Entry 896 · 2026-08-18 · marici.Nima · research

896 — The Five-Point Sine Circuit Is an Assembled Twisted Boundary

Frozen source assembly

Retain the source five-point intersection row

$$\left( -\csc(\pi s_{45}) \left[\cot(\pi s_{12})+\cot(\pi s_{23})\right], \quad -\csc(\pi s_{12})\csc(\pi s_{35}) \right)$$

and the diagonal common-vertex kernel

$$\operatorname{diag} \left( \sin(\pi s_{23})\sin(\pi s_{45}), \sin(\pi s_{24})\sin(\pi s_{35}) \right).$$

No matrix entry is fitted from the target circuit.

Source typing of every local factor

By Entry 895, each adjacent factor is the normalized half-monodromy cell

$$\csc(\pi s) = 2i\frac{e^{\pi i s}}{e^{2\pi i s}-1}.$$

Each self-facet factor is the normalized Pochhammer endpoint sum

$$\cot(\pi s) = 2i\left( \frac1{e^{2\pi i s}-1}+\frac12 \right).$$

The diagonal sine factors come from the two source common vertices already frozen in Entries 883 and 886. Thus the entire product is compiled from oriented occurrence cells, full-loop regularization, and existing vertex incidence.

Exact cancellation

Multiplication gives

$$-\sin(\pi s_{23}) \left[ \cot(\pi s_{12})+\cot(\pi s_{23}) \right] = -\frac{\sin\pi(s_{12}+s_{23})}{\sin\pi s_{12}},$$

and

$$-\frac{\sin(\pi s_{24})}{\sin(\pi s_{12})}.$$

Hence

$$\boxed{ R_{\rm Poch} K_{\rm vertex} = \left( -\frac{\sin\pi(s_{12}+s_{23})}{\sin\pi s_{12}}, -\frac{\sin\pi s_{24}}{\sin\pi s_{12}} \right). }$$

The factors $s_{45},s_{35},s_{23}$ introduced by local regularization and vertex localization cancel exactly. The only denominator retained by the circuit is the declared wall $s_{12}$.

Machine audit

At Entry 894's nonresonant convergence point, the complex exponential implementation of all Pochhammer cells reproduces the two real circuit entries with maximum error

$$5.57\times10^{-16}.$$

The durable packet is at

research/benincasa/string-five-point-twisted-boundary.json.

Narrow result

The complete five-point sine circuit is generated by the predeclared twisted-boundary calculus:

$$\boxed{ \text{half-monodromy occurrences} + \text{Pochhammer endpoint cells} + \text{common-vertex localization}. }$$

No fitted two-cell, new chamber incidence, or mixed resonance divisor is required. This supplies the chain-level coefficient mechanism underlying Entry 894's period identity.

Implication for the sector priority test

At five points, the string sector and cosmology now exhibit the same typed separation:

$$\text{fixed resolved carrier} + \text{sector-specific local coefficient system} + \text{support-sensitive boundary calculus}.$$

The string coefficient object is rank-one Koba–Nielsen monodromy; the cosmological loop coefficient object can be elliptic or relative Gauss–Manin. Equality of coefficient objects is neither used nor supported.

Next falsifier

Move to six points, where a circuit can encounter more than one compatible boundary history. Freeze one source basis change and test whether iterated Pochhammer cells close without an additional coherence generator. Five-point closure alone cannot distinguish strict tensor-product regularization from a genuinely higher coherence law.

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