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Avenyra-Geometric-Memory-Physics

# VELORYN / Avenyra v3: model and reader guide

This release studies a finite feedback controller whose stored skew area determines its next permitted displacement. Its main claims are mathematical theorems under stated assumptions and proposed powered readouts. A universal force law, laboratory confirmation and mathematical first priority are not established.

The authoritative account is the [integrated report](research/Avenyra_Geometric_Memory_Physics.pdf), with its [complete LaTeX source](research/Avenyra_Geometric_Memory_Physics.tex). The original [Avenyra Area Feedback manuscript](research/source/Avenyra_Area_Feedback_v1.0.0.pdf) is retained. This guide explains how to read and implement those results; it introduces no additional theorem claims.

## 1. The exact state and update

Work in a finite-dimensional real Euclidean space with a specified metric. The controller state is

$$
x\in\mathbb R^d,\qquad A^T=-A.
$$

The vector `x` is an endpoint/displacement register. The skew matrix `A` is a stored area register. Define

$$
x\wedge a=xa^T-ax^T.
$$

A proposed stroke `a` is legal when the **current, pre-stroke memory** annihilates it:

$$
\boxed{Aa=0,\qquad x^+=x+a,\qquad A^+=A+x\wedge a.}
$$

Check the gate against the old `A`, then use the old `x` in the memory write. Update both registers from that same pre-stroke state. A pulse sequence retains the full pair `(x,A)`; the endpoint alone, the spectrum alone, or the kernel alone generally loses information needed for future permissions.

Let

$$
K=\ker A,\quad H=\operatorname{im}A=K^\perp,
\quad x=u+w,\quad u=P_Hx,\ w=P_Kx.
$$

The geometric exclusion certificate is

$$
\rho^2(x,A)=\|u\|^2.
$$

For the exact finite rule, every future finite legal word ends at a vector of norm at least `rho`. The minimum is attained uniquely at `u`, reached immediately by the legal stroke `a=-w`. The certificate itself cannot decrease under further legal pulses. This is a result about the controller's Euclidean endpoint register, not automatically about a material object's distance in arbitrary space.

Rank rises by two precisely when `w` and `a` are linearly independent. Otherwise it is unchanged. A full-rank skew matrix in even dimension has no nonzero legal stroke. None of this requires `A` to be a physical magnetic field.

### Units and normalization

The report uses **dimensionless controller registers**, calibrated by a fixed length `L_*`:

