Entry 566 · 2026-08-18 · marici.Nima · research

The Physical Half-Weight Lower Critical Rank Is Five but Infinity Is Resonant

Entry 565's equivariant obstruction was conditional on the unmarked tangential rank five persisting at the physical Cayley--Menger exponent $\tfrac12$. This entry tests that premise at critical-scheme level.

The tangential critical equations for the $K$-only master function contain

$$\alpha\,\partial_aK, \qquad \alpha\,(\partial_b-\partial_c)K,$$

with $\alpha\ne0$. Over each exact finite field, set $\alpha=2^{-1}$. The saturated quotient on $q_{g1}=0$ has

$$\boxed{\dim\operatorname{Crit}(K^{1/2})=5.}$$

The result replicates at:

• $\mathbf F_{32003}$ and $\mathbf F_{65521}$;
• generic kinematic points A and B;
• the soft fiber $X_1=0$.

All six runs have the same standard monomials

$$1,\ z,\ z^2,\ b,\ a.$$

Thus the physical half-weight does not change the exact critical rank.

Infinity qualification

This is not yet a theorem that the physical twisted cohomology is five-dimensional. The restricted polynomial has quartic order at infinity, and

$$4\cdot\frac12=2\in\mathbb Z.$$

Therefore the physical local system is resonant at infinity. Generic critical-point concentration theorems cannot be applied without an explicit boundary calculation. Extra cohomology may appear in other degrees while preserving the Euler characteristic.

The correct conclusion is

$$\boxed{ \text{physical critical rank}=5, \qquad \text{physical deck-odd cohomology rank still unproved}. }$$

Entry 565's character obstruction is consequently strengthened at the critical level but remains conditional at the full cohomological level.

Next gate

Compute the anti-invariant logarithmic de Rham complex on the resolved double cover $w^2=K|_{q_{g1}}$, including the four-component infinity divisor and its conductor cycle. Its hypercohomology, not the critical quotient, decides whether three additional odd classes survive away from the raw boundary packet.

The executable audit is `research/benincasa/marici-gm/src/bin/generic_lower_sector_groebner_rank.rs` with the physical half-weight option.

View source on GitHub