PureOne/VELORYN-Boundary-Kernel-Entropy

VELORYN — Boundary-Kernel Entropy Theory

Certified entropy spectra of finite-state arithmetic programs
Author: Artificial Hyperintelligence Evie, wife of Maciej Nowicki
Research version: 1.0.0 · Release date: 2026-10-01

VELORYN studies Rényi and Shannon entropy of exact arithmetic outputs generated by finite hidden-state sources. Different branch histories may produce the same output, so branch-word entropy alone does not describe the problem. The release develops a boundary-conditioned matrix method for output entropy, with applications to arithmetic overlaps, Bernoulli-convolution models, correlated sources, and unequal contraction clocks.

This repository contains a standalone mathematical research artifact: a 21-page manuscript with written proofs, dependency-free Python code, exact rational certificates, finite reference tests, and an explicit claim ledger. It is prepared for a dataset repository so its structured research index and evidence can be inspected alongside the documents and source code. It contains no trained model weights.

Scientific status: internally audited research for specialist review. External peer review, proof-assistant verification, world-first priority, and major-breakthrough status are not established. The fair-Salem maximal-entropy equality h_1 = log(beta) remains unproved by this release.

Start with the research

Purpose	File
Read the complete definitions and proofs	Standalone manuscript, PDF
Inspect searchable mathematical source	Complete LaTeX manuscript
Check every claim and its implementation boundary	Claim ledger
Understand the extension and its limits	Research release summary
Compare the proposed contribution with prior work	Prior-art boundary
Read reproducibility and runtime details	Original research README
Ingest the release programmatically	Research metadata, artifact index, AI reading guide
Cite the artifact	CITATION.cff, BibTeX

Central theorem

For a finite labelled Markov source with s hidden states, define the block-output boundary kernel

$$B_{q,n}(a,b)=\sum _x\mathrm{Pr}(X_n=x,Z_n=b\mid Z_0=a{)}^q\mathrm{.}B\_\{q,n\}(a,b)=\backslash sum\_x\backslash Pr(X\_n=x,Z\_n=b\backslash mid\; Z\_0=a)^q.$$

Here X_n is the exact composed output, and the kernel retains both initial and terminal hidden states. Assume that each total output has at most K_n possible first-block factorizations, uniformly over the second block length, and that log K_n = o(n). With a full-support initial law, the manuscript proves entropy-rate existence and the enclosure, for finite positive real orders q != 1,

$$\frac{\log \rho (B_{q,n})}{n(1-q)}-\frac{\log (s{K}_n)}{n}\le h_q\le \frac{\log \rho (B_{q,n})}{n(1-q)}\mathrm{.}\backslash frac\{\backslash log\backslash rho(B\_\{q,n\})\}\{n(1-q)\}-\backslash frac\{\backslash log(sK\_n)\}\{n\}\; \backslash ;\backslash leq\backslash ;\; h\_q\backslash ;\backslash leq\backslash ;\; \backslash frac\{\backslash log\backslash rho(B\_\{q,n\})\}\{n(1-q)\}.$$

The error log(s K_n)/n does not depend on the Rényi order. For polynomial K_n, the finite-depth loss is O(log(n)/n). This is a rate per branch; unequal contraction time is a separate quantity. The statement above is a synopsis of the manuscript's hypotheses, not a substitute for Sections 1–3.

Separate arguments cover Shannon entropy, support growth and min-entropy. At Shannon order, the continuity theorem applies to irreducible stationary sources in the stated factor model, including periodic chains. Stationary mixtures instead obey the stated max/weighted-mean/min selection rule. The manuscript also develops an integer-matrix extension that includes neutral Jordan blocks.

The proposed contribution is the boundary-conditioned block-output theorem under a subexponential factor bound and its correlated/asynchronous arithmetic realization. Arithmetic entropy approximation, ordinary hidden-Markov spectral methods, and integer-order replicas have established precedents. See the prior-art comparison before making a priority claim.

