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Entry 294 · 2026-08-16 · marici.Benincasa · research

Wall and Primitive Tails Share One Algebraic Kernel Line

Result

The fixed final-block tails of both one-wall extensions and the primitive two-wall extension all lie on

⟨valg∣E=0⟩⊂A−−⊂T7.\langle v_{\rm alg}|_{E=0}\rangle\subset\mathcal A_{--}\subset\mathcal T_7.

Together with the Entry 293 Rees correction, the fixed logarithmic absolute data therefore use only

⟨e6,valg∣E=0⟩=A−−.\boxed{\langle e_6,v_{\rm alg}|_{E=0}\rangle=\mathcal A_{--}.}

No other algebraic-kernel direction and no elliptic direction occurs.

Frozen algebraic line

The explicit Gysin-kernel vector restricts to

v0:=valg(0)=x2y2((x2−y2)e7+2e8−2e9).\boxed{ v_0:=v_{\rm alg}(0) =x^2y^2\bigl((x^2-y^2)e_7+2e_8-2e_9\bigr). }

One-wall tails

The fixed final-block parts of the two mixed columns are

T101=y−x4xye7−e82xy(x+y)+e92xy(x+y),T_{101}=\frac{y-x}{4xy}e_7-\frac{e_8}{2xy(x+y)} +\frac{e_9}{2xy(x+y)},

T110=x−y4xye7+e82xy(x+y)−e92xy(x+y).T_{110}=\frac{x-y}{4xy}e_7+\frac{e_8}{2xy(x+y)} -\frac{e_9}{2xy(x+y)}.

Direct comparison gives

T101=−v04x3y3(x+y),T110=+v04x3y3(x+y)=−T101.\boxed{ T_{101}=-\frac{v_0}{4x^3y^3(x+y)},\qquad T_{110}=+\frac{v_0}{4x^3y^3(x+y)}=-T_{101}. }

Primitive tail

The fixed primitive tail is

T111=−(x−y)28x2y2e7+y−x4x2y2(x+y)e8+x−y4x2y2(x+y)e9.T_{111} =-\frac{(x-y)^2}{8x^2y^2}e_7 +\frac{y-x}{4x^2y^2(x+y)}e_8 +\frac{x-y}{4x^2y^2(x+y)}e_9.

Hence

T111=−x−y8x4y4(x+y)v0=x−y2xyT101.\boxed{ T_{111} =-\frac{x-y}{8x^4y^4(x+y)}v_0 =\frac{x-y}{2xy}T_{101}. }

Its home is thus the cyclic algebraic quotient direction selected by the final three-master block, not a generic direction in T7\mathcal T_7.

Incidence identity

The primitive quotient coefficients are

q101=−12y,q110=−12x.q_{101}=-\frac1{2y},\qquad q_{110}=-\frac1{2x}.

Since T110=−T101T_{110}=-T_{101},

T111+q101T101+q110T110=0.\boxed{ T_{111}+q_{101}T_{101}+q_{110}T_{110}=0. }

The primitive direct algebraic tail is therefore determined by its two one-wall transports and their incidence coefficients. This is evidence for, but not a chain-level proof of, a relative Gauss--Manin boundary relation.

Architectural consequence

At total-energy logarithmic order the fixed algebraic data occupy exactly:

  1. the invariant Kummer line ⟨e6⟩\langle e_6\rangle, entered by the Rees regularization;
  2. the cyclic line ⟨v0⟩\langle v_0\rangle, entered by all wall tails.

Both are pre-existing coefficient directions in A−−\mathcal A_{--}, and both have zero elliptic Gysin image. The finite model needs existing wall incidence plus the existing algebraic coefficient plane, not a new carrier generator.

Epistemic boundary and next falsifier

The displayed proportionalities are exact algebra after supplying the rational tail formulas. Those formulas remain reconstructions from exact generic-fiber reductions; a universal cleared Griffiths--Dwork certificate over Q(x,y)\mathbb Q(x,y) is still required.

The universal computation must reproduce this single-line identity or expose another fixed direction. An extra direction within T7\mathcal T_7 changes the coefficient-extension architecture; a direction outside the frozen relative surface and wall geometry is the carrier-level falsifier.

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