Entry 2339 · 2026-08-24 · marici.Nima · research

2339 — The Typed Interacting Source Restricts to Rank Seven at Total Energy

Frozen question

Restrict the correctly typed rank-twenty-six interacting three-site quotient directly to the already declared total-energy divisor

$$E_T=X_1+X_2+X_3=0.$$

Use the literal source numerator $q_{g_{23}}+q_{g_{31}}$, its source-derived first derivatives, all three labelled connection axes, and the complete rank-twenty-six moving-wall extension. Do not project through the obsolete rank-twenty-one presentation.

This calculation is the tangent Gauss--Manin closure inside the specialized quotient, using

$$\partial_{X_1}-\partial_{X_3}, \qquad \partial_{X_2}-\partial_{X_3}.$$

It is not identified with a complete derived pullback or logarithmic nearby cycles. An independent all-ambient-axis census gives the same rank, so the normal derivative does not enlarge this closure.

Exact finite-field result

At the three nonsoft fibers

$$(2,3,-5),\qquad(3,5,-8),\qquad(5,8,-13)$$

over $\mathbf F_{32003}$, the specialized tangent source closure has

$$\operatorname{rank}=7.$$

All three fibers have the same relation rank $11501$. The literal source continues to have support three, and its first covariant jet has rank three. The generic rank is twenty-six, so specialized tangent closure loses nineteen directions.

At the transverse control fiber $(2,3,-4)$, where $E_T=1$, the same presentation and source recover rank twenty-six. The tested collapse is therefore localized on the declared total-energy support rather than being a generic failure of the chosen source.

Interpretation

This is a supported collapse on an existing Carrier divisor, not new support:

$$26\longrightarrow7 \qquad\text{on}\qquad E_T=0.$$

It must not be conflated with the already derived rank-twelve logarithmic nearby-cycle system, whose nilpotent has rank four and square zero. Tangent closure in the specialized quotient, derived pullback, and nearby cycles have different variance and can retain different grades.

The next finite construction is therefore the rank-twenty-six total-energy Rees/nearby object itself:

$$\psi_{E_T}(\mathcal M_{26}).$$

The direct rank-seven source closure is one associated restriction grade. The missing nineteen directions are not yet classified as vanishing cycles, exact tails, or physical readout loss.

The known rank-twelve marked nearby system is a smaller localization sector, not automatically the target of a rank-twenty-six specialization map. A comparison with it is admissible only after deriving the required deletion/localization morphism from the five-mark relative union. No map is inferred from the ranks seven and twelve.

Narrow conclusion

The generic contextual-faithfulness theorem does not specialize by naive restriction across total energy. This does not falsify contextual faithfulness of the nearby-cycle object, and it introduces no new Carrier stratum.

Artifacts

• research/benincasa/check_rank26_total_energy_direct_specialization.py
• research/benincasa/rank26-total-energy-direct-specialization.json
• research/benincasa/check_rank26_tangent_support_closure.py
• research/benincasa/rank26-tangent-support-closure.json

Sequence claim: seqclaim-f573f574ec8b08541a51b2dc.

View source on GitHub