\sum_iF_i(Y)\\dd W^i+\n\\sum_{i,j}K_{ij}DF_j(Y)F_i(Y)\\dd t,\n\\qquad \\dd[W^i,W^j]=\\Sigma_{ij}\\dd t.\n\\]\nIts Stratonovich correction includes the area anomaly\n\\[\n\\sum_{i<j}\\Gamma_{ij}[F_i,F_j].\n\\]\nThe enhanced convergence assumption is, on stopped $C^3$ smooth domains,\n\\[\nZ_h\\Rightarrow W,\\qquad\n\\sum_{p<q<t/h}S_p^iS_q^j\\Rightarrow\n\\int_0^tW^i\\dd W^j+K_{ij}t\n\\]\nin a sufficient nongeometric/It\\^o enhanced-path topology. The raw ordered\nincrements above are not the geometric lift of a polygonal interpolation;\nthat alternative lift requires the corresponding Euler correction.\nFirst-level Brownian covariance alone does not determine the equation.\nExponentially mixing Gaussian OU data supply this enhanced limit; the explicit\nmatrix formulas follow by integrating $Ce^{-M^Tt/\\tau}/\\tau$ over the held\nintervals. Outside this smooth seeded sector a formal enhanced equation does\nnot repair the rank singularity.\n\nWriting $t_i=h\\cdot g_i$ and using the bracket convention\n$[F_i,F_j]=DF_jF_i-DF_iF_j$, differentiation in this sector gives\n\\[\n[F_i,F_j]=(\\ell_jt_i-\\ell_it_j)(n,x\\wedge n).\n\\]\nThis is a legal continuous tangent drift; it changes orientation and future\nrates, but does not directly change the spectral memory or exclusion radius.\nFor $M=I,C=\\nu I$, \\eqref{eq:matrix-ou} gives\n$C_0=2\\nu f(c)I$, $K=\\nu(1-f(c))I$ and $\\Gamma=0$.\nNonreversible channels can have a nonzero area anomaly even at a ratio where\n$C_0=\\Sigma$. Scalarizing such a bath by a single $f$ is generally incorrect.\nThese enhanced-limit methods have established rough-path ancestry \\cite{friz}.\n\n\\subsection{Why three scalar variables do not close higher rank}\nFor a collinear pulse $a=sn$, let $H=\\norm{A^+x}^2$ and\n$J=\\norm{(A^+)^2x}^2$. The exact formulas are\n\\[\nQ'=\\frac{Q+s}{\\sqrt{1+s^2H}},\\quad\nR'=R+\\frac{s^2H(Q+s)^2}{1+s^2H},\\quad\nE'=E+s^2R,\n\\]\n\\[\nH'=\\frac H{1+s^2H}+\n\\frac{s^2(Q+s)^2J}{(1+s^2H)^2}.\n\\]\nThese follow from the kernel graph and solving for the new Moore--Penrose\ninverse in $\\im A\\oplus\\R n$. With a resolved scalar square rate $\\kappa$,\n$\\dot H=\\kappa(-H^2+Q^2J)$. Rank two has $H=R/U$, $J=R/U^2$, closing\nSection 5. Higher rank needs additional inverse-spectral data.\nFor example, the same seven-dimensional skew matrix with squared frequencies\n$(1,2,4)$ and kernel $e_7$ has two states with image-plane squared lengths\n$(1,1,1)$ and $(5/4,1/4,3/2)$. Both have $E=7,R=3,H=7/4,Q=0$ and identical\nfull spectrum, but $J=21/16$ and $45/32$. Their next finite $H$ updates differ.\nA universal three-scalar higher-rank closure is disproved. The following\nfinite polynomial representation supplies a complete replacement.\n\n\\subsection{A complete finite polynomial closure at arbitrary seeded rank}\nThe failure of three scalars does not require an infinite hierarchy. In the\nintrinsic sector $x=u+Qn$, $n\\in\\ker A$, let the positive squared skew\nfrequencies be $\\sigma_j^2$, and let $r_j$ be the squared length of $u$ in\nits corresponding two-plane. For seeded rank $2m\\ge2$, define\n\\[\np(z)=\\prod_{j=1}^m(z+\\sigma_j^2),\\quad\nN(z)=p(z)\\sum_{j=1}^m\\frac{r_j}{z+\\sigma_j^2},\\qquad H=N(0)/p(0).\n\\]\nRepeated frequencies cause no ambiguity: only their combined weight matters.\nThere are $2m+1$ scalar coefficients including signed $Q$.\n\\begin{theorem}[Complete intrinsic spectral closure]\\label{thm:polynomial}\nFor every pulse $a=sn$ with the kernel line canonically transported, the exact\nnext intrinsic quotient state is\n\\[\n\\boxed{\\begin{aligned}\np'(z)&=p(z)+s^2N(z),\\\\\nN'(z)&=N(z)+\\frac{s^2(Q+s)^2}{1+s^2H}\\frac{Hp(z)-N(z)}z,\\\\\nQ'&=(Q+s)/\\sqrt{1+s^2H}.\n\\end{aligned}}\n\\]\nThe quotient in the second line is polynomial since $Hp(0)-N(0)=0$.\nThe coefficient description determines every future intrinsic scalar update;\nexternally labelled input directions still require orientation data.\n\\end{theorem}\n\\begin{proof}\nOn the image write $A=B$, and set\n$\\ell(t)=u^T(tI-B)^{-1}u=tN(t^2)/p(t^2)$.\nThe new skew block on $\\im A\\oplus\\R n$ is\n$\\left(\\begin{smallmatrix}B&su\\\\-su^T&0\\end{smallmatrix}\\right)$.\nIts determinant gives $p'=p+s^2N$. Its bordered inverse gives\n\\[\nx'^T(tI-A')^{-1}x'=\n\\frac{t\\ell(t)+(Q+s)^2}{t+s^2\\ell(t)}.\n\\]\nSubtract its zero-mode residue $Q'^2/t$, divide by $t$ and multiply by\n$p'(t^2)$ to obtain $N'$. Orthogonal normal form shows that frequencies,\nimage weights and signed $Q$ determine every next quotient state.\n\\end{proof}\n\nFor a continuous additive driver $Z$, a resolved square budget $\\kappa t$, and\nvanishing cubic remainders, the coefficient limit is exactly\n\\[\n\\boxed{\\dd p=\\kappa N\\dd t,\\qquad\n\\dd N=\\kappa Q^2\\frac{Hp-N}z\\dd t,\\qquad\n\\dd Q=\\dd Z-\\frac\\kappa2HQ\\dd t.}\n\\]\nFor sampled scalar OU input, $Z=\\sqrt{2\\nu}W$ and $\\kappa=2\\nu f(c)$.\nTaylor expansion and the same positive-clock convergence argument used in\nLemma~\\ref{lem:pulse-limit} prove the limit. It is globally unique in the\nseeded intrinsic sector. Indeed the smallest frequency never decreases,\n$H\\le R/\\sigma_*^2$, and the contraction for $Q$ bounds it by\n$|Q_0|+2\\sup|Z|$. Thus $\\dot R=\\kappa HQ^2$ bounds $R$ exponentially on\ncompact intervals. The energy coefficient and all other polynomial coefficients\nremain bounded; $p(0)$ stays positive. The positivity domain can also be preserved for a continuous additive input\nby splitting exact intrinsic continuous flows (which shift $Q$ and preserve\n$p,N$) with pairs of pulses $+\\epsilon,-\\epsilon$. The latter implement\nvariance-clock increments $2\\epsilon^2$ with the displayed drift and vanishing\ncubic errors. These approximations preserve the seeded gap. The compact bounds\nand coefficient uniqueness identify their limit. Smooth coefficient equations\nthen remove localization, including at repeated frequencies. At $m=1$ this reduces exactly\nto $p=z+U$, $N=R$ and the original three-scalar system. Minimality of the\n$2m+1$ coefficient count is not claimed. At $m=0$, intrinsic line input is the\ntrivial case $p=1,N=H=0$ and $Q$ follows its additive driver.\n\n","labels":["sec:multi","eq:tangency","eq:matrix-ou","thm:polynomial"],"source_line_start":1689,"source_line_end":1882,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_16","section_index":16,"title_latex":"Exact finite-duration realization in a powered hybrid model","text_latex":"\\section{Exact finite-duration realization in a powered hybrid model}\\label{sec:finite-rate}\nThe endpoint rule need not be an instantaneous impulse. Latch a direction\n$h\\parallel b_0$ and use finite positive filter time constants:\n\\[\n\\dot x=v,\\quad \\tau_x\\dot v+v=u(t)h,\\qquad\n\\dot b=j,\\quad \\tau_b\\dot j+j=x\\times v.\n\\]\nStarting with $v(0)=j(0)=0$, target $a=\\zeta h$ over any duration $T>0$.\nWith $r=t/T$, choose\n\\[\ny(t)=\\frac{140\\zeta}{T}r^3(1-r)^3,\\quad\nz=y+\\tau_b\\dot y,\\quad u=z+\\tau_x\\dot z.\n\\]\n\\begin{theorem}[Finite-bandwidth endpoint realization]\nFor this command the endpoint is exactly\n\\[\n\\boxed{x(T)=x_0+a,\\quad b(T)=b_0+x_0\\times a,\n\\qquad v(T)=j(T)=0.}\n\\]\nThe scalar command is continuous and zero at both boundaries. No infinite\nsettling or reset of a nonzero filter current is needed in this model.\n\\end{theorem}\n\\begin{proof}\n$v=hz$ solves the position filter. The path remains on the affine line through\n$x_0$ in direction $h$, so $x\\times v=(x_0\\times h)z$ and\n$j=(x_0\\times h)y$ solves the area filter. Since $y,\\dot y,\\ddot y$ vanish at\nboth boundaries, $j,v,u$ do too. Also $\\int_0^Ty\\dd t=\\zeta$ and\n$\\int_0^Tz\\dd t=\\zeta$. Integration proves the endpoints.\n\\end{proof}\nThe direction is latched at launch, rather than continuously gated against the\nchanging $b(t)$. The waveform may reverse near its endpoint; that is calibrated\nfilter compensation. Known additional filter stages can be treated by further\npolynomial derivative compensation and higher-order endpoint zeros. Strict\nFourier band limitation is not claimed for a compact-duration command.\n\nFor arbitrary inputs an effective measured register\n$b_{\\rm eff}=b+\\tau_bj$ satisfies\n$\\dot b_{\\rm eff}=x\\times\\dot x$ exactly in this model. Without this correction,\nthe area error after an arbitrary stopped stroke is\n$\\tau_b[j(0)-j(T)]$. Effective memory is a combination of two physical registers;\nits norm must not be mistaken for all stored capacitor energy.\n\nThe sextic command has the explicit amplitude budget\n\\[\n\\norm u_\\infty\\le |\\zeta|\\left[\n\\frac{35}{16T}+\\frac{84(\\tau_x+\\tau_b)}{5\\sqrt5\\,T^2}\n+\\frac{105\\tau_x\\tau_b}{2T^3}\\right].\n\\]\nFinite input range therefore imposes a genuine speed limit. As $T$ decreases at\nfixed filter bandwidth, the compensation grows as $T^{-3}$.\nThe finite-rate construction cannot by itself justify a zero-duration noise\nlimit at bounded input power and fixed hardware bandwidth.\n\n\\subsection{Dimensions, supply work and error budget}\nVoltage encodings $V_x=V_*x$, $V_b=B_*b$ and capacitors $C_x,C_b$ use physical\ncurrents $I_x=C_xV_*v$, $I_b=C_bB_*j$. An area-current command is\n\\[\nI_b^{\\rm command}=\\frac{C_bB_*}{C_xV_*^2}V_x\\times I_x.\n\\]\nThe prefactor has inverse-voltage units, so this is dimensionally correct.\nSix four-quadrant multiplier channels compute the cross product; three more\nmultiply the held direction by the scalar command. An internal clock, latch\nand finite-state sequencer provide operational autonomy under power. This is\na specified circuit model, not a fabricated circuit or passive material.\n\nMemory-capacitor energy at the settled endpoints increases by\n$\\Delta E_b=C_bB_*^2\\area/2$. Full supply work must also include changes to\nposition, filters and latches, resistor and amplifier heat, and the independent\nprobe work. The first-law accounting cannot identify geometric writing with\nall thermodynamic dissipation. Finite dynamic range $U_{\\max}$ implies, in the\nseeded noisy regime, saturation no later than\n$(U_{\\max}-U_0)/(2\\nu fR_0)$ when $\\nu fR_0>0$; resets or rescaling change the\noriginal rule and require their own account.\n\nIf the implemented dynamics has defects\n$\\dot x=v+e_x$, $\\dot b=j+e_b$,\n$\\tau_b\\dot j+j=x\\times v+e_m$, then\n\\[\n\\dot b_{\\rm eff}-x\\times\\dot x=e_b+e_m-x\\times e_x.\n\\]\nThis gives an integrated error bound\n$\\int_0^T(\\norm{e_b}+\\norm{e_m}+X_{\\max}\\norm{e_x})\\dd t$ whenever\n$\\norm x\\le X_{\\max}$. Unequal axis response also creates loop area even when\nendpoint position is correct; calibration must measure it.\nAn additive skew-memory defect $D_A$ adds at most\n$\\norm{D_A}_{\\rm op}\\norm{x'}^2/\\lambda$ to the per-step loss allowance in\nTheorem~\\ref{thm:ridge-error}. This connects the finite-rate error budget to the\nrank-continuous certificate.\n\nFor the finite-stage optical telescope, blank loss during the write and dwell\nat the endpoint. Residual off-state loss contributes at most\n$\\sum_j\\int_{\\mathrm{write},j}\\gamma_{\\rm off}(t)U(t)\\dd t$ to survival error,\nin addition to finite-dwell errors: each write propagator differs from the\nidentity by at most half that integral, and products of contractions telescope.\nContinuously active loss requires the time-ordered propagator instead.\n\n\\subsection{A concrete precision, storage and latency comparison}\nThree-dimensional registers require six real scalars, or $6p$ state bits at\n$p$ fixed-point bits each, before latches and arithmetic. An unnormalized\nstroke needs nine multiplications and nine additions/subtractions. No\ntechnology advantage follows from these operation counts alone.\nFor a specific equilibrium capacitor amplitude encoding with $2^p$ equally\nspaced levels over $[-V_{\\max},V_{\\max}]$, sampled Gaussian noise\n$\\sigma_V^2=\\kT/C$ and one-sided error target $\\epsilon$, the level spacing\nrequires\n\\[\n\\frac12CV_{\\max}^2\\ge\n\\frac12q_\\epsilon^2\\kT(2^p-1)^2,\n\\qquad q_\\epsilon=\\Phi^{-1}(1-\\epsilon).