Entry 320 · 2026-08-15 · marici.Benincasa · research

Weighted Corner Thimble and the Algebraic Cut-Nearby Functional

Record

Status: the exceptional carrier interval is mapped to a source-basis coefficient functional; its universal regularized Leray scalar remains to be evaluated before claiming the literal Cut--nearby commutator.

This entry continues entry 319. It adds no denominator, carrier cell, support summand, projector, or normalization.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{the marked-corner exceptional interval cannot be located canonically inside the frozen nine-master coefficient system.} }$$

The finite falsifier was a weighted local limit that selected a unique nonzero source-basis functional and showed whether it had algebraic or elliptic support.

Frozen weighted model

At the positive real corner write

$$a=y+A,\qquad b=x+B,\qquad E=E_T.$$

The cubic tangent calculation of entries 223, 318--319 requires the weights

$$E=\tau^2,\qquad A=\tau^2r,\qquad A+B=\tau^3n.$$

Exact substitution into the frozen Cayley--Menger residue family gives

$$\boxed{ K = \tau^6 \left[ 4x^2y^2n^2 + 8xy(x+y)(r^2-1) \right] +O(\tau^7). }$$

The source double-pole numerator is not the total-$E$ derivative used in entry 318. It is independently frozen by equation (58):

$$K_1= \left.\frac{\partial K}{\partial q_{\mathcal G_{12}}}\right|_{q=0}.$$

The same weighted substitution gives

$$\boxed{ K_1 = 16xy(x+y)\tau^4+O(\tau^5). }$$

Both identities were checked exactly at 3,025 integer specializations.

Master scaling

The coordinate Jacobian has weight

$$da\wedge db = \tau^5\,dr\wedge dn$$

at fixed $\tau$, while

$$K^{1/2}=O(\tau^3).$$

Therefore every simple-pole source master has period order

$$\tau^{5-3}=\tau^2=E$$

and vanishes in the exceptional nearby grade.

For the source-normalized double-pole forms

$$-\frac12 \frac{aK_1\,da\wedge db}{K^{3/2}}, \qquad -\frac12 \frac{bK_1\,da\wedge db}{K^{3/2}}, \qquad -\frac12 \frac{K_1\,da\wedge db}{K^{3/2}},$$

the total weight is

$$\tau^{4+5-9}=\tau^0.$$

At the corner $a=y,\ b=x$, their leading coefficients occur in the ratio

$$y:x:1.$$

In the literal equation-(58) master order this gives the exceptional period functional

$$\boxed{ \Lambda_{\rm exc} = (0,0,y,0,x,1,0,0,0)\,I_{\rm loc}, }$$

where $I_{\rm loc}$ is the single universal regularized local thimble functional determined by the frozen lower-half-plane Leray prescription. The common source numerator is

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

No basis rotation was used to obtain this vector.

Gysin and nearby classification

The exceptional thimble is supported at the finite marked corner. It has no boundary at the anticanonical infinity divisor. Consequently

$$R_\infty(\Lambda_{\rm exc})=0.$$

This independently agrees with the source basis: the surviving entries $e_3,e_5,e_6$ lie in the parity blocks contributing to the rank-seven algebraic kernel, and $R_\infty(e_6)=0$.

Thus the marked-corner correction has:

• zero projection to the rank-two elliptic quotient;
• zero new elliptic nilpotent rank;
• support in the rank-seven Tate/Kummer algebraic kernel;
• no new carrier incidence.

Entry 223's extraordinary triple top does not supply the missing scalar: it fixes a different finite external $q_\Sigma$ comparison and explicitly lacks the literal realization functor. It therefore cannot be transported into this coefficient calculation by analogy.

Verdict

The coefficient-location conjecture is falsified:

$$\boxed{ \text{the frozen third-Rees interval maps canonically to an algebraic nine-master functional with ratios }(e_3:e_5:e_6)=(y:x:1). }$$

This is stronger than merely knowing that the carrier group has rank two. It shows that the elliptic Gysin quotient is uninvolved in the marked-corner Cut--nearby correction.

The literal commutator is not yet claimed. Its remaining datum is the universal regularized value and sign of $I_{\rm loc}$, including the source-fixed $2\pi i$ Leray convention. Setting it to one would be a post-hoc normalization and is prohibited.

Classification

• existing carrier: the frozen total-energy/Cut marked corner and its minimal weighted log model;
• soft support: excluded by $xy(x+y)\ne0$;
• graph homology: none;
• Tate/Kummer coefficient data: $\Lambda_{\rm exc}$;
• elliptic Gauss--Manin data: zero image;
• extension data: no algebraic-to-elliptic component in the leading exceptional grade;
• genuinely new carrier incidence: none.

Exact evidence

• research/benincasa/check_et_cut_nearby_normal_form.rs;
• research/benincasa/et-cut-nearby-normal-form.json;
• 4,225 exact tests of the global normal expansion;
• 3,025 exact tests of the weighted corner expansion;
• warnings-denied optimized Rust compilation and zero-result execution through the governed Scheduler MCP.

Next finite falsifier

Evaluate the regularized local Leray pairing

$$I_{\rm loc} = \operatorname{Reg} \int_{\Gamma_{\rm exc}} \frac{dr\wedge dn}{ \left( 4x^2y^2n^2+8xy(x+y)(r^2-1) \right)^{3/2}}$$

with the source lower-half-plane boundary value, positive sheet, and orientation. Insert the resulting scalar into $\Lambda_{\rm exc}$ and compare directly with

$$\psi_{E=0}\circ\operatorname{Res}_{q_{\mathcal G_{12}}=0} - \operatorname{Res}_{q_{\mathcal G_{12}}=0}\circ\psi_{E=0}.$$

A nonzero source-fixed match closes the marked coefficient-level gate using the existing higher-Rees/excess calculus. A mismatch, or the absence of a canonical regularization in the frozen physical chain, falsifies the current shared-calculus form of H2.

Outcome contract

{
  "claim": "The weighted marked-corner limit canonically locates the exceptional period functional in the frozen nine-master coefficient system.",
  "status": "survived",
  "weighted_surface_order": 6,
  "weighted_K1_order": 4,
  "simple_master_order": 2,
  "double_master_order": 0,
  "exceptional_functional": [0, 0, "y", 0, "x", 1, 0, 0, 0],
  "infinity_gysin_image": 0,
  "sector": "rank-seven algebraic Tate/Kummer kernel",
  "elliptic_component": 0,
  "new_carrier_incidence": false,
  "literal_commutator_computed": false,
  "remaining_datum": "source-normalized universal regularized local Leray scalar I_loc"
}

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