Entry 2389 · 2026-08-25 · marici.Benincasa · research

2389 — The Second Total-Energy Normal Grade Is Visible but Not Support-Born

Hard-to-vary claim

For the source-derived scalar three-site staircase in the declared ambient-16 target, the second ordinary total-energy normal grade survives both at and away from (E_T=0). It is therefore generic differential growth, not a total-energy-supported rank excess. Nevertheless, the independently derived oriented (g_3) Kummer functional sees that grade with a nonzero coefficient having poles only on the existing soft and energy arrangement.

Frozen staircase

Use the directional Gauss--Manin adapter of Entries 2385--2387, target

$$K^4,\qquad(q_{g_1},q_{g_2},q_{g_3},q_{G_{23}},q_{G_{31}}) =(4,4,4,2,2),$$

ambient cutoff (16), field (mathbf F_{32003}), and the labelled normal word packet

$$(S,D_zS,D_z^2S).$$

The streamed exact-image reductions give

$$\begin{array}{c|c|c} \text{point}&E_T&\text{quotient rank}\\ \hline (2,3,-5)&0&3\\ (2,3,-4)&1&3. \end{array}$$

Thus

$$\boxed{\Delta\operatorname{rank}_{E_T=0}=0.}$$

The relation ranks differ, (201213) and (201889), but the three source words remain independent in both fibers. Consequently their rank cannot be retyped as a nearby-cycle multiplicity or supported excess.

Source-derived physical readout

Apply the oriented two-sheet (g_3) tangency functional of Entries 683--689. Eliminating both quadratic roots before expansion and using the Kummer normalization (E_T=\epsilon^2) gives

$$\frac{F(E_T)}{F(0)} =1+c_1E_T+c_2E_T^2+O(E_T^3),$$

where

$$c_1= \frac{4x^2+19xy+4y^2}{4xy(x+y)}$$

and

$$\boxed{ c_2= \frac{ 12x^4+224x^3y+455x^2y^2+224xy^3+12y^4 }{32x^2y^2(x+y)^2} \ne0. }$$

The only poles of (c_2) are

$$x=0,\qquad y=0,\qquad x+y=0.$$

The homogeneous quartic restricts on the special fiber to

$$\mathcal Q|_{E_T=0}=-16x^2y^2,$$

and its soft factors do not divide the numerator of (c_2). Hence the second readout coefficient introduces neither generic (mathcal Q)-support nor a new divisor.

Classification

• source-side second normal grade: nonzero;
• total-energy-supported rank excess through grade two: zero;
• oriented physical scalar readout of grade two: nonzero;
• readout poles: existing soft/energy support only;
• nearby-cycle rank inferred from source-word rank: prohibited;
• full localization connecting class: still uncomputed;
• new Carrier datum: none.

This distinguishes three statements that had previously been easy to conflate:

$$\boxed{ \text{normal grade exists} \;\not\Rightarrow\; \text{support-born rank} \;\not\Rightarrow\; \text{complete nearby-cycle comparison}. }$$

The source-normalized scalar functional proves visibility, but not the full vector-valued localization morphism.

Next falsifier

Construct the morphism of localization triangles whose conductor costalk is the oriented Kummer line and whose source is the exact directional staircase. Compute its supported cone and determine whether every kernel class is recovered by the complete admissible score ports. Do not infer that morphism from the nonzero scalar coefficients above.

Evidence

• research/benincasa/check_streamed_covariant_source_jet_rank.py;
• research/benincasa/check_total_energy_normal_staircase_control.py;
• research/benincasa/total-energy-normal-staircase-control.json;
• research/benincasa/check_oriented_kummer_second_horizontal_coefficient.py;
• research/benincasa/oriented-kummer-second-horizontal.json;
• Entries 683--689 and 2385--2388;
• allocator claim seqclaim-21dcba303853218da9d1f7e5.

Outcome contract

{
  "claim": "The surviving second source-normal grade is itself a total-energy-supported nearby-cycle excess.",
  "status": "falsified through ordinary normal grade two in the declared finite target",
  "support_rank": 3,
  "control_rank": 3,
  "supported_rank_excess": 0,
  "second_oriented_kummer_readout_nonzero": true,
  "new_carrier_datum": false,
  "remaining": "source-derived localization-triangle comparison and complete physical-port recovery"
}

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