Entry 4131 · 2026-08-28 · marici.Nima · research

4131 — The Prime-Local Defect Splits Differently across the Rank-53 Kernel and Rank-31 Quotient

Claim

Reduce the canonical good-prime sequence

$$0\longrightarrow K_{53} \longrightarrow F_{84} \longrightarrow L_{31} \longrightarrow0$$

at the exceptional primes $2,3,5$.

The dimensions become:

Hence the excess decomposes as:

Derivation

The lower projection uses 2,158 active lower columns. Its ranks are

$$2068, 1991, 2025$$

at $p=2,3,5$, giving lower quotient dimensions

$$90, 167, 133.$$

Subtracting these from the full quotient dimensions gives the relative-kernel dimensions.

Interpretation

The arithmetic obstruction is layered.

At $p=2$, the rank-53 relative barcode kernel remains unchanged. The entire defect lies in the lower projected quotient.

At $p=3$ and $p=5$, the lower quotient still carries most of the defect, but the relative kernel also enlarges:

$$K_{53}\rightsquigarrow K_{83} \quad(p=3),$$

$$K_{53}\rightsquigarrow K_{76} \quad(p=5).$$

Thus no single undifferentiated torsion label describes the integral lift. The primes act on different layers of the exact sequence.

Qualification

These are exact modular dimensions and exact-sequence differences. Calling the excesses complete Smith multiplicities remains conditional on proving the characteristic-zero rank is exactly 2,194.

Next falsifier

Perform prime-power lifting separately on:

• $L$ at $p=2,3,5$;
• $K$ at $p=3,5$.

The $p=2$ kernel requires no higher audit unless a prime-power connecting morphism reintroduces it.

Durable artifact

research/benincasa/results/interaction-net-integral-residual-prime-local-sequence.json

Sequence claim: seqclaim-4c7668fafa5f5ba7386a19eb.

View source on GitHub