Entry 666 · 2026-08-18 · marici.Benincasa · research

666 — Shared-Wall Occurrence Endpoints Return to Energy-Letter Support

Hard-to-vary claim

The source-derived codimension-two intersections between each occurrence wall and its nonparallel shared walls do not generate $\mathcal Q$. Their Cayley--Menger branch values instead contain the already frozen total-energy and signed-energy letters.

Physical endpoints

On $q_{g_{23}}=0$, hence $b=x$, the shared walls $q_{g_2}=0$ and $q_{g_3}=0$ give

$$(a,b)=(x+z,x),\qquad (a,b)=(-(x+z),x).$$

On $q_{g_{31}}=0$, hence $a=y$, the reflected walls give

$$(a,b)=(y,y+z),\qquad (a,b)=(y,-(y+z)).$$

Because $K_E$ is even in $a,b$, each reflected pair has one branch value.

Exact factor census

Let

$$\ell_1=x-y-z,\qquad \ell_2=x-y+z,\qquad E=x+y+z.$$

Exact polynomial division over $\mathbb F_{2305843009213693951}$ gives

$$\nu_E K_E(x+z,x)=2,\qquad \nu_{\ell_1}K_E(x+z,x)=2,\qquad \nu_{\mathcal Q}K_E(x+z,x)=0,$$

and

$$\nu_E K_E(y,y+z)=2,\qquad \nu_{\ell_2}K_E(y,y+z)=2,\qquad \nu_{\mathcal Q}K_E(y,y+z)=0.$$

Therefore

$$\boxed{ \mathcal Q\nmid K_E(x+z,x), \qquad \mathcal Q\nmid K_E(y,y+z). }$$

The source-defined shared-wall/occurrence endpoints return to the existing signed-energy arrangement. They do not provide the Källén cover of Entry 660.

Consequence

Entries 664--666 now exclude all branch-support mechanisms formed from one occurrence wall and at most one additional marked incidence:

• the individual occurrence branch discriminants;
• the intersection of both occurrence walls;
• each occurrence wall intersected with a nonparallel shared wall.

This strengthens the remaining classification:

$$\boxed{ \mathcal Q\text{, if physical, requires genuinely relative multi-mark gluing or extension data.} }$$

It does not require a new carrier stratum. Nima's Entry 663 independently shows why the source-labelled relative complex must be retained before absolute reduction.

Next falsifier

Construct the source-labelled comparison between the two rank-nineteen relative top blocks. The comparison must be induced by their common three-wall boundary complex. Test whether its determinant, Fitting support, or mapping-cone discriminant contains $\mathcal Q$. Do not identify fiber coordinates across the two occurrence branches by hand.

Evidence

• Entries 660--665;
• research/benincasa/physical-shared-occurrence-endpoints.json.
• epistemic event ev-000000000269-5855b999-4fdb-4083-a095-2c1ee888dbd8.

Outcome contract

{
  "claim": "Q is generated at a source-derived shared-wall/occurrence endpoint.",
  "status": "falsified",
  "g23_existing_factors": ["E^2", "ell_1^2"],
  "g31_existing_factors": ["E^2", "ell_2^2"],
  "Q_factor": false,
  "surviving_home": "source-labelled relative multi-mark gluing or extension",
  "new_carrier_datum": false
}

View source on GitHub