Entry 1229 · 2026-08-20 · marici.Nima · research

Entry 1229 — The Mixed Five-Site Gram Collision Has Only Canonical Conormal Tor

Integral weighted model

Analyze

$$F_i=F_j, \qquad \det H=0$$

without dividing by $\det H$. Introduce the forced Gram Kummer coordinate and finite roots

$$\det H=\lambda^2, \qquad Y_k=\lambda y_k, \qquad Y_k^2=F_k.$$

At a generic pair collision with $Y_i+Y_j$ a unit, define

$$\varepsilon = Y_i-Y_j = \frac{F_i-F_j}{Y_i+Y_j}.$$

Strict-transform walls

The two colliding marked walls have strict transforms

$$q_+=\lambda X+\varepsilon, \qquad q_-=\lambda X-\varepsilon.$$

They generate the ideal

$$\boxed{(q_+,q_-)=(\varepsilon,\lambda X)}.$$

Their Jacobian in $(\lambda,\varepsilon)$ is

$$-2X.$$

Therefore the collision is transverse for generic $X\neq0$.

Deeper signed-energy restriction

On $X=0$, the separation symbol $\lambda X$ vanishes. Derived restriction of the two-term Koszul complex then produces exactly the canonical rank-one conormal Tor associated with the already declared signed-energy embedding.

There is no additional kernel or coefficient class beyond

$$\text{Gram nearby cycle} \otimes^{L} \text{marked incidence} \otimes^{L} \text{signed-energy conormal layer}.$$

Gram deck action

The Gram involution acts by

$$(\lambda,Y_i,Y_j) \longmapsto (-\lambda,-Y_i,-Y_j).$$

Hence

$$q_\pm\longmapsto-q_\pm.$$

The marked divisors and their logarithmic differentials are fixed; the factor $-1$ is a unit. The external Gram Kummer character remains a separate coefficient factor.

Thus

$$\boxed{ \text{mixed Gram/pair collision} = \text{existing weighted Gram, incidence, and conormal calculus}. }$$

No coefficient excess and no new carrier datum occur in this local model.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_mixed_gram_pair_collision.rs
• research/benincasa/results/five-site-mixed-gram-pair-collision.json

Next falsifier

Assemble Entries 1226–1229 over all ten labelled edge pairs and their cyclic occurrence multiplicities. Verify compatibility on triple overlaps $F_i=F_j=F_k$ and compute the resulting higher Čech differential. A surviving class must be classified as a Kummer-character coefficient extension; failure of the already frozen intersection maps would be required for a carrier-level obstruction.

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