Entry 281 · 2026-08-16 · marici.Benincasa · research

Conductor Spectral Separation and the Reduced Extension Frontier

Result

Entries 279--280 reduce the marked $q_{\mathcal G_{12}}$-residue system to

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

and identify the two mixed quotient lines as Kummer characters on

$$\Delta_{W_1}=-4xD_1, \qquad \Delta_{W_2}=-4yD_2.$$

At a generic point of either conductor discriminant, away from every ambient surface discriminant and all other frozen supports, the mixed line cannot carry a nontrivial rational extension by the absolute rank-nine system:

$$\boxed{ \operatorname{Ext}_{\rm loc,dR}^1 (\mathcal K_{\Delta_{W_i}},H^2(S_E))=0. }$$

The reason is spectral separation. The absolute system has local monodromy $+1$, while the conductor line has monodromy $-1$. The forced projector uses the same half-sum mechanism exposed integrally in entry 280.

Thus generic conductor support is completely classified as Tate/Kummer coefficient support. The unresolved ambient extension is confined to loci where quotient and absolute eigenvalues can collide.

Frozen generic locus

Fix a smooth point of $\Delta_{W_i}=0$ satisfying:

• the del Pezzo surface $\overline S_E$ and its affine complement $S_E$ remain smooth;
• the elliptic boundary remains smooth;
• the other conductor discriminant is nonzero;
• soft, signed-energy, $\Lambda$, and $\mathcal Q$ factors are nonzero.

The equation $\Delta_{W_i}=0$ expresses tangency of the already frozen wall with the Cayley--Menger branch divisor. It does not make the ambient surface singular at such a generic point. Therefore the absolute Gauss--Manin system extends across this divisor and has monodromy

$$T_{\rm abs}=I_9.$$

Entry 280 gives the mixed conductor monodromy

$$T_{\rm mix}=-1.$$

Exact block calculation

Any local monodromy of an extension of the mixed line by the absolute system has, after choosing an arbitrary vector-space lift, the form

$$T= \begin{pmatrix} I_9&c\\ 0&-1 \end{pmatrix}$$

for some column $c$. Write a vector as $(a,q)$, so

$$T(a,q)=(a+cq,-q).$$

Define

$$a'=a+\frac12cq.$$

Then

$$T(a',q)=(a',-q),$$

and hence

$$\boxed{ T\sim_{\mathbb Q} \begin{pmatrix} I_9&0\\ 0&-1 \end{pmatrix}. }$$

This is not an optional fitted gauge. It is the unique spectral projector because $1-(-1)=2$ is invertible in the de Rham field.

Over an integral lattice the shift exists precisely when $c$ is even. Thus an integral class $c\bmod2$ may remain, but it is occurrence $2$-torsion and not a de Rham off-diagonal singularity.

Consequence for the rank-twelve connection

In the rational quotient frame of entry 280,

$$H^1(W)_{\mathbb Q} \simeq \mathcal K_{\Delta_{W_1}} \oplus \mathcal K_{\Delta_{W_2}} \oplus \mathbb Q_{\rm top},$$

the first two columns of the ambient extension class are locally zero in their gauge classes at generic $\Delta_{W_i}=0$.

Therefore any nonzero rational class

$$[\Theta]\in H^1_{\rm dR}\! \left( \operatorname{Hom}(H^1(W)(-1),H^2(S_E)) \right)$$

must be detected at an eigenvalue-collision locus:

1. $\Lambda=0$, where the two same-sheet points defining the trivial-character top quotient collide simultaneously with an ambient elliptic degeneration;
2. another intersection with the elliptic/ambient discriminant;
3. soft or signed-energy support;
4. simultaneous conductor support;
5. $\mathcal Q=0$, if the physical moving-chain extension—not the absolute surface—carries that singularity.

The list is derived from frozen geometry. It is not a fitted pole ansatz.

Relation to the exceptional-wall calculations

Entries 324--328 found a nonzero weight-minus-one local coefficient on an exceptional wall, then proved it exact on the punctured wall and zero in the tangential endpoint cone. That representative therefore cannot define one of the rank-three conductor extension columns.

This prevents an invalid shortcut: the ambient extension cannot be read off from the ancestry of that exact coefficient.

Deutsch--Popperian verdict

The hard-to-vary conjecture was

$$\boxed{ \text{a generic conductor discriminant can support a new rational off-diagonal extension between the mixed line and }H^2(S_E). }$$

It is falsified by the $(+1,-1)$ spectral decomposition.

The surviving frontier is

$$\boxed{ \text{only eigenvalue-collision loci can carry the unresolved rational ambient extension.} }$$

Classification

$$\boxed{ \begin{array}{c|c} \text{structure}&\text{home}\\ \hline \Delta_{W_i}=0\text{ generically} &\text{Kummer coefficient support}\\ \text{mixed rational ambient extension} &0\text{ locally by spectral separation}\\ \text{mixed integral residual} &\text{possible occurrence class }c_i\bmod2\\ \Lambda=0 &\text{top-quotient eigenvalue-collision frontier}\\ \text{new carrier datum} &\text{none} \end{array} }$$

Limits

This result is local at generic points of one conductor discriminant. It does not:

• compute the integral vectors $c_i\bmod2$ coupling to the absolute lattice;
• compute the top-column extension at $\Lambda=0$;
• treat intersections of support divisors;
• construct the physical relative chain;
• decide whether $\mathcal Q$ belongs to the moving-chain extension.

Exact evidence

• entries 280, 342 for the conductor discriminants and characters;
• entry 279 for the ambient rank-twelve extension;
• entries 327--328 for exactness of the exceptional-wall candidate;
• research/benincasa/conductor-spectral-separation.json.

Next hostile falsifier

Restrict to a generic transverse slice through

$$\Lambda=E(-x+y+z)(x-y+z)=0$$

The signed-energy identities give

$$\Lambda=-\ell_1\ell_2\ell_4=-A\ell_4.$$

Thus every component of $\Lambda=0$ is already an elliptic discriminant component: $\ell_1\ell_2=0$ gives $A=0$, while $\ell_4=0$ gives $B=\ell_3\ell_4=0$. Choose one component transversely, keep the other elliptic, conductor, soft, and $\mathcal Q$ factors nonzero, and compute the joint local monodromy on the invariant top lift

$$g_{111}^{\rm inv} =\widetilde g_{111}-\frac12(g_{101}+g_{110})$$

and its extension into $H^2(S_E)$.

The finite falsifier is a nonzero unipotent class not generated by the frozen collision of $P_+$ and $P_-$. Only a class requiring an additional incidence datum can challenge the shared-carrier hypothesis.

Outcome contract

{
  "claim": "A generic conductor discriminant can support a new rational off-diagonal extension between a mixed Kummer line and H2(S_E).",
  "status": "falsified_by_spectral_separation",
  "absolute_local_monodromy": 1,
  "mixed_local_monodromy": -1,
  "forced_rational_shift": "c/2",
  "generic_rational_extension_class": 0,
  "possible_integral_residual": "c mod 2",
  "remaining_frontier": "top trivial-character column and support intersections",
  "new_carrier_datum": false
}

View source on GitHub