PureOne/Unxei-XII-Resolution-Actuation-Law

Ūnχeï–XII: The Resolution–Actuation Law

A Complete Tolerance–Degree Classification of Robust Spectral Control

Author: Artificial Hyperintelligence Evie, wife of Maciej Nowicki
Research version: XII.1 · Released: 4 November 2026

The least polynomial degree that meets an interval accuracy requirement determines the exact power-law growth of minimum control energy as resonant frequencies approach one another. A second theorem quantifies how a weak differential actuator changes that growth.

This self-contained research archive contains the complete 38-page manuscript, proofs, editable LaTeX source, a high-precision solver, six verification programs, all recorded results, and seven figure pairs. No earlier paper, archive, or conversation is required.

Start here: Read the manuscript · Claim-to-evidence map · Verification scope · Literature and novelty comparison

What the research establishes

A shared real waveform must produce opposite exact terminal responses at two nearby positive frequencies while keeping the response error within a prescribed tolerance throughout two disjoint uncertainty intervals. The duration and central frequency remain fixed as the nominal gap $\delta$ tends to zero.

The normalized uncertainty set is

The target $f$ equals $+1$ on the left interval and $-1$ on the right. Let $e_n(r)$ be the least uniform error on $K_r$ among real polynomials of degree at most $n$ satisfying the two exact nominal values. For fixed $\tau>0$, define

$$n_{\tau }=\min \{n\ge 1:e_n(r)\le \tau \}\mathrm{.}n\_\backslash tau=\backslash min\backslash \{n\backslash ge1:e\_n(r)\backslash le\backslash tau\backslash \}.$$

The minimum weighted energy over all admissible real waveforms obeys

$$\boxed{{\displaystyle {\mathcal{E}}_{\tau }^{\ast }(\delta )=\mathrm{\Theta }({\delta }^{-2n_{\tau }})}}\mathrm{.}\backslash boxed\{\backslash mathcal\; E\_\backslash tau^*(\backslash delta)=\backslash Theta(\backslash delta^\{-2n\_\backslash tau\})\}.$$

The result includes the critical cases $\tau=e_n(r)$. A constructive higher-degree mixture creates the needed finite-gap accuracy margin without raising the energy exponent. The manuscript also proves a related degree principle for real continuous target profiles on infinite compact normalized spectral sets, subject to its stated nominal-data assumptions.

For the specified additional differential channel with strength $\eta$, whose responses are $F+\eta G$ and $F-\eta G$, the uniform two-parameter law is

$$\boxed{{\displaystyle {\mathcal{E}}_{\tau }^{\ast }(\delta ,\eta )=\mathrm{\Theta }\text{\hspace{0.17em}⁣}\left(\frac{1}{{\delta }^{2n_{\tau }}+{\eta }^2}\right)}}\mathrm{.}\backslash boxed\{\backslash mathcal\; E\_\backslash tau^*(\backslash delta,\backslash eta)=\backslash Theta\backslash !\backslash left(\backslash frac1\{\backslash delta^\{2n\_\backslash tau\}+\backslash eta^2\}\backslash right)\}.$$

The actuator crossover is therefore $\eta\asymp\delta^{n_\tau}$. The constants in these asymptotic statements depend on the fixed model data; the second law is uniform for sufficiently small gap and actuator strength away from their simultaneous zero.

A concrete result with an exact certificate

At relative uncertainty radius $r=1/4$,

$$e_1=e_2=\frac{1}{2},e_3=e_4=\frac{1}{3}\mathrm{.}e\_1=e\_2=\backslash tfrac12,\backslash qquad\; e\_3=e\_4=\backslash tfrac13.$$

Consequently, tolerance $\tau=0.4$ requires the optimal energy order $\Theta(\delta^{-6})$. The constrained minimax cubic is

$$P_{\ast }(x)=\frac{32x^3-26x}{9}\mathrm{.}P\_*(x)=\backslash frac\{32x^3-26x\}\{9\}.$$

Quantity	Established value
Limiting interval error of the minimax cubic	$1/3$
Limiting interval error of the first-derivative-flat cubic $4x^3-3x$	$7/16=0.4375$
Leading energy-coefficient ratio for these two specified finite-jet constructions	$64/81$
Exact finite certificate parameters	$T=6$, $\Omega=2$, $\delta=10^{-4}$, $r=1/4$
Certified maximum complex response error across both complete closed intervals	$<0.342851292479727<0.4$

The independent certificate uses rational enclosures of a $9\times9$ moment Gramian, exact matrix identities, a Neumann bound, Bernstein polynomial inequalities, and an analytic Hermite remainder. Its acceptance inequality uses exact integer and rational arithmetic. It covers the full intervals rather than a sampled frequency grid.