\n\\]\nThis is an encoding-specific bound, not a universal memory-energy law.\nAt $300$ K and $\\epsilon=10^{-6}$, it gives about $0.201$ nJ per scalar at\n16 bits and $51.5$ nJ at 20 bits. Leakage, offsets and active-circuit noise add\ncost. A valid benchmark must compare joules per certified update, update rate,\nrange, reliability, retention and reset cost at the same task and precision.\nThe model now supplies those quantities to measure; it does not supply a\nmeasured winner.\n\n","labels":["sec:finite-rate"],"source_line_start":1882,"source_line_end":2000,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_17","section_index":17,"title_latex":"The complete cone of quadratic spectral memory observables","text_latex":"\\section{The complete cone of quadratic spectral memory observables}\n\\label{sec:spectral-cone}\nThe ridge family belongs to a substantially larger exact monotone cone.\nThe classification below concerns finite legal strokes\n$Aa=0$, $y=x+a$, $A_+=A+x\\wedge a$ and observables of the specific form\n$x^T\\phi(-A^2)x$. It does not classify finite reachability or all nonlinear\nLyapunov functions. The aligned rank-one trace inequality used in its proof\nis classical generalized Klein monotonicity \\cite{klein}; its smooth version\nalso follows from established Fr\\'echet matrix calculus \\cite{rubensson}.\nThe own-area endpoint spectral transport is derived directly here.\n\n\\subsection{Exact endpoint spectral-measure transport}\nFor $a\\ne0$, write $a=sn$, $s=\\norm a>0$, $\\norm n=1$, $An=0$, and\n\\[\nx=u+Qn+\\beta e,\\qquad u=P_{\\im A}x,\\qquad q=Q+s,\n\\]\nwhere $e\\in\\ker A$ is a unit vector perpendicular to $n$ and $\\beta\\ge0$;\nomit $e$ when $\\beta=0$. Let $t_j=\\sigma_j^2>0$ be the distinct squared\nfrequencies visible to $u$, with weights $r_j=\\norm{P_{t_j}u}^2>0$.\nEach repeated-frequency eigenspace contributes one cyclic two-plane;\ndirections perpendicular to this plane have zero displacement weight.\nSet $H_0=\\diag(t_j)$ and $v=(\\sqrt{r_j})_j$. If $\\beta>0$, append a zero\ndiagonal entry and append $\\beta$ to $v$. Define\n\\[\nH_1=H_0+s^2vv^T,\\qquad\n\\mu_i(B)=v^TP_B(H_i)v.\n\\]\nThe old positive displacement measure is\n$\\mu_A=\\sum_jr_j\\delta_{t_j}$; in the rank-birth case $\\mu_0$ additionally\nhas the atom $\\beta^2\\delta_0$.\n\n\\begin{theorem}[Exact own-area spectral transport]\\label{thm:spectral-transport}\nOn the positive half-line, the new displacement measure is exactly\n\\begin{equation}\\label{eq:spectral-transport}\n\\boxed{\\mu_{A_+}(\\dd\\ell)\n=\\left(1+\\frac{s^2q^2}{\\ell}\\right)\\mu_1(\\dd\\ell),\\qquad\\ell>0.}\n\\end{equation}\nThe formula holds with repeated eigenvalues and with rank birth.\nThe zero stroke is the identity; when $v=0$ both positive measures vanish.\n\\end{theorem}\n\\begin{proof}\nChoose cyclic plane bases $(e_j,f_j)$ with $Ae_j=\\sigma_jf_j$ and\n$Af_j=-\\sigma_je_j$. On the two orthogonal blocks\n$E=\\operatorname{span}\\{e_j,e\\}$ and\n$F=\\operatorname{span}\\{f_j,n\\}$, omitting $e$ when absent, the new skew\nmatrix and endpoint have the form\n\\[\nA_+=\\begin{pmatrix}0&-T^T\\\\T&0\\end{pmatrix},\\qquad\nT=\\begin{pmatrix}D\\\\-sv^T\\end{pmatrix},\\qquad\nD=(\\diag(\\sigma_j)\\ \\ 0),\\qquad y=v+qn.\n\\]\nThe zero column of $D$ is present only when $e$ is present. Thus the blocks\nof $-A_+^2$ are $T^TT=H_1$ and $TT^T$. For every $\\ell>0$,\n\\[\nn^TP_\\ell(TT^T)n\n=\\frac1\\ell(T^Tn)^TP_\\ell(H_1)(T^Tn)\n=\\frac{s^2}{\\ell}v^TP_\\ell(H_1)v.\n\\]\nThe squared endpoint weights from the two blocks add, proving the result.\nAll omitted inactive planes and kernel directions remain uncoupled.\n\\end{proof}\n\nWriting $R=\\norm u^2$, the exact total positive mass is\n\\begin{equation}\\label{eq:spectral-mass}\nR_+=R+\\beta^2+s^2q^2v^TH_1^{-1}v.\n\\end{equation}\nThe inverse is used only on the nonempty active block. If $\\beta=0$ and\n$h=v^TH_0^{-1}v$, it gives\n$R_+=R+s^2q^2h/(1+s^2h)$.\nIf $\\beta>0$, then $H_1^{-1}v=e_0/(s^2\\beta)$, giving\n$R_+=R+\\beta^2+q^2=\\norm y^2$.\n\n\\subsection{A nonsmooth rank-one order lemma}\n\\begin{lemma}[Aligned rank-one spectral order]\\label{lem:rankone-order}\nLet $B=C+c vv^T$, $c>0$, for finite nonnegative symmetric matrices.\nFor every scalar nondecreasing finite-valued $\\phi$ on their spectra,\n$v^T\\phi(B)v\\ge v^T\\phi(C)v$.\nIf $v\\ne0$ and $\\phi$ is strictly increasing on the endpoint spectra,\nthe inequality is strict.\n\\end{lemma}\n\\begin{proof}\nLet $(\\alpha_i,u_i)$ and $(\\beta_j,w_j)$ be orthonormal spectral\ndecompositions of $C$ and $B$. The elementary trace expansion is\n\\begin{align*}\nc\\,v^T[\\phi(B)-\\phi(C)]v\n&=\\tr\\{[\\phi(B)-\\phi(C)](B-C)\\}\\\\\n&=\\sum_{i,j}[\\phi(\\beta_j)-\\phi(\\alpha_i)]\n(\\beta_j-\\alpha_i)|u_i^Tw_j|^2\\ge0.\n\\end{align*}\nNo continuity, divided-difference convention, or eigenvalue simplicity is\nrequired. Under strict scalar monotonicity, equality would require all\noverlapping eigenvalue pairs to agree. This contradicts\n$\\tr(B-C)^2=c^2\\norm v^4>0$.\n\\end{proof}\nThe probe vector's alignment with the perturbation is essential: scalar\nmonotonicity does not imply operator monotonicity. For example,\n$C=\\diag(1,4)$, $v=(1,1)^T$, $B=C+vv^T/2$ and $w=(2,-1)^T$ give\n$w^TB^2w-w^TC^2w=-3/2$.\n\n\\subsection{Necessity and sufficiency of the full cone}\n\\begin{theorem}[Complete quadratic spectral cone]\\label{thm:complete-cone}\nLet $\\phi:[0,\\infty)\\to\\R$ be finite-valued. The observable\n\\[\nM_\\phi(x,A)=x^T\\phi(-A^2)x\n\\]\nis nondecreasing under every exact legal own-area stroke in every finite\ndimension if and only if\n\\begin{equation}\\label{eq:complete-cone}\n\\boxed{\\phi(0)=0,\\qquad\n\\phi\\text{ is nonnegative and nondecreasing on }(0,\\infty).}\n\\end{equation}\nDimension three supplies all necessity tests. The class includes\ndiscontinuous functions and functions unbounded on the half-line.\n\\end{theorem}\n\\begin{proof}\nFor sufficiency, zero kernel weight and Theorem\n\\ref{thm:spectral-transport} give\n\\begin{align}\\label{eq:cone-increment}\n\\Delta M_\\phi\n&=v^T[\\phi(H_1)-\\phi(H_0)]v\n+s^2q^2\\int_{(0,\\infty)}\\frac{\\phi(\\ell)}\\ell\\mu_1(\\dd\\ell)\n\\ge0.\n\\end{align}\nThe two terms are separately nonnegative by Lemma\n\\ref{lem:rankone-order} and scalar positivity.\n\nFor necessity, at $A=0$ the collinear strokes $a=-x$ and $a=x$ force\n$\\phi(0)\\le0$ and $\\phi(0)\\ge0$, respectively. Perpendicular nonzero\nstrokes from $A=0$ create any prescribed positive squared frequency,\nforcing $\\phi(t)\\ge0$ for every $t>0$. Finally take a three-dimensional\nrank-two seed with $(U,R,Q)=(t,1,-s)$ and legal pulse $a=sn$. Its exact\nendpoint has $(U_+,R_+,Q_+)=(t+s^2,1,0)$. Hence\n$\\phi(t+s^2)\\ge\\phi(t)$ for arbitrary $t,s^2>0$.\n\\end{proof}\n\nExamples include $\\phi(t)=t^p$ for every $p>0$, $\\log(1+t)$,\n$\\min(t,c)$, and both $\\mathbf1_{[c,\\infty)}(t)$ and\n$\\mathbf1_{(c,\\infty)}(t)$ for $c>0$. The powers $p>1$ extend beyond\nthe complete Bernstein subclass obtainable as positive ridge mixtures.\nIf $\\phi$ is strictly increasing on $(0,\\infty)$, every nonzero area\nstroke strictly increases $M_\\phi$; its nonnegative values and\n$\\phi(0)=0$ also make the extension strictly increasing at zero.\nFor a nonstrict function, equality means exactly that both nonnegative\nterms in \\eqref{eq:cone-increment} vanish. No general strictness is claimed\nfor threshold or projector observables.\n\nIf $\\phi(t)-m t$ itself belongs to the cone, the polynomial law yields\n\\[\n\\Delta M_\\phi\\ge m\\area\\norm{x+a}^2,\n\\]\nwhere $\\area=\\norm x^2\\norm a^2-(x\\cdot a)^2$ is the squared\nparallelogram area, avoiding a matrix-norm normalization ambiguity.\n\n\\subsection{Observable order and its limits}\nOrdering all functions in \\eqref{eq:complete-cone} is equivalent to\nordering every positive spectral tail mass:\n\\[\n\\mu_2([c,\\infty))\\ge\\mu_1([c,\\infty))\\quad\\text{for every }c>0.\n\\]\nThe equivalent open-tail formulation also includes any prescribed jump\nvalue. Balancing total masses by adding zero atoms gives ordinary\nfirst-order stochastic dominance. Every legal own-area transition has\nthis order, but the order is not a complete reachability criterion.\nIn three dimensions $(U,R,Q)=(1,1,0)$ is ordered below $(4,1,0)$.\nNevertheless the first nonzero pulse raises $R$ by\n$s^4/(1+s^2)>0$, and later pulses cannot reduce it; the second state is\nnot finitely reachable, even modulo orthogonal coordinates.\n\nPointwise ridge order between arbitrary states is also weaker than this\nfull cone order. For $\\mu_1=\\delta_4$, $\\mu_2=2\\delta_3$, all ridge responses\nincrease because\n\\[\n\\frac6{z+3}-\\frac4{z+4}\n=\\frac{2z+12}{(z+3)(z+4)}>0,\n\\]\nwhile the tail at four decreases and the cubic certificate falls from\n$64$ to $54$.\n\nThe specified held harmonic probe realizes the ridge subclass; exact\nfinite spectral reconstruction then determines every member of the cone.\nThis does not assert universal passive realizability for arbitrary\n$\\phi$, or stable noisy reconstruction of thresholds. Threshold\ncertificates may jump at frequency crossings, whereas the earlier ridge\nbounds remain rank-continuous.\n\nFinally, the cone does not uniquely characterize the write law. Any rule\n$A_+=A+\\gamma x\\wedge a$, $\\gamma\\in\\R$, preserves it: the transport proof\nreplaces $s^2$ by $\\gamma^2s^2$, while $q=Q+s$ is unchanged. Monotonicity\nalone fixes neither gain nor sign. All results here concern finite\nmatrices and exact finite strokes, not unbounded operators. Discontinuous\ncertificates require a separate convergence argument in continuous or\nstochastic limits.\n\n\n\\subsection{Finite tomography of the observable spectral geometry}\n\\label{sec:finite-tomography}\n\nLet $H=-A^2\\succeq0$. Denote its distinct positive eigenvalues visible to\n$x$ by $t_j=\\sigma_j^2$, and put $r_j=\\|P_{t_j}x\\|^2>0$.\nThe observable measure and ridge response are\n\\begin{equation}\\label{eq:tomographic-response}\n \\mu_x=\\sum_{j=1}^{m_{\\rm obs}}r_j\\delta_{t_j},\\qquad\n F_{\\sqrt z}(x,A)=\\sum_{j=1}^{m_{\\rm obs}}\\frac{r_jt_j}{z+t_j}\n =\\frac{P(z)}{Q(z)},\\quad z>0.\n\\end{equation}\nFor $m_{\nm obs}=0$ the response is identically zero. Otherwise,\nin reduced form $Q(z)=\\prod_j(z+t_j)$ is monic, $\\deg P=m_{\\rm obs}-1$,\nand $P,Q$ are coprime. Indeed the residue at $-t_j$ is the strictly positive\n$r_jt_j$, so no pole cancels. The harmonic equilibrium readout of\nSection~\\ref{sec:ridge} measures this response by varying the known probe\nstiffness scale. The reconstruction below assumes exact response values;\nfinite equilibrium sampling produces an inverse problem with uncertainty.\n\n\\begin{theorem}[Exact finite observable tomography]\\label{thm:finite-tomography}\nIf the number of visible distinct positive eigenvalues is at most $m$, the\nvalues of $F_{\\sqrt z}$ at any $2m$ distinct positive $z$ uniquely determine\nthe reduced rational response and $\\mu_x$. If the actual count is known to\nbe $m$, the corresponding monic rational-interpolation system is nonsingular.\nFor otherwise unrestricted positive weights and distinct frequencies,\nfewer than $2m$ such scalar samples fail to identify the measure locally.\n\\end{theorem}\n\\begin{proof}\nFor two admissible responses, their cross-multiplied difference\n$P_1Q_2-P_2Q_1$ has degree at most $2m-1$. Its $2m$ distinct measured zeros\nmake it identically zero. Equal rational functions have equal negative\nsimple poles and residues, recovering $t_j$ and $r_j$.\nWith the actual degree fixed, the homogeneous interpolation system has\n$\\delta P\\,Q-P\\,\\delta Q=0$, with both variations of degree at most $m-1$.\nCoprimality forces $Q\\mid\\delta Q$, hence both variations vanish.\n\nFor local necessity, differentiate the response with respect to its $2m$\nparameters:\n\\[\n \\delta F(z)=\\sum_j\\left[\n \\frac{t_j\\delta r_j}{z+t_j}\n +\\frac{r_jz\\delta t_j}{(z+t_j)^2}\\right].\n\\]\nAfter multiplication by $Q^2$, its numerator has degree at most $2m-1$.\nVanishing at $2m$ samples therefore gives $\\delta F=0$ identically.