Entry 1232 · 2026-08-20 · marici.Nima · research

Entry 1232 — Total-Energy Specialization Adds Only the Canonical $A_2$ Conormal Layer

Input from Entry 1231

The 80 labelled triple overlaps split into:

• 30 whose three energy walls already have rank two;
• 50 whose three energy walls have rank three and span $E_T$.

The root-normal Čech map is the constant row

$$(1,-1,1)$$

with rank-two $A_2$ kernel.

Derived specialization

For the 50 rank-three triples, quotienting by $E_T$ lowers the energy-wall rank from three to two. Hence it creates exactly one relation among the three generators. The resulting conormal/Koszul factor has ranks

$$(1,1).$$

For the other 30 triples, the rank-one relation already exists before total-energy specialization; no new nearby relation is created.

After $E_T=0$, all 80 triples therefore have energy-relation nullity one, but only 50 carry the new nearby-cycle conormal class.

Compatibility with the marked complex

The $A_2$ row is constant over the energy base, so specialization commutes strictly with the Čech differential:

$$\boxed{ \psi_{E_T}(1,-1,1) = (1,-1,1)\psi_{E_T}. }$$

Tensoring the rank-two $A_2$ kernel with the new conormal factor gives ranks

$$\boxed{(2,2)}$$

per newly resonant labelled overlap. Across all 50 overlaps, the new nearby ranks are

$$\boxed{(100,100)}.$$

Since the 50 labels form ten free $C_5$-orbits, each nearby degree is

$$20\,\mathbb Q[C_5].$$

Rees order

All source energy walls in this subsystem are linear. Therefore their deformation has no second ordinary/Rees grade:

$$\operatorname{gr}^{(2)}_{E_T}=0$$

for the marked-incidence subsystem considered here. This does not constrain separate integrated coefficient periods, whose first nontrivial deformation may begin at higher order.

Verdict

$$\boxed{ \text{new }E_T\text{ layer} = \text{existing }A_2\text{ kernel} \otimes \text{canonical Cartier conormal factor}. }$$

There is no specialization commutator, coefficient excess, or new carrier datum.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_triple_total_energy_nearby.rs
• research/benincasa/results/five-site-triple-total-energy-nearby.json

Next falsifier

The marked-incidence branch is now closed through triple overlaps. Return to the coefficient layer: determine whether the physical five-site period system has any non-Kummer discriminant or higher-normal deformation not generated by the rank-32 Kummer connection, Gram density, and the supported incidence objects above. Freeze an actual source period block before computing.

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