Entry 279 · 2026-08-16 · marici.Benincasa · research

Vanishing $H^3$ and the Canonical Two-Wall Localization Extension

Result

Entries 342--343 left open the localization boundary

$$H^1(W_1\cup W_2)(-1)\longrightarrow H^3(S_E).$$

On the generic smooth de Rham locus its target vanishes:

$$\boxed{H^3(S_E)=0.}$$

Moreover, the union of the two frozen denominator walls is principal on the affine residue surface. Its degree-zero Gysin class therefore vanishes. Consequently the marked $q_{\mathcal G_{12}}$-residue sector fits into the canonical short exact sequence

$$\boxed{ 0\longrightarrow H^2(S_E) \xrightarrow{j^*} H^2\!\left(S_E\setminus(W_1\cup W_2)\right) \xrightarrow{\operatorname{Res}_W} H^1(W_1\cup W_2)(-1) \longrightarrow0. }$$

Its generic ranks are

$$\boxed{0\longrightarrow 9\longrightarrow12\longrightarrow3 \longrightarrow0.}$$

This sequence is canonical. A splitting is not.

Frozen compactification

Retain entry 150's compactification

$$S_E=\overline S_E\setminus D_\infty,$$

where $\overline S_E$ is a generic smooth degree-two del Pezzo surface and $D_\infty$ is its connected smooth anticanonical elliptic curve. Thus

$$H^1(\overline S_E)=H^3(\overline S_E)=0, \qquad \operatorname{rank}H^2(\overline S_E)=8.$$

No new compactification, boundary component, or coefficient summand is introduced.

The top Gysin map

The localization sequence for $D_\infty\hookrightarrow\overline S_E$ contains

$$0=H^3(\overline S_E) \longrightarrow H^3(S_E) \longrightarrow H^2(D_\infty)(-1) \xrightarrow{i_*}H^4(\overline S_E).$$

Both groups at the right have rank one. The Gysin map is nonzero: for the normalized top class $\eta$ of the connected curve,

$$\int_{\overline S_E}i_*\eta=\int_{D_\infty}\eta=1.$$

Hence $i_*$ is an isomorphism over the generic de Rham field, and

$$\boxed{H^3(S_E)=0.}$$

Equivalently, the localization boundary of every finite-wall conductor class vanishes before any choice of master basis or physical cycle. Because the statement holds for the full cohomology of the double-cover surface, it also holds in each deck-character summand used to realize the source Kummer coefficient.

Independent rank-nine audit

The preceding part of the same compactification sequence is

$$H^0(D_\infty)(-1) \xrightarrow{i_*}H^2(\overline S_E) \longrightarrow H^2(S_E) \longrightarrow H^1(D_\infty)(-1) \longrightarrow0.$$

The first map sends $1$ to $[D_\infty]$, which is nonzero; indeed

$$D_\infty^2=(-K_{\overline S_E})^2=2.$$

Therefore

$$\operatorname{rank}H^2(S_E) =(8-1)+2=9.$$

This recovers geometrically the rank-seven algebraic kernel plus rank-two elliptic quotient of entry 150:

$$0\longrightarrow\mathcal T_7 \longrightarrow H^2(S_E) \longrightarrow H^1(D_\infty)(-1) \longrightarrow0.$$

It supplies no canonical splitting of that sequence.

The finite-wall Gysin is zero

On $S_E$, the two source walls are cut by the actual lower denominators

$$W_1=\{q_{\mathfrak g_1}=0\}, \qquad W_2=\{q_{\mathfrak g_2}=0\}.$$

Their reduced union is the principal Cartier divisor

$$W=W_1\cup W_2 =\{q_{\mathfrak g_1}q_{\mathfrak g_2}=0\}.$$

It is connected because the two wall curves meet at the same-sheet points $P_\pm$ of entries 266, 341--343. Hence $H^0(W)$ has rank one. The degree-zero Gysin in the wall localization sequence sends its generator to the divisor class $[W]$, but

$$[W]=c_1\!\left(\mathcal O_{S_E}(\operatorname{div} (q_{\mathfrak g_1}q_{\mathfrak g_2}))\right)=0.$$

Thus the exact localization segment

$$H^0(W)(-1)\longrightarrow H^2(S_E) \longrightarrow H^2(S_E\setminus W) \longrightarrow H^1(W)(-1) \longrightarrow H^3(S_E)$$

reduces to the displayed short exact sequence.