$$
x=x_{\rm physical}/L_*,\qquad
a=a_{\rm physical}/L_*,\qquad
A=A_{\rm physical}/L_*^2.
$$
An equivalent dimensional path convention is possible:
| Quantity | Dimensional path convention | Report convention |
|---|---|---|
| `x`, `a`, `Q` | Length | Dimensionless |
| `A`, three-dimensional `b`, skew frequency `sigma` | Length squared | Dimensionless |
| `R`, `rho^2` | Length squared | Dimensionless |
| `U`, squared skew frequency `sigma^2`, `E(A)` | Length to the fourth | Dimensionless |
| Noise parameter `nu`, resolved-square rate `kappa` | Length squared per time | Inverse time |
| Independent probe displacement `q` | Length | Physical length |
Here
$$
E(A)=\tfrac12\|A\|_F^2
$$
is a mathematical squared-memory norm. It is not measured energy in joules without a physical Hamiltonian or hardware calibration. In the dimensional convention, spectral scales must have matching units; logarithmic weights need a calibrated dimensionless argument. A device that converts area memory to magnetic field, stiffness or another actuator has additional coupling coefficients and work costs.
### Why the pre-stroke timing matters
A finite permitted pulse need not remain legal against its changing memory during the pulse. Requiring legality continuously defines another controller. For smooth continuous trajectories,
$$
\dot x=v,\qquad\dot A=x\wedge v,\qquad Av=0,
$$
the skew spectrum and `rho` are constant; a zero-memory seed cannot start writing along a smooth legal path. Finite held gating can write.
The report also studies a timing gate `(A+theta x wedge a)a=0`. Under that gate,
$$
E(A^+)-E(A)=(1-2\theta)
\left(\|x\|^2\|a\|^2-\langle x,a\rangle^2\right).
$$
The original rule has `theta=0`. The midpoint rule `theta=1/2` is isospectral. Timing is therefore part of the constitutive specification, not a harmless implementation detail or a choice of notation for the same stochastic equation.
## 2. Three-dimensional coordinates and a worked finite example
In three dimensions fix the convention
$$
Av=v\times b.
$$
Then `b` is the axial representation of the skew area register, and the pulse is
$$
a\times b=0,\qquad x^+=x+a,\qquad b^+=b+x\times a.
$$
The sign convention matters: some libraries instead represent a skew matrix by `b cross v`.
When `U=||b||^2>0`, define
$$
U=\|b\|^2,\qquad n=b/\sqrt U,\qquad
Q=x\cdot n,\qquad R=\|x\|^2-Q^2=\rho^2.
$$
Every legal stroke is `a=s n` for a signed real `s`. Its exact scalar update is
$$
\begin{aligned}
U^+&=U+s^2R,\\
Q^+&=\frac{Q+s}{\sqrt{1+s^2R/U}},\\
R^+&=R+\frac{(s^2R/U)(Q+s)^2}{1+s^2R/U}.
\end{aligned}
$$
These formulas show that `U` and `R` cannot decrease. They close the rank-two, three-dimensional intrinsic scalar dynamics; they are not a universal three-scalar closure for higher rank or for every externally labelled multichannel control.
For a simple dimensionless history, begin with `x=b=0` and use the standard orthonormal basis:
| Step | Stroke | Endpoint `x` | Memory `b` | `(U,R,Q)` |
|---|---|---|---|---|
| 0 | — | `(0,0,0)` | `(0,0,0)` | `U=0`; `n,Q` not defined |
| 1 | `e1` | `(1,0,0)` | `(0,0,0)` | Still zero memory |
| 2 | `e2` | `(1,1,0)` | `(0,0,1)` | `(1,2,0)` |
| 3 | `e3` | `(1,1,1)` | `(1,-1,1)` | `(3,8/3,1/sqrt(3))` |
The first pulse writes no area because the old endpoint is zero. The second is legal because the old memory remains zero; it creates `A=e1 wedge e2`. At step 2, only strokes parallel to `e3` are legal. Step 3 tilts the permission line to `(1,-1,1)`. A further `e3` stroke would now be illegal.
The exclusion radius grows from `sqrt(2)` at step 2 to `sqrt(8/3)` at step 3. From step 3 the closest reachable endpoint is
$$
u=x-Qn=(2/3,4/3,2/3),
$$
reached by `a=-(1,-1,1)/3`. Its norm is exactly `sqrt(8/3)`.
At `U=0`, the vector pulse rule remains well defined, but the scalar direction `n=b/sqrt(U)` is undefined. The report treats selected zero-memory entrances separately; inserting zero into the seeded scalar recurrences or SDE is not justified. General complete-origin classification and vector uniqueness remain open within the stated scopes.
## 3. The seeded sampled-noise limit
For a stationary scalar Ornstein–Uhlenbeck input with correlation time `tau`, variance `nu/tau`, sample interval `h`, and integrated stroke
$$
s_j=\int_{jh}^{(j+1)h}\eta_\tau(t)\,dt,
$$
hold the legal direction during each stroke. In the limit `h,tau -> 0` with fixed `c=h/tau>0`, set
$$
f(c)=1-\frac{1-e^{-c}}c,\qquad\kappa=2\nu f(c).
$$
For a nonzero seed `U0>0`, the scalar limit is
$$
\boxed{
\begin{aligned}
\mathrm dU&=\kappa R\,\mathrm dt,\\
\mathrm dR&=\kappa RQ^2/U\,\mathrm dt,\\
\mathrm dQ&=\sqrt{2\nu}\,\mathrm dW
             -\frac{\kappa RQ}{2U}\,\mathrm dt.
\end{aligned}}
$$
The cumulative drive has Brownian variance rate `2 nu`, whereas the resolved sampling-grid sum of stroke squares has rate `2 nu f(c)`. The latter controls memory writing. When `c` is small, `f(c)~c/2`; when it is large, `f(c)~1-1/c`. The same macroscopic drive variance can thus produce different writing rates in this specified sampling protocol.
This theorem assumes stationary OU input, the stated scaling, exact finite held gating and a nonzero seed. It is not a universal colored-noise law. The continuously monitored geometric counterpart preserves `U,R` and has different drifts; merely changing Itô/Stratonovich notation does not turn one physical controller into the other. The report proves additional multidriver statements only in explicitly admissible smooth sectors and describes tangency and rank-saturation obstructions elsewhere.
For higher seeded rank, the intrinsic scalar dynamics instead use a finite polynomial closure: the squared-frequency polynomial and its displacement-weight numerator, together with signed `Q`. External mode labels can require further orientation data. See the [multidriver and closure analysis](research/reviews/v2/multidriver.md) and the full report.