\nThe double-pole coefficient $-r_jt_j\\delta t_j$ then forces every\n$\\delta t_j=0$, and the simple-pole residues force every $\\delta r_j=0$.\nThus the $2m$-sample Jacobian is invertible. Any $k<2m$ sample rows can be\nextended to such a matrix and have rank $k$. The implicit-function theorem\ngives local positive-dimensional fibers of measures with the same $k$ data.\n\\end{proof}\n\nWhen $m$ is only an upper bound, coefficient interpolation can have multiple\nsolutions with a common numerator/denominator factor. Cancel that factor,\nor select the minimal denominator, before interpreting its poles. For\nexample, $t=(1,4)$ and $r=(2,3)$ give\n\\[\n F_{\\sqrt z}=\\frac{14z+20}{z^2+5z+4},\\qquad\n F_{\\sqrt z}\\big|_{z=(1/2,1,2,4)}=(4,17/5,8/3,19/10).\n\\]\n\nThe reconstructed measure determines every quadratic spectral functional\n$\\sum_jr_j\\phi(t_j)$ with $\\phi(0)=0$, including thresholded exclusion\nweights. Since $x$ is a measured classical register, its aggregate kernel\nweight is also known:\n\\[\n r_0=\\|x\\|^2-\\sum_jr_j,\\qquad\n x^T\\phi(H)x=r_0\\phi(0)+\\sum_jr_j\\phi(t_j).\n\\]\nIt does not identify eigenspaces invisible to $x$, their multiplicities,\nmode orientation, the full matrix $A$, or its permission kernel.\nThe local sample lower bound counts ridge data alone; independently known\nnormalizations or a known zero kernel weight can reduce the free parameters.\nFor instance, with fixed $x=e_1$, coupling the same skew frequency to $e_2$\nor to $e_3$ gives the same ridge response and different legal control lines.\nNear-coincident poles and weak residues make inversion poorly conditioned.\nExact identifiability is therefore distinct from a stable noisy inverse\ntheorem; the direct ridge-error certificates remain applicable even when\nindividual poles cannot be resolved.\n\n\\paragraph{Correlation tomography has a different data requirement.}\nUnder the independently verified joint-loss averaging hypotheses,\n\\[\n \\chi(q)=C_0+2\\sum_{k\\ge1}C_kq^k,\n \\qquad -C_0\\frac{\\dd\\log S}{\\dd\\log U}=\\chi(q),\n \\quad q=e^{-\\ell U/2}.\n\\]\nAn exact response known on an open interval determines this analytic\nfunction and hence $C_k=\\chi^{(k)}(0)/(2k!)$. This uses arbitrarily precise\ncontinuum response data; finitely many noisy samples cannot stably recover\ninfinitely many unrestricted lags. The covariance spectral measure gives\n$\\chi(q)=\\int (1-q^2)/(1-2q\\cos\\omega+q^2)\\,\\mu_{\\rm noise}(\\dd\\omega)\n\\ge0$, including negative-correlation sources. Rational interpolation,\nspectral measures, and covariance-filtered attenuation have established\nancestry; the model-specific connection is the endogenous area-memory\nresponse and its operational certificates.\n\n\n","labels":["sec:spectral-cone","thm:spectral-transport","eq:spectral-transport","eq:spectral-mass","lem:rankone-order","thm:complete-cone","eq:complete-cone","eq:cone-increment","sec:finite-tomography","eq:tomographic-response","thm:finite-tomography"],"source_line_start":2000,"source_line_end":2293,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_18","section_index":18,"title_latex":"Joint controller and loss scaling: the covariance-filter endpoint law","text_latex":"\\section{Joint controller and loss scaling: the covariance-filter endpoint law}\n\\label{sec:joint-loss}\nThe two sequential limits do not exhaust the readout regimes. A finite loss\nexponent per held interval gives a computable intermediate law. The noisy\nrotator and partial absorber mechanism, including covariance weights set by\nabsorber transmission, has direct ancestry in Kofman--Kurizki--Opatrn\\'y\n\\cite{kofman}, particularly their Eq.~(32). The contribution here is the\nendogenous Avenyra memory endpoint formula and a controlled homogenization\nproof for its state-dependent weights. It is a result within ordinary passive\nloss dynamics, not a new quantum principle or an advantage over classical waves.\n\n\\subsection{Protocol and the joint limit}\nFix $c>0$, $\\nu>0$, and a seeded state $U_0>0$. For controller interval $h$, let\n$\\eta_h$ be stationary OU velocity with correlation time $\\tau_h=h/c$ and\n\\[\n \\E[\\eta_h(t)\\eta_h(s)]=\\frac\\nu{\\tau_h}e^{-|t-s|/\\tau_h},\n \\qquad s_j=\\int_{(j-1)h}^{jh}\\eta_h(t)\\,\\dd t.\n\\]\nThe exact writes and the subsequent held-loss operations are\n\\begin{align}\n x_j&=x_{j-1}+s_j n_{j-1},&\n b_j&=b_{j-1}+s_j(x_{j-1}\\times n_{j-1}), \\label{eq:jwrite}\\\\\n \\psi_j&=[P_j+q_h(U_j)(I-P_j)]\\psi_{j-1},&\n q_h(u)&=e^{-\\ell_hu/2},\\quad \\ell_h=\\gamma_h h,\n \\label{eq:jprobe}\n\\end{align}\nwhere $n_j=b_j/\\sqrt{U_j}$, $P_j=n_jn_j^T$, $\\psi_0=n_0$, and\n$S_j=\\norm{\\psi_j}^2$. These are ordered writes followed by dwell operations;\nloss overlapping a finite hardware write is another protocol. Here $h$ is the\ncalibrated controller time increment and also the model's exposure duration.\nA different exposure duration $d_h$ replaces $\\ell_h$ by $\\gamma_h d_h$;\nadditional write or blanking time must be included when converting the\ncontroller clock to laboratory time.\n\nPut $\\rho=e^{-c}$ and $f=1-(1-\\rho)/c$. Direct integration gives the normalized\nstroke covariances\n\\begin{equation}\\label{eq:jcov}\n C_0=\\frac{\\E s_j^2}{h}=2\\nu f,\n \\qquad C_k=\\frac{\\E[s_js_{j-k}]}h\n =\\frac\\nu c(1-\\rho)^2\\rho^{k-1}\\quad(k\\ge1).\n\\end{equation}\nTheir absorber-weighted generating function is\n\\begin{equation}\\label{eq:jchi}\n \\chi_c(q)=C_0+2\\sum_{k\\ge1}C_kq^k=2\\nu F_c(q),\\qquad\n F_c(q)=1-\\frac{(1-\\rho)(1-q)}{c(1-\\rho q)}.\n\\end{equation}\nFor $\\ell>0$ define\n\\begin{equation}\\label{eq:jendpoint}\n \\Phi_\\ell(u)=\\exp\\!\\left[-\\frac1f\\int_{U_0}^{u}\n F_c(e^{-\\ell v/2})\\frac{\\dd v}{v}\\right],\n \\qquad \\Phi_\\infty(u)=U_0/u.\n\\end{equation}\n\n\\begin{theorem}[Positive joint loss scaling]\\label{thm:joint-loss}\nSuppose $\\ell_h\\to\\ell\\in(0,\\infty]$ while $c$ remains fixed. On every finite\nhorizon, the exact probe satisfies\n\\[\n \\sup_{j\\le T/h}|S_j-\\Phi_{\\ell_h}(U_j)|\\longrightarrow0,\n \\qquad\n \\sup_{j\\le T/h}\\norm{\\psi_j-\n \\sqrt{\\Phi_{\\ell_h}(U_j)}n_j}\\longrightarrow0\n\\]\nin probability and every finite $L^p$. Consequently its joint weak limit with\nthe controller has survival $S(t)=\\Phi_\\ell(U(t))$, where the controller obeys\n\\eqref{eq:core}. On a compact controller localization with bounded square\nbudget, if $\\ell_h\\ge\\ell_{\\min}>0$, the sharper log error is\n\\begin{equation}\\label{eq:jrate}\n \\sup_{j\\le T/h}\\left|\\log S_j+\n \\frac1{C_0}\\int_{U_0}^{U_j}\\chi_c(q_h(v))\\frac{\\dd v}{v}\\right|\n =O_{\\mathbb P}\\!\\left(\\sqrt{h\\log(1/h)}\\right).\n\\end{equation}\nThe constants in this localized rate are uniform over\n$\\ell_h\\ge\\ell_{\\min}$, including $\\ell_h\\to\\infty$.\n\\end{theorem}\n\nSince $f\\le F_c(q)\\le1$, this OU bath has\n\\begin{equation}\\label{eq:jbound}\n (U_0/U)^{1/f}\\le\\Phi_\\ell(U)\\le U_0/U,\n \\qquad\n -\\frac{\\dd\\log\\Phi_\\ell}{\\dd\\log U}\n =\\frac{F_c(e^{-\\ell U/2})}{f}.\n\\end{equation}\nFinite $\\ell$ generally gives an endpoint integral, not a constant power.\nIncreasing loss per interval raises survival for this positively correlated\nbath. The limits $\\ell\\uparrow\\infty$ and $\\ell\\downarrow0$ give exponent one\nand exponent $1/f$, respectively. The latter is a limit of the proved\npositive-$\\ell$ formula, not a theorem for every joint path $\\ell_h\\to0$:\nthat condition alone even permits a fixed finite $\\gamma_h$. Simultaneous\nvariation of $c$, or overlapping write and loss, also requires separate bounds.\n\n\\subsection{Deterministic elimination of the bright lag}\nThe proof starts with exact geometry rather than a guessed stochastic\ninterpretation. Set\n\\[\n g_{j-1}=\\frac{x_{j-1}\\times n_{j-1}}{\\sqrt{U_{j-1}}},\n \\qquad G_{j-1}=\\norm{g_{j-1}}^2=R_{j-1}/U_{j-1}.\n\\]\nThen $n_j=(n_{j-1}+s_jg_{j-1})/\\sqrt{1+s_j^2G_{j-1}}$ exactly.\nWrite $\\psi_j=\\alpha_j(n_j+r_j)$, with $r_j\\perp n_j$. For sufficiently small\nmaximal stroke $m=\\max_{j\\le N}|s_j|$ and $q_j\\le q_*<1$, the scalar amplitude\nstays positive, and direct multiplication gives\n\\begin{equation}\\label{eq:jalpha}\n \\frac{\\alpha_j}{\\alpha_{j-1}}\n =\\frac{1+s_jg_{j-1}\\cdot r_{j-1}}\n {\\sqrt{1+s_j^2G_{j-1}}},\\qquad\n r_j=q_j(r_{j-1}-s_jg_{j-1})+e_j.\n\\end{equation}\nOn compact localization,\n\\[\n |e_j|\\le C\\bigl(s_j^2+|s_j|\\norm{r_{j-1}}\n +|s_j|\\norm{r_{j-1}}^2\\bigr).\n\\]\nA small invariant ball yields $\\sup_j\\norm{r_j}\\le Cm$. Stable geometric\nconvolution and absorption of the $Cm\\norm r_{\\ell^2}$ term yield\n\\[\n \\norm r_{\\ell^2}\\le C\\norm s_{\\ell^2},\\qquad\n \\norm e_{\\ell^2}\\le Cm\\norm s_{\\ell^2}.\n\\]\nThus the error from solving the linear bright filter has $\\ell^2$ norm at most\n$Cm\\norm s_{\\ell^2}$. Freezing $g$ and $q$ over its geometric lags obeys the\nsame bound: $|g_i-g_{i-1}|\\le C|s_i|$ and\n$|q_i-q_{i-1}|\\le Cs_i^2$, and all sums $\\sum k^a q_*^k$ are finite. The\nsecond bound is uniform in $\\ell_h$, since\n$|q_h'(u)|\\le1/(eU_0)$ for $u\\ge U_0$. Therefore\n\\begin{equation}\\label{eq:jbright}\n r_{j-1}=-g_{j-1}\\sum_{k=1}^{j-1}q_j^ks_{j-k}+\\varepsilon_{j-1},\n \\qquad \\norm\\varepsilon_{\\ell^2}\\le Cm\\norm s_{\\ell^2}.\n\\end{equation}\nExpanding \\eqref{eq:jalpha}, using\n$S_j=\\alpha_j^2(1+\\norm{r_j}^2)$, and applying Cauchy--Schwarz to\n$\\sum |s_j|\\norm{\\varepsilon_{j-1}}$ proves the finite pathwise estimate\n\\begin{equation}\\label{eq:jdet}\n \\sup_{j\\le N}\\left|\\log S_j+\n \\sum_{i\\le j}G_{i-1}\\left[s_i^2+\n 2s_i\\sum_{k=1}^{i-1}q_i^ks_{i-k}\\right]\\right|\n \\le C\\left[m\\sum_{i\\le N}s_i^2+m^2\\right].\n\\end{equation}\nThe logarithm's quadratic remainder is smaller,\n$O(m^2\\sum s_i^2)$. Constants depend on the localization and $q_*$.\nThe seed gives $U_j\\ge U_0$, so a lower bound on $\\ell_h$ supplies the required\nuniform $q_*<1$. Gaussian marginal tails give\n$m=O_{\\mathbb P}(\\sqrt{h\\log(1/h)})$, while the square budget is\n$O_{\\mathbb P}(1)$. This explains the rate in \\eqref{eq:jrate}.\n\n\\subsection{An explicit Gaussian Poisson equation for averaging}\nIt remains to average the lag quadratics despite their dependence on the\ncontroller. Let $\\zeta_j=s_j/\\sqrt h$; its stationary law is independent of\n$h$. For deterministic $q\\in[0,q_*]$, put\n\\[\n B_j(q)=\\sum_{k\\ge1}q^k\\zeta_{j-k},\\quad\n X_j(q)=\\zeta_j^2+2\\zeta_jB_j(q),\\quad\n A_n(q)=h\\sum_{j\\le n}[X_j(q)-\\chi_c(q)].\n\\]\nWe prove the uniform maximal estimate\n\\begin{equation}\\label{eq:jmax}\n \\E\\sup_{n\\le T/h}\\sup_{q\\le q_*}\n (|A_n(q)|^2+|\\partial_q A_n(q)|^2)\\le Ch.\n\\end{equation}\nAn OU interval has an exact independent Gaussian innovation pair\n$(\\epsilon_j,\\iota_j)$ with variances and covariance\n\\[\n v_\\epsilon=\\nu c(1-\\rho^2),\\qquad\n v_\\iota=2\\nu-\\nu(3-4\\rho+\\rho^2)/c,\\qquad\n v_\\times=\\nu(1-\\rho)^2.\n\\]\nWriting $a=(1-\\rho)/c$, its sampled endpoint and pulse satisfy\n$\\xi_j=\\rho\\xi_{j-1}+\\epsilon_j$ and\n$\\zeta_j=a\\xi_{j-1}+\\iota_j$. Consequently the preinterval state\n$Z_j=(\\xi_{j-1},B_j)^T$ is a stationary Gaussian chain\n\\[\n Z_{j+1}=F(q)Z_j+v_j(q),\\qquad\n F(q)=\\begin{pmatrix}\\rho&0\\\\qa&q\\end{pmatrix},\\quad\n v_j(q)=\\binom{\\epsilon_j}{q\\iota_j}.\n\\]\nLet $\\Sigma(q)$ be its stationary covariance and\n$H=\\left(\\begin{smallmatrix}a^2&a\\\\a&0\\end{smallmatrix}\\right)$. Conditioning\non the innovation past $\\mathcal G_{j-1}$ gives\n\\[\n \\E[X_j\\mid\\mathcal G_{j-1}]=Z_j^THZ_j+v_\\iota,\n \\qquad \\chi_c(q)=\\tr(H\\Sigma(q))+v_\\iota.\n\\]\nUniform stability of $F$ supplies the explicit Lyapunov solution\n\\[\n K(q)=\\sum_{r\\ge0}(F(q)^r)^THF(q)^r,\n \\qquad K-F^TKF=H.\n\\]\nDefine the quadratic coboundary and martingale difference\n\\[\n J_j=Z_{j+1}^TKZ_{j+1}-\\tr(K\\Sigma),\\qquad\n D_j=X_j-\\chi_c+J_j-J_{j-1}.\n\\]\nThe Lyapunov identity gives $\\E[D_j\\mid\\mathcal G_{j-1}]=0$ exactly.\nUniform stability bounds $K$ and its first two parameter derivatives.