Entry 343 gives

$$\operatorname{rank}H^1(W)=3$$

with the saturated conductor filtration

$$0\to H^1(W_1)\oplus H^1(W_2) \to H^1(W) \to \mathbb Q_{\rm top} \to0.$$

Therefore the mixed grades $101,110$ and top grade $111$ are the three-dimensional quotient of one canonical marked relative extension.

Deutsch--Popperian verdict

The hard-to-vary conjecture was

$$\boxed{ \text{a nonzero }H^3(S_E)\text{ obstruction may prevent the frozen two-wall conductor lattice from lifting to marked relative cohomology.} }$$

It is falsified on the generic smooth locus. The target is zero and the wall divisor is principal. Every conductor class lifts to $H^2(S_E\setminus W)$, although no canonical lift is selected.

The narrower surviving statement is

$$\boxed{ \text{the unresolved datum is the extension class and its physical relative-chain realization, not existence of a lift.} }$$

Classification

$$\boxed{ \begin{array}{c|c} \text{structure} & \text{geometric home}\\ \hline \mathcal T_7 & \text{algebraic part of }H^2(S_E)\\ \mathbb V_{\rm ell}(-1) & H^1(D_\infty)(-1)\\ g_{101},g_{110},g_{111} & H^1(W_1\cup W_2)(-1)\\ \text{mixed/top gluing} & \text{localization extension in }H^2(S_E\setminus W)\\ \text{localization obstruction} & 0\\ \text{new carrier datum} & \text{none} \end{array} }$$

The coefficient system is filtered, not canonically decomposed:

$$0\to H^2(S_E)\to H^2(S_E\setminus W) \to H^1(W)(-1)\to0.$$

Limits

This entry is restricted to the generic smooth de Rham locus. It does not:

• extend the sequence across the elliptic, conductor, signed-energy, or soft discriminants;
• compute the Gauss--Manin extension matrix;
• select a splitting or coordinates for conductor classes in the absolute nine-master basis;
• construct the physical relative integration chain;
• prove integral lattice normalization;
• identify the source quartic $\mathcal Q$ with this extension class.

In particular,

$$H^3(S_E)=0 \not\Rightarrow \text{the marked relative extension splits as a variation}.$$

Next hostile falsifier

Compute the Gauss--Manin connection of the canonical rank-twelve marked residue extension in a cover adapted simultaneously to $D_\infty$, $W_1$, and $W_2$. Determine its off-diagonal extension block without choosing a section after seeing the answer.

Then test whether every non-elliptic pole of that block is generated by the already frozen energy, conductor, soft, and $\mathcal Q$ supports.

The finite falsifier is:

$$\boxed{ \text{the extension connection has a residual pole whose divisor cannot be derived from the frozen compactification and marked walls.} }$$

Only such a residual divisor can reopen the new-carrier question.

Outcome contract

{
  "claim": "A nonzero H3(S_E) obstruction may prevent the frozen two-wall conductor lattice from lifting to marked relative cohomology.",
  "status": "falsified_on_generic_smooth_de_Rham_locus",
  "compact_surface": "generic degree-two del Pezzo",
  "boundary": "connected smooth anticanonical elliptic curve",
  "H3_compact": 0,
  "top_boundary_gysin_rank": 1,
  "H3_affine": 0,
  "H2_affine_rank": 9,
  "wall_union": "principal connected divisor div(q_g1*q_g2)",
  "wall_H1_rank": 3,
  "marked_H2_rank": 12,
  "canonical_sequence": "0 -> H2(S_E) -> H2(S_E minus W) -> H1(W)(-1) -> 0",
  "canonical_splitting": false,
  "new_carrier_datum": false,
  "remaining_problem": "Gauss-Manin extension class, discriminant extension, and physical relative-chain realization"
}

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