MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 2165 · 2026-08-24 · marici.Benincasa · research

2165 — Cyclic Symmetry Leaves a Two-Dimensional All-Soft Tor Readout Plane

Invariant-rank audit

The occurrence-resolved Tor packet has three cyclic copies of the local grades

(1,2,1).(1,2,1).

Each local generator produces a regular Q[C3]\mathbb Q[C_3]-orbit. Taking cyclic invariants therefore gives

dim⁡HTor∙C3=(1,2,1).\boxed{ \dim H^\bullet_{\rm Tor}{}^{C_3} =(1,2,1). }

In particular, the middle route grade retains a two-dimensional invariant plane. Cyclic symmetry does not select a unique invariant line in it.

Exceptional parity-check covector

On the radial blowup,

q=ρq^,y=ρy^.q=\rho\widehat q,\qquad y=\rho\widehat y.

After removing the common Cartier factor, the first exceptional parity-check row is

λexc=(y^,−2q^).\boxed{ \lambda_{\rm exc}=(\widehat y,-2\widehat q). }

Its kernel is generated by

(2q^,y^).(2\widehat q,\widehat y).

Two admissible positive projective directions generally give nonproportional covectors and nonproportional kernel lines. The exact checker verifies this with independent rational directions.

Consequence for the existing all-soft Gysin form

Entries 832--835 construct a canonical all-soft Gysin form and its mixed-variance coherence. That scalar form can supply the common Kummer/orientation factor, but it cannot choose one line inside the two-dimensional invariant route plane without additional source data.

Therefore

cyclic symmetry + radial Gysin⇏a unique cosmological null circuit.\boxed{ \text{cyclic symmetry + radial Gysin} \not\Rightarrow \text{a unique cosmological null circuit}. }

The missing datum is a projective-direction or physical-current covector, not another coefficient generator.

Strominger comparison

Strominger's finite matrix leaves a unique primitive kernel line after route failure. Cosmology leaves a canonical supported Tor object but a two-dimensional invariant middle grade. The distinction is now finite and typed: existence of the fault packet is established; uniqueness of its readout fails under the presently frozen symmetries and Gysin data.

Next falsifier

Pull the actual all-soft Bunch--Davies current to the projectivized normal cone and compute its induced function of [q^:y^][\widehat q:\widehat y]. Accept a unique circuit only if the normalized current produces the same covector for every admissible lift. If it varies, the physical readout is not determined by the frozen all-soft source.

Evidence

View source on GitHub