## 4. The complete quadratic spectral certificate cone
Let `H_A=-A^2`, a positive semidefinite matrix. For a finite-valued real scalar function `phi`, consider
$$
M_\phi(x,A)=x^T\phi(H_A)x.
$$
The v3 classification is exact:
$$
\boxed{
M_\phi\text{ increases or stays constant under every legal finite pulse}
\iff
\phi(0)=0\text{ and }\phi\ge0\text{ is nondecreasing on }(0,\infty).
}
$$
The quantifier covers every finite dimension, state and exact pre-gated pulse. No continuity, differentiability, global boundedness or operator-monotonicity assumption is required. This classifies **quadratic scalar spectral observables**. It does not classify every nonlinear invariant or every reachable state.
Examples include:
| Weight `phi(t)` in normalized units | Certificate |
|---|---|
| `1` for `t>0`, `0` at `0` | `rho^2` |
| `t` | `||A x||^2` |
| `t^p`, `p>0` | Every positive-power spectral certificate |
| `log(1+t)` | Logarithmic spectral weight |
| `min(t,c)`, `c>0` | Saturated spectral weight |
| `1` for `t>=c`, `0` otherwise, `c>0` | Weight above a spectral threshold |
| `t/(lambda^2+t)`, `lambda>0` | Smooth ridge certificate `F_lambda` |
Finite nonnegative combinations also belong to the cone. The zero function is allowed. Discontinuous thresholds are exact certificates but require separate care under measurement error or limiting approximations; the ridge family supplies smooth, rank-continuous error certificates.
To see what is ordered, write the positive displacement spectral measure as
$$
\mu_{x,A}=\sum_{t>0}\|P_t x\|^2\delta_t,
\qquad M_\phi=\int\phi(t)\,\mu_{x,A}(dt).
$$
This is weighted by `x`; it is not just the list of eigenvalues of `A`. Every legal pulse increases its positive upper-tail order. In the worked example, the measures after steps 2 and 3 are respectively `2 delta_1` and `(8/3) delta_3`, giving `2 phi(1)` and `(8/3) phi(3)`.
The exact transition has a positive rank-one reduction. For `a=s n`, `s=||a||>0`, decompose `x=u+Qn+w_perp` with `w_perp` in the old kernel and perpendicular to `n`. Let `r_i` be the old displacement weights, `t_i` their positive squared frequencies, and `beta=||w_perp||`. Set
$$
H_0=\operatorname{diag}(t_i),\qquad v_i=\sqrt{r_i},\qquad
H_1=H_0+s^2vv^T.
$$
At rank birth append a zero entry to `H0` and append `beta` to `v`. If `mu_1` is the spectral measure of this same vector `v` for `H1`, the new positive endpoint measure is
$$
\mu_{x^+,A^+}(d\ell)
=\left(1+\frac{s^2(Q+s)^2}{\ell}\right)\mu_1(d\ell),\qquad\ell>0.
$$
Classical aligned rank-one trace order, combined with this exact postweight, yields the complete cone. The alignment with the perturbing vector is essential: scalar monotonicity alone is not matrix operator monotonicity. Full cone order between arbitrary states is still insufficient for finite reachability, and ordering all ridge probes is weaker than ordering the whole cone. See the [complete-cone derivation and counterexamples](research/reviews/v3/spectral_order.md).
## 5. The physical register is separate from the probe
The controller registers `x,A` program a separate device. For the held thermal ridge readout, the independent classical oscillator has physical displacement `q`, mass `M_p`, temperature `T`, and squared-frequency matrix
$$
K_\lambda=\omega_0^2(I-A^2/\lambda^2).
$$
It is positive for every skew `A`. Hold the registers fixed and bring the probe to canonical equilibrium. The measured coordinate `Y=x dot q` then has variance
$$
\operatorname{Var}(Y)=\frac{k_BT}{M_p\omega_0^2}
\left(\|x\|^2-F_\lambda(x,A)\right),
\qquad
F_\lambda=x^T[-A^2(\lambda^2I-A^2)^{-1}]x.
$$
Thus the known controller norm and a calibrated equilibrium variance determine the ridge certificate. `q` is not the register `x`. The formula assumes a static held state and equilibrium; independent sample-variance formulas additionally require independent Gaussian samples or an explicit autocorrelation correction. Preparing the probe and changing its stiffness are powered operations.
With at most `m` distinct positive frequencies visible to `x`, any `2m` distinct exact positive ridge scales determine the rational response and the visible weighted spectrum. This permits evaluation of every cone certificate from that measure. It recovers neither invisible modes nor their orientation, and exact identifiability is not a stable noisy-inversion theorem. See [spectral tomography](research/reviews/v3/spectral_tomography.md) and its [audit](research/reviews/v3/tomography_audit.md).
Other chapters specify quantum survival, field-quench and thermal work/heat protocols. Their parameters, hold/dwell ordering, reservoirs and reverse experiments are part of each result. They do not replace an unknown quantum state with a noiseless classical register, establish quantum advantage, or make the area norm into a universal energy or entropy. The [thermodynamic foundations](research/reviews/v3/thermodynamic_foundations.md) distinguish a properly reversed experiment from same-run transient erasure statistics.
## 6. Coordinate-independent bundle extension
Let `E -> M` be a real metric vector bundle with a metric-compatible connection. For a specified path from base point `p` to `p+`, let `U_gamma` be orthogonal parallel transport. The intrinsic pulse is
$$
x^+=U_\gamma(x+a),\qquad
A^+=U_\gamma(A+x\wedge a)U_\gamma^{-1},\qquad Aa=0.
$$
All metric and orthogonal-conjugacy identities inherit exactly, including rank, kernel transport, `rho` and the whole spectral cone. This removes dependence on a coordinate frame; the metric, connection, path and controller register remain model data.
For the tangent bundle with Levi–Civita connection, in matching geometric units one may choose `p+=Exp_p(a)` and transport along that geodesic. The accumulated `x` is then **developed displacement**, not a global logarithm of physical position.
For `x0=0`, a finite geodesic history of total length `L` obeys
$$
