Entry 302 · 2026-08-16 · marici.Benincasa · research

Relative Stokes Pairing Replaces Master-Coordinate Contraction

Type correction (Entry 306). The coordinate-contraction falsifier in this entry remains valid, but the replacement below was stated in the wrong cohomological type. The source system is the complement $S_E\setminus W_E$, governed by logarithmic residue and Leray-tube duality, not the ordinary relative pair $H^\bullet(S_E,W_E)$. The meromorphic $\eta$'s of Entries 296--297 are surface-complement primitives, not forms on $W_E$. Read all claims below about a mapping-cone boundary primitive through the corrected formulation of Entry 306.

Result

The remaining comparison between the logarithmic marked-extension columns and the enhanced higher-Rees functionals cannot be obtained by contracting their equation-(58) master coordinates.

That proposed contraction is not invariant under the exact-lift gauge. The obstruction is source-visible:

• Entry 291 proves that the one-wall class is defined modulo a five-dimensional exact-lift variation whose freely varying coordinates include $e_3,e_5,e_6$;
• Entries 226, 320 prove that the exceptional functional is supported precisely on $e_3,e_5,e_6$.

Therefore a representative change that leaves the relative cohomology class fixed can change the naive coordinate contraction. No matrix obtained by dotting the displayed source vectors is canonical.

The proposed ordinary mapping-cone pairing was

$$\boxed{ \left\langle [(\Omega,\eta)], [(\Gamma,\gamma)] \right\rangle_{\rm rel} = \int_\Gamma\Omega - \int_\gamma\eta, \qquad \partial\Gamma=i_*\gamma. }$$

This formula is valid for an ordinary relative pair, but it is not the complex that defines the frozen marked source system. Entries 296--297 provide exact gauges in the surface complement; they do not provide the $\eta$-component of this cone.

Entry 306 replaces this proposed home by

$$\boxed{ \text{logarithmic residue/Leray-tube duality on the frozen marked complement}, }$$

not a new coefficient direction or carrier stratum.

Superseded ordinary-pair model

The following ordinary-pair cone was used in the original argument:

$$i:W_E\hookrightarrow S_E,$$

use the de Rham mapping cone

$$\operatorname{Cone} \left( \Omega^\bullet(S_E) \xrightarrow{i^*} \Omega^\bullet(W_E) \right)[-1].$$

A degree-two relative cocycle is represented by

$$(\Omega,\eta), \qquad d\Omega=0, \qquad i^*\Omega=d\eta.$$

A relative cycle is represented by

$$(\Gamma,\gamma), \qquad \partial\Gamma=i_*\gamma, \qquad \partial\gamma=0.$$

It is retained here only to expose the type error. The complement instead uses the residue triangle

$$0\longrightarrow\Omega^\bullet_{S_E} \longrightarrow\Omega^\bullet_{S_E}(\log W_E) \xrightarrow{\operatorname{Res}} \Omega^{\bullet-1}_{W_E}\longrightarrow0$$

on a smooth wall, and the normalization/conductor totalization for the actual reducible wall.

Gauge invariance

A relative coboundary change generated by a one-form $\xi$ has

$$(\Omega,\eta) \longmapsto (\Omega+d\xi,\eta+i^*\xi)$$

up to the lower boundary gauge on $W_E$. The pairing changes by

$$\int_\Gamma d\xi - \int_\gamma i^*\xi = \int_{\partial\Gamma}\xi - \int_{i_*\gamma}\xi =0.$$

Hence the relative Stokes expression descends to cohomology and homology.

By contrast, the absolute term alone changes by

$$\int_\Gamma d\xi = \int_\gamma i^*\xi,$$

which is generically nonzero for the enhanced thimble because it has the source-selected exceptional boundary.

This is the exact reason that the master-coordinate contraction fails.

Finite falsifier of the naive contraction

At a generic fiber, Entry 291 finds that the exact-lift variation projects with rank five onto

$$\langle e_1,e_2,e_3,e_5,e_6\rangle.$$

The physical enhanced functional of Entry 226 is

$$\Lambda_{\rm exc} = \left( 0,0,-\frac{2\pi^2}{x}, 0,-\frac{2\pi^2}{y}, -\frac{2\pi^2}{xy}, 0,0,0 \right).$$

Its restriction to the gauge-variation subspace is nonzero. Indeed it is nonzero on each of the $e_3,e_5,e_6$ coordinate axes. Thus there exist cohomologous exact-lift representatives with different values under the naive source-coordinate dot product.

This is a finite linear-algebra falsifier:

$$\boxed{ \Lambda_{\rm exc}\cdot\Theta \quad\text{is not a function of }[\Theta]_{\rm rel}. }$$

The same issue applies to the symmetric wall and to the primitive top lift, whose ordinary representative also carries exact-lift freedom.

Available central-fiber exact-gauge data

Entry 296 gives explicit rational primitives

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

in the corresponding wall complements.

Entry 297 gives the assembled two-wall primitive

$$\eta_{111}^{\rm rat} = \eta_{\rm inc} +\frac1{2y}\eta_{101} +\frac1{2x}\eta_{110},$$

while the filtered survivor is $e_6/[8(x+y)]$.

These are frozen exact gauges in the complement:

$$(\Theta_{101}^{\rm fix},\eta_{101}), \quad (\Theta_{110}^{\rm fix},\eta_{110}), \quad (\Theta_{111}^{\rm filt},\eta_{111}^{\rm rat}),$$

and the four source-oriented enhanced Leray-tube cycles.

They are not forms on the wall and therefore are not directly integrated over a wall cycle. The actual boundary/coefficient objects are the logarithmic residues on the moving source walls; Entry 304 computes their exceptional endpoint logarithms.

Classification

Deutsch--Popperian update M2.45

The hard-to-vary claim

$$\text{the source master coordinates alone canonically pair the enhanced and logarithmic algebraic layers}$$

is falsified.

The smaller surviving theorem, corrected in Entry 306, is

$$\boxed{ \text{their canonical comparison is logarithmic residue/Leray duality for the frozen marked complement.} }$$

This strengthens the six-functor form of H2 while prohibiting a basis-dependent shortcut.

Scope boundary

This entry falsifies coordinate contraction; Entry 306 identifies the correct pairing mechanism. It does not yet evaluate the resulting $3\times3$ matrix. Entry 303 shows that the central-fiber primitives cannot simply be restricted to the four moving source-oriented exceptional intervals.

Next hostile test

Compute the Gauss--Manin connection on the logarithmic residue triangle, using the normalization/conductor totalization and the orientation-twisted integral frame of Entry 305. Verify

$$\operatorname{Res}\circ\nabla = \nabla_W\circ\operatorname{Res},$$

then pair the wall classes of Entry 304 with the enhanced classes through Leray tubes. A forced singular divisor outside the frozen energy, conductor, Cayley--Menger, soft, and $\mathcal Q$ coefficient supports is the next carrier-level falsifier.

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