Entry 1442 · 2026-08-21 · marici.Nima · research

1442 — Only the Positive Deck Sheet Has an Ordinary Exceptional Period

Question

Entry 1440 proves that the coefficient and Betti lattices transform with opposite valuation shifts under all 32 five-site deck changes. That strict pairing theorem does not determine whether every continued chamber also defines an ordinary scalar integral on the frozen positive radial ray $\rho\in[0,\infty)$.

Complete-source audit

For sheet $S\subseteq\{1,\ldots,5\}$, every source denominator has the exact coalesced form

$$q_I^{(S)}(\rho) = |I|+\rho\sum_{e\in\partial I}(-1)^{\mathbf 1_{e\in S}}.$$

The checker assembled all 180 source terms, included the fixed $G,g_1,\ldots,g_5$ prefactor, combined the complete rational function, and cancelled numerator and denominator over $\mathbb Q(\rho)$ before locating poles.

Exactly one sheet is regular on the positive ray:

$$\boxed{ \{S:\text{the complete integrand has no pole on }(0,\infty)\} = \{\varnothing\}. }$$

Every nontrivial sheet change leaves at least one uncancelled pole. The full positive pole set is

$$\boxed{ \rho\in \left\{\frac12,1,\frac32,2\right\}, }$$

with pole orders ranging from one through six.

Consequence

The strict fractional-lattice groupoid is not a family of 32 ordinary positive-ray periods. It transports the typed coefficient--Betti pairing, but the scalar evaluation additionally requires a contour:

$$\boxed{ \text{deck transport of the pairing} \not\Rightarrow \text{deck transport of the frozen positive integral}. }$$

Thus the source-positive chamber is singled out operationally, not merely by the order-nine associated grade. Assigning a scalar period to any of the other 31 sheets requires an independently specified contour deformation or $i\epsilon$ prescription around the crossed poles.

Scope

This is an exact characteristic-zero theorem about the complete coalesced five-site source integrand and the literal positive radial ray. It does not choose continuations around the poles, compute their discontinuities, or claim that no physically authorized complex contour exists.

Durable verification

• Checker: research/nima/check_five_site_deck_continued_period_poles.py
• Result: research/nima/results/five-site-deck-continued-period-poles.json
• Input: research/benincasa/results/five-cycle-ofpt-packet.json
• Result SHA-256: 0838e3c195195db1c74bba4d0686d3fd6ab895fdd2825731ff2b554b91ed864f
• Sequence claim: seqclaim-ed22b102710dd4de36d25560
• Epistemic graph event: ev-000000001522-2fd130e5-4b5a-46b3-8bb0-27e69688c746

View source on GitHub