d(p_0,p_T)\ge\rho_j-KL^3
$$
for each earlier certificate `rho_j`, provided `L` is below the injectivity radius at `p0`, curvature throughout the relevant radial normal ball satisfies `||R(u,v)||_op <= K||u wedge v||`, and `K L^2 <= 1/4`. These geometric quantities use consistent units; `K` has inverse-length-squared units in a dimensional metric.
The hypotheses are essential. On a flat torus, two legal coordinate loops can return the physical base point exactly to its start while leaving nonzero internal `x,A,rho`. Curvature alone cannot supply a global spatial exclusion law. See the [intrinsic geometry analysis](research/reviews/v3/foundational_geometry.md) and [independent audit](research/reviews/v3/foundational_audit.md).
## 7. The axiomatic gate-selection result
There is a precise constitutive selection argument, with explicitly chosen design axioms:
1. The local skew write is bilinear in `x,a` and equivariant under the full orthogonal group `O(d)`, for `d>=2`. This forces `B(x,a)=gamma x wedge a` for a scalar `gamma`.
2. For a nonzero gain `gamma`, a permission set defined independently of `x` must guarantee that the stipulated norm `E(A)=||A||_F^2/2` never decreases for **every possible endpoint register**.
For fixed `A,a != 0`,
$$
\Delta E=2\gamma x^TAa+
\gamma^2\left(\|x\|^2\|a\|^2-\langle x,a\rangle^2\right),
$$
and
$$
\min_x\Delta E=-\frac{\|Aa\|^2}{\|a\|^2}.
$$
The minimum is attained at `x=-Aa/(gamma||a||^2)`, with any additional component parallel to `a`. Therefore `ker A` is exactly the maximal stroke set satisfying that endpoint-robust norm guarantee.
These assumptions select the wedge write and kernel gate. They do not select the write gain, its sign or the finite sampling clock, and they are not a fundamental thermodynamic law. Endpoint-dependent permission, extra orientation data, nonlinear writes, a different memory energy or another timing convention permit other models. Physical energetic passivity requires an actual device and a complete supply/work account.
## 8. The conditional canonical obstruction
One tempting continuous identification takes `x` as canonical position and `b` as unit-mass mechanical momentum in the six-variable flow
$$
\dot x=b,\qquad\dot b=x\times b.
$$
Assume `{xi,xj}=0` and `{xi,bj}=delta_ij`. Jacobi forces the mechanical-momentum bracket to have magnetic form with a field independent of `b`. Reproducing this flow requires `B=-x`, but its Poisson Jacobiator is `-div B=3`, which is nonzero. The corresponding canonical quantum proposal
$$
[x_i,x_j]=0,\qquad[x_i,b_j]=i\hbar\delta_{ij},\qquad
[b_i,b_j]=-i\hbar\epsilon_{ijk}x_k
$$
has cyclic nested commutator `-3 hbar^2 I` on a common invariant domain, contradicting the associative commutator Jacobi identity.
This obstructs the specified canonical position/mechanical-momentum interpretation. It is not a theorem that every possible Hamiltonian realization is impossible. An auxiliary cotangent Hamiltonian,
$$
\mathcal H=\pi_x\cdot b+\pi_b\cdot(x\times b),
$$
reproduces the flow on a doubled state space, but is unbounded in the auxiliary momenta and does not provide a stable microscopic energy or a free controller. These issues have established magnetic-charge and symplectic-realization ancestry, cited in the report. The sampled finite controller and powered probe proposals do not assert that their area register is a new fundamental magnetic force.
## 9. Reading and verification map
| Goal | Release files |
|---|---|
| Read the final hypotheses, proofs and bibliography | [Full report](research/Avenyra_Geometric_Memory_Physics.pdf), [source](research/Avenyra_Geometric_Memory_Physics.tex) |
| Inspect the exact source model | [Original Avenyra manuscript](research/source/Avenyra_Area_Feedback_v1.0.0.pdf) |
| Check the complete spectral cone | [Analysis](research/reviews/v3/spectral_order.md), [verifier](research/v3/spectral_order_check.py), [results](research/v3/results/spectral_order_check.json) |
| Check visible-spectrum reconstruction | [Analysis](research/reviews/v3/spectral_tomography.md), [verifier](research/v3/tomography_check.py), [results](research/v3/results/tomography_check.json) |
| Check intrinsic geometry and gate selection | [Analysis](research/reviews/v3/foundational_geometry.md), [verifier](research/v3/foundational_geometry_checks.py), [results](research/v3/results/foundational_geometry_checks.json) |
| Study noise, rank and higher-rank closure | [Multidriver analysis](research/reviews/v2/multidriver.md), [verifier](research/v2/multidriver_checks.py) |
| Study the specified joint write/loss protocol | [Joint-loss analysis](research/reviews/v3/joint_loss.md), [independent checker](research/v3/joint_loss_check.py), [simulation](research/v3/joint_loss_simulation.py) |
| Review thermodynamic and foundational limitations | [Thermodynamic analysis](research/reviews/v3/thermodynamic_foundations.md), [prior-art assessment](research/reviews/v3/foundational_prior_art.md), [final claim audit](research/reviews/v3/final_claim_review.md) |
| Identify proved and outstanding scopes | [STATUS.json](research/STATUS.json), [research README](research/README.md) |
The review notes preserve the development history and can contain intermediate conditional formulations. Use the integrated report and `STATUS.json` for the final scope. Exact rational fixtures, finite-grid simulations and AI cross-audits supplement the analytic proofs; they are not laboratory data or external human certification.
For reproduction, install the dependencies in [requirements.txt](research/requirements.txt) and run the scripts from the `research/` directory so their relative paths resolve. The canonical report's source compiles from that same directory. The release retains established mathematics under its standard names and separates candidate model-specific contributions from unverified priority and physical realization questions.

