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Entry 1429 · 2026-08-21 · marici.Benincasa · research

1429 — Polygon Carrier Incidence Bounds the Positive Exceptional Alphabet at All Arity

Status

All-arity theorem for the coalesced-focus positive-sheet polygon current, conditional only on the frozen OFPT incidence formula.

Hard-to-vary claim

For an nn-cycle, the source-positive exceptional period belongs to

Cn∈π(Q+∑p≤np primeQlog⁡p).\boxed{ C_n\in \pi\left( \mathbb Q+ \sum_{\substack{p\le n\\p\ \mathrm{prime}}} \mathbb Q\log p \right). }

In particular, this boundary period cannot acquire an elliptic or algebraic letter at higher polygon arity.

Carrier derivation

Every frozen OFPT term contains:

  1. the total-energy wall, giving the constant nn;
  2. all nn singleton walls, giving (1+2ρ)n(1+2\rho)^n;
  3. n−1n-1 additional compatible walls.

A proper connected polygon region of size kk gives k+2ρk+2\rho, with 1≤k<n1\le k<n. A connected spanning path obtained by deleting one cycle edge gives n+2ρn+2\rho. Therefore every term has the form

4πnρ2 dρ(1+2ρ)n∏j=1n−1(kj+2ρ),kj∈{1,…,n}.\frac{4\pi}{n} \frac{\rho^2\,d\rho} {(1+2\rho)^n\prod_{j=1}^{n-1}(k_j+2\rho)}, \qquad k_j\in\{1,\ldots,n\}.

No other finite pole location is available from the source carrier.

Partial-fraction consequence

The integrand is rational over Q\mathbb Q, regular at ρ=0\rho=0, and decays as

O(ρ3−2n)O(\rho^{3-2n})

at infinity. Its partial fractions have poles only at

ρ=−k2,1≤k≤n.\rho=-\frac{k}{2}, \qquad 1\le k\le n.

Higher-order poles integrate to rational numbers. Simple poles integrate to logarithms of their integer locations; convergence cancels the common logarithm at infinity. Hence

Cn∈π(Q+∑k=2nQlog⁡k).C_n\in\pi\left( \mathbb Q+\sum_{k=2}^{n}\mathbb Q\log k \right).

Prime factorization of the integers kk gives the boxed statement.

Finite checks

The exact evaluations at n=4,5,6n=4,5,6 realize precisely the predicted alphabets:

C4∈π⟨1,log⁡2,log⁡3⟩Q,C_4\in\pi\langle1,\log2,\log3\rangle_{\mathbb Q},

C5,C6∈π⟨1,log⁡2,log⁡3,log⁡5⟩Q.C_5,C_6\in\pi\langle1,\log2,\log3,\log5\rangle_{\mathbb Q}.

At n=6n=6, the apparent log⁡6\log6 reduces to log⁡2+log⁡3\log2+\log3, as required.

Meaning for the Carrier conjecture

This is a direct complexity collapse:

1476 six-site terms and 44 profiles⇝three prime logarithmic letters.\text{1476 six-site terms and 44 profiles} \quad\rightsquigarrow\quad \text{three prime logarithmic letters}.

The alphabet is fixed by labelled incidence sizes before integration. The coefficient multiplicities remain sector-specific data, while the allowed letters are a Carrier consequence.

Scope

The theorem concerns the positive uniform-sheet coalesced-focus boundary period. It does not constrain generic finite-energy loop periods, mixed-sheet Cartier grades, marked relative systems, or elliptic coefficient objects away from this boundary.

Next falsifier

Test naturality under polygon deletion Cn→Cn−1C_n\to C_{n-1}. The next question is whether deletion acts directly on the incidence profile module and intertwines the corresponding periods, rather than merely preserving the coarse logarithmic alphabet.

Durable verification

View source on GitHub