Entry 309 · 2026-08-16 · marici.Benincasa · research

Central Algebraic Tails Are the Shadow of Half-Kummer Transport

Result

The three central algebraic tails computed in Entries 291--294 are not independent ambient extension data. Their incidence coefficients are exactly the total-energy specialization of the primitive half-Kummer connection derived in Entry 308.

Consequently the $v_{\rm alg}$ tail cancels when the primitive top class is transported horizontally. The surviving central ambient datum is the filtered $e_6/[8(x+y)]$ Rees class of Entry 297, not a new $\mathcal Q$-supported line.

Half-Kummer coefficients at total energy zero

Entry 307 gives

$$a_1=-\frac12d\log\Delta_1, \qquad a_2=-\frac12d\log\Delta_2.$$

Along the total-energy normal,

$$a_1(\partial_E)\big|_{E=0}=-\frac1y, \qquad a_2(\partial_E)\big|_{E=0}=-\frac1x.$$

Entry 308 proves

$$\nabla\widetilde g_{111} = \frac12a_1\,g_{101} +\frac12a_2\,g_{110}.$$

Therefore its central mixing coefficients are

$$\boxed{ \frac12a_1(\partial_E)\big|_{E=0} =-\frac1{2y}=q_{101}, \qquad \frac12a_2(\partial_E)\big|_{E=0} =-\frac1{2x}=q_{110}. }$$

These are exactly the coefficients reconstructed independently in Entry 294.

Exact tail cancellation

Write $s=x+y$ and

$$v_0 = x^2y^2\bigl[ (x^2-y^2)e_7+2e_8-2e_9 \bigr].$$

The frozen central reductions give

$$T_{101} = -\frac{v_0}{4x^3y^3s}, \qquad T_{110} = \frac{v_0}{4x^3y^3s},$$

$$T_{111} = -\frac{x-y}{8x^4y^4s}v_0.$$

Direct substitution gives

$$\boxed{ T_{111} +\frac12a_1(\partial_E)\big|_{E=0}T_{101} +\frac12a_2(\partial_E)\big|_{E=0}T_{110} =0. }$$

Thus the incidence identity

$$T_{111}+q_{101}T_{101}+q_{110}T_{110}=0$$

is precisely the central specialization of horizontality for $\widetilde g_{111}$. It is not an accidental rational relation among three fitted columns.

Role of the complement primitives

Entry 296 proves

$$\Theta_{101}^{\rm fix}=d\eta_{101}, \qquad \Theta_{110}^{\rm fix}=d\eta_{110}$$

using meromorphic primitives with only the frozen wall poles. Entry 297 constructs the corresponding two-wall primitive. In the corrected type of Entry 306, these are complement exact gauges, not forms on a pair boundary.

Their central algebraic tails therefore record how ordinary specialization of a moving logarithmic class fails to commute with residue transport. Once the quotient Kummer connection is restored, the primitive $v_{\rm alg}$ tail cancels by the displayed identity.

This does not imply that the full ambient localization extension splits. The filtered survivor

$$\boxed{ \frac{e_6}{8(x+y)} }$$

is not removed by the ordinary complement primitive. It is a genuine higher-normal/Rees coefficient class in the algebraic kernel.

Classification

Deutsch--Popperian update M2.52

The hard-to-vary claim

$$\text{the three }v_{\rm alg}\text{ tails are independent mixed ambient extension columns}$$

is falsified. The smaller surviving theorem is

$$\boxed{ \text{they are the central shadow of the half-Kummer conductor transport and cancel in the horizontal primitive top combination.} }$$

Consequence for the ambient frontier

At the central associated grade, the ambient localization problem has been reduced from three algebraic tails to one filtered survivor on the $e_6$ line. The next hostile test is to transport that survivor through second Rees order and determine its extension with the generic $\langle e_6,v_{\rm alg}\rangle$ algebraic plane.

Entry 299 already falsifies the simplest candidate

$$\frac12d\log(-\mathcal Q)$$

for its canonical $e_6$ coordinate. What remains is to decide whether the survivor defines a nontrivial extension with $v_{\rm alg}$, or is removed by a residue-compatible higher-Rees gauge. A forced pole outside the frozen supports remains the carrier-level falsifier.

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