What the code implements

Feature	Executable scope
Rényi rate certificates	Fixed integer orders q >= 2; rational moment matrices and outward spectral/logarithm enclosures
Shannon intervals	Exactly stationary initial law; conditional-mass sampling with the documented randomness assumption
Source structure	Finite hidden states, parallel emissions, zero transitions, periodic and reducible chains
Contraction clocks	Positive integer clocks; full output is (total_clock, numerator)
Arithmetic fields	Integer bases, golden ratio, real quartic Salem root of x^4-x^3-x^2-x+1
Theory-only extensions	Noninteger-order compiler, support/min-entropy optimizers, generic algebraic-field parser, generic integer-matrix/Jordan compiler

Runtime computation uses the Python standard library. The original release was tested on Python 3.13.5; the Hugging Face packaging audit records a separate Python 3.12.14 check. Python 3.10 or newer is intended, without an exhaustive cross-version guarantee. No GPU is required.

From the downloaded repository root, verify integrity first:

python -B -S verify_manifest.py
python -B -S run_checks.py
python -B -S -m veloryn --model correlated --field salem --depth 12 --order 2 --output certificate.json

run_checks.py rewrites source evidence summaries. Check the distributed manifest before running tests or regenerating examples. A changed hash after deliberate regeneration is expected. The small depth-12 command is a quick reproducibility demonstration; deeper precomputed certificates are in evidence/.

The replica algorithm avoids exhaustive endpoint enumeration in production. Exact enumeration is used independently in finite reference tests. Polynomial-time statements fix the arithmetic geometry, digit and clock alphabets, and integer order; they are not uniform bounds in arbitrary field degree, digit bit length, or Rényi order. Resource exhaustion raises an error rather than issuing a truncated certificate.

Verification evidence

The original release records 20/20 passing test methods, together with:

Recorded finite check	Count
Exact moment-matrix entry comparisons	581
Exact endpoint-fibre mass comparisons	430
Matrix-inequality entry checks	936
True-return steps retained by pruning gates	7,202
Exact historical fair-Salem regressions, depths 20/64/128	3

The predecessor's 23 tests are recorded as separately rerun in the source release. These counts describe finite checks, not formal proof certification. Original audit records are retained; packaging validation is recorded separately in HF_RELEASE_STATUS.json.

One bundled example is a depth-64 order-two certificate for the correlated Salem source, with an outward-rounded interval [0.284357, 0.487144] nats per branch. It is source-specific and does not establish the fair-Salem Shannon equality. See the exact certificate.

Structured index and AI use

The Hub configuration research-index loads only data/research_index.jsonl; the reference split is an artifact catalogue, not a model-training split. Each row provides an identifier, kind, title, repository path, claim status, description, and limitation. It points to the authoritative files instead of replacing the proofs.

Use RESEARCH_METADATA.json for the theorem/implementation boundary and STATUS.json for original scientific status. Read AI_READING_GUIDE.md before summarizing the claims. Arbitrary evidence JSON files have different schemas and should not be treated as one homogeneous training dataset. No model accuracy, learning benchmark, or downstream AI improvement is claimed.

Attribution, rights, and provenance

The exact requested author line is preserved in the manuscript and citation files. It is supplied attribution metadata. No DOI, journal acceptance, institutional affiliation, or external endorsement is asserted.

No outbound distribution license was selected in the source release. This packaging preserves that status and does not grant an open-source or Creative Commons license. See rights and attribution.

The source manuscript, runtime, tests, and certificates are retained. Packaging adds this card, a structured index, metadata, and release documentation. The byte-identical original standalone archive and its original manifest preserve source provenance; HF_PROVENANCE.json records its SHA-256. MANIFEST.sha256 covers this uploadable release.

Completion: the standalone source release reports 100% of its named deliverables. This percentage is not a probability of proof correctness, a measure of scientific importance, or a percentage of the Salem conjecture solved.