# Verification of the evolute area formulas

This report accompanies *A Telescoping Formula for Evolute Areas in Elliptic Billiards* by Alper Ferudun. The written proof establishes the inner and outer evolute area ratios for physical elliptic billiard orbits whose **least period is N>4**. An explicit polynomial identity, cyclic summation and geometric denominator arguments establish the general result. Exact finite orbits are regression and boundary checks, not a proof by sampling.

The report records the author's mathematical self-audit and the exact executions completed on 30 September 2026. It does not claim independent human review, proof-assistant formalization, exhaustive historical priority, a finalized PDF layout, or a public deposit.

## Precise theorem scope

The billiard boundary is x²/u+y²/v=1 with u>v>0. Its nondegenerate confocal elliptic caustic is x²/A+y²/B=1, where A=u−λ, B=v−λ and 0<λ<v. The contact polygon Q follows the caustic contact points in trajectory order. The outer polygon T consists of intersections of the boundary tangents at successive bounce vertices. It is not the bounce polygon obtained by intersecting caustic tangents.

Every area is a signed shoelace area, including for self-intersecting star trajectories. The discrete evolute Ev(P) is the cyclic polygon of circumcenters of consecutive triples of P; the source's perpendicular-bisector order changes this only by a cyclic shift. Set

    Δ=u−v=A−B,
    d=λ(1/v−1/u),
    e=−1+λ(1/u+1/v),
    ν=d²+2e−1.

The proved formulas are

    area(Q)/area(Ev(Q)) = 8AB(1−d²)/(Δ²ν),
    area(T)/area(Ev(T)) = 8AB(1−d²)/(Δ²νe²).

For the stated physical least-period domain, the required circumcenters and both area quotients exist. The statement does not cover hyperbolic caustics, circular billiards, unsigned areas of filled-in regions, arbitrary permutations of the vertices, or the literal extension that admits repeated primitive four-cycles. The linked inner and outer assertions, source invariants k703 and k702, are two consequences in one manuscript. The neighboring worker-owned k701 record is not classified or counted here.

## Load bearing arguments in the written proof

1. Intersecting consecutive caustic tangents and requiring their intersection to lie on the outer ellipse gives the biquadratic contact relation F(x,y)=(1−dx²)y²+2exy+x²−d=0 for unit complex parameters x and y.
2. The physical inequalities 0<d<1 and |e|<1−d exclude equal and antipodal successive contacts and repeated roots. The non-retracing billiard branch identifies the two roots as the previous and next contacts. Three consecutive contacts are therefore distinct and noncollinear.
3. Restricting the equation of the existing Euclidean circumcircle to the caustic yields a quartic. Comparing its cubic coefficient gives the actual circumcenter, not merely a candidate satisfying a formal coefficient relation.
4. The rational circumcenter parameter R and rational function H satisfy the explicit cleared-denominator certificate

       T E=−(x²−y²)F(x,y)[U(xy)−V(xy)(x²+y²)],

   where E is the local evolute-area expression minus the constant multiple of the contact-area expression and minus H(y)−H(x). All factors, including U, V and T, are written in the proof. Expansion is an exact identity over the integer polynomial ring in x,y,d,e.
5. Physical denominator bounds allow division on each orbit edge when e≠0. Summing the resulting local identity on a closed cycle cancels the H terms. The signed determinant of the map from the complex circumcenter parameter to the real circumcenter supplies the inner formula.
6. An oriented tangent-side argument makes all contact-edge cross-products strictly of one sign, including for allowed star trajectories. Thus area(Q) cannot vanish. This is a geometric argument, not an assumption drawn from the finite examples.
7. The exceptional equation e=0 forces least contact period four. The equation ν=0, together with a separate polynomial identity for the contact recurrence, forces least period three. Contact and bounce sequences reconstruct each other, so least period N>4 excludes both exceptions and establishes nonzero evolute area.
8. Polarity gives T=M Q for M=diag(u/A,v/B). On this fixed conic, the actual consecutive circumcenters transform as γM⁻¹C, with γ=1−λ²/(AB). This is not general affine covariance of Euclidean circumcenters. Since γ/det(M)=−e, the outer quotient is the inner quotient divided by e², and its denominator is nonzero in the same domain.

These arguments, rather than the number of test orbits, are what establish the all-period theorem. In particular, no unchecked premise that sibling evolute ratios were already proved is used.

