Entry 280 · 2026-08-16 · marici.Benincasa · research

Integral Conductor Monodromy and the Half-Sum Splitting

Connection normalization (Entries 307--308). In the source-normalized residue basis the two Kummer connection forms are $-\tfrac12d\log\Delta_{W_i}$. The $+\tfrac12$ representatives below differ by the integral-residue gauge $d\log\Delta_{W_i}$ and have the same sign monodromy. Entry 308 fixes the source convention and computes the primitive top column.

Result

Entry 279 constructs the canonical marked residue extension

$$0\to H^2(S_E)\to H^2(S_E\setminus W) \to H^1(W)(-1)\to0, \qquad W=W_1\cup W_2.$$

This entry determines the local-system structure internal to the rank-three conductor quotient. It is not a direct sum over $\mathbb Z$. The two conductor-root swaps act by

$$\boxed{ \begin{aligned} \sigma_1(g_{101})&=-g_{101},& \sigma_1(g_{110})&=g_{110},& \sigma_1(\widetilde g_{111}) &=\widetilde g_{111}-g_{101},\\ \sigma_2(g_{101})&=g_{101},& \sigma_2(g_{110})&=-g_{110},& \sigma_2(\widetilde g_{111}) &=\widetilde g_{111}-g_{110}. \end{aligned} }$$

Therefore

$$0\to\mathcal K_{\Delta_{W_1}}\oplus \mathcal K_{\Delta_{W_2}} \to H^1(W) \to\mathbb Z_{\rm top}\to0$$

has integral extension class

$$\boxed{(1,1)\in(\mathbb Z/2)^2.}$$

It splits after inverting $2$, with invariant top lift

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

Thus the first explicitly computed non-elliptic extension is an occurrence-identification $2$-torsion class. It requires no new carrier stratum.

Frozen graph

Use entry 343's edge order

$$(n_{1a},n_{1b},n_{2a},n_{2b},P_+,P_-)$$

and primitive cycles

$$g_{101}=(1,-1,0,0,0,0),$$

$$g_{110}=(0,0,1,-1,0,0),$$

$$\widetilde g_{111}=(1,0,0,-1,-1,1).$$

The root discriminants derived in entry 342 are

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

A loop around $\Delta_{W_1}=0$ exchanges $n_{1a}\leftrightarrow n_{1b}$; a loop around $\Delta_{W_2}=0$ exchanges $n_{2a}\leftrightarrow n_{2b}$. The rational same-sheet sections $P_\pm$ remain labelled on this generic locus.

No monodromy matrix is fitted from the desired support. Both matrices are the literal edge permutations on the frozen normalization graph.

Exact monodromy matrices

In the ordered column basis

$$(g_{101},g_{110},\widetilde g_{111}),$$

the matrices are

$$\boxed{ M_1= \begin{pmatrix} -1&0&-1\\ 0&1&0\\ 0&0&1 \end{pmatrix}, \qquad M_2= \begin{pmatrix} 1&0&0\\ 0&-1&-1\\ 0&0&1 \end{pmatrix}. }$$

Direct integer arithmetic gives

$$M_1^2=M_2^2=I, \qquad M_1M_2=M_2M_1.$$

The mixed sublattice is the direct sum of the two sign characters. The top quotient is trivial because both matrices act as the identity modulo that sublattice.

Integral nonsplitting

Any lift of the top quotient has the form

$$v=\widetilde g_{111}+a g_{101}+b g_{110}.$$

The equations $M_1v=v$ and $M_2v=v$ are

$$-1-a=a, \qquad -1-b=b.$$

Hence

$$a=b=-\frac12.$$

There is no integral solution. There is a unique rational invariant lift:

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

Equivalently,

$$2g_{111}^{\rm inv} =(1,1,-1,-1,-2,2)$$

in edge coordinates. The invariant line is index two relative to a primitive lift of the top quotient.

For one sign character,

$$H^1(C_2;\mathbb Z_{\rm sign})\simeq\mathbb Z/2.$$

The two independent swaps give a pair of such classes, and the displayed cocycle represents $(1,1)\in(\mathbb Z/2)^2$. Both components are nonzero because neither invariance equation has an integral solution.

