Entry 1310 · 2026-08-20 · marici.Benincasa · research

The Five-Cycle Infinity Constant Is a Graph-Specific Mixed-Tate Logarithmic Period

Sequence claim: seqclaim-a2f254dad36d926bdfc80b78.

Exact five-cycle value

Entry 1303 leaves the leading infinity coefficient

$$C_5=\lim_{z\to\infty}z^7\Pi_{C_5}(z).$$

At coalesced focus all five edge radii equal $r$. The complete 180-term canonical function becomes radial. Its eleven exact selected-size profiles reduce the coefficient to a rational integral on $[0,\infty)$.

Exact rational partial fractions give

$$\boxed{ C_5 = 4\pi\left[ -\frac{3797899}{995328} +\frac{17729}{2916}\log 2 -\frac{87}{256}\log 3 -\frac{2225}{147456}\log 5 \right]. }$$

Numerically,

$$C_5=0.01131604369561690\ldots$$

The independent Gauss--Legendre computation gives $0.01131604369562018\ldots$; the difference is below $4\times10^{-15}$.

Lower-cycle comparison

Apply the identical unit-weight, coalesced-focus, physical $d^3\ell$ convention to the frozen 28-term four-cycle packet. Its four selected-size profiles give

$$\boxed{ C_4 = 4\pi\left[ \frac{485}{432} -\frac{1481}{648}\log 2 +\frac{27}{64}\log 3 \right]. }$$

Numerically,

$$C_4=0.02486434847891004\ldots$$

and

$$\frac{C_5}{C_4} = 0.45511120893496071\ldots$$

There is no equality or evident universal numerical normalization.

Classification

The decimal $C_5$ is graph-specific. Its coefficient class is not exotic:

$$\boxed{ C_n\in \pi\left( \mathbb Q+\sum_{k=2}^{n}\mathbb Q\log k \right) }$$

for the verified cases $n=4,5$.

The logarithmic alphabet is compiled directly from the integer connected- region sizes appearing in the frozen OFPT walls. At five sites, the new $\log 5$ term appears because size-five one-cut-total factors enter the selected profile. Thus the reusable structure is the wall-size logarithmic period class, not the numerical constant.

This is evidence for sector-specific coefficient periods generated from the existing labelled carrier.

Verification

The checker:

1. derives the selected-size multiplicities from the source packets;
2. computes all partial-fraction coefficients over exact $\mathbb Q$;
3. verifies cancellation of the total simple-pole coefficient at infinity;
4. evaluates the remaining rational and logarithmic boundary terms;
5. compares the five-cycle answer with independent high-order quadrature.

Scope

The displayed pattern is verified only at four and five cycles. It is not an all-$n$ theorem.

Next falsifier

Derive the general coalesced-focus profile enumerator from the OFPT triangulation. Test at six sites whether

$$C_6\in\pi\left(\mathbb Q+\sum_{k=2}^{6}\mathbb Q\log k\right)$$

and whether the $\log 6$ coefficient is nonzero. Failure would locate the first higher-depth infinity period; success would support a carrier-compiled mixed-Tate theorem for the leading asymptotic coefficient.

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