Datasets:

PureOne
/

VELORYN-Boundary-Kernel-Entropy

---
pretty_name: "VELORYN — Boundary-Kernel Entropy Theory"
language:
  - en
tags:
  - mathematics
  - information-theory
  - renyi-entropy
  - shannon-entropy
  - hidden-markov-models
  - bernoulli-convolutions
  - self-similar-measures
  - exact-overlaps
  - salem-numbers
  - pisot-numbers
  - arithmetic-dynamics
  - certified-computation
  - research-artifact
  - reproducibility
configs:
  - config_name: research-index
    data_files:
      - split: reference
        path: data/research_index.jsonl
---
# 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](manuscript/VELORYN_Manuscript.pdf) |
| Inspect searchable mathematical source | [Complete LaTeX manuscript](manuscript/VELORYN_Manuscript.tex) |
| Check every claim and its implementation boundary | [Claim ledger](docs/CLAIM_LEDGER.md) |
| Understand the extension and its limits | [Research release summary](docs/RELEASE_SUMMARY.md) |
| Compare the proposed contribution with prior work | [Prior-art boundary](docs/PRIOR_ART.md) |
| Read reproducibility and runtime details | [Original research README](docs/ORIGINAL_RESEARCH_README.md) |
| Ingest the release programmatically | [Research metadata](RESEARCH_METADATA.json), [artifact index](data/research_index.jsonl), [AI reading guide](AI_READING_GUIDE.md) |
| Cite the artifact | [CITATION.cff](CITATION.cff), [BibTeX](CITATION.bib) |
## Central theorem
For a finite labelled Markov source with `s` hidden states, define the block-output boundary kernel
$$
B_{q,n}(a,b)=\sum_x\Pr(X_n=x,Z_n=b\mid Z_0=a)^q.
$$
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`,
$$
\frac{\log\rho(B_{q,n})}{n(1-q)}-\frac{\log(sK_n)}{n}
\;\leq\; h_q\;\leq\;
\frac{\log\rho(B_{q,n})}{n(1-q)}.
$$
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](docs/PRIOR_ART.md) 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:
```bash
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](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](evidence/salem_correlated_q2.json).
## Structured index and AI use
The Hub configuration `research-index` loads only [data/research_index.jsonl](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](RESEARCH_METADATA.json) for the theorem/implementation boundary and [STATUS.json](STATUS.json) for original scientific status. Read [AI_READING_GUIDE.md](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](RIGHTS_AND_ATTRIBUTION.md).
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](provenance/VELORYN_v1.0.0_Complete_Research_Package.zip) and [its original manifest](provenance/ORIGINAL_MANIFEST.sha256) preserve source provenance; [HF_PROVENANCE.json](HF_PROVENANCE.json) records its SHA-256. [MANIFEST.sha256](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.

	---
	pretty_name: "VELORYN — Boundary-Kernel Entropy Theory"
	language:
	- en
	tags:
	- mathematics
	- information-theory
	- renyi-entropy
	- shannon-entropy
	- hidden-markov-models
	- bernoulli-convolutions
	- self-similar-measures
	- exact-overlaps
	- salem-numbers
	- pisot-numbers
	- arithmetic-dynamics
	- certified-computation
	- research-artifact
	- reproducibility
	configs:
	- config_name: research-index
	data_files:
	- split: reference
	path: data/research_index.jsonl
	---

	# 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](manuscript/VELORYN_Manuscript.pdf) \|
	\| Inspect searchable mathematical source \| [Complete LaTeX manuscript](manuscript/VELORYN_Manuscript.tex) \|
	\| Check every claim and its implementation boundary \| [Claim ledger](docs/CLAIM_LEDGER.md) \|
	\| Understand the extension and its limits \| [Research release summary](docs/RELEASE_SUMMARY.md) \|
	\| Compare the proposed contribution with prior work \| [Prior-art boundary](docs/PRIOR_ART.md) \|
	\| Read reproducibility and runtime details \| [Original research README](docs/ORIGINAL_RESEARCH_README.md) \|
	\| Ingest the release programmatically \| [Research metadata](RESEARCH_METADATA.json), [artifact index](data/research_index.jsonl), [AI reading guide](AI_READING_GUIDE.md) \|
	\| Cite the artifact \| [CITATION.cff](CITATION.cff), [BibTeX](CITATION.bib) \|

	## Central theorem

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

	$$
	B_{q,n}(a,b)=\sum_x\Pr(X_n=x,Z_n=b\mid Z_0=a)^q.
	$$

	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`,

	$$
	\frac{\log\rho(B_{q,n})}{n(1-q)}-\frac{\log(sK_n)}{n}
	\;\leq\; h_q\;\leq\;
	\frac{\log\rho(B_{q,n})}{n(1-q)}.
	$$

	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](docs/PRIOR_ART.md) 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:

	```bash
	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](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](evidence/salem_correlated_q2.json).

	## Structured index and AI use

	The Hub configuration `research-index` loads only [data/research_index.jsonl](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](RESEARCH_METADATA.json) for the theorem/implementation boundary and [STATUS.json](STATUS.json) for original scientific status. Read [AI_READING_GUIDE.md](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](RIGHTS_AND_ATTRIBUTION.md).

	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](provenance/VELORYN_v1.0.0_Complete_Research_Package.zip) and [its original manifest](provenance/ORIGINAL_MANIFEST.sha256) preserve source provenance; [HF_PROVENANCE.json](HF_PROVENANCE.json) records its SHA-256. [MANIFEST.sha256](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.