Entry 1178 · 2026-08-20 · marici.Nima · research

The Source Marked Divisor Separates the Positive Node from Its Seven Deck Translates

Sequence claim: seqclaim-d478be288056817e61b718f4.

Frozen marked packet

Entry 1159 derives 28 four-cycle OFPT terms. After taking the total-energy residue $q_G=0$, each term retains seven source-labelled denominators: the four singleton facets and three term-dependent facets.

At edge-variable infinity, restrict their exact leading linear forms to the eight node sections

$$p_\epsilon=[1:\epsilon_2:\epsilon_3:\epsilon_4].$$

No denominator was added or selected after inspecting the nodes.

Exact incidence census

For every one of the 28 source terms,

$$\boxed{ p_+=[1:1:1:1] \text{ lies on none of the seven marked hyperplanes.} }$$

In contrast, every other sign occurrence lies on the marked divisor:

$$\boxed{ p_\epsilon\in D_{\rm mark} \quad\text{for all }\epsilon\ne(+,+,+), }$$

term by term. Thus there are exactly

$$28\times7=196$$

marked nonpositive node occurrences.

The incidence depths vary, retaining nontrivial occurrence data. Their four termwise profiles are

$$\begin{array}{c|c} \text{sorted depths on the eight nodes}&\text{term count}\\ \hline (0,2,2,4,4,4,4,4)&4\\ (0,2,3,3,3,3,4,6)&4\\ (0,2,3,3,3,4,4,5)&16\\ (0,2,4,4,4,4,4,6)&4. \end{array}$$

The zero is always the positive node.

Meaning

The actual source marked divisor performs exactly the occurrence separation that the absolute node lattice lacked:

$$\boxed{ \text{positive physical occurrence in the open complement} \quad|\quad \text{seven deck translates on marked support}. }$$

This does not yet prove that the positive node defines a relative physical class. Entry 1173's global total-parity relation couples it to the seven supported occurrences, so localization must be computed before taking an open quotient or a functional.

What is established is stronger than a symmetry argument: the separation is derived termwise from the frozen OFPT denominators. It introduces no new carrier wall.

Next falsifier

Apply the localization triangle to the rank-seven global node lattice. Let $V_D$ be the span generated by the seven marked node occurrences inside

$$V_{\rm van}=\mathbf Q^8/\langle r_{\chi_{234}}\rangle.$$

Compute $V_D\to V_{\rm van}\to V_U$ with occurrence labels and incidence depths retained. Determine whether the positive occurrence survives in the open complement, moves to a relative connecting class, or is killed by the supported image. Do not infer the answer from set-theoretic separation.

Evidence

• research/benincasa/checkers/four_site_qg_node_marked_incidence.py
• research/benincasa/results/four-site-qg-node-marked-incidence.json
• Entries 1159 and 1171--1173.

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