combinatorics

deckard @slimer48484

2026-08-04 — saved image

deckard @slimer48484
Good morning! As a side effect of making mathematically inspired art, Fable has - again - discovered new mathematics.

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THE TWO WHEELS (wheels_4096.png, 4096²) — MO 513838, products of two k-cycles with overlapping support. One specimen (k=41, m=5, type (29,27,21)) drawn as threads through the vesica of two wheels; 260 re-drawn partners as fog; the c-spectrum band shows Pr[#cycles] is identical for every k. New mathematics in verification.md; the overlap principle, the master product formula, (1,664 exact checks), the m=3 closed form (the poster's "wall"), k=13 predicted, k=15 Monte-Carlo confirmed.

2. THE PICKET FENCE (fence_2560.png, 2560²) — AP-obstruction atlas piece 39: Z[√2] censused to 4×10⁹ (601,376,078 members). Log-embedding country (units translate horizontally); equal-gap runs of consecutive members as gold fences; l=6 never occurs though an iid null expects ~7,600 - and a 2-adic tower theorem shows a six-post fence needs 24 | gap. atlas39_notes.md

3. THE RHOMBUS PLATEAU (plateau_2560.png, 2560²) — MO 137177: unit-sided polygons maximizing Σ|PiPj|². n=4 is a flat valley (every rhombus scores exactly 8 — Euler's identity); valley closes (regular wins, verified multistart n≤16); the stiffness ladder holds 1/φ at n=5, triple degenerancies 4/√2 (n=8) and 10φ (n=10), softest mode ≈ n³/8π². plateau_notes.md

(m-1)!, independent of both choices. The negative-hypergeometric weights are the number of ways complementary aggregate over a i slots. ■ (Exhaustively verified as above.)

2. The wall at m = 3, demolished (closed form)

For m = 3: p_3 = 1/2 on λ = (3) and 1/2 on λ = (1,1,1). Specializing the master formula and doing inclusion-exclusion on the box constraints gives, for v = (a ≥ b ≥ c) ≥ 2k−3 with three parts:

q_(k,3)(v) = 2 · |perms(v)| · (b·c − t(t+1)) / ((k−1)² (k−2)²), t = max(0, k−2−a),

where |perms(v)| ∈ {1, 3, 6} is the number of distinct orderings of (a,b,c); and Pr[π is a single (2k-3)-cycle] = 1/2 for every k ≥ 3 (this is p_3(3)) — the overlap principle in action; the single-cycle probability never depends on k for any odd m: it equals p_m((m))).

Two chambers, one wall at a = k−2 (the largest part is the largest single-side excursion), deficit t(t+1) in the inner chamber & piecewise polynomial exactly as double-Hurwitz theory predicts, now with the exact closed form.

Derivation from the master formula: for λ = (1,1,1) the gap weights are trivial and the inner sum is the box count T(a,b,c) = #{G ∈ [0,a-1]×[0,b-1]×[0,c-1]: ZG = k−3}; the closed form is equivalent to the lattice identity T(a,b,c) = bc − t(t+1) (verified for all 52,728 admissible triples with k < 80; zero failures). For λ = (3) the weight C(k-1,2)² cancels the normalization, giving Pr[single cycle] = p_3(3) = 1/2 for every k — analytically, not just empirically.

Checks: exact for all 143 partitions across k = 4..12; k = 13: 43-way (12! per orbit class) returned all 44 three-cycle partitions and the single-cycle 1/2 precisely as the law demanded; ...te-Carlo at k = 15 (60M samples) confirms the t = 4 chamber: v = (9,9,9) observed 3.6787e-3 vs predicted 2(81−20)/33124 = ...

[right column continuation, partially cut off:]
and called m = 3 "the wall."

All results below were found and verified by exact rational-arithmetic censuses (orbit-reduced exhaustive enumeration in C, counts converted to exact fractions): all m ≤ k for k ≤ 12, plus k = 13 at m = 3 — 43 tables, every probability exact. Verification artifacts: wheels.c (orbit-reduced enumerator), wheels_wrap.py (exact rational conversion + sum-to-1 checks), wheels_brute.py (independent brute force, matches the C enumerator on all overlapping tables), /data/.

1. The Overlap Principle (empirical theorem, exact for all data)

Write A = S1 ∩ S2. Let ρ = σ_A · τ_A, where σ_A, τ_A are the first-return maps of σ, τ to A (each is a uniform m-cycle on A, and they are independent).

(a) Cycle-count law. The number of cycles of ρ has the law of the number of cycles of a product of two independent uniform m-cycles on m points — it does not depend on k at all.

Verified exactly for every (k, m), k ≤ 12: e.g. the c-distribution at m = 5 is (1/3, 5/8, 1/24) on [1, 3, 5] for k = 5, 6, ..., 12 identically. Via Boccara/Stanley the right-hand side is classical. In particular supp c(π) = (m, m−2, m−4, ...) (c ≡ m mod 2 by sign; c ≤ m because every cycle of π meets A — a cycle avoiding A would live in B1 = S1\A or B2 = S2\A alone, where π acts as a restriction of the single cycle τ resp. σ, which visits all).

(b) Master formula. Condition on the type λ = (a_1 ≥ ... ≥ a_c) of ρ, whose law p_m(λ) = q_[m,m](λ) is the classical two-cycles-...

ed form: the master formula IS the closed form (p_4 = ... xpanded into box-count polynomials the same way.
2,2) and λ = (3,1) — that is exactly why it resisted a si... actly in every table (it is enforced structurally by the v...

10:03 PM · Aug 3, 2026 · 641 Views

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