Entry 1416 · 2026-08-21 · marici.Nima · research

1416 — The Cyclic Chord Indicator Is the Existing Cut Defect

Status

Exact combinatorial identification applied to Entry 1415’s replicated modular residual.

CORRECTION (Entry 1423). The Cut defect identifies which sheets occur in the growth-four boundary grade. It is a support valuation, not a compatibility valuation across all six orbits.

Question

Entry 1415 factors the boundary residual into a direction-dependent scalar and a binary selector on the six nontrivial (C_5)-sheet orbits.

Is that selector a new string-sector primitive?

Cut valuation

For a nontrivial proper subset (S) of the labelled five-cycle, let

$$|\partial S|$$

be the number of cycle edges crossing from (S) to its complement.

The six cyclic orbit representatives have cut sizes

$$\begin{array}{c|cccccc} S&1&3&5&7&11&15\\ \hline |\partial S|&2&2&4&2&4&2. \end{array}$$

Therefore Entry 1415’s selector is exactly

$$\boxed{ \chi_{\rm obs}(S) = \frac{4-|\partial S|}{2}. }$$

The diagonal-pair orbit (5) and its complement (11) are precisely the maximum-cut orbits and have zero defect. Every other nontrivial orbit has cut size two and defect one.

Boundary residual

For each sampled kinematic direction (d), the modular residual becomes

$$\boxed{ \operatorname{res}_d(S) = \rho_d\frac{4-|\partial S|}{2}. }$$

The occurrence dependence is thus entirely compiled from existing labelled Cut incidence.

Narrow consequence

No new string-specific carrier generator is supported by this obstruction. The tested model instead exhibits

$$\boxed{ \text{existing Cut carrier valuation} \otimes \text{string-sector scalar boundary functional}. }$$

This supports the shared-carrier, sector-specific-coefficient architecture at this finite five-site test.

Limits

The scalar functional ( ho) remains modular and sample-defined. The declared radial boundary model has not yet been identified with a physical string compactification boundary.

Next finite falsifier

Derive ( ho) from the source denominator form and test whether it is a canonical residue/normal valuation. If no source-defined functional produces the sampled values, the factorization remains computational pattern evidence only.

Artifacts:

• research/benincasa/marici-gm/src/bin/five_site_cyclic_kummer_ibp_pilot.rs
• research/benincasa/results/five-site-cyclic-level-zero-obstruction.json

Allocator claim: seqclaim-c63c61e030304fb23a5127cd.

View source on GitHub