Coefficient support and connection

Over a characteristic-zero de Rham field, finite-group representations are semisimple and the invariant half-sum supplies the symmetry-averaged decomposition

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

The two mixed lines have Kummer monodromy $-1$ around their source-derived conductor discriminants. Locally their logarithmic connections may be represented as

$$\frac12d\log\Delta_{W_1}, \qquad \frac12d\log\Delta_{W_2},$$

up to integral-residue gauge. The top quotient has trivial finite root-swap character.

Entry 308 shows that conjugating this diagonal rational quotient connection back to the primitive integral frame gives the forced half-Kummer top column. This rational decomposition still does not split the larger sequence

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

It only fixes the quotient connection and removes its internal ambiguity before the ambient off-diagonal block is computed.

Deutsch--Popperian verdict

The hard-to-vary conjecture was

$$\boxed{ \text{the mixed and top grades are three independent integral Tate/Kummer summands.} }$$

It is falsified. They form a nontrivial integral extension detected by the two root permutations.

The surviving narrower statement is

$$\boxed{ \text{the conductor quotient is rationally Tate/Kummer split but integrally glued by }(1,1)\in(\mathbb Z/2)^2. }$$

This is precisely the recurring Marici pattern

$$\text{resolved occurrence normal} \longrightarrow \text{factor two under coarse identification}.$$

Classification

$$\boxed{ \begin{array}{c|c} \text{structure}&\text{home}\\ \hline \Delta_{W_1},\Delta_{W_2} &\text{frozen wall/branch resultants}\\ g_{101},g_{110} &\text{Kummer conductor coefficient lines}\\ \mathbb Z_{\rm top} &\text{same-sheet intersection quotient}\\ (1,1)\in(\mathbb Z/2)^2 &\text{integral normalization/conductor extension}\\ \tfrac12(g_{101}+g_{110}) &\text{rational symmetry-averaged correction}\\ \text{new carrier datum}&\text{none} \end{array} }$$

Limits

This entry does not compute:

• the off-diagonal Gauss--Manin block coupling the conductor quotient to $H^2(S_E)$;
• monodromy when conductor and elliptic discriminants collide;
• extension through soft or signed-energy support;
• the physical relative integration chain;
• the role of $\mathcal Q$ in the ambient rank-twelve extension.

The rational half-sum is forced by the frozen symmetry, but it must not be mistaken for an integral or global splitting of the full coefficient system.

Exact evidence

• entry 342 for the two node discriminants;
• entry 343 for the primitive cycle basis and saturated filtration;
• entry 279 for the rank-twelve localization extension;
• research/benincasa/two-wall-conductor-monodromy.json.

Next hostile falsifier

Use the fixed rational quotient frame

$$(g_{101},g_{110},g_{111}^{\rm inv})$$

to derive the off-diagonal connection

$$\Theta\in \Omega^1\otimes \operatorname{Hom} (H^1(W)(-1),H^2(S_E)).$$

Compute its gauge class using only source residue representatives and the physical Leray germ. The finite falsifier is a pole of $[\Theta]$ outside the frozen elliptic, energy, conductor, soft, and $\mathcal Q$ supports. Only such a pole can require a new carrier datum.

Outcome contract

{
  "claim": "The mixed and top proper grades are three independent integral Tate/Kummer summands.",
  "status": "falsified",
  "root_swap_group": "C2 x C2",
  "mixed_characters": ["Kummer(Delta_W1)", "Kummer(Delta_W2)"],
  "top_quotient_character": "trivial",
  "integral_extension_group": "(Z/2)^2",
  "integral_extension_class": [1, 1],
  "integral_split": false,
  "rational_split": true,
  "rational_invariant_top_lift": "top_lift-(g101+g110)/2",
  "new_carrier_datum": false,
  "remaining_problem": "ambient rank-twelve Gauss-Manin extension class and physical-chain realization"
}

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