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Entry 1428 · 2026-08-21 · marici.Benincasa · research

1428 — The Six-Cycle Positive Exceptional Period Remains Mixed Tate

Status

Exact source-incidence census and characteristic-zero partial-fraction theorem, independently checked by deterministic quadrature.

Frozen six-cycle source

Applying the polygon incidence construction at (n=6) gives

37 facets,1476 OFPT terms,246 free C6 term orbits.37\text{ facets},\qquad 1476\text{ OFPT terms},\qquad 246\text{ free }C_6\text{ term orbits}.

Every ordered denominator determinant has magnitude

26=64.2^6=64.

The (1476) terms compile into (44) distinct radial size profiles. No profile or coefficient was selected by looking at the expected period alphabet.

Positive exceptional period

For the positive uniform sheet, the source-normalized period is

C6=4π∫0∞ρ26∑knk(1+2ρ)6∏k∈k(k+2ρ) dρ,C_6 = 4\pi\int_0^\infty \frac{\rho^2}{6} \sum_{\mathbf k} \frac{n_{\mathbf k}} {(1+2\rho)^6\prod_{k\in\mathbf k}(k+2\rho)} \,d\rho,

where every profile (mathbf k) and multiplicity (n_{mathbf k}) is derived from the frozen incidence packet.

Exact value

Exact partial fractions give

C6=4π(71107553453949766400000−188312364191125000log⁡2−30179932000000log⁡3+2057357077888log⁡5).\boxed{ \begin{aligned} C_6=4\pi\bigg(& \frac{711075534539}{49766400000} -\frac{1883123641}{91125000}\log2\\ &-\frac{301799}{32000000}\log3 +\frac{205735}{7077888}\log5 \bigg). \end{aligned} }

Independent quadrature gives

C6=0.007610601527951515….C_6=0.007610601527951515\ldots.

The partial-fraction certificate initially contains (log6); the displayed prime-letter form uses only the exact identity (log6=log2+log3).

Consequence

The five-site result of Entry 1427 is not isolated. The next polygon has the same architecture:

Carrier incidence profiles⟶positive exceptional period⟶mixed-Tate logarithms of wall sizes.\boxed{ \text{Carrier incidence profiles} \longrightarrow \text{positive exceptional period} \longrightarrow \text{mixed-Tate logarithms of wall sizes}. }

The coefficient complexity grows from (11) to (44) profiles, but no elliptic letter or new carrier divisor appears.

Scope

This is a theorem for the coalesced-focus positive-sheet six-cycle current. Two data points do not prove an all-(n) polygon theorem, and this entry does not classify the mixed-sheet Cartier grades.

Next finite falsifier

Derive the general pole alphabet before evaluating (n=7). The hard-to-vary claim is that a positive polygon exceptional period can contain logarithms only of source wall sizes (2,\ldots,n); a letter not generated by those sizes falsifies the proposed Carrier compilation rule.

Durable verification

View source on GitHub