# Verification report

Title: A Pentagonal Counterexample to a Poncelet Squared-Side Criterion

Author: Alper Ferudun[^author]

[^author]: Mercury Software GmbH. alper@mercurycodelab.com; https://github.com/AlperTheKing.

Internal investigation identity: AMR-050-0053; frozen record 5100053; source invariant k811. Adjacent source: Murad 2026, Section 6, Conjecture 1. English preprint, version 1.0, manuscript date 2 October 2026.

This work is AI-assisted, originating-researcher self-audited, and unrefereed. No independent human review, formal proof-assistant verification, novelty certification, or absolute-priority claim is made.

## Accepted scope

The accepted theorem has two carefully separated parts.

First, for a strictly nested bicentric Poncelet family with coherent tangent normals, both explicit reciprocal constructions

    D_i = C + n_i/r,
    I_i = C + (Q_i-C)/|Q_i-C|^2

have phase-independent sums of squared consecutive side lengths. `D` is the unit inversion of the pedal feet of the polar polygon, while `I` is the unit inversion of the polar vertices. Their traces and constant values differ. The `D` result is an elementary consequence of Roitman, Garcia and Reznik's published bicentric cosine theorem. The `I` result is an explicitly mapped application of Khare, Lakshminarayan and Sukhatme's published MI-II cyclic identity.

Second, the paper constructs a strictly nested circle/ellipse Poncelet family of primitive convex pentagons with constant squared-side sum even though the circumcenter is neither the ellipse center nor either focus. This is a complete counterexample to the necessity direction of Murad 2026, Section 6, Conjecture 1 as displayed for a fixed `n` and fixed circle/conic family.

The originating researcher accepted this scope after detailed self-audit on 2 October 2026. The acceptance record sets `proof_accepted=true`, `theorem_scope_resolved=true`, `adjacent_conjecture_refuted=true`, `original_problem_resolved=false`, `whole_source_record_resolved=false`, `new_result=false`, and novelty `UNDETERMINED_AFTER_BOUNDED_PRIMARY_SEARCH`. There are no known mathematical gaps in the explicitly stated theorem. The negative original-record flags retain k811's source-definition ambiguity; they are not a gap in the adjacent counterexample.

## General argument actually accepted

Let `Q` be a polygon between two strictly nested circles, with inner center `C`, radius `r`, and coherently oriented outward tangent normals `n_i`. Solving consecutive tangent equations gives

    I_i = C + (n_i+n_{i+1})/(2r) = (D_i+D_{i+1})/2.

Therefore the squared-side trace of `I` is `(N-S_2)/(2r^2)`, where `S_2` is the two-step normal cosine sum. In Jacobi's bicentric parametrization, set `E=cn+i sn` and `H=1/E`. The complexified trace becomes a finite sum of reciprocal quartets in shifted `E` values. Replacing denominators by `H` makes each term a degree-four shifted Jacobi polynomial with the required `2K` and `2iK'` periods.

For `N>=4`, the possible singularities are simple. The four local pole coefficients cancel pairwise; equivalently, the published MI-II master identity has no principal-part coefficient of order at least two in this case. The sum is an entire elliptic function and hence constant. The colliding-shift case `N=3` reduces to the published adjacent-normal cosine invariant. Concentric pairs, reversal, and fixed repeated traversals are treated directly.

The projectivity

    T(x,y) = (x,y)/(2dx+q),  q=R^2-d^2,

agrees with unit vertex inversion on the original outer circle. Its denominator is positive on the complete outer disk, so it preserves the full nested configuration, actual tangent segments, strict convexity, and the fixed tangent branch. It maps the inner circle to an explicit noncircular ellipse and the outer circle to another circle.

For `R=1`, `d=1/5`, and `t` the unique root in `(1/10,1/5)` of `25t^3-50t^2-40t+8`, set `r=4/5-2t^2`. Five explicitly displayed outer-circle vertices form a primitive convex pentagon tangent to the inner circle on the actual edge segments. Poncelet closure supplies the complete fixed five-periodic family. After applying `T`, its squared-side sum is

    175/18 - (625/48)t.