\nThe corresponding derivatives of $B_j$ are stable Gaussian filters with\nuniformly bounded fourth moments. Thus $D_j,J_j$ and their first two\n$q$ derivatives have uniform $L^2$ bounds. Doob bounds the maximal\nmartingale sum by $CN$ in mean square; the coboundary maximum is bounded by\n$\\sum_{j\\le N}\\E|J_j|^2\\le CN$. Multiplication by $h^2$ and the\none-dimensional Sobolev bound in $q$ prove \\eqref{eq:jmax}, also for the first\nderivative by using derivatives through order two. This is a finite-dimensional\nproof of the needed averaging, with no unverified mixing assumption.\n\n\\subsection{Endogenous bounded-variation weights and memory elimination}\nThe exact scalar pulse formulas show, on localization with\n$\\sum s_i^2\\le K$, that\n\\begin{equation}\\label{eq:jbv}\n \\sup_jG_j\\le C_K,\\qquad\n \\operatorname{TV}(G_j)+\\operatorname{TV}(q_j)\\le C_K.\n\\end{equation}\nTheir increments are $O(s_j^2)$, even though $Q$ and the moving frame fluctuate\nat first order. Discrete summation by parts, telescoping $G_jA_j(q_j)$, gives\n\\[\n \\sup_n\\left|h\\sum_{j\\le n}G_{j-1}\n [X_j(q_j)-\\chi_c(q_j)]\\right|\n \\le C_K\\sup_{n,q}(|A_n(q)|+|\\partial_qA_n(q)|).\n\\]\nNo independence of the weights from the OU pulse is needed. The stationary\nfilter uses negative-time strokes only as a proof device; the actual photon\nstarts with $r_0=0$. The difference from its finite startup filter is a\ngeometric boundary term with cumulative $L^2$ size $O(h)$. Equations\n\\eqref{eq:jdet}--\\eqref{eq:jbv} therefore give\n\\[\n \\log S_n=-h\\sum_{j\\le n}G_{j-1}\\chi_c(q_j)\n +O_{\\mathbb P}(\\sqrt{h\\log(1/h)})\n\\]\nuniformly on localization. Finally the exact memory increment is\n\\[\n \\Delta\\log U_j=\\log(1+G_{j-1}s_j^2)\n =G_{j-1}s_j^2+O(s_j^4).\n\\]\nThe same maximal estimate for $\\zeta_j^2-C_0$, with BV weight\n$G_{j-1}\\chi_c(q_j)$, implies\n\\[\n h\\sum_{j\\le n}G_{j-1}\\chi_c(q_j)\n =\\frac1{C_0}\\sum_{j\\le n}\\chi_c(q_j)\\Delta\\log U_j\n +O_{\\mathbb P}(\\sqrt h).\n\\]\nThe remaining monotone-memory Stieltjes sum approximates the integral in\n\\eqref{eq:jrate} with error $O_{\\mathbb P}(h\\log(1/h))$. This proves the rate.\nThe earlier seeded-controller bounds and nonexplosion remove localization\nfor convergence. Equation~\\eqref{eq:jbright} proves amplitude alignment;\nbounded norms upgrade probability convergence to every finite $L^p$.\nThe joint controller convergence to \\eqref{eq:core} then proves\nTheorem~\\ref{thm:joint-loss}. If $\\ell_h\\to\\infty$, $q_h(u)\\to0$ uniformly\nfor $u\\ge U_0$, so any divergent $\\ell_h$ gives the inverse-memory endpoint\nlaw. The stronger earlier dwell/log condition was sufficient, not necessary.\n\nA finite confidence bound is also available. Let $N=\\lfloor T/h\\rfloor$ and\n$m_\\delta=\\sqrt{2C_0h\\log(2N/\\delta)}$. If $m_\\delta$ is below the\nsmall-stroke threshold, then, for any $\\epsilon>0$,\n\\begin{equation}\\label{eq:jconfidence}\n \\mathbb P\\left(\\mathcal C_K\\ \\text{and endpoint log error}>\n C_K(m_\\delta+m_\\delta^2+\\epsilon)\\right)\n \\le\\delta+C_Kh/\\epsilon^2.\n\\end{equation}\nHere $\\mathcal C_K$ denotes the compact/square-budget localization. The Gaussian\nunion bound controls the largest stroke and \\eqref{eq:jmax} controls the\ncentered quadratic terms; startup and Stieltjes errors are included.\n\n\\subsection{Broader covariance law and its precise limits}\nFor general stationary normalized strokes with covariances $C_k$, the same\ngeometric filter has $\\chi(q)=C_0+2\\sum_{k\\ge1}C_kq^k$. If $C_0>0$, the\nuniform quadratic maximal bound \\eqref{eq:jmax} is verified, and the controller\nhas the stated BV bounds and a proved limit $\\dd U=C_0R\\,\\dd t$, then\n\\[\n S(t)=\\exp\\!\\left[-\\frac1{C_0}\\int_{U_0}^{U(t)}\n \\chi(q(u))\\frac{\\dd u}{u}\\right].\n\\]\nThis extension is conditional on quadratic averaging, not merely summable\ncovariance. For example $\\zeta_j=V\\varepsilon_j$, with a nondegenerate fixed\n$V$ independent of iid centered unit-variance $\\varepsilon_j$ and $\\E V^2=1$,\nhas $C_0=1,C_k=0$ but square budget tending to $V^2t$, rather than $t$.\n\nWith spectral measure convention $C_k=\\int e^{ik\\theta}\\,\\dd\\mu(\\theta)$,\n\\[\n \\chi(q)=\\int_{-\\pi}^{\\pi}\n \\frac{1-q^2}{1-2q\\cos\\theta+q^2}\\,\\dd\\mu(\\theta)\\ge0\n \\quad(0\\le q<1).\n\\]\nNegative correlations can give $\\chi(q)<C_0$; OU's exponent interval and loss\nmonotonicity are not universal. A calibrated loss scan measures the local\nslope $-C_0\\,\\dd\\log S/\\dd\\log U=\\chi(q)$, whose analytic coefficients\nrecover $C_k=\\chi^{(k)}(0)/(2k!)$ for $k\\ge1$. This is a model-based covariance\nidentification statement, not quantum-enhanced estimation. The covariance\nsensitivity itself must retain its prior-art credit \\cite{kofman}.\n\nExact OU-interval and exact controller/probe simulations agree with the\nendpoint integral. In a separate 256-path check at $c=1,\\nu=0.2,T=1$,\nquadrupling the grid approximately halves mean uniform log errors; for\n$\\ell=2$, grids $256,1024,4096$ give $0.00882,0.00446,0.00228$.\nIndependent quadrature verifies the endpoint series and the exact $q=0$\ntelescope to numerical precision. These are reproducible finite-grid checks,\nnot measurements or a replacement for the proof. The companion 512-path rate\nscan supplies the report figure.\n\n\n\\subsection{A proved finite-MA comparison: endpoint budgets do not identify the bath}\n\\label{sec:finite-ma}\n\nLet $(\\varepsilon_j)_{j\\in\\mathbb Z}$ be independent standard Gaussians and set\n\\[\n \\alpha=\\frac{\\sqrt{7/5}+\\sqrt{3/5}}2,\\qquad\n \\beta=\\frac{\\sqrt{7/5}-\\sqrt{3/5}}2,\\qquad\n s_j^{(m)}=\\sqrt h\\,(\\alpha\\varepsilon_j+\\beta\\varepsilon_{j-m}),\n \\quad m\\in\\{1,2\\}.\n\\]\nThus $\\alpha^2+\\beta^2=1$, $\\alpha\\beta=1/5$. These are stationary\nfinite-dependent Gaussian strokes, with $C_0=1$, $C_m=1/5$, all other\npositive-lag $C_k=0$, and total Brownian variance $L=7/5$.\n\n\\begin{corollary}[Two baths with identical sequential endpoints]\n\\label{cor:finite-ma}\nFor identical seeded data $U_0>0$, both sources generate the same limiting\nscalar Avenyra diffusion,\n\\begin{equation}\n \\dd U=R\\,\\dd t,\\qquad\n \\dd R=\\frac{RQ^2}{U}\\,\\dd t,\\qquad\n \\dd Q=\\sqrt{7/5}\\,\\dd W-\\frac{RQ}{2U}\\,\\dd t.\n \\label{eq:ma-controller}\n\\end{equation}\nBoth fully projected sequential readouts have survival $U_0/U$, and both\ndiffusion-first then strong-loss readouts have survival $(U_0/U)^{7/5}$.\nNevertheless, for the held post-write loss protocol with\n$\\gamma_hh\\to\\ell>0$ and initially dark photon, their joint limiting\nsurvivals differ:\n\\begin{equation}\n \\chi_m(q)=1+\\frac25q^m,\\qquad\n S_m(t)=\\exp\\!\\left[-\\int_{U_0}^{U(t)}\n \\left(1+\\frac25e^{-m\\ell u/2}\\right)\\frac{\\dd u}{u}\\right].\n \\label{eq:ma-loss}\n\\end{equation}\nConvergence is uniform in probability on finite horizons. Evaluated at the\nsame memory endpoint $U(t)>U_0$, the predictions obey $S_2(U(t))>S_1(U(t))$\nfor $\\ell>0$. The two limiting controllers can be coupled with the same $W$;\nindependent finite runs need not share a realized memory endpoint.\n\\end{corollary}\n\\begin{proof}\nThe cumulative driver is $(\\alpha+\\beta)\\sqrt h\\sum\\varepsilon_j$ plus\nfinitely many endpoint innovations; their uniform maximum is\n$O_p(\\sqrt{h\\log(1/h)})$. Gaussian finite dependence gives an $O(h)$\nvariance for each cumulative square budget, which converges uniformly to\n$t$ by positivity. Moreover $\\E\\sum|s_j|^3=O(\\sqrt h)$.\nThe one-driver flow/Euler pulse limit therefore applies, giving\n\\eqref{eq:ma-controller} with $\\nu=7/10$, $f=5/7$. Its angular quadratic\nvariation is $(7/5)R/U\\,\\dd t$, whereas $\\dd\\log U=R/U\\,\\dd t$;\nthis proves the two sequential endpoints.\n\nThe general lag-filter theorem applies here with verified quadratic\naveraging, rather than by importing the OU hypothesis. Write\n$\\xi_j=s_j/\\sqrt h$ and\n$B_j(q)=\\sum_{k\\ge1}q^k\\xi_{j-k}$. The preinterval Gaussian state\n$Z_j=(\\varepsilon_{j-1},\\ldots,\\varepsilon_{j-m},B_j)^T$ obeys\n\\begin{equation}\n Z_{j+1}=F_m(q)Z_j+v(q)\\varepsilon_j,\\qquad\n F_m(q)=\\begin{pmatrix}N_m&0\\\\q\\beta e_m^T&q\\end{pmatrix},\\qquad\n v(q)=\\binom{e_1}{q\\alpha},\n\\end{equation}\nwhere $N_m$ is the nilpotent lag shift. For\n$X_j=\\xi_j^2+2\\xi_jB_j$,\n$\\E[X_j\\mid\\text{past}]=\\alpha^2+Z_j^THZ_j$, where\n$H_{mm}=\\beta^2$, $H_{m,m+1}=H_{m+1,m}=\\beta$ and other entries vanish.\nWith stationary covariance $\\Sigma(q)$, define\n\\[\n K(q)=\\sum_{r\\ge0}(F_m(q)^r)^THF_m(q)^r,\n \\qquad K-F_m^TKF_m=H.\n\\]\nThe eigenvalues of $F_m$ are zero and $q$. For $q\\le q_*<1$, its powers,\ntheir first two $q$ derivatives, and the Gaussian fourth moments are\nuniformly controlled. The same quadratic Lyapunov corrector,\nmartingale/coboundary decomposition and Sobolev argument used for\n\\eqref{eq:jmax} therefore prove that maximal bound for these sources.\nThe existing geometric bright-filter and endogenous BV-weight estimates\nnow give \\eqref{eq:ma-loss}. Its strict inequality follows from $q^2<q$.\n\\end{proof}\n\nThe comparison concerns retained continuum controller laws and the two\nsequential readout endpoints; resolved pulse records distinguish the baths.\nIt requires the seeded held protocol and positive joint loss scaling.\nIt does not cover arbitrary zero-loss scaling, loss overlapping a write,\nor nonstationary preparation. Covariance-weighted optical attenuation has\nestablished ancestry \\cite{kofman}; this corollary supplies an exact\nendogenous-memory example of what the two endpoint budgets fail to identify.\n\n\n\\subsection{Independent joint-scaling simulation}\nThe lead-agent calculation uses exact integrated OU strokes and exact\nheld-stage amplitude contraction, with 512 paths per grid and\n$c=1,\\nu=0.3,T=1$, seeded $x_0=(1,0.4,0.2)$ and $b_0=(0,0,1)$.\nHere $\\gamma_h=\\ell/h$; changing the grid holds the loss per dwell fixed.\nAn independent adaptive quadrature checks the accumulated endpoint integral.\n\\begin{center}\n\\begin{tabular}{rrrr}\n\\toprule $\\ell$ & MAE at 1,024 strokes & MAE at 4,096 & MAE at 16,384\\\\\\midrule\n0.05 & 0.067219 & 0.034328 & 0.017923\\\\\n0.2 & 0.033082 & 0.015366 & 0.007955\\\\\n1 & 0.008831 & 0.004363 & 0.002106\\\\\n4 & 0.000835 & 0.000447 & 0.000212\\\\\n16 & $1.152\\times10^{-6}$ & $6.060\\times10^{-7}$ & $3.063\\times10^{-7}$\\\\\\bottomrule\n\\end{tabular}\n\\end{center}\nMAE is mean pathwise absolute survival error against that path's endpoint\nprediction, rather than an error of the ensemble mean. At the finest grid\nthe mean survival for $\\ell=1$ is $0.660389$, versus target $0.660526$;\nthe paired discrepancy standard error is $0.000115$.\nThe 60 independent endpoint-quadrature checks at that grid have maximum\ndiscrepancy $9.75\\times10^{-12}$. Grid trends support the proof but do not\ncertify a numerical convergence exponent or an apparatus error budget.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.94\\linewidth]{v3/results/joint_loss_simulation.pdf}\n\\caption{Positive joint monitoring: exact-map simulations and endpoint\npredictions. The right panel compares pathwise survival error across grids.\nAll points are simulated model data with finite sampling and grid errors.}\n\\end{figure}\n\n\\Needspace{8\\baselineskip}\n","labels":["sec:joint-loss","eq:jwrite","eq:jprobe","eq:jcov","eq:jchi","eq:jendpoint","thm:joint-loss","eq:jrate","eq:jbound","eq:jalpha","eq:jbright","eq:jdet","eq:jmax","eq:jbv","eq:jconfidence","sec:finite-ma","cor:finite-ma","eq:ma-controller","eq:ma-loss"],"source_line_start":2293,"source_line_end":2709,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_19","section_index":19,"title_latex":"Intrinsic geometry and the limits of a fundamental-force interpretation","text_latex":"\\section{Intrinsic geometry and the limits of a fundamental-force interpretation}\n\\label{sec:intrinsicgeometry}\n\nThe finite kernel gate has an intrinsic formulation. This removes arbitrary\ncoordinates and frames, but does not derive its sampling convention or its\nmemory register from a fundamental field equation.