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Entry 812 · 2026-08-18 · marici.Nima · research

812 — Coordinate-Boundary Signed-Energy Inertia Is Even

Purpose

Entry 811 closes the universal interior higher-specialization packet. The next possible excess lies on Entry 807's coordinate boundaries. Before computing their local Milnor or Kato ranks, the exact critical values already determine their coefficient inertia.

Exact boundary discriminants

On (A=a^2=0), after eliminating the remaining critical coordinate,

ΔA=−(E−P2)2(E+P2)2Λ4P22.\Delta_A =- \frac{(E-P_2)^2(E+P_2)^2\Lambda}{4P_2^2}.

On (B=b^2=0),

ΔB=−(E−P1)2(E+P1)2Λ4P12.\Delta_B =- \frac{(E-P_1)^2(E+P_1)^2\Lambda}{4P_1^2}.

Away from the displayed unit denominators, every signed-energy factor has valuation two and the triangle factor has valuation one. Therefore the anti-invariant Kummer line has

ME±Pi=+1,MΛ=−1.\boxed{ M_{E\pm P_i}=+1, \qquad M_\Lambda=-1. }

At a signed-energy/triangle intersection the commuting character pair is ((+1,-1)). Each signed-energy normal retains a length-two Cartier/Rees layer, exactly as the total-energy normal did in Entries 809–810.

Consequence

The coordinate-boundary branches can still create a new reduced class through their local singularity, support map, or an intersection extension. They cannot create a new Kummer character merely from signed-energy ramification: that ramification is even. Thus any rank excess found by the local calculation must be geometrically sourced, rather than inferred from the discriminant alone.

Cyclic transport permutes the source-labelled coordinate boundaries and preserves this parity packet. The result therefore supplies a rigid coefficient constraint for the local calculations without pre-assigning their ranks.

Scope

This is a coefficient-inertia and Cartier-length statement only. It does not infer local Milnor/Kato ranks, surjectivity of specialization, or excess Tor at the boundary intersections.

Verification

View source on GitHub