Exact inequalities exclude coincidence of the transformed circumcenter with the ellipse center or either focus.

## Actual originating exact executions

The standard-library checker `problems/AMR-050-0053/algebra_lane/check_projective_pentagon.py` has SHA256 `0fa142da31e90987bd8dd40e52b43f0515b723b20ce6724a3ece937cc80c7737`. Its retained actual root-run output completed at 2026-10-02T08:38:37.744634+00:00, has SHA256 `255342b31f22c065230dd3f86f640485954ced28b699ba675bb5d182601c0042`, and reports `PASS_EXACT_PROJECTIVE_PRIMITIVE_PENTAGON`.

That execution checks the isolated cubic root, its positive radical, five distinct vertices, strict convexity, primitive period five, original circle incidence, original and transformed actual segment tangencies, the projective horizon, transformed circle and ellipse equations, strict nesting, excluded center/focus coincidences, and the exact displayed squared-side sum. It uses no floating point, external request, third-party dependency, or repository import. It explicitly reports `all_family_constancy_proved_by_this_finite_checker=false`; the analytic trace proof and Poncelet porism remain theorem dependencies.

The second standard-library checker `problems/AMR-050-0053/algebra_lane/check_dual_squares.py` has SHA256 `c41ed96a5e35626b59f48d9e9515a8d098436f55d0c6e80b647b49b69bf7f594`. Its retained actual root-run output completed at 2026-10-02T07:58:23.609842+00:00, has SHA256 `a0d074de075eada5a5906cd1bdd6fcca6793b8972c540d3aae04350f91a4ead2`, and reports `PASS_EXACT_DUAL_CONSTRUCTION_CONTROLS`.

That execution checks ten exact fixtures: four period-three focal cases, four period-four cases, and both windings of a circular period-five case. It verifies boundary reflection, confocal caustic contact on actual segments, the distinction between pedal reciprocal and vertex inverse, the midpoint relation, both trace formulas, reversal and repetition. It also supplies a control showing that the original focal polar's squared-side trace varies between two period-four phases. It explicitly reports `all_N_proved_by_finite_controls=false` and `source_ambiguity_resolved_by_checker=false`.

## Configured portable replay

The source archive is required to contain byte-identical copies of both checkers and both retained originating JSON outputs. From a fresh extraction root, the required commands are:

```sh
python3 -I -B reproducibility/check_projective_pentagon.py
python3 -I -B reproducibility/check_dual_squares.py
```

The `-I` flag isolates the interpreter from user and repository import paths, while `-B` prevents bytecode writes. Neither checker has a non-standard dependency. Do not use `-O` or `PYTHONOPTIMIZE`, because exact assertions are part of the certificate.

Each execution emits a truthful new timestamp. The replay harness requires zero exit status, empty stderr, exact script hashes, the expected result status, and exact equality of every parsed output field except `actual_execution_completed_at`. It records the raw stdout/stderr hashes and the normalized payload hash. No portable replay is claimed until the package-stage receipt exists and records those actual executions from a fresh extraction.

## Source scope, prior art, and build boundary

The original k811 prose and figure do not identify a single consistent dual operation. Proving both explicit interpretations does not authorize silently rewriting the frozen source. The original record therefore remains unresolved as an unqualified proposition and contributes no frozen-record closure.

The adjacent Murad conjecture is a separate 2026 source outside the frozen corpus. The manuscript refutes only its necessity direction and does not challenge its proved triangle results. The load-bearing analytic ingredients are expressly credited to the 2021 bicentric cosine theorem and the 2003 MI-II identity. A bounded search found no matched prior pentagonal counterexample, but that is not an exhaustive novelty or priority certificate.

Native compilation, PDF export, warning inspection, all-page visual QA, source archive construction, and fresh-extraction replay require separate actual receipts. File existence is not evidence that these operations occurred. No paper-readiness, DOI, external publication, arXiv submission, or guaranteed indexing is asserted by this report.