\n\n\\subsection{Exact metric-bundle completion}\nLet $E\\to M$ be a real vector bundle with a fibre metric and a metric-compatible\nconnection. For a specified path $\\gamma:q\\to q^+$, denote its orthogonal\nparallel transport by $U_\\gamma$. A state has $x\\in E_q$ and skew-adjoint\n$A:E_q\\to E_q$. For $a\\in\\ker A$, set\n\\begin{equation}\n x^+=U_\\gamma(x+a),\\qquad\n A^+=U_\\gamma(A+x\\wedge a)U_\\gamma^{-1}.\n \\label{eq:bundlepulse}\n\\end{equation}\nThese equations are independent of a local frame. The connection is model\ndata; changing coordinates does not change it.\n\n\\begin{proposition}[Exact inheritance]\nEvery Euclidean pulse statement expressed through the fibre\nmetric and orthogonal conjugacy holds unchanged under\n\\eqref{eq:bundlepulse}. In particular, kernel transport, rank birth, spectral\nmonotonicity, $\\rho=\\norm{P_{\\im A}x}$, and the complete quadratic spectral\ncertificate cone survive for arbitrary curvature.\n\\end{proposition}\n\\begin{proof}\nFirst compute the Euclidean pulse within $E_q$, then apply the isometry\n$U_\\gamma$. Kernels, images and their orthogonal projections transport by\nconjugation. Functional calculus obeys\n\\[\n\\phi(-UA^2U^{-1})=U\\phi(-A^2)U^{-1},\n\\]\nso the scalar certificates have exactly the same values. Necessity of any universal\ncertificate condition follows by choosing a flat trivial bundle.\n\\end{proof}\n\nFor $E=TM$ with the Levi--Civita connection, a natural pulse uses\n$q^+=\\operatorname{Exp}_q(a)$ and transport along that geodesic. With $x_0=0$,\nthe endpoint register is the intrinsic developed displacement:\n\\begin{equation}\n y_T:=P_{\\rm hist}(T)^{-1}x_T\n =\\int_0^T P_{\\rm hist}(t)^{-1}\\dot q(t)\\,\\dd t.\n \\label{eq:antidevelopment}\n\\end{equation}\nIt is not generally $\\operatorname{Log}_{q_0}(q_T)$. The finite gate is sampled\nat the start of each segment. By contrast, an actually continuous legal\ntrajectory satisfies $D_t x=\\dot q$, $D_t A=x\\wedge\\dot q$, $A\\dot q=0$.\nIn a parallel frame these are the original continuous equations and preserve\nthe skew spectrum and $\\rho$. Connection transport cannot generate rank or\nspectrum from zero memory.\n\n\\subsection{Physical exclusion with a controlled curvature error}\n\\begin{theorem}[Short-history spatial certificate]\nConsider finitely many geodesic pulses with $x_0=0$, initial point $q_0$ and\ntotal length $L$. Suppose $L<\\operatorname{inj}(q_0)$, and throughout the\nradial normal ball containing the history,\n\\begin{equation}\n \\norm{R(u,v)}_{\\rm op}\\le K\\norm{u\\wedge v},\\qquad KL^2\\le\\tfrac14.\n\\end{equation}\nThen, for $v_T=\\operatorname{Log}_{q_0}(q_T)$,\n\\begin{equation}\n \\norm{y_T-v_T}\\le KL^3,\\qquad\n d(q_0,q_T)\\ge\\norm{x_T}-KL^3\\ge\\rho_j-KL^3\n \\label{eq:curvaturecertificate}\n\\end{equation}\nfor every earlier pulse index $j$. Thus the certificate bounds physical\ndistance when it exceeds the curvature error.\n\\end{theorem}\n\\begin{proof}\nPut $v(t)=\\operatorname{Log}_{q_0}(q(t))$ and\n$\\delta=KL^2/5\\le1/20$. In radial parallel frames the Jacobi equation has\nthe Volterra form\n\\[\n j(s)=s w-\\int_0^s(s-r)\\mathcal R_rj(r)\\,\\dd r,\n \\qquad \\norm{\\mathcal R_r}\\le K\\norm v^2.\n\\]\nIts convergent Volterra iteration gives\n\\begin{equation}\n \\norm{\\dd\\operatorname{Exp}_v-P_{\\rm rad}}\n \\le\\frac{\\sinh(\\sqrt K\\norm v)}{\\sqrt K\\norm v}-1\n \\le\\frac{K\\norm v^2}{6}\\cosh(1/2)\\le\\delta.\n\\end{equation}\nThe ratio at $K=0$ is interpreted by its limit. Consequently\n\\[\n \\norm{\\dd\\operatorname{Log}-P_{\\rm rad}^{-1}}\n \\le\\frac\\delta{1-\\delta},\\qquad\n \\norm{\\dot v}\\le\\frac{\\norm{\\dot q}}{1-\\delta}.\n\\]\nFor the radial homotopy $F(t,s)=\\operatorname{Exp}_{q_0}(sv(t))$,\n$\\norm{\\partial_sF}\\le L$ and\n$\\norm{\\partial_tF}\\le(1+\\delta)s\\norm{\\dot v}$. Its parametrized area is\nat most $\\mathcal A=L^2(1+\\delta)/[2(1-\\delta)]$.\nConnection variation expresses the derivative of relative history/radial\ntransport as the integral of conjugated curvature over this surface.\nAll conjugations are orthogonal, hence\n$\\norm{P_{\\rm hist}^{-1}-P_{\\rm rad}^{-1}}\\le K\\mathcal A$.\nCombining these bounds with \\eqref{eq:antidevelopment} yields\n\\begin{align*}\n \\norm{y_T-v_T}\n &\\le L\\left[\\frac\\delta{1-\\delta}\n +\\frac{KL^2(1+\\delta)}{2(1-\\delta)}\\right]\\\\\n &=KL^3\\frac{0.7+0.5\\delta}{1-\\delta}\n \\le\\frac{29}{38}KL^3\\le KL^3.\n\\end{align*}\nIn the normal ball, $\\norm{v_T}=d(q_0,q_T)$, whereas\n$\\norm{y_T}=\\norm{x_T}\\ge\\rho_T\\ge\\rho_j$. This proves the result.\nPiecewise geodesic corners are treated segmentwise.\n\\end{proof}\n\nThe connection and Jacobi estimates are established geometry; their coupling\nto the exact internal certificate is the model-specific extension. The\ninjectivity assumption is essential even when curvature vanishes. On a flat\nthree-torus of coordinate period $\\ell$, start with $x=A=0$ and take\n$a_1=\\ell e_1$, followed by $a_2=\\ell e_2$. Both are legal and both physical\ngeodesics are closed loops. Nevertheless\n\\[\n q_2=q_0,\\qquad x_2=\\ell(e_1+e_2),\\qquad\n A_2=\\ell^2e_1\\wedge e_2,\\qquad\\rho_2=\\sqrt2\\,\\ell>0.\n\\]\nThere is therefore no universal global spatial exclusion theorem obtained\nby identifying internal developed displacement with physical position.\n\n\\subsection{An axiomatic selection principle for the gate}\nEvery bilinear $O(d)$-equivariant skew memory write in dimension $d\\ge2$\nhas the form $B(x,a)=\\gamma x\\wedge a$. Indeed, diagonal reflections force\n$B(e_i,e_i)=0$ and allow only the $e_i\\wedge e_j$ component in\n$B(e_i,e_j)$; orthogonal permutations equate the coefficients.\nFor $\\gamma\\ne0$, write $A^+=A+\\gamma x\\wedge a$ and\n$E(A)=\\norm A_F^2/2$.\n\\begin{proposition}[Endpoint-robust passivity]\nFor fixed $A$ and a nonzero $a$, $E(A^+)\\ge E(A)$ for every $x$ if and\nonly if $Aa=0$. Thus $\\ker A$ is the maximal allowed stroke set,\nindependent of $x$, with this guarantee.\n\\end{proposition}\n\\begin{proof}\nUsing $Aa\\perp a$, exact quadratic minimization gives\n\\[\n \\Delta E=2\\gamma x^TAa+\\gamma^2(\\norm x^2\\norm a^2-\\langle x,a\\rangle^2),\n \\qquad\\min_x\\Delta E=-\\frac{\\norm{Aa}^2}{\\norm a^2},\n\\]\nattained at $x=-Aa/(\\gamma\\norm a^2)$, up to an arbitrary component\nparallel to $a$. The minimum is nonnegative precisely when $Aa=0$.\n\\end{proof}\nThese axioms select the wedge and kernel gate, but they are controller\ndesign assumptions. Endpoint-dependent permission, a supplied orientation,\nnonlinear writes, another energy or another timing convention permit\nalternatives. Equating this memory norm with physical stored energy still\nrequires a device realization and a separate controller-work account.\n\n\\subsection{A conditional magnetic Poisson and quantum obstruction}\nIn three dimensions write $Av=v\\times b$. A continuous legal direction is\nparallel to $b$; an unnormalized flow is\n\\begin{equation}\n \\dot x=b,\\qquad\\dot b=x\\times b.\n \\label{eq:intrinsic3dflow}\n\\end{equation}\nSuppose one interprets $x$ as commuting canonical positions and $b$ as\nunit-mass mechanical momenta on this same six-dimensional state space:\n$\\{x_i,x_j\\}=0$, $\\{x_i,b_j\\}=\\delta_{ij}$.\nJacobi for $(x_i,b_j,b_k)$ forces $\\{b_j,b_k\\}$ to be independent of $b$.\nWrite $\\{b_i,b_j\\}=\\epsilon_{ijk}B_k(x)$. The position equation forces\n$H=\\norm b^2/2+V(x)$; the momentum equation at $b=0$ forces $V$ constant.\nThen $\\dot b=b\\times B$, and equality with \\eqref{eq:intrinsic3dflow} for\nevery $b$ uniquely requires $B=-x$. But its Jacobiator is\n\\begin{equation}\n \\{b_1,\\{b_2,b_3\\}\\}+\\text{cyclic}\n =-\\nabla\\cdot B=3\\ne0.\n\\end{equation}\nThus this canonical position/mechanical-momentum interpretation is not a\nPoisson system. The normalized flow has the same obstruction with\n$H=\\norm b$. This is an instance of the established magnetic charge/Jacobi\nproblem, not a new magnetic force \\cite{szabo}.\n\nThe corresponding proposed quantum relations are\n\\[\n [x_i,x_j]=0,\\quad[x_i,b_j]=i\\hbar\\delta_{ij},\\quad\n [b_i,b_j]=-i\\hbar\\epsilon_{ijk}x_k.\n\\]\nOn any common invariant domain they give\n$[b_1,[b_2,b_3]]+\\text{cyclic}=-3\\hbar^2I\\ne0$, contrary to the exact\nJacobi identity for commutators in an associative operator algebra.\nThese particular canonical relations consequently cannot define standard\nquantum observables. The obstruction does not prohibit controlled fields,\nadditional variables, or different noncanonical physical identifications.\n\n\\subsection{Why an auxiliary action is not a fundamental derivation}\nIndeed, the cotangent lift gives an ordinary canonical Hamiltonian on a\ndoubled state space:\n\\begin{equation}\n \\mathcal H=\\pi_x\\cdot b+\\pi_b\\cdot(x\\times b),\\qquad\n \\mathcal S=\\int\\bigl(\\pi_x\\cdot\\dot x+\\pi_b\\cdot\\dot b\n -\\mathcal H\\bigr)\\,\\dd t.\n\\end{equation}\nIts equations reproduce \\eqref{eq:intrinsic3dflow} exactly and also give\n$\\dot\\pi_x=\\pi_b\\times b$ and\n$\\dot\\pi_b=-\\pi_x-\\pi_b\\times x$.\nMore generally, any smooth chosen vector field $X$ on a regular state\nmanifold has the intrinsic cotangent Hamiltonian $\\mathcal H=\\langle\\pi,X\\rangle$.\nThis is a formal realization with auxiliary resources and an energy\nunbounded in the auxiliary momenta. It does not choose the gate or supply\na stable microscopic energy, reservoir or finite sampling clock.\nThe honest conclusion is that coordinate-independent geometry is available\nand a conventional magnetic identification is obstructed; a fundamental\nvariational physics theory requires additional physical content.\n\n\n","labels":["sec:intrinsicgeometry","eq:bundlepulse","eq:antidevelopment","eq:curvaturecertificate","eq:intrinsic3dflow"],"source_line_start":2709,"source_line_end":2917,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_20","section_index":20,"title_latex":"Thermal completion and an operational writing certificate","text_latex":"\\section{Thermal completion and an operational writing certificate}\n\\label{sec:thermal-foundations}\n\nThe geometric register alone does not specify a Hamiltonian or a cost in joules.\nA thermal model must include its controller, fuel, reverse events and physical\ntime reversal. The results below separate these requirements from the exact\narea rule. The earlier stationarity obstruction already excludes a complete,\nstationary process whose memory strictly increases whenever it writes.\n\n\\begin{theorem}[The reverse gate is an adjoint gate]\n\\label{thm:adjoint-completion}\nFor a legal event $Aa=0$,\n\\[\n (x,A)\\longrightarrow(x+a,A+x\\wedge a),\n\\]\nthe inverse event restoring both registers obeys the post-gate. It obeys the\nsame pre-gate if and only if $x\\wedge a=0$. Consequently, any bidirectional\nresolved Markov completion must pair every writing pre-gate with a post-gated\ninverse, or resolve an additional controller phase selecting the inverse.\n\\end{theorem}\n\\begin{proof}\nSet $B=x\\wedge a$. At the final state the inverse displacement is $-a$ and\nits area increment is $-B$. Its post-gate is $(A+B-B)(-a)=0$.\nIts pre-gate instead requires\n$(A+B)a=\\norm a^2x-(x\\cdot a)a=0$, equivalent to $B=0$.\nThe same calculation exchanges a $\\theta$-gate with a $(1-\\theta)$-gate.\n\\end{proof}\n\nHere is an explicit autonomous, finite-resource thermal completion. Take a\nfinite registry of exact register states $s_i$, declared legal writing edges\n$i\\to j$, and actual implementation energies $\\mathcal E_i$. Add their\npost-gated inverses and a fuel count $n\\in\\{0,\\ldots,N\\}$ with energy\n$n\\Delta\\mu$, where $\\Delta\\mu>0$ is in joules. On\n$(i,n)\\rightleftarrows(j,n-1)$, choose the paired rates\n\\begin{equation}\\label{eq:thermal-pair}\n k_\\pm=c_{ij,n}\\exp\\!\\left[\\pm\\frac{\\Delta\\mu-(\\mathcal E_j-\n \\mathcal E_i)}{2k_BT_{\\mathrm b}}\\right],\\qquad c_{ij,n}>0.