96 lines
3.4 KiB
Markdown
96 lines
3.4 KiB
Markdown
# Ramanujan Challenge, Problem 2.8
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This package proves, for each of the four official columns,
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\[
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\lim_{N\to\infty}\frac{P_{N,j}}{Q_{N,j}}
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=\frac{\sqrt{10005}}{\pi},
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\qquad
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\lim_{N\to\infty}\frac{Q_{N,j}}{P_{N,j}}
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=\frac{\pi}{\sqrt{10005}}.
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\]
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The first display is the orientation requested in Problem 2.8; the second
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is its reciprocal consequence.
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## Contents
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- `solution.pdf` — the complete proof.
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- `solution.tex` — its LaTeX source.
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- `certificates/p28_standalone_equations.py` — mandatory, dependency-free
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expansion of the four cleared Ore identities, tail coefficient equations,
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terminating base/generic/top identities, ascension, and the
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hypergeometric differential equation. It uses rational coefficient
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dictionaries only: no division algorithm, simplifier, factorizer, special
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function library, root finder, or sample values.
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- `certificates/p28_dominant_product_algebra.py` — mandatory,
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dependency-free verification of the balanced limit, characteristic
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polynomial, root-separation inequalities, left-eigenvector identity, and
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four positive-coordinate rewrites.
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- `certificates/STANDALONE_EQUATION_CERTIFICATES.md` — the same Ore and
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terminating identities in a human-readable, denominator-cleared equation
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sheet.
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- `certificates/p28_full_closure_certificate.wl` — optional independent
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symbolic cross-check of the differential gauge and closure.
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- `certificates/p28_full_closure_certificate.PASS.txt` — transcript of a
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stateless Wolfram Language run (22 exact checks plus the consolidated
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conclusion).
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- `certificates/p28_convergence_constants.py` and
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`certificates/p28_rank_ode_bound_verifier.py` — dependency-free exact
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rational checks for the fixed-point convergence bound.
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- `certificates/p28_kernel_contiguity_certificate.sage`,
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`certificates/p28_lattice_hypotheses_certificate.sage`, and
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`certificates/all_four_columns_certificate.sage` — independent exact
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SageMath cross-checks.
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- `certificates/p28_parametric_pade_probe.py` — finite exact regression,
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included as a diagnostic only and not used as proof.
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- `ADVERSARIAL_AUDIT.md` — defect ledger, repairs, replay evidence, and
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the exact trust boundary.
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## Reproduction
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From this directory, run:
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```sh
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./run_checks.sh
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```
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The mandatory proof path is Python-standard-library only. The Wolfram
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cross-check can also be run directly:
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```sh
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wolframscript -file certificates/p28_full_closure_certificate.wl
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```
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The Python checks use only the standard library:
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```sh
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python3 certificates/p28_rank_ode_bound_verifier.py
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python3 certificates/p28_convergence_constants.py
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python3 certificates/p28_standalone_equations.py
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python3 certificates/p28_dominant_product_algebra.py
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```
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For the independent SageMath checks:
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```sh
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sage certificates/p28_kernel_contiguity_certificate.sage
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sage certificates/p28_lattice_hypotheses_certificate.sage
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sage certificates/all_four_columns_certificate.sage
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```
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To rebuild the manuscript:
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```sh
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latexmk -pdf solution.tex
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```
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## Trust boundary
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No numerical enclosure is used to infer equality. The recurrence proof is
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expanded into explicit equations and an elementary stable-graph contraction.
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The sole imported mathematical theorem is the classical Chudnovsky formula,
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identified precisely in `solution.tex` with a reference to a complete
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modular/CM derivation. Thus the package is self-contained relative to that
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published theorem; it does not claim to reconstruct the entire theory of the
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Chudnovsky formula from first principles.
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