PureOne/goormaghtigh-equation-frontier-synthesis-v11

Goormaghtigh Equation Frontier Synthesis v11.0.0

Two-Chart Reconstruction, Radical Bounds, Exact Fibres, and Proof-Carrying Computation

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Release date: 28 September 2026
Status: Partial result. The unrestricted Goormaghtigh conjecture is not proved.
Verification status: all declared computational scopes in this release are replayable from source; the work is not externally peer reviewed or proof-assistant formalized.

This repository is a self-contained research release on the Goormaghtigh equation

[ R_m(x)=R_n(y),\qquad R_s(t)=1+t+\cdots+t^{s-1},\qquad x>y\ge2,\quad 3\le m<n. ]

The project combines exact arithmetic kernels, low-dimensional coordinate compression, primitive abc triples, finite inverse-fibre algorithms, polynomial coefficient certificates, and independent replay code. Its principal global contribution is an unconditional two-chart reduction: every Goormaghtigh solution is mapped to at least one explicitly reconstructible primitive additive triple satisfying a strong radical-quality inequality. Each fixed triple has a finite, exactly computable inverse fibre.

That does not make the global candidate set finite. The unresolved step is to prove that all additional high-quality reconstructed triples have empty fibres, or otherwise obtain an unconditional global height/descent bound. No claim of a complete solution, historical priority, or world-record computation is made.

Main mathematical result in this release

Every solution is represented in at least one of two arithmetic charts.

Chart A — general cofactor chart

With

[ g=\gcd(x,y),\quad a=x/g,\quad b=y/g,\quad D=(x-y)/g^m, ]

and

[ h=\gcd(D,y-1), ]

the construction gives a primitive additive triple

[ U=\frac{a^m(y-1)}h,\qquad V=\frac Dh,\qquad W=\frac{y^n(x-1)}{g^m h},\qquad U+V=W. ]

For the proved parameter region the manuscript derives an explicit radical inequality strong enough to place all hypothetical solutions in a high-quality abc class. A complete inverse algorithm reconstructs every Goormaghtigh solution represented by any fixed (U,V,W).

Chart B — quadratic branch

For m=3, completing the square gives

[ (y-1)(2x+1)^2+(3y+1)=4y^n. ]

After primitive normalization this supplies a second additive triple and a corresponding finite inverse routine. The second chart is needed because the general radical estimate degenerates in the quadratic branch.

Together the two charts yield the release's global reduction. The exact constants, hypotheses, boundary cases, and proofs are in MANUSCRIPT.pdf / MANUSCRIPT.md.

What is proved vs. what remains open

Proved inside this release:

• the stated two-chart reconstruction maps and inverse-fibre algorithms;
• the declared radical-quality inequalities and their explicit boundary handling;
• exact adjacent-length certificates retained from the earlier kernel line;
• replayable finite-height and fixed-input computations with independent implementations;
• integrity, claim, and provenance records for the public package.

Not proved:

• that the two known identities are the only Goormaghtigh solutions;
• a global unconditional bound on the reconstructed triple height;
• that all high-quality primitive triples outside the observed ranges have empty inverse fibres;
• historical novelty of the general normalization or use of abc ideas, which have relevant prior art.

The two reference identities are

[ (5,2,3,5),\qquad(90,2,3,13), ]

with common values 31 and 8191.

Executed verification scope

Scope	Verified count / bound
General-chart normalized height	10^12
Canonical states checked by two enumerators	1,162,887
Independent raw D cells	81,892,702
Quadratic-chart height columns	1,279
Radical-boundary columns	21
Complete adjacent-length slice certificates	11
Original-equation finite remainder checks	1,391,146
Exact polynomial-identity evaluations	3,894
Fixed-input columns replayed	324,675
Software tests	50

These scopes overlap. They are not a percentage of the unrestricted conjecture and are not advertised as a search over every repunit value below 10^12.

Reproduce the release

Requires Python 3.10+ and the standard library for verification:

python verify_release.py

For a fast byte-integrity check only:

python verify_release.py --integrity-only

Windows users can run:

VERIFY_WINDOWS.bat

The full verifier writes generated replay output only to local_verification/.

Useful individual commands:

python src/abc_fibres.py inverse 125 3
python src/abc_fibres.py scan --height 1000000000000
python src/verify_abc_independent.py --height 1000000000000
python src/quadratic_abc.py --output-dir local_verification

Repository map

• MANUSCRIPT.pdf — typeset paper.
• MANUSCRIPT.md / MANUSCRIPT.tex — editable source.
• CLAIMS.json — unconditional, conditional, computational, and unresolved claims separated explicitly.
• STATUS.json — machine-readable research status.
• AI_CONTEXT.md, AGENTS.md, llms.txt — concise machine/agent orientation.
• src/ — exact arithmetic and reconstruction kernels.
• certificates/ — proof-carrying polynomial certificates.
• results/ — committed verification records.
• tests/ — software and malformed-certificate tests.
• docs/PRIOR_ART.md — prior-art and overlap notes.
• docs/PROOF_BOUNDARY.md — exact boundary between proved and unresolved claims.
• docs/REPRODUCIBILITY.md — reproducibility instructions.
• provenance/ — source-access and transfer records.
• CITATION.cff — citation metadata.
• HF_RELEASE_NOTES.md — Hugging Face publication metadata and suggested repository settings.

Suggested citation

Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Goormaghtigh Equation Frontier Synthesis v11.0.0: Two-Chart Reconstruction, Radical Bounds, Exact Fibres, and Proof-Carrying Computation. Research release, 28 September 2026.

Use CITATION.cff for structured citation metadata.

Prior art and scientific positioning

The release explicitly distinguishes its results from earlier Goormaghtigh work. The general normalization and abc direction have important prior art, including work of Laishram–Shorey. Modern literature also contains strong effective and special-case results due to multiple authors. The present package should therefore be described as a partial, independently replayable synthesis and reduction, not as a solved conjecture.

See docs/PRIOR_ART.md and the manuscript bibliography before making novelty claims.

License and publication rights

No open-source or open-content license has been selected automatically. See LICENSE_NOTICE.md. Public visibility on Hugging Face is not equivalent to granting an open license.