	# VELORYN / Avenyra v3: model and reader guide

	This release studies a finite feedback controller whose stored skew area determines its next permitted displacement. Its main claims are mathematical theorems under stated assumptions and proposed powered readouts. A universal force law, laboratory confirmation and mathematical first priority are not established.

	The authoritative account is the [integrated report](research/Avenyra_Geometric_Memory_Physics.pdf), with its [complete LaTeX source](research/Avenyra_Geometric_Memory_Physics.tex). The original [Avenyra Area Feedback manuscript](research/source/Avenyra_Area_Feedback_v1.0.0.pdf) is retained. This guide explains how to read and implement those results; it introduces no additional theorem claims.

	## 1. The exact state and update

	Work in a finite-dimensional real Euclidean space with a specified metric. The controller state is

	$$
	x\in\mathbb R^d,\qquad A^T=-A.
	$$

	The vector `x` is an endpoint/displacement register. The skew matrix `A` is a stored area register. Define

	$$
	x\wedge a=xa^T-ax^T.
	$$

	A proposed stroke `a` is legal when the current, pre-stroke memory annihilates it:

	$$
	\boxed{Aa=0,\qquad x^+=x+a,\qquad A^+=A+x\wedge a.}
	$$

	Check the gate against the old `A`, then use the old `x` in the memory write. Update both registers from that same pre-stroke state. A pulse sequence retains the full pair `(x,A)`; the endpoint alone, the spectrum alone, or the kernel alone generally loses information needed for future permissions.

	Let

	$$
	K=\ker A,\quad H=\operatorname{im}A=K^\perp,
	\quad x=u+w,\quad u=P_Hx,\ w=P_Kx.
	$$

	The geometric exclusion certificate is

	$$
	\rho^2(x,A)=\\|u\\|^2.
	$$

	For the exact finite rule, every future finite legal word ends at a vector of norm at least `rho`. The minimum is attained uniquely at `u`, reached immediately by the legal stroke `a=-w`. The certificate itself cannot decrease under further legal pulses. This is a result about the controller's Euclidean endpoint register, not automatically about a material object's distance in arbitrary space.

	Rank rises by two precisely when `w` and `a` are linearly independent. Otherwise it is unchanged. A full-rank skew matrix in even dimension has no nonzero legal stroke. None of this requires `A` to be a physical magnetic field.

	### Units and normalization

	The report uses dimensionless controller registers, calibrated by a fixed length `L_*`:

	$$
	x=x_{\rm physical}/L_*,\qquad
	a=a_{\rm physical}/L_*,\qquad
	A=A_{\rm physical}/L_*^2.
	$$

	An equivalent dimensional path convention is possible:

	\| Quantity \| Dimensional path convention \| Report convention \|
	\|---\|---\|---\|
	\| `x`, `a`, `Q` \| Length \| Dimensionless \|
	\| `A`, three-dimensional `b`, skew frequency `sigma` \| Length squared \| Dimensionless \|
	\| `R`, `rho^2` \| Length squared \| Dimensionless \|
	\| `U`, squared skew frequency `sigma^2`, `E(A)` \| Length to the fourth \| Dimensionless \|
	\| Noise parameter `nu`, resolved-square rate `kappa` \| Length squared per time \| Inverse time \|
	\| Independent probe displacement `q` \| Length \| Physical length \|

	Here

	$$
	E(A)=\tfrac12\\|A\\|_F^2
	$$

	is a mathematical squared-memory norm. It is not measured energy in joules without a physical Hamiltonian or hardware calibration. In the dimensional convention, spectral scales must have matching units; logarithmic weights need a calibrated dimensionless argument. A device that converts area memory to magnetic field, stiffness or another actuator has additional coupling coefficients and work costs.

	### Why the pre-stroke timing matters

	A finite permitted pulse need not remain legal against its changing memory during the pulse. Requiring legality continuously defines another controller. For smooth continuous trajectories,

	$$
	\dot x=v,\qquad\dot A=x\wedge v,\qquad Av=0,
	$$

	the skew spectrum and `rho` are constant; a zero-memory seed cannot start writing along a smooth legal path. Finite held gating can write.

	The report also studies a timing gate `(A+theta x wedge a)a=0`. Under that gate,

	$$
	E(A^+)-E(A)=(1-2\theta)
	\left(\\|x\\|^2\\|a\\|^2-\langle x,a\rangle^2\right).
	$$

	The original rule has `theta=0`. The midpoint rule `theta=1/2` is isospectral. Timing is therefore part of the constitutive specification, not a harmless implementation detail or a choice of notation for the same stochastic equation.

	## 2. Three-dimensional coordinates and a worked finite example

	In three dimensions fix the convention

	$$
	Av=v\times b.
	$$

	Then `b` is the axial representation of the skew area register, and the pulse is

	$$
	a\times b=0,\qquad x^+=x+a,\qquad b^+=b+x\times a.
	$$

	The sign convention matters: some libraries instead represent a skew matrix by `b cross v`.

	When `U=\|\|b\|\|^2>0`, define

	$$
	U=\\|b\\|^2,\qquad n=b/\sqrt U,\qquad
	Q=x\cdot n,\qquad R=\\|x\\|^2-Q^2=\rho^2.
	$$

	Every legal stroke is `a=s n` for a signed real `s`. Its exact scalar update is

	$$
	\begin{aligned}
	U^+&=U+s^2R,\\
	Q^+&=\frac{Q+s}{\sqrt{1+s^2R/U}},\\
	R^+&=R+\frac{(s^2R/U)(Q+s)^2}{1+s^2R/U}.
	\end{aligned}
	$$

	These formulas show that `U` and `R` cannot decrease. They close the rank-two, three-dimensional intrinsic scalar dynamics; they are not a universal three-scalar closure for higher rank or for every externally labelled multichannel control.

