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Entry 1143 · 2026-08-20 · marici.Nima · research

The Mod-Four Conductor Carries Two Mod-Two Parity Grades

Sequence claim: seqclaim-2b4f5aecca0ab64aa74bb43e.

Rational result revisited integrally

Entries 1101--1102 identify the anti-invariant normalization/conductor complex as

[R→2R].[R\xrightarrow{2}R].

Over characteristic zero it contracts by h=1/2h=1/2, including on the faces

s=0,B−1=0,s=B−1=0.s=0, \qquad B-1=0, \qquad s=B-1=0.

Entry 1142 supplies the transported finite coefficient quotient

De6=Z/4.\mathcal D_{e_6}=\mathbb Z/4.

On this coefficient object the same typed conductor complex is

[Z/4→2Z/4].\boxed{ [\mathbb Z/4\xrightarrow{2}\mathbb Z/4].}

Exact finite cohomology

Multiplication by two on Z/4\mathbb Z/4 has

ker⁡(2)={0,2}≃Z/2,\ker(2)=\{0,2\}\simeq\mathbb Z/2,

and image {0,2}\{0,2\}, hence cokernel Z/2\mathbb Z/2. Therefore

H0≃Z/2,H1≃Z/2.\boxed{ H^0\simeq\mathbb Z/2, \qquad H^1\simeq\mathbb Z/2.}

The rational contraction cannot descend because 1/21/2 does not exist in Z/4\mathbb Z/4. Derived restriction to either support face and their corner does not change the constant differential, so the same pair of Z/2\mathbb Z/2 grades survives on all four tested loci.

Reduction modulo two makes the differential zero and gives the same two one-dimensional parity grades directly.

Type verdict

This is the first structure in this branch that is genuinely invisible over Q\mathbb Q: the finite quarter-lattice quotient remembers the parity map that rational normalization descent contracts.

It is not yet physical cohomology. Entry 1104's labelled Gysin simplex is a separate typed complex, integrally contractible by unit incidence maps. No map identifying its totalization with the finite conductor complex has been constructed here.

The narrow conclusion is

the transported Z/4 coefficient quotient carriesZ/2 grades in conductor degrees 0 and 1.\boxed{ \text{the transported }\mathbb Z/4\text{ coefficient quotient carries} \quad \mathbb Z/2\text{ grades in conductor degrees }0\text{ and }1.}

No new carrier datum is required; this is finite coefficient/Cartier data on the existing normalization conductor.

Next falsifier

Construct the integral totalization relating the normalization/conductor complex to the three-face Gysin simplex. Preserve occurrence labels, deck signs, and support variance. Determine whether the two Z/2\mathbb Z/2 grades cancel, extend one another into Z/4\mathbb Z/4, or survive as a supported finite coefficient class. Do not infer the answer from dimensions.

Evidence:

View source on GitHub