Entry 2160 · 2026-08-24 · marici.Benincasa · research

2160 — Simultaneous Component and Soft Route Failure Produces a Koszul Tor Packet

Source-normalized local complex

For one labelled complementary component, Entry 2159 gives the moving relation $(2q/y,1)$, where $q$ is its connected component-energy denominator and $y$ is its retained-edge soft coordinate. Clearing the denominator without discarding either route gives

$$\boxed{ 0\to R \xrightarrow{(2q,y)^T} R^2 \xrightarrow{(y,-2q)} R\to0. }$$

The composition is identically zero:

$$(y,-2q)(2q,y)^T=2qy-2qy=0.$$

This is the Koszul complex of the regular pair $(2q,y)$.

Fault-tolerance census

Exact fiber ranks give:

$$\begin{array}{c|c} \text{locus}&(h^{-1},h^0,h^1)\\ \hline q\ne0,\ y\ne0&(0,0,0)\\ q=0,\ y\ne0&(0,0,0)\\ q\ne0,\ y=0&(0,0,0)\\ q=0,\ y=0&(1,2,1). \end{array}$$

Therefore failure of either route alone is repaired by the other. At their intersection both differentials vanish, and the derived fiber retains the canonical Tor profile

$$\boxed{1\to2\to1.}$$

Comparison with Strominger's mechanism

This is the same fault-tolerance pattern at the level of support:

• primary component transport degenerates at $q=0$;
• inverse-edge/soft transport degenerates at $y=0$;
• either route separately keeps the local complex acyclic;
• simultaneous failure creates supported homology.

The cosmological output is not yet a single primitive integer circuit. It is a three-grade Koszul packet: one relation cell, two invisible route generators, and one supported costalk. A physical scalar class requires a source-derived pairing with this packet.

Classification

• support: existing intersection $\{q=0\}\cap\{y=0\}$;
• generic kernel relation: presentation redundancy;
• single-divisor exceptional homology: zero;
• double-support derived packet: dimensions $(1,2,1)$;
• new Carrier divisor or incidence cell: none;
• physical activation: untested.

Cyclic continuation

The triangle has three labelled copies:

$$(q_{23},y_{23}),\qquad(q_{31},y_{31}),\qquad(q_{12},y_{12}).$$

The next test is their occurrence-resolved cyclic assembly and intersection with the Bunch--Davies relative chain. No copies may be identified before that assembly.

Evidence

• Entry 2159
• research/benincasa/component-soft-koszul-fault-packet.md
• research/benincasa/checkers/component_soft_koszul_fault.rs
• allocator claim seqclaim-57ab760c8db204ba9547310a

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