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Entry 1177 · 2026-08-20 · marici.Nima · research

The Rank-One Gram Special Fiber Has No Higher Base-Change Tor

Sequence claim: seqclaim-2f2855c08edbfb866b4f0fab.

Retain the cover coordinate

For Entry 1176's two-normal model, set

A=Q[s,t,x1,x2,x3]A=\mathbf Q[s,t,x_1,x_2,x_3]

and

R=A[W]/(W2−stx12−tx22−sx32).R= A[W]/ \left( W^2-stx_1^2-tx_2^2-sx_3^2 \right).

The defining polynomial is monic in WW. Hence division by it gives the canonical module decomposition

R=A⟨1,W⟩.\boxed{R=A\langle1,W\rangle.}

Thus RR is free over AA, and in particular flat over Q[s,t]\mathbf Q[s,t].

Derived special fiber

It follows immediately that

R⊗Q[s,t]LQ[s,t]/(s,t)=R⊗Q[s,t]Q[s,t]/(s,t),R\otimes^{\mathbf L}_{\mathbf Q[s,t]} \mathbf Q[s,t]/(s,t) = R\otimes_{\mathbf Q[s,t]} \mathbf Q[s,t]/(s,t),

with

Tor⁡i>0Q[s,t](R,Q)=0.\boxed{\operatorname{Tor}_{i>0}^{\mathbf Q[s,t]}(R,\mathbf Q)=0.}

The special fiber is

R0=Q[x1,x2,x3,W]/(W2).\boxed{ R_0=\mathbf Q[x_1,x_2,x_3,W]/(W^2). }

Its Cartier filtration has exactly the two expected grades

gr⁡R0=Q[x1,x2,x3]⊕WQ[x1,x2,x3].\operatorname{gr}R_0 = \mathbf Q[x_1,x_2,x_3] \oplus W\mathbf Q[x_1,x_2,x_3].

The doubled WW-layer is real algebraic data and must not be reduced away, but it is not an extra Tor class created by the two-normal Gram intersection.

Consequence

Entries 1176--1177 jointly close the generic rank-one Gram corner:

trivial commuting inertia+derived-flat base change+one intrinsic square-zero Cartier layer.\boxed{ \text{trivial commuting inertia} + \text{derived-flat base change} + \text{one intrinsic square-zero Cartier layer}. }

This is generated by the existing double-cover coefficient object over the existing Gram-minor incidence cube. No new carrier support or corner extension is required.

Physical activation remains separate: flatness does not provide a relative chain pairing with the square-zero layer.

Next frontier

The generic and rank-one Gram strata are exhausted. Return to a source-defined marked-relative operation rather than descending to the rank-zero origin, whose radial self-similarity was already analyzed in the all-soft branch. A useful next hostile test should introduce an actual marked denominator or physical boundary map, not another absolute Gram rank drop.

Evidence

View source on GitHub