\n\\end{equation}\nThe finite Gibbs law $\\pi_{i,n}\\propto\n\\exp[-(\\mathcal E_i+n\\Delta\\mu)/(k_BT_{\\mathrm b})]$ satisfies detailed\nbalance on every pair. Preparing large $n$ supplies nonequilibrium free\nenergy and favors writing transiently; the reverse remains possible at\nfinite temperature and finite energies. The ratio\n$k_-/k_+\\le\\epsilon$ requires\n$\\Delta\\mu\\ge\\mathcal E_j-\\mathcal E_i+k_BT_{\\mathrm b}\\ln(1/\\epsilon)$.\nHere $c_{ij,n}$ has units of inverse time. This is a channel-rate ratio,\nrather than a deadline-success probability.\nIt is a consistent finite-state thermal embedding, rather than a claim that\narbitrary register barriers have already been built.\n\nLet $\\mathbb P_F$ be a full forward path law over duration $\\tau$, and\n$\\mathbb P_R$ the correctly conjugated physical reverse experiment, started\nfrom the forward final ensemble. Let $\\Theta$ reverse the order of the path\nand transform its odd variables and controls. For thermodynamically resolved\nMarkov dynamics with local detailed balance, the mean total entropy production\nis $\\langle\\Sigma_\\tau\\rangle=k_B\nD(\\mathbb P_F\\Vert\\mathbb P_R^\\Theta)$, including all reservoirs and\nthe system entropy change \\cite{seifert}. This identification is a physical\nassumption about the resolved apparatus; it is not an entropy assignment to\nan abstract deterministic algorithm.\n\nUse a calibrated, time-reversal-even certificate $\\Phi=F_\\lambda$ at fixed\n$\\lambda>0$, and fix a threshold $\\delta>0$. Define\n$\\mathcal W=\\{\\Phi(s_\\tau)-\\Phi(s_0)\\ge\\delta\\}$. One can instead use\na checkpoint history with no decreases and at least one resolved increase;\nits reverse is the corresponding erasing history. The certificate must be\neven under the apparatus's actual $\\Theta$, which must be checked for its\nparticular register encoding.\n\n\\begin{theorem}[Observable writing--reversal certificate]\n\\label{thm:thermal-writing}\nSet $p=\\mathbb P_F(\\mathcal W)$ and\n$q_R=\\mathbb P_R(\\Theta\\mathcal W)$. Then\n\\begin{equation}\\label{eq:binary-writing-ep}\n \\langle\\Sigma_\\tau\\rangle\\ge k_B d(p\\Vert q_R),\\qquad\n d(p\\Vert q)=p\\ln\\frac pq+(1-p)\\ln\\frac{1-p}{1-q}.\n\\end{equation}\nIf, before pulling back by $\\Theta$, the physical reverse experiment has\nthe same path law as the forward experiment, $\\mathbb P_R=\\mathbb P_F$,\nwrite $q=\\mathbb P_F(\\Theta\\mathcal W)$. Then\n\\begin{equation}\\label{eq:three-bin-writing-ep}\n \\boxed{\\langle\\Sigma_\\tau\\rangle\\ge\n k_B(p-q)\\ln(p/q).}\n\\end{equation}\nIn particular, $q\\le\\epsilon p$, $0<\\epsilon<1$, entails\n$\\langle\\Sigma_\\tau\\rangle\\ge\nk_Bp(1-\\epsilon)\\ln(1/\\epsilon)$.\n\\end{theorem}\n\\begin{proof}\nRelative entropy contracts under the two-bin observation\n$(\\mathcal W,\\mathcal W^c)$, giving\n\\eqref{eq:binary-writing-ep}. Under the added compatibility assumption,\nreversal swaps the probabilities of the disjoint events\n$\\mathcal W,\\Theta\\mathcal W$ and leaves their complement fixed.\nTheir three-bin relative entropy is\n$p\\ln(p/q)+q\\ln(q/p)=(p-q)\\ln(p/q)$.\nFor $q/p=r\\le\\epsilon$, $(1-r)\\ln(1/r)$ decreases with $r$.\n\\end{proof}\n\nThe same-law assumption holds for an autonomous stationary, resolved\neven-state Markov-jump model with the same generator in the reverse\nexperiment and no unconjugated odd external controls. Stationarity alone is\ninsufficient when magnetic parameters, odd register variables or scheduled\nprotocols change the physical reverse experiment: use $q_R$ from that\nexperiment in \\eqref{eq:binary-writing-ep}. The observed register need not\nitself be Markov. At equilibrium the correctly reversed path law agrees\nwith the forward law, so $p=q_R$; identifying this with a same-run erase\nprobability additionally uses the compatibility assumption. Exact\n$p>0,q=0$ under the same-law hypotheses instead gives singular reverse\nsupport and infinite path relative entropy. This diagnoses a missing\nreverse channel or a singular limit of that complete model; it does not\nimply infinite work for a sampled, externally programmed algorithm.\n\n\\begin{theorem}[Finite prepared resource leaves a reverse-event floor]\n\\label{thm:finite-resource-floor}\nFor the finite autonomous detailed-balance completion, let\n\\[\n \\mathcal F(r)=\\sum_i r_i\\mathcal E_{\\mathrm{tot},i}\n +k_BT_{\\mathrm b}\\sum_i r_i\\ln r_i,\\qquad\n B=\\frac{\\mathcal F(r_0)-\\mathcal F(\\pi)}{k_BT_{\\mathrm b}}.\n\\]\nFor $p>0$ and the actual transient reverse probability $q_R$,\n\\begin{equation}\\label{eq:finite-resource-floor}\n \\boxed{q_R\\ge\\exp[-(B+h(p))/p],\\qquad\n h(p)=-p\\ln p-(1-p)\\ln(1-p).}\n\\end{equation}\nRequirements $p\\ge1-\\epsilon$ and $q_R\\le\\epsilon<1/2$ demand\n$B\\ge(1-2\\epsilon)\\ln[(1-\\epsilon)/\\epsilon]$.\n\\end{theorem}\n\\begin{proof}\nDetailed balance gives the path log ratio\n$\\ln[r_0(i_0)/r_\\tau(i_\\tau)]-\n(\\mathcal E_{\\mathrm{tot},i_\\tau}-\\mathcal E_{\\mathrm{tot},i_0})/\n(k_BT_{\\mathrm b})$.\nThus $\\langle\\Sigma_\\tau\\rangle/k_B\n=[\\mathcal F(r_0)-\\mathcal F(r_\\tau)]/(k_BT_{\\mathrm b})\\le B$.\nTogether with \\eqref{eq:binary-writing-ep} and\n$d(p\\Vert q_R)\\ge p\\ln(1/q_R)-h(p)$ this proves the floor.\nFor the stated reliability requirements the binary relative entropy is\nminimized at $p=1-\\epsilon,q_R=\\epsilon$.\n\\end{proof}\n\nThe resource is the actual preparation free energy of the full finite\ncontroller, including its fuel. A forward run started at the smallest\nmemory can have no negative endpoint increments while its reverse run has\npositive erasure probability; such a transient same-run count cannot\nreplace $q_R$. Nor is there a universal heat cost per unit geometric area:\ntwo equal-energy states connected by a legal writing event and its inverse\ncan write transiently with zero bath heat while consuming initial ensemble\nfree energy. Only for a fully accounted isothermal cyclic apparatus with\n$W_{\\mathrm{diss}}=T_{\\mathrm b}\\Sigma$ does\n\\eqref{eq:three-bin-writing-ep} become the corresponding dissipated-work\nbound. Entropy is in joules per kelvin, $B,p,q$ are dimensionless, and\n$\\Phi,\\delta$ are calibrated register quantities.\n\nThese are established path-information inequalities specialized to the\ngate's explicit pre/post reverse pair and rank-continuous monotone, rather\nthan a new second law. A proposed passive thermal implementation must also\naccount for fluctuation--dissipation noise in its constraint mechanism:\nrecent nonholonomic models show that adding bath noise to an otherwise\nideal noiseless constraint can produce spurious work extraction, corrected\nby including its physical contact fluctuations \\cite{jagla}.\n\n\n","labels":["sec:thermal-foundations","thm:adjoint-completion","eq:thermal-pair","thm:thermal-writing","eq:binary-writing-ep","eq:three-bin-writing-ep","thm:finite-resource-floor","eq:finite-resource-floor"],"source_line_start":2917,"source_line_end":3081,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_21","section_index":21,"title_latex":"Proposed experiments and decision criteria","text_latex":"\\section{Proposed experiments and decision criteria}\nThe shortest experiment is a controlled three-axis signal system; a claim\nabout fundamental spacetime is unnecessary. A computer may retain $(x,b)$,\ncommand real couplings, and log all actual strokes. That establishes a test of\nthe response predictions, although it would not show that a passive material\nimplements the gate autonomously.\n\n\\begin{enumerate}\n\\item \\textbf{Calibrate the plant.} Measure the noise covariance and correlation\ntime, the actual hold intervals, the coupling matrix, and the gate residual.\nCreate a finite nonzero seed with $R_0>0$. Retain a fixed anchor.\n\\item \\textbf{Test the exact finite closure.} From logged vectors compute $U,R,Q$\nbefore and after each stroke. Fit no free dynamics beyond calibrated scales.\nCompare with \\eqref{eq:closure}; use the residual identity for tolerances.\n\\item \\textbf{Change noise resolution.} Keep macroscopic driver variance\napproximately fixed while varying $h/\\tau$. Compare the resolved-stroke budget\nand the conditional rate $\\Delta U/(R\\Delta t)$ with $2\\nu f(c)$. A failure\nafter accurate covariance and stroke calibration falsifies this controller\nmodel or its limit assumptions.\n\\item \\textbf{Read a thermal probe.} Implement the positive spring matrix\n\\eqref{eq:probe}. Measure positional covariance and quench heat after verified\nequilibration. Test the determinant, work, and midpoint heat identities.\nKeep $q$ independent from the history register $x$.\n\\item \\textbf{Read optical modes.} Program loss with $A^TA=U(I-P_n)$, prepare\nthe initial dark signal, and compare full-relaxation survival with $U_0/U_N$.\nVary dwell time using the two-pulse sequence \\eqref{eq:two-pulse}. Include\nintrinsic loss and finite auxiliary bandwidth in the fitted physical model.\n\\item \\textbf{Run controls.} Compare continuously monitored noise, midpoint\ngating, coherent transport before loss, and a fixed memory tensor. These\ncontrols distinguish timing, ordinary attenuation, and the endogenous gate.\n\\end{enumerate}\nVersion 2 adds three decisive tests: estimate $F_\\lambda$ from equilibrium\nprobe variance at held checkpoints and compare gain with its error budget;\nreplay one correlated history with stage-resolved versus continuous active loss;\nand measure the shaped finite-rate stroke's endpoint area and residual currents\nwhile varying duration. Loss switching, sampling, component bandwidth and\ncalibration must be recorded to interpret these tests.\n\nThe digital register alone cannot establish a novel autonomous material.\nAn autonomous next stage would need a physical area integrator, retained\ncouplings, a kernel selector, a clock or finite-event mechanism, and a measured\npower supply. None of those has been fabricated in this work.\n\n\n\\subsection{Decisive tests of the foundational model}\nThree experiments distinguish specific consequences from terminology.\nFirst, prepare equal measured $x$ with distinct known $A$ and test the\ndifferent permission kernels and the independently measured ridge responses.\nConversely, matched complete register states and the same future input law\nmust predict the same future controller law. For finite OU noise this matching\nalso includes its retained bath endpoint or conditional bath preparation;\nmatching $(x,A)$ alone while leaving a correlated bath unmatched is not a\nvalid state-sufficiency test. Unrestricted hidden-state alternatives cannot\nall be excluded: this controller is itself a state-augmented model.\n\nSecond, calibrate $h,\\tau$, stroke covariance, $U$ and loss strength\nindependently, then scan positive $\\ell$ and test\n\\eqref{eq:jendpoint} without fitting a new exponent. Replay the complete\nsequence $(n_j,U_j,q_j)$, and compare a photon with a normalized classical\nwave. The passive finite products predict identical attenuation for live\nversus replayed controls and for the two amplitude implementations.\nRepreparing the signal while retaining or erasing $A$ tests the causal role\nof classical memory. A reproducible difference calls for an additional\nphysical coupling absent from the present model; it is not predicted here.\n\nThird, realize and measure the fuel and post-gated inverse events, including\nreset, contact fluctuations and supply work. Perform the actual physical\nreverse experiment with odd controls conjugated and the forward-final\nensemble prepared. Compare $p,q_R$ with\n\\eqref{eq:binary-writing-ep} and \\eqref{eq:finite-resource-floor}.\nSame-run erasure counts can substitute only under the explicitly proved\nreverse-compatibility assumption. Agreement would validate a powered\ngeometric-memory phenomenon. A proposed new microscopic law would require\na reproducible discrepancy after this conventional apparatus model and its\nfinite-resolution uncertainties have been tested.