Entry 318 · 2026-08-15 · marici.Benincasa · research

Total-Energy Cut Normal Form and the Marked-Corner Excess Gate

Record

Status: exact frozen-polynomial normal-form theorem and finite falsification of depth-two Rees sufficiency for the physical Cut--nearby comparison at the nonsoft marked corners.

This entry continues entries 161, 180, 316, and 220. It uses the literal source normalization

$$q_{\mathcal G_{12}}=E_T+y_{12}, \qquad dq_{\mathcal G_{12}}=dy_{12},$$

the canonical Leray germ of entry 180, and the frozen Cayley--Menger family. It adds no denominator, support summand, projector, normalization, marked section, or carrier cell.

Deutsch--Popperian claim tested

The hard-to-vary claim was:

$$\boxed{ \text{the predeclared nearby/Gysin/excess calculus through second }E_T\text{-Rees grade controls the Cut specialization at every nonsoft marked intersection.} }$$

The finite falsifier was a source-forced marked intersection at which both the first and second normal coefficients vanish while a higher coefficient is nonzero.

Frozen Cut family

Take the source nonseparating Cut

$$\operatorname{Cut}_{12} =\operatorname{Res}_{q_{\mathcal G_{12}}=0}.$$

On the residue surface set

$$x=X_1, \qquad y=X_2, \qquad E=E_T, \qquad a=y_{23}, \qquad b=y_{31}.$$

The frozen Cayley--Menger family is

$$S_E:\quad w^2=K_E(a,b).$$

At total energy zero,

$$K_0=R^2, \qquad R=xa^2+yb^2-xy(x+y).$$

Thus the central surface has the two source-defined components

$$w=R \qquad\text{and}\qquad w=-R.$$

Exact first normal coefficient

Direct expansion of the frozen determinant gives

$$\boxed{ [E]K_E =-2(x+y)(a^2-y^2)(b^2-x^2). }$$

With

$$U=w-R, \qquad V=w+R,$$

the generic conductor therefore has local form

$$UV=E\cdot\text{unit}+O(E^2)$$

away from

$$a=\pm y \qquad\text{or}\qquad b=\pm x.$$

Hence generic Cut--nearby base change is the ordinary semistable nodal model. Every failure of first-order transversality is confined to four already frozen axial marked divisors. No new carrier support appears.

The four nonsoft marked corners

The axial divisors meet the reduced conductor at

$$(a,b)=(\pm y,\pm x).$$

For generic

$$xy(x+y)\ne0,$$

these are nonsoft points. At all four corners the exact specialization is

$$\begin{aligned} K_E(\pm y,\pm x) =E^3\big[&-8xy(x+y)\\ &+(5x^2+14xy+5y^2)E\\ &-6(x+y)E^2+2E^3\big]. \end{aligned}$$

Consequently

$$[E]K_E=[E^2]K_E=0, \qquad \boxed{[E^3]K_E=-8xy(x+y)\ne0.}$$

The first nontrivial smoothing datum at the marked Cut corner is third normal order. Neither the first jet nor the second Rees grade sees it.

Relation to the quartic $\mathcal Q$

The established coefficient identity remains

$$\operatorname{gr}^{(2)}_{E_T}\mathcal Q=-8xy.$$

It concerns the source algebraic-letter quartic. The new coefficient

$$-8xy(x+y)$$

is instead the third normal coefficient of the Cayley--Menger Cut family at the marked physical-chain corner. Their common factor does not identify their types or make $\mathcal Q$ support.

Verdict

The tested depth-two claim is falsified:

$$\boxed{ \operatorname{Rees}_{\le2} \text{ is insufficient for the full marked physical Cut--nearby comparison.} }$$

The stronger carrier falsifier does not fire. The entire excess locus is the pre-existing union

$$\{a=\pm y\}\cup\{b=\pm x\},$$

and its four pairwise corners. Therefore:

• existing carrier: total-energy normal, Cut divisor, axial marked sections, reduced conductor, and their flagged intersections;
• soft support: excluded by $xy(x+y)\ne0$;
• graph homology: no new class derived;
• Tate/Kummer data: unchanged;
• Legendre/Gauss--Manin data: unchanged rank-two nodal quotient;
• extension/excess data: first possible marked-corner contribution occurs in third normal grade;
• genuinely new carrier incidence: none;
• falsified mechanism: truncation of the shared comparison envelope at second Rees order.

Thus H2 survives only after deleting the unsupported depth-two bound. This is a reduction of the surviving conjecture, not permission to fit a third-grade correction to the target answer.

Exact evidence

• research/benincasa/check_et_cut_nearby_normal_form.rs;
• research/benincasa/et-cut-nearby-normal-form.json;
• 4,225 exact integer specializations of the complete frozen polynomial;
• warnings-denied optimized Rust compilation and zero-result execution through the governed Scheduler MCP.

Next hostile falsifier

Freeze the third normal coefficient just derived and construct the minimal log blowup of the local marked-corner model. Compute the relative can--variation map of the canonical Leray chain and test whether its class is the functorial third-Rees excess of the existing axial-flag square.

Admissible outcomes:

1. the class is generated canonically by the frozen flagged carrier and the ordinary higher-Rees/excess calculus, strengthening unbounded-grade H2; or
2. the class cannot be produced without an additional support-switch object, falsifying the current shared-calculus form of H2 at this corner.

No third-grade correction may be chosen from the desired commutator.

Outcome contract

{
  "claim": "Nearby/Gysin/excess comparison through second E_T-Rees grade controls the source-defined nonseparating Cut at every generic nonsoft marked intersection.",
  "status": "falsified_at_four_nonsoft_marked_corners",
  "cut": "Res_{q_G12=0}, q_G12=E_T+y12, Jacobian 1",
  "central_fiber": "K_0=R^2",
  "first_normal": "-2*(x+y)*(a^2-y^2)*(b^2-x^2)",
  "generic_cut_nearby_model": "semistable",
  "excess_support": ["a=y", "a=-y", "b=x", "b=-x"],
  "corner_gr1": 0,
  "corner_gr2": 0,
  "corner_gr3": "-8*x*y*(x+y)",
  "corner_nonsoft_open": "x*y*(x+y)!=0",
  "depth_two_sufficient": false,
  "new_carrier_incidence": false,
  "surviving_hypothesis": "shared frozen carrier plus sector-specific coefficients and unbounded higher-Rees/excess calculus",
  "next_experiment": "Compute the third-Rees relative can-var class on the minimal log blowup of one marked corner."
}

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