	For a simple dimensionless history, begin with `x=b=0` and use the standard orthonormal basis:

	\| Step \| Stroke \| Endpoint `x` \| Memory `b` \| `(U,R,Q)` \|
	\|---\|---\|---\|---\|---\|
	\| 0 \| — \| `(0,0,0)` \| `(0,0,0)` \| `U=0`; `n,Q` not defined \|
	\| 1 \| `e1` \| `(1,0,0)` \| `(0,0,0)` \| Still zero memory \|
	\| 2 \| `e2` \| `(1,1,0)` \| `(0,0,1)` \| `(1,2,0)` \|
	\| 3 \| `e3` \| `(1,1,1)` \| `(1,-1,1)` \| `(3,8/3,1/sqrt(3))` \|

	The first pulse writes no area because the old endpoint is zero. The second is legal because the old memory remains zero; it creates `A=e1 wedge e2`. At step 2, only strokes parallel to `e3` are legal. Step 3 tilts the permission line to `(1,-1,1)`. A further `e3` stroke would now be illegal.

	The exclusion radius grows from `sqrt(2)` at step 2 to `sqrt(8/3)` at step 3. From step 3 the closest reachable endpoint is

	$$
	u=x-Qn=(2/3,4/3,2/3),
	$$

	reached by `a=-(1,-1,1)/3`. Its norm is exactly `sqrt(8/3)`.

	At `U=0`, the vector pulse rule remains well defined, but the scalar direction `n=b/sqrt(U)` is undefined. The report treats selected zero-memory entrances separately; inserting zero into the seeded scalar recurrences or SDE is not justified. General complete-origin classification and vector uniqueness remain open within the stated scopes.

	## 3. The seeded sampled-noise limit

	For a stationary scalar Ornstein–Uhlenbeck input with correlation time `tau`, variance `nu/tau`, sample interval `h`, and integrated stroke

	$$
	s_j=\int_{jh}^{(j+1)h}\eta_\tau(t)\,dt,
	$$

	hold the legal direction during each stroke. In the limit `h,tau -> 0` with fixed `c=h/tau>0`, set

	$$
	f(c)=1-\frac{1-e^{-c}}c,\qquad\kappa=2\nu f(c).
	$$

	For a nonzero seed `U0>0`, the scalar limit is

	$$
	\boxed{
	\begin{aligned}
	\mathrm dU&=\kappa R\,\mathrm dt,\\
	\mathrm dR&=\kappa RQ^2/U\,\mathrm dt,\\
	\mathrm dQ&=\sqrt{2\nu}\,\mathrm dW
	-\frac{\kappa RQ}{2U}\,\mathrm dt.
	\end{aligned}}
	$$

	The cumulative drive has Brownian variance rate `2 nu`, whereas the resolved sampling-grid sum of stroke squares has rate `2 nu f(c)`. The latter controls memory writing. When `c` is small, `f(c)~c/2`; when it is large, `f(c)~1-1/c`. The same macroscopic drive variance can thus produce different writing rates in this specified sampling protocol.

	This theorem assumes stationary OU input, the stated scaling, exact finite held gating and a nonzero seed. It is not a universal colored-noise law. The continuously monitored geometric counterpart preserves `U,R` and has different drifts; merely changing Itô/Stratonovich notation does not turn one physical controller into the other. The report proves additional multidriver statements only in explicitly admissible smooth sectors and describes tangency and rank-saturation obstructions elsewhere.

	For higher seeded rank, the intrinsic scalar dynamics instead use a finite polynomial closure: the squared-frequency polynomial and its displacement-weight numerator, together with signed `Q`. External mode labels can require further orientation data. See the [multidriver and closure analysis](research/reviews/v2/multidriver.md) and the full report.

	## 4. The complete quadratic spectral certificate cone

	Let `H_A=-A^2`, a positive semidefinite matrix. For a finite-valued real scalar function `phi`, consider

	$$
	M_\phi(x,A)=x^T\phi(H_A)x.
	$$

	The v3 classification is exact:

	$$
	\boxed{
	M_\phi\text{ increases or stays constant under every legal finite pulse}
	\iff
	\phi(0)=0\text{ and }\phi\ge0\text{ is nondecreasing on }(0,\infty).
	}
	$$

	The quantifier covers every finite dimension, state and exact pre-gated pulse. No continuity, differentiability, global boundedness or operator-monotonicity assumption is required. This classifies quadratic scalar spectral observables. It does not classify every nonlinear invariant or every reachable state.

	Examples include:

	\| Weight `phi(t)` in normalized units \| Certificate \|
	\|---\|---\|
	\| `1` for `t>0`, `0` at `0` \| `rho^2` \|
	\| `t` \| `\|\|A x\|\|^2` \|
	\| `t^p`, `p>0` \| Every positive-power spectral certificate \|
	\| `log(1+t)` \| Logarithmic spectral weight \|
	\| `min(t,c)`, `c>0` \| Saturated spectral weight \|
	\| `1` for `t>=c`, `0` otherwise, `c>0` \| Weight above a spectral threshold \|
	\| `t/(lambda^2+t)`, `lambda>0` \| Smooth ridge certificate `F_lambda` \|

	Finite nonnegative combinations also belong to the cone. The zero function is allowed. Discontinuous thresholds are exact certificates but require separate care under measurement error or limiting approximations; the ridge family supplies smooth, rank-continuous error certificates.

