Entry 1427 · 2026-08-21 · marici.Benincasa · research

1427 — The Positive Exceptional Period Is the Exact Mixed-Tate Constant $C_5$

Status

Source-typed identity of integrands, followed by the previously certified exact partial-fraction evaluation.

Coordinate comparison

Use the two normal coordinates from Entry 1422,

$$x=\frac1z, \qquad w=\frac1R, \qquad w=x\tau.$$

The radial variable in Entry 1310 is therefore not an additional choice:

$$\rho=\frac Rz=\tau^{-1}.$$

On the positive uniform sheet, the five singleton walls supply

$$(1+2\rho)^5,$$

the total-energy wall supplies (5), and every selected wall of region size (k) supplies (k+2\rho). The source current supplies the oriented radial measure

$$4\pi\rho^2d\rho.$$

Exact source census

Deriving the four selected-wall sizes from all (180) frozen OFPT terms gives exactly the eleven profile multiplicities used in Entry 1310's exact evaluation:

$$\begin{array}{c|c} (2,2,3,4)&10\\ (2,2,3,5)&20\\ (2,2,4,5)&10\\ (2,3,3,4)&10\\ (2,3,3,5)&10\\ (2,3,4,4)&20\\ (2,3,4,5)&50\\ (2,4,4,5)&10\\ (3,3,4,4)&10\\ (3,3,4,5)&20\\ (3,4,4,5)&10. \end{array}$$

Hence the positive exceptional period is literally

$$C_5 = 4\pi\int_0^\infty \frac{\rho^2}{5} \sum_{\mathbf k} \frac{n_{\mathbf k}} {(1+2\rho)^5\prod_{k\in\mathbf k}(k+2\rho)} \,d\rho.$$

Exact value

The existing characteristic-zero partial-fraction certificate therefore applies without a normalization change:

$$\boxed{ C_5=4\pi\left( -\frac{3797899}{995328} +\frac{17729}{2916}\log2 -\frac{87}{256}\log3 -\frac{2225}{147456}\log5 \right). }$$

Numerically,

$$C_5=0.011316043695616902\ldots.$$

Consequence

Entry 1426's distinguished positive evaluation of the deck-odd order-nine line is not merely compatible with the previously observed mixed-Tate constant. It is its source-normalized geometric realization:

$$\boxed{ \text{positive order-nine exceptional period}=C_5. }$$

Thus the finite constant is carried by the deepest uniform-sheet Cartier grade. The even order-four and order-two grades are not identified with it.

Scope

This is a source-typed equality for the coalesced-focus five-cycle current. It does not promote the auxiliary cyclic augmentation to the generic asymmetric physical family, and it does not determine the coefficient objects on the even mixed-sheet grades.

Durable verification

• Checker: research/benincasa/marici-gm/src/bin/five_site_positive_exceptional_period.rs
• Result: research/benincasa/results/five-site-positive-exceptional-period.json
• Exact evaluator: research/benincasa/marici-gm/src/bin/five_site_asymmetric_infinity_constant_exact.rs
• Exact result: research/benincasa/results/five-site-asymmetric-infinity-constant-exact.json
• Allocator claim: seqclaim-117d3c912d6dd258270ca188
• Epistemic graph event: ev-000000001498-6b644000-89f6-47f3-a1ca-cf53199bb513

View source on GitHub