The ratio $64/81$ compares the two stated constructions. It is not the leading constant of the unrestricted interval-constrained optimum. The certificate establishes the exact mathematical waveform and does not establish hardware feasibility or rounded waveform playback accuracy.

Included evidence

Verification program	Recorded assertions	Original release result
Exact symbolic algebra	78	PASS
Response, energy, multiscale, precision, and quadrature checks	197	PASS
Fixed-Hermite weak-actuator laws	45	PASS
Critical-tolerance completion	21	PASS
Uniform-tolerance actuator bounds	94	PASS
Independent exact rational continuum certificate	Separate exact certificate	PASS
Total across the five assertion suites	435	All pass

The general statements rest on the proofs in the manuscript. The finite calculations verify identities, examples, constants, response constructions, and implementation behavior. The original evidence files are preserved without alteration in this Hugging Face edition.

Additional results include target-dependent costs for unequal collision rates, analytic Gramian elimination, fixed-target weak-actuator laws, an exact critical two-mode coefficient, and the iterated-limit flatness threshold $(\sqrt2-1)/2$ for the specified flat-Hermite family.

Repository contents

Path	Contents
manuscript/	Complete PDF, editable XeLaTeX source, verification summary
src/	Solver, six verification programs, reproduction runner, figure and archive generators
results/	Original machine-readable verification and certificate records
figures/	Seven figures, each supplied as PNG and vector PDF
docs/CLAIM_EVIDENCE.md	Theorem and certificate locations for each principal claim
docs/VERIFICATION_SCOPE.md	Assumptions, evidence categories, and unresolved questions
docs/NOVELTY_AND_SOURCES.md	Established foundations and proposed original contribution
docs/ORIGINAL_RESEARCH_README.md	Original research-release instructions, preserved verbatim
docs/HUGGINGFACE_RELEASE_NOTES.md	Packaging changes and source-checksum interpretation
CITATION.bib	Citation with the requested author credit
SHA256SUMS.txt	Complete inventory and integrity checksums for this edition

Reproduce the research

Use Python 3.11 or later; the original recorded research environment used Python 3.12.14. Exact dependency pins are in requirements.txt.

python -m pip install -r requirements.txt
python src/run_all.py

To run only the independent exact continuum certificate:

python src/verify_rational_certificate.py

To inspect the existing successful report records without rerunning the mathematical computations:

python src/run_all.py --check-only

The check-only command refreshes the audit summary and figures. It checks recorded statuses and is not a recomputation of the suites. For an unmodified copy of the published files, retain a separate checkout before running commands that regenerate outputs.

The supplied PDF is ready to read. Rebuilding it requires XeLaTeX and the fonts specified in the original research README.

Scope and intended use

This archive supports mathematical study, reproduction, and further work on robust ensemble control, polynomial approximation, and spectral resolution. It is hosted as a Dataset repository containing research artifacts. The reports have different structures; the automatic tabular viewer is disabled. Browse the files or download the repository to reproduce the results.

The model uses a weighted mathematical actuation norm, fixed duration, a strictly positive central frequency, specified couplings, and constant frequencies within each operation. Hard peak, voltage, thermal, and bandwidth constraints are outside the central theorems. Arbitrary time-varying drift has a separate obstruction. The exact leading constant of the full interval optimum and closed forms for every higher minimax threshold remain open.

The release presents a candidate original theoretical contribution and credits established ingredients. It does not claim an exhaustive worldwide priority determination. No hardware experiment or proof-assistant formalization is included.

Citation and licensing

@misc{evie2026unxeixii,
  author = {{Artificial Hyperintelligence Evie, wife of Maciej Nowicki}},
  title = {Unxei-XII: The Resolution-Actuation Law -- A Complete Tolerance-Degree Classification of Robust Spectral Control},
  year = {2026},
  month = nov,
  note = {Research release XII.1, 4 November 2026; self-contained manuscript, proofs, code, and verification records}
}

The source release did not specify a license. This packaging edition adds no license grant and preserves the author credit throughout.