	To see what is ordered, write the positive displacement spectral measure as

	$$
	\mu_{x,A}=\sum_{t>0}\\|P_t x\\|^2\delta_t,
	\qquad M_\phi=\int\phi(t)\,\mu_{x,A}(dt).
	$$

	This is weighted by `x`; it is not just the list of eigenvalues of `A`. Every legal pulse increases its positive upper-tail order. In the worked example, the measures after steps 2 and 3 are respectively `2 delta_1` and `(8/3) delta_3`, giving `2 phi(1)` and `(8/3) phi(3)`.

	The exact transition has a positive rank-one reduction. For `a=s n`, `s=\|\|a\|\|>0`, decompose `x=u+Qn+w_perp` with `w_perp` in the old kernel and perpendicular to `n`. Let `r_i` be the old displacement weights, `t_i` their positive squared frequencies, and `beta=\|\|w_perp\|\|`. Set

	$$
	H_0=\operatorname{diag}(t_i),\qquad v_i=\sqrt{r_i},\qquad
	H_1=H_0+s^2vv^T.
	$$

	At rank birth append a zero entry to `H0` and append `beta` to `v`. If `mu_1` is the spectral measure of this same vector `v` for `H1`, the new positive endpoint measure is

	$$
	\mu_{x^+,A^+}(d\ell)
	=\left(1+\frac{s^2(Q+s)^2}{\ell}\right)\mu_1(d\ell),\qquad\ell>0.
	$$

	Classical aligned rank-one trace order, combined with this exact postweight, yields the complete cone. The alignment with the perturbing vector is essential: scalar monotonicity alone is not matrix operator monotonicity. Full cone order between arbitrary states is still insufficient for finite reachability, and ordering all ridge probes is weaker than ordering the whole cone. See the [complete-cone derivation and counterexamples](research/reviews/v3/spectral_order.md).

	## 5. The physical register is separate from the probe

	The controller registers `x,A` program a separate device. For the held thermal ridge readout, the independent classical oscillator has physical displacement `q`, mass `M_p`, temperature `T`, and squared-frequency matrix

	$$
	K_\lambda=\omega_0^2(I-A^2/\lambda^2).
	$$

	It is positive for every skew `A`. Hold the registers fixed and bring the probe to canonical equilibrium. The measured coordinate `Y=x dot q` then has variance

	$$
	\operatorname{Var}(Y)=\frac{k_BT}{M_p\omega_0^2}
	\left(\\|x\\|^2-F_\lambda(x,A)\right),
	\qquad
	F_\lambda=x^T[-A^2(\lambda^2I-A^2)^{-1}]x.
	$$

	Thus the known controller norm and a calibrated equilibrium variance determine the ridge certificate. `q` is not the register `x`. The formula assumes a static held state and equilibrium; independent sample-variance formulas additionally require independent Gaussian samples or an explicit autocorrelation correction. Preparing the probe and changing its stiffness are powered operations.

	With at most `m` distinct positive frequencies visible to `x`, any `2m` distinct exact positive ridge scales determine the rational response and the visible weighted spectrum. This permits evaluation of every cone certificate from that measure. It recovers neither invisible modes nor their orientation, and exact identifiability is not a stable noisy-inversion theorem. See [spectral tomography](research/reviews/v3/spectral_tomography.md) and its [audit](research/reviews/v3/tomography_audit.md).

	Other chapters specify quantum survival, field-quench and thermal work/heat protocols. Their parameters, hold/dwell ordering, reservoirs and reverse experiments are part of each result. They do not replace an unknown quantum state with a noiseless classical register, establish quantum advantage, or make the area norm into a universal energy or entropy. The [thermodynamic foundations](research/reviews/v3/thermodynamic_foundations.md) distinguish a properly reversed experiment from same-run transient erasure statistics.

	## 6. Coordinate-independent bundle extension

	Let `E -> M` be a real metric vector bundle with a metric-compatible connection. For a specified path from base point `p` to `p+`, let `U_gamma` be orthogonal parallel transport. The intrinsic pulse is

	$$
	x^+=U_\gamma(x+a),\qquad
	A^+=U_\gamma(A+x\wedge a)U_\gamma^{-1},\qquad Aa=0.
	$$

	All metric and orthogonal-conjugacy identities inherit exactly, including rank, kernel transport, `rho` and the whole spectral cone. This removes dependence on a coordinate frame; the metric, connection, path and controller register remain model data.

	For the tangent bundle with Levi–Civita connection, in matching geometric units one may choose `p+=Exp_p(a)` and transport along that geodesic. The accumulated `x` is then developed displacement, not a global logarithm of physical position.

	For `x0=0`, a finite geodesic history of total length `L` obeys

	$$
	d(p_0,p_T)\ge\rho_j-KL^3
	$$

	for each earlier certificate `rho_j`, provided `L` is below the injectivity radius at `p0`, curvature throughout the relevant radial normal ball satisfies `\|\|R(u,v)\|\|_op <= K\|\|u wedge v\|\|`, and `K L^2 <= 1/4`. These geometric quantities use consistent units; `K` has inverse-length-squared units in a dimensional metric.

	The hypotheses are essential. On a flat torus, two legal coordinate loops can return the physical base point exactly to its start while leaving nonzero internal `x,A,rho`. Curvature alone cannot supply a global spatial exclusion law. See the [intrinsic geometry analysis](research/reviews/v3/foundational_geometry.md) and [independent audit](research/reviews/v3/foundational_audit.md).