## Exact algebra and orbit checks

The compact symbolic checker verifies the quotient certificate by expansion with zero remainder. It also checks that the compact and original fractions for R and H agree and that H has the required reciprocal antisymmetry. These are exact symbolic equalities, not floating-point residual bounds.

The two preserved six-periodic fixtures have opposite traversal signs and each gives the inner quotient −640/729. For the new primitive eight-cycle with (u,v,λ)=(280,105,24), exact rational computations give

    area(Q)=2016/5,
    area(Ev(Q))=−8575/48,
    area(Q)/area(Ev(Q))=−13824/6125.

The eight-cycle checker separately verifies outer-ellipse membership, directional reflection, segment-interior tangency, closure, least period, and the actual consecutive circumcenters. No primitive five- or seven-period numerical example is claimed. Their absence is not a restriction of the analytic proof.

## Preserved failure and four period controls

The omitted-coboundary shortcut is false: in the preserved exact edge example, an edgewise density ratio is 5/3 rather than 1, and the nonzero coboundary difference is 4i√5/9. The theorem requires cyclic telescoping, not edgewise proportionality.

The formal substitution e=0 into the coefficient c is also invalid as an extension of the local identity: H has a pole there. An axis four-cycle has actual normalized evolute-area multiplier 1, whereas the formal coefficient limit is 1/2.

Two physical four-cycles on the same conic pair, (u,v,λ)=(336,105,80), have the exact inner quotients shown below. Each row is supported by exact reflection, tangency, closure and circumcenter checks.

| Four-cycle | area(Q) | area(Ev(Q)) | Inner quotient |
| --- | --- | --- | --- |
| Axis contacts | 160 | −53361/160 | −25600/53361 |
| Oblique contacts | 1792/13 | −23905728/1373125 | −422500/53361 |

Repeating either cycle twice doubles both signed areas and leaves its quotient unchanged, giving different quotients for two eight-term lists. This is a counterexample to the broader literal interpretation in which N denotes list length and repeated primitive four-cycles are allowed. It is **not** a counterexample to the intended source-standard least-period N>4 assertion. A second exact four-cycle pair in the source-bridge checker corroborates this boundary distinction; its parameters and quotients are different and are not substituted for the manuscript's displayed pair.

General affine transport of circumcenters is not used. The outer transfer is proved from differences of squared norms on the particular conic, and cannot be replaced by an arbitrary affine invariance assertion.

## Recorded execution evidence

The fresh execution receipt is `submissions/checkpoints/20260930-evolute-closure/execution-1/receipt.json`, verified at 2026-09-30T12:45:16.426387+00:00. All three executions exited with code 0 and empty standard error. Their exact recorded statuses are:

| Original checker | Fresh execution result |
| --- | --- |
| `problems/AMR-050-0041/attacks/telescoping/check_telescoper.py` | `PASS_EXACT_TELESCOPER_AND_CONTROLS` |
| `problems/AMR-050-0041/attacks/additional-controls/check_controls.py` | `PASS_COMPACT_IDENTITY_AND_EXACT_ADDITIONAL_CONTROLS` |
| `problems/AMR-050-0041/attacks/source-scope-bridge/check_bridge.py` | `PASS_EXACT_SOURCE_BRIDGE_CONTROLS` |

The original checker SHA256 values bound by that receipt are:

    telescoping:
    dc33c71cb0d720ca587f6172d858acfbf4330f2f8e6dc616d4764f320da99f33
    additional controls:
    0cb5b5a493bf0663e0aa41abe9f66aef2e62c84309984ee5437fa14c01502e3e
    source scope bridge:
    f9f092f24826b3c56d9902967a64725152004660ded6c6a5f51baa00b6743c0b

Earlier receipts also record byte-identical exact outputs on Python 3.13.5 and 3.9.6 for the telescoping and source-bridge checks. Running the same calculations in two runtimes is a reproducibility check, not independent mathematical review.

The public reproducibility runner is intended to preserve these original checker bytes under `reproducibility/attacks/` and supply the preserved six-periodic fixture under `reproducibility/fixtures/`. Its extracted-archive execution, the final PDF compilation and visual inspection, and final package hashes must be checked at release assembly; they are not certified by the earlier repository executions listed here.

AI assistance was used in mathematical exploration, exact symbolic computation, literature searching and manuscript preparation. The mathematical claims are the author's responsibility. This is an unrefereed preprint, and publication or a DOI does not certify correctness or novelty.