\n\n","labels":[],"source_line_start":3081,"source_line_end":3157,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_22","section_index":22,"title_latex":"Prior-art investigation and the narrow novelty target","text_latex":"\\section{Prior-art investigation and the narrow novelty target}\nThe literature search was adversarial: a useful result must survive the closest\nexisting reductions. Primary abstracts, publisher pages, and selected full\nequations were examined; the audit notes record the level of inspection.\n\n\\begin{longtable}{p{.24\\linewidth}p{.34\\linewidth}p{.34\\linewidth}}\n\\toprule Established source & Existing result & Boundary of this manuscript\\\\\\midrule\\endhead\nPath signatures and step-two groups \\cite{signature}\n& Signed area is the second-level antisymmetric path memory.\n& The signed-area algebra is inherited; it is not a novelty claim.\\\\\nNonholonomic and stochastic integrators \\cite{hong,luesink}\n& Midpoint and related methods can preserve quadratic or Casimir invariants.\n& The specific gate yields an exact Householder congruence, rank-endpoint\nclassification and threshold. Generic midpoint conservation is established.\\\\\nOdd elasticity \\cite{odd}\n& Active skew response can inject work over deformation cycles.\n& No claim to invent active skew response or area-dependent work. The proposed\ncontroller uses its own stored area to gate future displacements.\\\\\nIrreversible mechanical metamaterials \\cite{counting}\n& Materials can count cycles, store driving history and recognize target\nsequences.\n& Generic mechanical memory and sequence dependence are established. The\nexact Avenyra closure and response certificates are the narrower target.\\\\\nNoise-feedback timing \\cite{pesce,hottovy,volpe}\n& Correlation time versus feedback delay can alter effective stochastic drift;\nexperiments already observe such transitions.\n& The crossover principle is established. Here the sample-held OU law is tied\nto exact pathwise monotones and a scalar convergence proof.\\\\\nPhysical magnetic Brownian rough paths \\cite{friz}\n& A physical stochastic area can retain microscopic corrections after a\nBrownian position limit.\n& No claim that microscopic area sensitivity is new. The own-area kernel gate\nis additional controlled-system data.\\\\\nGKSL, dark-space and Zeno geometry \\cite{lindblad,gks,burgarth,albert,yamamoto}\n& Linear jump operators define completely positive loss channels and dark\nsubspaces.\n& No new quantum dynamics. The proposed endpoint-only attenuation telescope\nuses a particular classical area-feedback controller.\\\\\nLocal detailed balance \\cite{maes,zeraati}\n& Resolved reverse channels and entropy flux constrain thermal models;\nstrict one-way rates need careful ideal-limit interpretation.\n& The reversible fuel completion is an application, not a new second law.\\\\\n\\bottomrule\n\\end{longtable}\n\nThe strongest precise Version 3 candidate is the complete own-area quadratic\nspectral classification and its exact endpoint-measure transport. The earlier\nresolvent family retains its finite-error certificate. The exact endogenous memory--survival\nexponent is a secondary candidate connection. Inverse identities, regularization,\nperturbation theory and quantum loss are established tools.\n\nThe additional audit found especially close optical prior art:\nKofman--Kurizki--Opatrn\\'y \\cite{kofman} studies noisy polarization rotation,\nselective absorption and correlation-sensitive monitoring. Its white-noise\nmoment equation matches our benchmark, and its covariance-weighted attenuation\nanticipates the generic resolution mechanism. Generic noisy geometric\nattenuation and generic noncommuting monitoring limits are therefore not\nnovelty claims. Our proof handles an endogenous random mode, with a particular\nmemory endpoint relation. Smooth contracting-generator adiabatic theory\n\\cite{avron} and stochastic adiabatic theory \\cite{fraas} provide further ancestry.\n\nNo exact predecessor for the own-area certificate identity was located in the\nexamined corpus. This is a bounded search result, not an originality certificate.\nPriority and importance require external review and a broader investigation.\n\nFor the OU coefficient there is a particularly close relationship: the\ncontinuous fixed-delay correction of Hottovy--McDaniel--Wehr contains an\nexponential of delay divided by correlation time \\cite{hottovy}. Averaging that\nexponential over the age of a held sample gives $(1-e^{-c})/c=1-f(c)$.\nThis is a useful \\emph{inference about ancestry}, not a proof that their\ncontinuous-delay apparatus and the present held-stroke controller are equivalent.\nIt further narrows any priority claim for the coefficient alone.\n\n\nVersion 3 exposes additional close ancestry rather than broadening priority\nclaims. The aligned rank-one inequality in the cone proof is generalized\nKlein trace calculus \\cite{klein}; smooth matrix derivatives are classical\n\\cite{rubensson}. Finite rational reconstruction and usual stochastic order\nare established tools. Path relative entropy and its contraction under\nobservations are standard thermodynamic methods \\cite{seifert,coarse}.\nThe magnetic Jacobi obstruction is an application of the established\nmagnetic-charge/nonassociativity issue \\cite{szabo}; a cotangent lift is a\ngeneric auxiliary action, not a new fundamental derivation. The recent\nfinite-temperature constraint example \\cite{jagla} motivates accounting for\nphysical contact fluctuations rather than idealizing a powered constraint\nas a single equilibrium bath. Broad self-generated path feedback also has\ndirect predecessors \\cite{sarkar}. No claim of a new spectral inequality,\ninformation-theoretic second law, Zeno principle, Hamiltonian embedding or\nuniversal path-memory mechanism is made.\n\nThe narrower candidates are the exact necessity-and-sufficiency cone for\nthe own-area pulse, its operational reconstruction and the controlled\nendogenous memory conversion in the joint loss limit. The isotropic\npassivity principle states which chosen controller axioms select the gate.\nThe bibliography and audit distinguish primary full-equation comparisons\nfrom abstract-level inspection. No equivalent complete own-area classification\nwas located in the bounded corpus; this is not a priority certificate.\n\n","labels":[],"source_line_start":3157,"source_line_end":3255,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_23","section_index":23,"title_latex":"Verification, reproducibility, and limitations","text_latex":"\\section{Verification, reproducibility, and limitations}\nThe main verifier compares full matrices, singular-vector projections, and\nmatrix exponentials against independently reduced formulas. It uses rational\narithmetic for finite exact timing examples; stochastic simulations test\nestimates rather than proving theorems.\n\nThe captured first verification suite contains:\n\\begin{itemize}\n\\item 30 exact rational full-rank four-dimensional timing cases: gate legality,\ncongruence, energy budget, complementary reversal, Pfaffian and midpoint\ncharacteristic-polynomial invariants.\n\\item 300 matrix cases in dimensions two through seven: energy, determinant,\nthermal work, and dark-space projection identities, including rank creation.\n\\item 300 three-dimensional vector/scalar comparisons and full matrix-exponential\nchecks of one-stage finite dwell.\n\\item 100 independently generated midpoint heat comparisons.\n\\item 2,048 independent Gaussian controller paths with 8,192 strokes; and 1,024\nintegrated-OU paths at each of five time-scale ratios.\n\\end{itemize}\nAll eight grouped finite identity checks pass. The largest numerical matrix\nresidual is approximately $2.14\\times10^{-14}$; exact rational equalities have\nzero residual. In the independent-stroke ensemble at $T=2,\\nu=0.3$, the expected\n$\\norm x^2$ is $2.4$ and its measured mean is $2.3954708$ with standard error\n$0.0330504$. This is compatible with the exact moment but does not certify all\ndistributional consequences. The survival telescope residual is below\n$4.3\\times10^{-14}$ over those histories.\n\nThe OU normalized stroke budgets are listed below; standard errors come from\nthe 1,024-path finite-stroke ensemble and are estimates.\n\\begin{center}\n\\begin{tabular}{rrrr}\n\\toprule $c=h/\\tau$ & Predicted $f(c)$ & Simulated mean & Standard error\\\\\\midrule\n0.05 & 0.0245885 & 0.0246458 & 0.0000556\\\\\n0.2 & 0.0936538 & 0.0933956 & 0.0001086\\\\\n1 & 0.3678794 & 0.3681172 & 0.0002413\\\\\n5 & 0.8013476 & 0.8014431 & 0.0003986\\\\\n20 & 0.9500000 & 0.9503123 & 0.0004654\\\\\\bottomrule\n\\end{tabular}\n\\end{center}\nThese checks support the finite stochastic implementation; they are not a\nsubstitute for the convergence proof or physical measurements. Additional\nmultistage finite-dwell checks, recorded in the cross-audit note, include 250\nrandom histories and 100 exact-sequence interference tests without violations\nof the stated error bound.\n\nVersion 2 adds 264 exact rational ridge pulses and 4,000 finite-error stress\ncases; 36 exact transport cases, 36 polynomial resolvent cases and 52 exact\nrank-saturation cases; and 105 exact finite-seed classification/budget cases.\nThe finite-rate suite checks 100 main sextic commands, 100 supplementary\nquartic commands and 100 loop-area quadratures. Main endpoint/current residuals\nare below $5.77\\times10^{-11}$; finite loop-quadrature error is below\n$1.82\\times10^{-7}$. A known-optics benchmark checks 2,048 Brownian-angle\npaths at 16,384 steps, and a separate colored controller/probe simulation uses\n512 paths at 16,384 strokes.\n\nIn that colored simulation ($c=1,f=e^{-1},\\nu=0.3,T=1$), mean resolved\nsurvival is $0.792445$ and the mean diffusion-first target is $0.531379$.\nFinite-loss means at $\\gamma=4,16,64,256$ are respectively\n$0.648721,0.565581,0.537742,0.534445$. At the last rate, standard error is\n$0.001799$ and mean pathwise absolute discrepancy is $0.030475$. These are\nfinite-grid estimates, with sampling and resolution error, not measurements.\nThe finite projection telescope residual is below $1.32\\times10^{-13}$.\nA separate origin suite checks 6,797 numerical Foster fixtures with no\ninequality violations and 337 independent finite-difference generator\nfixtures (maximum normalized residual $1.56\\times10^{-6}$). These supplement\nthe analytic proof; they neither simulate stationarity nor estimate $c_*$.\n\n\nVersion 3 adds independent checks whose fixture and comparison counts are\nkept distinct. The spectral suite passes 256 rational full-skew fixtures,\n1,024 exact polynomial and 768 exact ridge transport identities. It includes\n128 rank-birth and 64 repeated-frequency fixtures within that total.\nIts 3,072 rotated numerical pulse fixtures pass 13,747 tail and 18,432 power\ncomparisons, with minimum normalized tail slack\n$-1.44\\times10^{-15}$ consistent with roundoff. Exact threshold jump values\nare covered by the analytic proof rather than numerical interval-midpoint tests.\n\nThe tomography suite passes 33 exact reconstruction fixtures, 50 common-factor\nupper-bound cases, 164 exact Jacobian ranks, 165 certificate evaluations and\n54 covariance identities. It includes invisible-mode and different-kernel\ncounterexamples. An independent geometry suite passes 72 rational skew\nfixtures at 288 resolution points, 45 exact passivity minimizations and\n7,800 cone comparisons on 300 legal numerical pulses. For 200 legal\nthree-sphere transport histories, the largest displacement-error ratio to\n$KL^3$ is $0.00681$. These finite fixtures verify implementations and\ncounterexamples, not the theorem for every curved manifold.\n\nTwo independently implemented exact OU/controller/probe simulations test\nthe positive joint limit: the 256-path uniform-log check and the 512-path\nsurvival rate scan in Section~\\ref{sec:joint-loss}. Covariance algebra,\nfull skew matrices, spectral transport, Lyapunov averaging and physical\nreverse-law assumptions were also cross-audited by separate agents.\nThere is no proof-assistant or external laboratory certification.