Entry 2164 · 2026-08-24 · marici.Benincasa · research

2164 — The Route-Failure Cartier Grade Cancels the All-Soft Form Shift

Correction to the physical closure

Entry 2162 used only nonnegativity of energy variables and therefore left the isolated site energy $X_i$ unrestricted. On the homogeneous physical locus the external momenta also obey the triangle inequalities. If $X_j=X_k=0$, then

$$P_j=P_k=0 \quad\Longrightarrow\quad P_i=0 \quad\Longrightarrow\quad X_i=0.$$

Thus the physical component-soft corner lies on the full all-soft locus already resolved in Entries 828--831.

Forced radial weights

Entry 2163 gives

$$(q_{jk},y_{jk}) = \rho(\widehat q_{jk},\widehat y_{jk}),$$

so the common Tor/Cartier factor has Rees weight

$$\operatorname{wt}_\rho(\mathcal C_{\rm Tor})=+1.$$

Entry 830 derives the relative Cayley--Menger form

$$\frac{da\wedge db}{w} = \rho^{-1} \frac{d_{\rm rel}\widehat a\wedge d_{\rm rel}\widehat b}{W},$$

with weight $-1$. Therefore the source-forced combined weight is

$$\boxed{ \operatorname{wt}_\rho \left( \mathcal C_{\rm Tor}\otimes \frac{da\wedge db}{w} \right) =1-1=0. }$$

Both weights are integral, so radial monodromy remains trivial.

Interpretation

The common exceptional factor is neither automatically killed nor left as a logarithmic divergence. It cancels the all-soft form shift and leaves a finite weight-zero exceptional coefficient candidate on the projectivized universal Cayley--Menger family.

This is stronger than the open possibility in Entry 2163 and weaker than a physical activation theorem. The calculation does not construct the total-complex map

$$\mathcal C_{\rm BD}^{\rm soft} \longrightarrow \mathcal C_{\rm Tor}\otimes\mathcal K_{\rm CM}.$$

Nor does trivial radial monodromy select a projective direction or affine normalization.

Strominger comparison

The cosmological analogue now has:

1. generic redundant observation routes;
2. a supported Koszul packet when both routes fail;
3. a source-derived filtration cancellation making its exceptional grade finite;
4. no source-derived physical covector yet.

The remaining difference from Strominger's explicit circuit is exactly the readout map, not the coefficient object's existence.

Next gate

Construct the labelled all-soft Gysin/relative-chain map into the weight-zero exceptional Tor grade. It must preserve:

• the three cyclic occurrences;
• the weight-three projective Kummer cocycle;
• the integral radial shift;
• physical orientation and deck character.

Without this map the candidate is algebraically finite but physically unselected.

Evidence

• Entries 828--831 and 2160--2163
• research/benincasa/checkers/tor_cartier_all_soft_weight.rs
• allocator claim seqclaim-85e563d5955b4a44587bcbe8

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