PureOne/Qeira-II-Carry-Holonomy-Rank-Law

Qeïra II — The Carry–Holonomy Rank Law

An exact multivariate first-carry rank theorem and a correction to an odd-prime matrix-digit conjecture.

Research author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Release: 1.1.0 · 24 October 2026 · research preprint with reproducible code

This repository presents an exact rank formula for the second base- $p$ digit of standard-digit dot products, for every odd prime $p$ and latent dimension $r$. The manuscript proves the upper bound in Elsheikh–Novocin–Giesbrecht, Conjecture 17, gives a sharp replacement for the proposed generic value, and identifies degree-sensitive deficits indexed by irregular Bernoulli numbers.

Priority position: the rank-one specialization, including the irregular-prime deficit, agrees with classical Stickelberger-ideal mathematics. The complete multivariate formula is the candidate contribution; its historical priority remains unconfirmed. See PRIOR_ART.md.

Start here

Reader	Entry point
Researcher	Complete manuscript, PDF
AI agent or text search	Full manuscript, Markdown, AGENTS.md, research.json
Theorem reviewer	Claim and dependency ledger, PROOF_STATUS.md
Reproducer	VERIFICATION.md, fresh result summary
First experiment	examples/quickstart.py
Citation or attribution	CITATION.cff, references.bib, RIGHTS.md

Exact statement

For $x,y\in\mathbb F_p^r$, take the standard integer lifts $\widetilde x_i,\widetilde y_i\in{0,\ldots,p-1}$ and define

$$C_{p,r}(x,y)=\lfloor \frac{\sum _i{\tilde{x}}_i{\tilde{y}}_i}{p}\rfloor p\mathrm{.}C\_\{p,r\}(x,y)=\backslash left\backslash lfloor\backslash frac\{\backslash sum\_i\backslash widetilde\; x\_i\backslash widetilde\; y\_i\}\{p\}\backslash right\backslash rfloor\backslash bmod\; p.$$

If $f_p(T)=\sum_{d=1}^{p-1}c_dT^d$ is the reduced polynomial agreeing with $(a^p-a)/p\bmod p$ on the standard digits, the universal matrix indexed by all $x,y\in\mathbb F_p^r$ has rank

$$R_p(r)=\left(\genfrac{}{}{0px}{}{r+p-1}{p}\right)-r+\sum _{\begin{array}{c}2\le d\le p-1\\ c_d\mathrm{\ne }0\end{array}}\left(\genfrac{}{}{0px}{}{r+d-1}{d}\right)\mathrm{.}R\_p(r)=\backslash binom\{r+p-1\}\{p\}-r+\; \backslash sum\_\{\backslash substack\{2\backslash leq\; d\backslash leq\; p-1\backslash \backslash c\_d\backslash ne0\}\}\backslash binom\{r+d-1\}\{d\}.$$

An explicit factorization $C=\Phi D\Phi^{\mathsf T}$ supplies independent reduced-polynomial features and an invertible diagonal $D$. A left inverse of $\Phi$ proves the matching rank lower bound over the finite field. The sharp maximum is attained with invertible factors satisfying the original conjecture's constraints. A separate random-factor theorem states the corrected asymptotic generic limit with $p,r$ fixed.

At $p=37$, the formula gives

Latent dimension	Proposed generic value	Exact maximum
1	18	17
2	413	407
3	5,827	5,806

The rank-one example is a useful exact certificate of the failure of the proposed generic value. Its underlying irregularity phenomenon is classical. For general $r$, the deficit is $\binom{r+4}{5}$ at this prime.

Reproduce in one command

Only Python's standard library is needed for the prime-37 certificate:

python tools/run_verification.py --quick

For the full numerical verification suite:

python -m pip install -r requirements.txt
python tools/run_verification.py

The runner works in a temporary copy and writes new reports to verification_local/; the packaged reports stay intact. The fresh packaged run passed on Python 3.12.14, NumPy 2.3.5 and SymPy 1.14.0: 46 primes, 20 complete carry matrices, 606,396 feature-factorization entries, 13,056 fixed-frame comparisons, and the rank-17 certificate with minor determinant 14 modulo 37. These finite checks support the implementation and examples. Universal and infinite statements rely on the manuscript's proofs.

python tools/verify_integrity.py
python examples/quickstart.py

Formal transport extensions

The paper also gives a Heisenberg commutator realization, a single fixed-frame response for a known square-zero central sector, an arithmetic Frobenius visibility threshold, and a support criterion for reconstruction modulo a bornology. These are formal mathematical results under their stated observation hypotheses. They do not establish an experimental sensor, quantum speedup, general AI improvement, or unrestricted graph reconstruction.

Scope of the data

data/rank_examples.jsonl contains a small table of formula evaluations, including the 20 cases checked by full-matrix elimination. Each row records whether the full matrix was actually checked. The $p=521,r=2$ rank of 68,899 is a formula evaluation: its 271,441-square matrix was not constructed. This is a mathematical research archive with example data, rather than a trained model or an empirical benchmark.

The first-carry digit is the coefficient of $p$, also called digit index 1. The result does not classify all higher digit ranks or the full Smith spectrum. Standard lifts matter. Linear probe bounds apply to the stated linear observable quotient, not arbitrary nonlinear encodings of a single input.

Sources and rights

The prior-art audit credits Skula (1986, 1989), Cikánek (1991), Sturtivant–Frandsen (1993), Elsheikh–Novocin–Giesbrecht (2014), and Kaji–Maeno–Nuida–Numata (2016 revision). Source PDFs are linked rather than redistributed. No exhaustive historical-priority certification or proof-assistant certification is claimed.

The supplied source package declared no license. This release preserves that status; RIGHTS.md explains it. No new reuse license is silently assigned by the packaging process.

Upload and build

For Windows, extract the archive and double-click UPLOAD_TO_HF.bat. The suggested destination is PureOne/Qeira-II-Carry-Holonomy-Rank-Law as a Hugging Face dataset repository. Authentication uses the standard Hugging Face flow. See UPLOAD.md; preparing this package does not publish it.

To rebuild the PDF locally, run pdflatex twice on Qeira_II_Carry_Holonomy.tex from this directory. The bibliography is embedded and compilation requires no network. MANUSCRIPT.md is a complete text conversion of this revised source; the PDF/LaTeX controls if conversion formatting differs.