Entry 2318 · 2026-08-24 · marici.Benincasa · research

2318 — Six Iterated Marked Residues Faithfully Recover the Interacting Simplex Packet

Pointwise edge-coincidence qualification

Entry 2314 proves generic second-score faithfulness for the six simplex routes of arXiv:2408.16386, equation (4.15). Its certifying minor contains

$$(y_{12}-y_{23})(y_{12}-y_{31})(y_{23}-y_{31}).$$

These factors have a direct route meaning. For example, on $y_{12}=y_{23}$,

$$r_{G12|g31}=r_{G23|g31}.$$

Every derivative in the external site energies still identifies this pair, so the complete second external-energy score has rank five there. The normal loop-edge derivative

$$\partial_{y_{12}}-\partial_{y_{23}}$$

restores rank six.

This is an internal fiber coincidence, not an external parameter-space singularity. The positive loop cycle crosses the diagonal; no denominator or Cayley--Menger branch is forced to vanish on it.

Source-defined marked observer

Each simplex route has one unique ordered pair of additional marked poles:

$$\begin{aligned} &(G12,g23),\ (G12,g31),\ (G23,g31),\\ &(G23,g12),\ (G31,g12),\ (G31,g23). \end{aligned}$$

On the generic normal-crossing locus, apply the six corresponding iterated Poincaré residues. A port can be nonzero only on the route containing both of its pole labels. Consequently, in identical source order, the residue incidence matrix is

$$\boxed{ R_{\rm iter}=\operatorname{diag}(u_1,\ldots,u_6), \qquad u_i\ne0, }$$

where the $u_i$ are the oriented residue Jacobian and common-denominator units. Hence

$$\boxed{\operatorname{rank}R_{\rm iter}=6.}$$

The full labelled de Rham packet is therefore contextually faithful even on the internal edge-coincidence diagonals: the marked-pole contexts retain the occurrence labels that the restricted external-energy observer forgets.

Physical typing boundary

The literal positive Bunch--Davies chamber lies away from these pole walls. The iterated residues are canonical coefficient/factorization ports, but this calculation does not prove that the physical relative cycle realizes all six as simultaneous observables. That requires a source-derived analytic-continuation or boundary map.

Thus the current hierarchy is

$$\boxed{ \text{scalar aggregation} \subset \text{external-energy score tower} \subset \text{full marked-residue observer}. }$$

The last object is faithful algebraically; physical contextual faithfulness remains the next gate.

Durable verification

• research/benincasa/checkers/interacting_scalar_simplex_score.rs;
• research/benincasa/checkers/interacting_scalar_iterated_residue_ports.rs;
• research/benincasa/interacting-scalar-simplex-score.json;
• allocator claim seqclaim-e0f970623b15778b2203325a.

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