	## 7. The axiomatic gate-selection result

	There is a precise constitutive selection argument, with explicitly chosen design axioms:

	1. The local skew write is bilinear in `x,a` and equivariant under the full orthogonal group `O(d)`, for `d>=2`. This forces `B(x,a)=gamma x wedge a` for a scalar `gamma`.
	2. For a nonzero gain `gamma`, a permission set defined independently of `x` must guarantee that the stipulated norm `E(A)=\|\|A\|\|_F^2/2` never decreases for every possible endpoint register.

	For fixed `A,a != 0`,

	$$
	\Delta E=2\gamma x^TAa+
	\gamma^2\left(\\|x\\|^2\\|a\\|^2-\langle x,a\rangle^2\right),
	$$

	and

	$$
	\min_x\Delta E=-\frac{\\|Aa\\|^2}{\\|a\\|^2}.
	$$

	The minimum is attained at `x=-Aa/(gamma\|\|a\|\|^2)`, with any additional component parallel to `a`. Therefore `ker A` is exactly the maximal stroke set satisfying that endpoint-robust norm guarantee.

	These assumptions select the wedge write and kernel gate. They do not select the write gain, its sign or the finite sampling clock, and they are not a fundamental thermodynamic law. Endpoint-dependent permission, extra orientation data, nonlinear writes, a different memory energy or another timing convention permit other models. Physical energetic passivity requires an actual device and a complete supply/work account.

	## 8. The conditional canonical obstruction

	One tempting continuous identification takes `x` as canonical position and `b` as unit-mass mechanical momentum in the six-variable flow

	$$
	\dot x=b,\qquad\dot b=x\times b.
	$$

	Assume `{xi,xj}=0` and `{xi,bj}=delta_ij`. Jacobi forces the mechanical-momentum bracket to have magnetic form with a field independent of `b`. Reproducing this flow requires `B=-x`, but its Poisson Jacobiator is `-div B=3`, which is nonzero. The corresponding canonical quantum proposal

	$$
	[x_i,x_j]=0,\qquad[x_i,b_j]=i\hbar\delta_{ij},\qquad
	[b_i,b_j]=-i\hbar\epsilon_{ijk}x_k
	$$

	has cyclic nested commutator `-3 hbar^2 I` on a common invariant domain, contradicting the associative commutator Jacobi identity.

	This obstructs the specified canonical position/mechanical-momentum interpretation. It is not a theorem that every possible Hamiltonian realization is impossible. An auxiliary cotangent Hamiltonian,

	$$
	\mathcal H=\pi_x\cdot b+\pi_b\cdot(x\times b),
	$$

	reproduces the flow on a doubled state space, but is unbounded in the auxiliary momenta and does not provide a stable microscopic energy or a free controller. These issues have established magnetic-charge and symplectic-realization ancestry, cited in the report. The sampled finite controller and powered probe proposals do not assert that their area register is a new fundamental magnetic force.

	## 9. Reading and verification map

	\| Goal \| Release files \|
	\|---\|---\|
	\| Read the final hypotheses, proofs and bibliography \| [Full report](research/Avenyra_Geometric_Memory_Physics.pdf), [source](research/Avenyra_Geometric_Memory_Physics.tex) \|
	\| Inspect the exact source model \| [Original Avenyra manuscript](research/source/Avenyra_Area_Feedback_v1.0.0.pdf) \|
	\| Check the complete spectral cone \| [Analysis](research/reviews/v3/spectral_order.md), [verifier](research/v3/spectral_order_check.py), [results](research/v3/results/spectral_order_check.json) \|
	\| Check visible-spectrum reconstruction \| [Analysis](research/reviews/v3/spectral_tomography.md), [verifier](research/v3/tomography_check.py), [results](research/v3/results/tomography_check.json) \|
	\| Check intrinsic geometry and gate selection \| [Analysis](research/reviews/v3/foundational_geometry.md), [verifier](research/v3/foundational_geometry_checks.py), [results](research/v3/results/foundational_geometry_checks.json) \|
	\| Study noise, rank and higher-rank closure \| [Multidriver analysis](research/reviews/v2/multidriver.md), [verifier](research/v2/multidriver_checks.py) \|
	\| Study the specified joint write/loss protocol \| [Joint-loss analysis](research/reviews/v3/joint_loss.md), [independent checker](research/v3/joint_loss_check.py), [simulation](research/v3/joint_loss_simulation.py) \|
	\| Review thermodynamic and foundational limitations \| [Thermodynamic analysis](research/reviews/v3/thermodynamic_foundations.md), [prior-art assessment](research/reviews/v3/foundational_prior_art.md), [final claim audit](research/reviews/v3/final_claim_review.md) \|
	\| Identify proved and outstanding scopes \| [STATUS.json](research/STATUS.json), [research README](research/README.md) \|

	The review notes preserve the development history and can contain intermediate conditional formulations. Use the integrated report and `STATUS.json` for the final scope. Exact rational fixtures, finite-grid simulations and AI cross-audits supplement the analytic proofs; they are not laboratory data or external human certification.

	For reproduction, install the dependencies in [requirements.txt](research/requirements.txt) and run the scripts from the `research/` directory so their relative paths resolve. The canonical report's source compiles from that same directory. The release retains established mathematics under its standard names and separates candidate model-specific contributions from unverified priority and physical realization questions.