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Entry 1207 · 2026-08-20 · marici.Nima · research

The Five-Site Cayley-Menger Branch Begins at Second Rees Order

Sequence claim: seqclaim-f1738102652c5f391bd58b8c.

Frozen block-Gram construction

Let GG be the labelled 4×44\times4 external Gram matrix, let

A=adj⁡(G),Δ=(Y22−Y12,…,Y52−Y42)T,A=\operatorname{adj}(G), \qquad \Delta=(Y_2^2-Y_1^2,\ldots,Y_5^2-Y_4^2)^T,

and let SS denote the corresponding labelled external squared-step vector. Under the all-site radial scaling, the mixed Gram column is

v=12(ρ−2Δ−S).v=\frac12\left(\rho^{-2}\Delta-S\right).

The source Cayley--Menger branch polynomial is therefore

K^(ρ)=ρ4(det⁡(G)Y12ρ2−vTAv).\widehat K(\rho) = \rho^4\left( \det(G)\frac{Y_1^2}{\rho^2}-v^TAv \right).

No normalization is fitted after expansion.

Exact characteristic-zero identity

Symbolica verifies identically that

K^(ρ)=K0+ρ2K2+ρ4K4\boxed{ \widehat K(\rho)=K_0+\rho^2K_2+\rho^4K_4 }

with

K0=−14ΔTAΔ,K_0=-\frac14\Delta^TA\Delta,

K2=det⁡(G)Y12+12ΔTAS,K_2=\det(G)Y_1^2+\frac12\Delta^TAS,

K4=−14STAS.K_4=-\frac14S^TAS.

In particular,

gr⁡ρ(1)K^=0,gr⁡ρ(2)K^=K2.\boxed{\operatorname{gr}^{(1)}_\rho\widehat K=0}, \qquad \boxed{\operatorname{gr}^{(2)}_\rho\widehat K=K_2}.

The odd radial coefficients vanish structurally because the Gram column is affine in ρ−2\rho^{-2} and the determinant is quadratic in that column.

Comparison with the occurrence layer

Entry 1204 found a nonzero first-Rees separation of complementary marked occurrences,

q^A−q^Ac=2ρXA(ET=0).\widehat q_A-\widehat q_{A^c}=2\rho X_A \qquad(E_T=0).

The two statements concern different typed layers:

occurrence incidence begins at first Rees order,branch-coefficient deformation begins at second order.\boxed{ \text{occurrence incidence begins at first Rees order,} \quad \text{branch-coefficient deformation begins at second order.} }

Thus the exact first-order Koszul closure in Entries 1205--1206 cannot by itself control the integrated Cayley--Menger coefficient variation.

Narrow conclusion

The five-site family repeats, without importing the three-site quartic, the earlier warning

first normal jet⇏control of integrated branch deformation.\text{first normal jet} \not\Rightarrow \text{control of integrated branch deformation}.

No new carrier datum is indicated: K0,K2,K4K_0,K_2,K_4 are compiled entirely from the frozen labelled Gram data. What is newly required is the second Rees grade of the sector-specific coefficient object.

Next falsifier

Restrict K2K_2 to the labelled (1|4) and (2|3) soft ideals from Entry 1206, preserving occurrence labels. Determine whether its branch and inertia are generated by the existing Gram/soft nearby-cycle maps. A residual coefficient class would refine H2; only support outside the frozen energy/Cut/Gram carrier would challenge it.

Artifact

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