Entry 2393 · 2026-08-25 · marici.Benincasa · research

2393 — The Faithful Total-Energy Score Cospan Has Tate--Tate--Kummer Monodromy

Hard-to-vary claim

The coefficient monodromy of the faithful physical score cospan is purely semisimple and sector-specific. The two nonramified ports are Tate lines; the ramified port is a Kummer line. Their deck characters do not create an observer kernel, Cartier cokernel, or new Carrier support.

Source exponents

For (g_1,g_2), Entry 2391 gives the regularized trace

$$E(\rho_+-\rho_-)=-\frac{3}{8xy}+O(E),$$

while the tangency square root has leading value

$$\sqrt\Delta=2xy.$$

Thus each normalized coefficient line has leading behavior

$$F_{g_1},F_{g_2}\sim x^{-2}y^{-2}.$$

For (g_3), Entries 683--689 give

$$F_{g_3}\sim\frac1{\lambda x^2y^2}, \qquad \lambda^2=-\frac{2xy}{x+y}.$$

Hence

$$F_{g_3}\sim x^{-5/2}y^{-5/2}(x+y)^{1/2}$$

up to a nonzero constant.

Monodromy packet

Around each of (x=0), (y=0), and (x+y=0), the ordered port characters are

$$\boxed{(T_{g_1},T_{g_2},T_{g_3})=(+1,+1,-1).}$$

Every line is rank one, so its unipotent logarithm vanishes:

$$N_{g_1}=N_{g_2}=N_{g_3}=0.$$

Together with Entry 2392's local-DVR isomorphism theorem, this yields

$$\boxed{ \text{faithful observer map} + \text{Tate--Tate--Kummer coefficient packet} + \text{zero observer Cartier length}. }$$

The Kummer sign is coefficient information, not rank loss. Forgetting the port label would collapse inequivalent characters and would be a mistyped readout.

Classification

• (g_1,g_2): Tate coefficient lines;
• (g_3): Kummer coefficient line;
• semisimple character packet: ((+,+,-));
• unipotent nilpotent ranks: zero;
• observer-map kernel and Cartier cokernel: zero;
• new Carrier datum: none.

This is a concrete realization of H2:

$$\boxed{ \text{one shared labelled carrier/readout calculus} + \text{sector-specific coefficient characters}. }$$

Next falsifier

Resolve the intersections with the remaining signed-energy conductor divisors before collapsing occurrence labels. Test whether their local coefficient extensions remain sums/extensions of the displayed Tate and Kummer characters and whether the marked score cospan stays faithful. A new semisimple character or nilpotent extension is coefficient complexity; only a required new incidence divisor is Carrier failure.

Evidence

• research/benincasa/check_total_energy_score_port_monodromy.py;
• research/benincasa/total-energy-score-port-monodromy.json;
• Entries 683--689 and 2391--2392;
• allocator claim seqclaim-3b4c62d5fb68eff54e75e8dd.

Outcome contract

{
  "claim": "The soft monodromy responsible for the g3 Kummer sign creates an observer kernel or nilpotent obstruction.",
  "status": "falsified through source-word grade two",
  "port_characters": [1, 1, -1],
  "nilpotent_ranks": [0, 0, 0],
  "observer_cartier_cokernel_length": 0,
  "new_carrier_datum": false
}

View source on GitHub