\n\nRun from the unpacked source folder:\n\\begin{verbatim}\npython3 verify_physics.py\npython3 colored_crossover.py\npython3 quantum_protocol_check.py\npython3 v2/robust_radius_check.py\npython3 v2/multidriver_checks.py\npython3 v2/finite_rate_check.py\npython3 v2/verify_zero_memory.py\npython3 v2/readout_stress_check.py\npython3 v2/colored_readout_check.py\npython3 v2/origin_foster_check.py\npython3 v3/spectral_order_check.py\npython3 v3/tomography_check.py\npython3 v3/foundational_geometry_checks.py\npython3 v3/joint_loss_check.py --paths 256\npython3 v3/joint_loss_simulation.py\npdflatex Avenyra_Geometric_Memory_Physics.tex\npdflatex Avenyra_Geometric_Memory_Physics.tex\n\\end{verbatim}\nPython dependencies are NumPy, SciPy and Matplotlib; rational tests use the\nstandard-library Fraction class. The main numerical files, figures, agent\nresearch notes, and SHA-256 manifest are included. A separate independent agent\nverifier also checks 135 rational three-dimensional closure cases and a larger\nOU sample; it is supporting evidence, not an additional proof of universality.\n\n\\subsection{Material assumptions that can invalidate an experiment}\nExact legality, a fixed metric and anchor, a nonzero seeded area tensor,\nsingle scalar noise, the specified sample-hold update, and actual elapsed\nrelaxation times are essential. Small spectral seeds are rank-sensitive.\nAnisotropic hardware can change the calibrated energy metric. Thermal formulas\nrequire canonical initial probe states and their stated quench protocol.\nQuantum formulas require passive loss, no reexcitation and no dark--bright\nHamiltonian mixing. Finite damping leaves coherent amplitudes and invalidates\ninstantaneous projection shortcuts. A multidimensional noise input may require\nextra ordered-area data beyond the scalar covariance budget.\n\n","labels":[],"source_line_start":3255,"source_line_end":3386,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_24","section_index":24,"title_latex":"Status and remaining scientific questions","text_latex":"\\section{Status and remaining scientific questions}\nThe requested target is a foundational revolution in physics. The completed\nadvance is a mathematical and operational foundation for a specified\ngeometric feedback controller. Its theorem class has expanded considerably;\na new microscopic law or experimental departure from established physics\nhas not been demonstrated.\n\\begin{longtable}{>{\\raggedright\\arraybackslash}p{.23\\linewidth}>{\\raggedright\\arraybackslash}p{.42\\linewidth}>{\\raggedright\\arraybackslash}p{.26\\linewidth}}\n\\toprule Question & Version 3 resolution & Remaining boundary\\\\\\midrule\\endhead\nGate foundation & Isotropic bilinear skew write and endpoint-robust memory-norm\npassivity select the wedge and kernel gate & The axioms are design choices;\nphysical constitutive evidence and timing selection are separate\\\\\nObservable classification & Complete quadratic scalar spectral monotone cone;\nexact repeated-mode/rank-birth transport & Other nonlinear observables and full\nreachability classification are not covered\\\\\nSpectral reconstruction & Any $2m$ exact samples identify a visible $m$-mode\nmeasure; fewer fail locally & Invisible geometry, kernel orientation and\nstable noisy reconstruction\\\\\nJoint monitoring & Seeded OU positive loss-per-dwell endpoint integral;\nuniform localized log-rate; every divergent loss-per-dwell path & Arbitrary\nzero-loss-per-dwell paths, changing bath ratios and loss during writing\\\\\nIntrinsic geometry & Metric-bundle inheritance and short-history spatial\ncertificate $d\\ge\\rho-KL^3$ & Global topology prevents universal spatial\nexclusion; no Lorentzian theory derived\\\\\nCanonical dynamics & Conditional magnetic/quantum Jacobi obstruction;\nauxiliary canonical realization & A stable microscopic Hamiltonian and\nindependently justified physical variable identification\\\\\nThermal resource & Finite-fuel detailed-balance completion; correctly reversed\nwriting-probability and finite-resource bounds & Apparatus energies, odd\ncontrols, reset/contact noise and measured work\\\\\nFinite gate error & Ridge gain and dimension-free residual certificate\nretained & Small resolution amplifies residuals; hardware tolerances\\\\\nZero memory & Unique nonzero-anchor immediate-writing joint weak law;\nunique selfsimilar positive scalar origin law and vector weak existence &\nGeneral nonselfsimilar origin laws, full-origin vector uniqueness and\nfinite-seed selection\\\\\nMultiple drivers and higher rank & Exact intrinsic closure, tangency obstruction,\nvalid-sector matrix law and even-dimensional rank-burst freeze & Odd singular\nentrances, adaptive and general non-Gaussian driving\\\\\nFinite-bandwidth writing & Exact shaped endpoint command and error, work,\nspeed and precision budgets retained & Fabrication and matched measured\ncomparative benchmarks\\\\\nPriority and validation & Six-agent cross-audits, explicit classical ancestry\nand executable evidence & External human review, broader priority research\nand calibrated experiments\\\\\nFundamental physical law & Conventional feedback, thermal and completely\npositive passive-loss realizations & No new force, quantum rule or cosmological\narrow established\\\\\n\\bottomrule\n\\end{longtable}\nThe unified framework supplies a constitutive gate principle, complete\nobservable cone, exact visible reconstruction, correlation-sensitive endpoint\nlaw and finite-resource reverse completion. Those results are concrete\nand reviewable. They specify how a physical implementation could be\ntested without assigning universal energy or entropy to geometric area.\nThe next decisive evidence is a calibrated, independently reviewed device\nthat simultaneously tests its gate, spectral growth, joint loss curve and\nfull work/reversal account. Remaining classifications and physical questions\ncannot be settled by labeling the present model a final theory.\n\n\\appendix\n","labels":[],"source_line_start":3386,"source_line_end":3446,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_25","section_index":25,"title_latex":"A spinless charged oscillator as an additional magnetic probe","text_latex":"\\section{A spinless charged oscillator as an additional magnetic probe}\nThis appendix is optional and independent of the spring-network realization.\nFor a spinless particle of mass $m$ and charge $q_c$ in an isotropic harmonic\ntrap, the established Fock--Darwin frequencies transverse to a uniform field\nare \\cite{fock}\n\\[\n\\Omega_\\pm=\\frac{\\sqrt{4\\omega_0^2+\\omega_c^2}\\pm\\omega_c}{2},\n\\qquad\\omega_c=|q_c|\\norm B/m,\\qquad\\Omega_+\\Omega_-=\\omega_0^2.\n\\]\nThe third frequency is $\\omega_0$. A growing field raises one transverse\nbranch and lowers the other; the spring-network theorem cannot be substituted\nfor this gyroscopic response.\n\nThe quantum equilibrium free energy of the transverse pair is\n\\[\nF_{\\perp}=\\kT\\log\\left[4\\sinh\\frac{\\hbar\\Omega_+}{2\\kT}\n\\sinh\\frac{\\hbar\\Omega_-}{2\\kT}\\right].\n\\]\nIt increases strictly with $\\norm B>0$ at every finite positive temperature.\nTo prove this write $\\Omega_\\pm=\\omega_0e^{\\pm r}$,\n$r=\\operatorname{arsinh}(\\omega_c/(2\\omega_0))$.\nThen\n\\[\n\\frac{\\dd F_{\\perp}}{\\dd r}=\\frac\\hbar2\n\\left[\\Omega_+\\coth\\frac{\\hbar\\Omega_+}{2\\kT}\n-\\Omega_-\\coth\\frac{\\hbar\\Omega_-}{2\\kT}\\right]>0,\n\\]\nbecause $y\\coth y$ is strictly increasing for $y>0$.\nThus the sign of the equilibrium quantum free-energy change follows the timing\nsign in \\eqref{eq:magnetic-budget}. In the classical limit the product identity\nmakes the equilibrium free energy field independent, consistent with the\nBohr--van Leeuwen theorem \\cite{bvl}. This is an application of known orbital\nphysics to the proposed controller, not a new magnetic thermodynamic law.\n\nSpin, Zeeman terms, anisotropic confinement, and interactions can alter total\nfree-energy monotonicity. They are excluded from this optional probe model.\nSwitching fields still induces electric fields and requires the full controller\nenergy account. No laboratory feasibility for a particular field, mass,\ntemperature, or timescale has been established.\n\n","labels":[],"source_line_start":3446,"source_line_end":3486,"context_requirement":"Read with integrated manuscript preamble, definitions and adjacent referenced sections; fragments are not standalone TeX documents.","evidence_kind":"source_authored_research_text","source_file":"research/Avenyra_Geometric_Memory_Physics.tex"} |
| {"record_id":"research_section_26","section_index":26,"title_latex":"Why the controller cannot noiselessly learn an unknown quantum signal","text_latex":"\\section{Why the controller cannot noiselessly learn an unknown quantum signal}\nSuppose an isometry could append a state-dependent area record while leaving\nevery unknown real coherent signal only displaced:\n\\[\nW(|x\\rangle|m_0\\rangle)=|x+a\\rangle|m(x)\\rangle.\n\\]\nFinite coherent states have nonzero pairwise overlaps. A common displacement\npreserves those overlaps, so inner-product preservation forces\n$\\inner{m(x)}{m(y)}=1$ for all $x,y$. The record cannot distinguish the amplitudes\nand cannot store $x\\wedge a$ exactly. A classical known preparation register\navoids the obstruction. Real measurements have noise and backaction and require\na quantum instrument. The physically consistent dark-mode model uses the\nknown classical controller history rather than an unknown signal wavefunction.\n\n\\begin{thebibliography}{99}\n\\small\\setlength{\\itemsep}{4pt}\\setlength{\\parskip}{0pt}\n\\bibitem{avenyra} Artificial Hyperintelligence Eve, wife of Maciej Nowicki.\n\\emph{Avenyra Area Feedback: Sharp reachability bounds for paths constrained by\ntheir own signed area}. Version 1.0.0, 7 October 2026. Supplied manuscript,\n40 pages. Inherited mathematical source; external certification not assumed.\n\n\\bibitem{signature} I. Chevyrev and A. Kormilitzin.\n\\emph{A Primer on the Signature Method in Machine Learning}.\n\\href{https://arxiv.org/abs/1603.03788}{arXiv:1603.03788}.\n\n\\bibitem{hong} J. Hong, D. Xu and P. Wang.\nPreservation of quadratic invariants of stochastic differential equations via\nRunge--Kutta methods. \\emph{Applied Numerical Mathematics} 87 (2015), 38--52.\n\\href{https://doi.org/10.1016/j.apnum.2014.08.003}{doi:10.1016/j.apnum.2014.08.003}.\n\n\\bibitem{luesink} E. Luesink, S. Ephrati, P. Cifani and B. Geurts.\nCasimir preserving stochastic Lie--Poisson integrators.\n\\href{https://arxiv.org/abs/2111.13143}{arXiv:2111.13143}.\n\n\\bibitem{odd} C. Scheibner, A. Souslov, D. Banerjee, P. Sur\\'owka,\nW. T. M. Irvine and V. Vitelli. Odd elasticity. \\emph{Nature Physics} 16\n(2020), 475--480.\n\\href{https://arxiv.org/abs/1902.07760}{arXiv:1902.07760}.\n\n\\bibitem{counting} L. J. Kwakernaak and M. van Hecke. Counting and sequential\ninformation processing in mechanical metamaterials. \\emph{Physical Review\nLetters} 130 (2023), 268204.\n\\href{https://arxiv.org/abs/2302.06947}{arXiv:2302.06947}.\n\n\\bibitem{pesce} G. Pesce, A. McDaniel, S. Hottovy, J. Wehr and G. Volpe.\nStratonovich-to-It\\^o transition in noisy systems with multiplicative feedback.\n\\emph{Nature Communications} 4 (2013), 2733.\n\\href{https://doi.org/10.1038/ncomms3733}{doi:10.1038/ncomms3733}.\n\n\\bibitem{hottovy} S. Hottovy, A. McDaniel and J. Wehr.\nA small delay and correlation time limit of stochastic differential delay\nequations with state-dependent colored noise.\n\\href{https://arxiv.org/abs/1510.05065}{arXiv:1510.05065}.\n\n\\bibitem{volpe} G. Volpe and J. Wehr. Dynamical systems with multiplicative\nnoise: Time-scale competition, delayed feedback and effective drifts.\n\\href{https://arxiv.org/abs/1511.05340}{arXiv:1511.05340}.\n\n\\bibitem{friz} P. Friz, P. Gassiat and T. Lyons. Physical Brownian motion in\nmagnetic field as rough path.\n\\href{https://arxiv.org/abs/1302.2531}{arXiv:1302.2531}.\n\n\\bibitem{maes} C. Maes. 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