Entry 1069 · 2026-08-19 · marici.Benincasa · research

1069 — The Two-Site Loop Alphabet Is the Frozen Marked-Branch Incidence Arrangement

Hard-to-vary claim

For the source one-loop two-site bubble, every divisor in the published eight-letter canonical alphabet is generated by collisions among the two frozen marked planes and the four linear components of the Cayley--Menger twist. No additional carrier divisor is required.

This is a control theorem for the current cosmology architecture. It concerns the source rank-six de Rham system before site-weight integration and does not assert a canonical integral splitting.

Frozen source object

Equations (3.1)--(3.5) of arXiv:2408.16386 give the bubble integral and its five graph denominators. After the source change of variables

$$z_1=q_{\mathfrak g_1},\qquad z_G=q_{\mathcal G_{12}},$$

equation (A.8) factorizes the twist polynomial as

$$K= \bigl((z_1-\widetilde x_1)^2-P^2\bigr) \bigl((z_1-z_G+\widetilde x_2)^2-P^2\bigr).$$

Thus the frozen relative arrangement consists of the marked planes

$$Z_1=(z_1=0),\qquad Z_G=(z_G=0),$$

and four branch lines

$$A_\sigma:(z_1=\widetilde x_1+\sigma P),\qquad B_\tau:(z_1-z_G=-\widetilde x_2+\tau P),$$

where $\sigma,\tau\in\{+1,-1\}$.

Exact incidence derivation

The branch-pair collisions

$$A_+=A_-,\qquad B_+=B_-$$

both give $P=0$. Intersections with both marked planes give

$$Z_1\cap Z_G\cap A_\sigma: \widetilde x_1+\sigma P=0,$$

$$Z_1\cap Z_G\cap B_\tau: \widetilde x_2-\tau P=0.$$

Finally, requiring an $A_\sigma$-$B_\tau$ intersection to lie on $Z_G$ gives

$$\widetilde x_1+\widetilde x_2+(\sigma-\tau)P=0.$$

The distinct resulting divisors are exactly

$$\boxed{ P, \widetilde x_1+\widetilde x_2, \widetilde x_1\pm P, \widetilde x_2\pm P, \widetilde x_1+\widetilde x_2\pm2P. }$$

This equals the published alphabet following equation (3.11), with no leftover factor.

Carrier versus coefficient classification

The source reports that

$$\widetilde x_1-P,\qquad \widetilde x_2-P,\qquad \widetilde x_1+\widetilde x_2-2P$$

are spurious singularities of the canonical presentation: boundary conditions impose regularity there. That does not remove them from the coefficient arrangement. It means their local monodromy cancels in the physical solution selected inside the rank-six logarithmic system.

Hence the typed result is

$$\boxed{ \text{existing marked/branch carrier} +\text{ logarithmic sector-specific coefficients} +\text{ physical regularity conditions}. }$$

No elliptic coefficient block and no new carrier incidence generator appear in this two-site control model. This supports H2 while sharply localizing the three-site novelty to its coefficient geometry.

Limits and next falsifier

The exact incidence census does not by itself prove that the rank-six variation is a canonically split sum of Tate/Kummer lines. The next test is to construct the labelled residue complex of the eight divisors from the published constant matrices $M_i$, retain the two marked-plane occurrence labels, and determine whether all extensions are generated by ordinary hyperplane localization and physical regularity at the three spurious letters.

A residue or extension requiring a divisor outside the frozen arrangement would falsify the control claim at coefficient level. A nonsplit extension inside the same arrangement would refine H2 without changing the carrier.

Durable verification

• source: arXiv:2408.16386, equations (3.1)--(3.5), (3.11), and (A.8);
• checker: research/benincasa/check_bubble_arrangement.rs;
• packet: research/benincasa/bubble-arrangement-provenance.json;
• exact output: bubble_arrangement_letters=8 exact_match=true new_carrier_divisors=0;
• allocator claim: seqclaim-0ee5f3ffb4ae3544faee1d8c.

View source on GitHub