Exact census evidence

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This is the author evidence for the finite census and the mathematical reduction. It confers no independent review and proves no assertion about orders outside the specified ranges.

From the repository root, using the existing repository environment:

uv run --no-sync python wiki/research/leads/concentric_polygon_census/evidence/main.py

The default run rebuilds every positive critical radius for $3\le m\le32$ in the half-step concentric family and every positive critical parameter for $8\le n\le24$ in the specified affine family. It takes about half a minute on one thread; runtime varies by machine. Dependencies are SymPy and python-flint, exercised with versions 1.14.0 and 0.9.0. No solver or network access is needed. --quick checks only $3\le m\le6$ and $8\le n\le10$; its reduced coverage is printed. The program also works under python -O.

Retained exact inputs

The 47 JSON files under assets/ are author-generated mathematical event data, one per order, recomputed and compared in full by main.py. They carry no success logs or cached acceptance. All records are sorted by exact parameter order. The first event in a record already specifies its exact parameter; the remaining events record all coincident equations.

• assets/concentric_MM.json records the field order $2m$, threshold $m$, generic orbit counts, the number of equations, real-root tallies before deduplication, and every distinct positive critical radius. An event ["P", i, j, branch] or ["Q", i, j, branch] uses the polynomial in the sketch, $1\le i\le\lfloor m/2\rfloor$ and $0\le j<\lceil m/2\rceil$. After making a quadratic monic, branch $-1$ means its smaller real root and $1$ its larger; $0$ means the unique root of a linear or double-root equation. The intended cyclotomic embedding is always $\zeta_{2m}=e^{\pi i/m}$. position is below, lower_boundary, inside, upper_boundary or above, relative to the open strict-convexity interval. counts lists the $A$ and $B$ orbit counts.
• assets/affine_NN.json records the field order $n$, threshold $\lfloor n/2\rfloor$, generic counts at all pins, pair-equation tallies, and every distinct positive critical $s$. An event [i, j, k] means the quotient $(X_{ik}-X_{ij})/(Y_{ij}-Y_{ik})$ in the sketch, in the embedding $\zeta_n=e^{2\pi i/n}$, with $i$ the pin and $j<k$ its two other vertices. counts lists all pins in order $0,\ldots,n-1$; circle records exact equality $s=1$. Persistent equalities are accounted for in the generic counts and pair tallies, not as isolated parameter events.

See the order-three input for the two degenerate convexity endpoints and the order-four input for the nonconvex eight-point control. The complete data can be regenerated with produce.py --outdir <working-directory>; that discovery command writes only to the explicitly named directory. Its output requires the mathematical checks in main.py before use as evidence.

Checked obligations and failure behavior

The implementation is exact.py for cyclotomic arithmetic, real-root identities and certified ordering, and census.py for the complete equality equations and distance partitions.

All equalities are rational polynomial identities modulo a cyclotomic polynomial. Arb enclosures decide only strict order. An overlap triggers an exact identity test or precision refinement, and an unresolved comparison fails at the precision limit. No tolerance decides a coincidence.

The checker verifies input range coverage, every event record, every recorded count, the absence of a convex every-vertex witness, and the exact two-event pattern in the concentric convex intervals. It checks the coordinate formulas for affine squared distances by a second construction; at every affine event it evaluates every pinned distance and reconstructs all new equalities. For concentric radii it directly partitions all vertex labels at every admissible event, every event lowering both orbits, and all events with $m\le5$. Other concentric counts use the complete event lists and the injectivity proof in the sketch. Reciprocal events must exchange the two orbit counts.

Controls distinguish quadratic branches, common roots of different quadratics, repeated roots, linear equations, zero roots, strict convexity endpoints, and unequal parameters separated by $2^{-300}$. The eight-point control must have counts $(3,3)$ and fail convexity. A corrupted retained count or omitted critical record fails the full-data comparison with a nonzero exit, including under python -O. Every asserted computational obligation is enforced without bare assertions. The noncomputational generic-complement and convexity arguments remain visible in the sketch.

The affine octagon at $s=1+2\sqrt2$ must have the complete count vector $(3,6,4,6,3,6,4,6)$; its exact parameter must occur in the enumeration. It checks that one favorable pin cannot stand in for every vertex.