allaun
3eded12fe1
Documents the radix formulation from MathPunch-FiniteState
(coq/MathPunchFiniteStateAudit/Radix.v, MathPunchFiniteState/Radix.lean), which
is the correctness obligation underneath the encoder method.
The defect, proved rather than asserted: positional evaluation is not injective
on digit strings. Both collisions are machine-checked in Coq and Lean --
eval_digits 8 [0;1] = eval_digits 8 [1], and eval_digits 8 [] = eval_digits 8
[0]. Quantified: in base 8 over lengths 0-3, 585 strings collapse onto 512
values.
Two repairs, both BIJECTIONS onto correctly stated codomains:
framing : digit strings <-> U_n {n} x [0, b^n)
verified exhaustively base 8, n=0..4 (1,8,64,512,4096; no gaps)
DFA : canonical numerals <-> N+
verified base 8 to length 5 (32767 strings, values exactly [1,8^5))
Bijectivity is the operative property, not injectivity: injectivity says
encodings do not collide, bijectivity says decoding is TOTAL on the valid
codomain. Stating the codomain as N x N would make framing merely injective;
stating it correctly makes it bijective.
Distinctiveness assessed fairly: the content is classical numeration-system
material (regular numeral languages, canonical numeration, Cobham). The
distinctive move is making it an explicitly proved prerequisite of an encoding
pipeline, with the collisions exhibited as theorems in two proof assistants.
Also flags a maintenance hazard: the DFA and toDigits exist only in Radix.lean;
Radix.v has eval_digits and framed_value alone. Lean is regenerated from coq/*.v
in that repo, so regeneration would silently erase the automaton.
Corrects NOTATION_AND_BORROWED_TERMINOLOGY.md: the blanket 'no biology claim
anywhere' disclaimer was too strong and is now scoped to this submission, since
a Lean probe elsewhere deliberately models expanded genetic alphabets (4/8/12
letters) and proves an optimality statement about them.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
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|---|---|---|
| .. | ||
| certificates | ||
| ADVERSARIAL_AUDIT.md | ||
| ADVERSARIAL_REVIEW_AGAINST_CHALLENGE_RULES.md | ||
| NOTATION_AND_BORROWED_TERMINOLOGY.md | ||
| ramanujan_challenge_problem_2_8.zip | ||
| README.md | ||
| run_checks.sh | ||
| solution.pdf | ||
| solution.tex | ||
| THE_ENCODER_APPROACH.md | ||
Ramanujan Challenge, Problem 2.8
This package proves, for each of the four official columns,
[ \lim_{N\to\infty}\frac{P_{N,j}}{Q_{N,j}} =\frac{\sqrt{10005}}{\pi}, \qquad \lim_{N\to\infty}\frac{Q_{N,j}}{P_{N,j}} =\frac{\pi}{\sqrt{10005}}. ]
The first display is the orientation requested in Problem 2.8; the second is its reciprocal consequence.
Contents
solution.pdf— the complete proof.solution.tex— its LaTeX source.certificates/p28_standalone_equations.py— mandatory, dependency-free expansion of the four cleared Ore identities, tail coefficient equations, terminating base/generic/top identities, ascension, and the hypergeometric differential equation. It uses rational coefficient dictionaries only: no division algorithm, simplifier, factorizer, special function library, root finder, or sample values.certificates/p28_dominant_product_algebra.py— mandatory, dependency-free verification of the balanced limit, characteristic polynomial, root-separation inequalities, left-eigenvector identity, and four positive-coordinate rewrites.certificates/STANDALONE_EQUATION_CERTIFICATES.md— the same Ore and terminating identities in a human-readable, denominator-cleared equation sheet.certificates/p28_full_closure_certificate.wl— optional independent symbolic cross-check of the differential gauge and closure.certificates/p28_full_closure_certificate.PASS.txt— transcript of a stateless Wolfram Language run (22 exact checks plus the consolidated conclusion).certificates/p28_convergence_constants.pyandcertificates/p28_rank_ode_bound_verifier.py— dependency-free exact rational checks for the fixed-point convergence bound.certificates/p28_kernel_contiguity_certificate.sage,certificates/p28_lattice_hypotheses_certificate.sage, andcertificates/all_four_columns_certificate.sage— independent exact SageMath cross-checks.certificates/p28_parametric_pade_probe.py— finite exact regression, included as a diagnostic only and not used as proof.ADVERSARIAL_AUDIT.md— defect ledger, repairs, replay evidence, and the exact trust boundary.
Reproduction
From this directory, run:
./run_checks.sh
The mandatory proof path is Python-standard-library only. The Wolfram cross-check can also be run directly:
wolframscript -file certificates/p28_full_closure_certificate.wl
The Python checks use only the standard library:
python3 certificates/p28_rank_ode_bound_verifier.py
python3 certificates/p28_convergence_constants.py
python3 certificates/p28_standalone_equations.py
python3 certificates/p28_dominant_product_algebra.py
For the independent SageMath checks:
sage certificates/p28_kernel_contiguity_certificate.sage
sage certificates/p28_lattice_hypotheses_certificate.sage
sage certificates/all_four_columns_certificate.sage
To rebuild the manuscript:
latexmk -pdf solution.tex
Trust boundary
No numerical enclosure is used to infer equality. The recurrence proof is
expanded into explicit equations and an elementary stable-graph contraction.
The sole imported mathematical theorem is the classical Chudnovsky formula,
identified precisely in solution.tex with a reference to a complete
modular/CM derivation. Thus the package is self-contained relative to that
published theorem; it does not claim to reconstruct the entire theory of the
Chudnovsky formula from first principles.