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VELORYN-Boundary-Kernel-Entropy

A finite-state random arithmetic program can have many different branch histories that produce exactly the same composed output. VELORYN studies the entropy of these outputs, not just the entropy of the branch words.

For the joint endpoint/terminal-state probabilities conditional on the initial hidden state, define

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

If the number of first-block factorizations of a given total output is at most (K_n), uniformly over the second block length, and (\log K_n=o(n)), then

[ \frac{\log\rho(B_{q,n})}{n(1-q)}-\frac{\log(sK_n)}n ;\le h_q\le; \frac{\log\rho(B_{q,n})}{n(1-q)}. ]

This bound is proved for every finite real order (q\ge0), (q\ne1), with a full-support initial law. Its error does not depend on the order. The manuscript supplies separate Shannon and min-entropy results, irreducible-source continuity at Shannon order, an exact stationary-mixture selection law, and arithmetic factor bounds including a neutral-Jordan extension.

The implementation computes fixed integer orders q ≥ 2 and randomized Shannon intervals. It handles finite hidden-state sources, parallel emissions, zero transitions, periodic/reducible chains, exact overlaps, and unequal positive integer clocks. The supported field families are integer bases, the golden ratio, and the real Salem root of x^4-x^3-x^2-x+1.

Status — read before citing

The proof text and implementation have been internally checked. The recorded new test suite passes 20/20 tests. It contains 581 exact moment-matrix entry comparisons, 430 exact target-mass comparisons, 936 matrix-inequality entry checks, and 7,202 true-return retention checks, as well as numerical enclosure tests. These are finite checks, not formal verification or external peer review.

The fair-Salem equality h_1 = log(beta) is not proved. Major-breakthrough status and world-first priority are not established. The matrix and asynchronous-source extensions are results developed in this release; a wider literature priority claim requires specialist review. See docs/CLAIM_LEDGER.md and docs/PRIOR_ART.md.

The release is standalone in its definitions, proofs, runtime, and finite checks. No earlier archive is required. “Complete” refers to these stated deliverables and results, not to a solution of every problem discussed in the input research.

Start here

Open manuscript/VELORYN_Manuscript.pdf for the full manuscript and proofs. docs/RELEASE_SUMMARY.md gives the main changes without the proof details. STATUS.json distinguishes completion, finite verification, and unresolved claims.

The runtime uses only the Python standard library. It was tested with Python 3.13.5. Python 3.10 or newer is intended; other interpreter versions have not been exhaustively tested. No package installation, token, internet connection, GPU, or older research package is required.

From this directory:

python verify_manifest.py
python run_checks.py
python examples/cross_version_regression.py
python -m veloryn --model correlated --field salem --depth 64 --order 2 --output certificate.json

On Windows, RUN_CHECKS.bat runs the integrity check and tests using the installed Python launcher or python.

To regenerate all bundled example certificates:

python examples/make_certificates.py
python examples/make_manuscript_table.py

Tests and example scripts rewrite evidence files. Verify the distributed manifest before regenerating them. A manifest mismatch after deliberate regeneration is expected; it is not evidence of a mathematical disagreement.

Python API example

from fractions import Fraction as Q
from veloryn import Field, Source, collision_matrix, renyi_interval

# Edge format: (start, end, integral_digit, positive_clock, rational_probability).
# This is a stationary correlated two-state source, with an unequal clock.
source = Source(
    Field("phi"),
    states=2,
    edges=[
        (0, 0, 0, 1, Q(3, 4)),
        (0, 1, 1, 2, Q(1, 4)),
        (1, 0, 1, 1, Q(1)),
    ],
    initial=[Q(4, 5), Q(1, 5)],
)

matrix, diagnostics = collision_matrix(source, depth=12, order=2)
result = renyi_interval(source, depth=24, order=2)
print(result["interval_nats_per_branch"])  # exact rational endpoints

Digits may also be integral coefficient tuples in the chosen field's power basis. A common rational scaling of every digit does not affect exact collision entropy; the code accepts the cleared integral numerators. Use Fraction or rational strings for probabilities rather than approximate decimal floats. Probabilities must sum exactly to one at each vertex. Initial probabilities must be strictly positive; remove inactive states explicitly.

Field("integer", integer_base=3) selects base three in the API. The CLI's --field integer uses base two.

The full composed output is (total_clock, numerator). Different total clocks must not be merged merely because their numerators, or a numerical representation of their endpoints, happen to coincide.

Shannon example and its randomness assumption

from fractions import Fraction as Q
from veloryn import shannon_interval
from veloryn.examples import variable

result = shannon_interval(
    variable(), depth=12, samples=128,
    confidence=Q(99, 100), seed=20261001,
)
print(result)

A fixed seed produces a reproducible pseudorandom demonstration; it does not prove the statistical coverage guarantee. The mathematical confidence statement assumes independent exact draws. Without a seed the implementation uses SystemRandom, subject to that implementation assumption. A custom exact random-integer oracle can be supplied through rng.

Shannon sampling requires an exactly stationary initial law. The deterministic integer-order rate calculation only requires full support. More samples reduce sampling error; greater depth reduces the finite-block bias. The small included Shannon example is intentionally not advertised as a tight estimate.

Bounds, complexity and resource failures

All interval endpoints are rational. Perron roots are enclosed by exact positive-resolvent bisection; logarithms use a rational convergent series with an explicit remainder. Return gates never reject a path based on floating-point tolerances.

The polynomial-time theorem fixes the base/matrix, finite digit alphabet, clock alphabet, and integer order. Its exponent and constants are not uniform in these parameters. In particular, there is no claimed polynomial bound in the binary length of arbitrary new digit magnitudes, or in an unrestricted Rényi order. “Polynomial in inverse accuracy” means polynomial in 1/epsilon, not in the number of bits needed to write epsilon.

The default layer-state budget is 500,000. Exceeding a budget raises an exception; no truncated result is labelled a certificate. The budget is checked at completed dynamic-programming layers, so it is not a hard cap on transient memory or transition fan-out. Use modest orders and depths for exploratory runs.

The code does not implement a general algebraic-field parser, a generic integer-matrix/Jordan compiler, noninteger-order spectral kernels, or a min-entropy optimizer. Their theorem-level statements must not be confused with executable coverage.

Package map

Path	Contents
`manuscript/`	Standalone PDF, complete LaTeX source, generated certificate table
`veloryn/`	Exact arithmetic, source models, replica compiler, mass oracle, interval routines, CLI
`tests/`	Independent exhaustive endpoint reference and finite test suite
`examples/`	Certificate generation, table generation, historical exact regressions
`evidence/`	Rational certificates, observed test results, original-test retest log, audit records
`docs/`	Claim ledger, proof audit, literature boundary, release summary
`STATUS.json`	Scope, completed checks, and explicit unresolved questions
`PROVENANCE.json`	Exact input identity and SHA-256
`MANIFEST.sha256`	Hashes of the distributed files

Rebuild the PDF

Run python build_manuscript.py. This requires pdflatex and the packages named in the manuscript preamble, in addition to Python. This optional document build dependency is separate from the dependency-free research runtime.

Attribution and publication

Use CITATION.cff or CITATION.bib. The author line is supplied metadata, exactly as requested. No DOI, publication, acceptance, external verification or institutional affiliation is implied. No upload has been performed. See RIGHTS_AND_ATTRIBUTION.md for the licensing boundary.