A Global-Bit Obstruction for Shift-Dependent Cumulants

Abstract

A single radial tensor-power parameter cannot encode every shift-dependent cumulant.

Theorem 1.1 (One radial tensor square misses an explicit shift pattern).

$$\begin{array}{c}{R}_{ij}(c,\delta )=c{\delta }_i{\delta }_j,\\ K_{01}=K_{12}=1,K_{02}=0,\\ (R_{01}\ne 0\wedge R_{12}\ne 0)\Rightarrow R_{02}\ne 0,\\ \neg \exists c,\delta ,R(c,\delta )=K.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/GlobalBitCumulantObstruction.one_global_bit_cannot_encode_all_pair_cumulants (✓ std3). ∎

Source. Repository-derived.

Commentary.

A one-bit radial model produces pair cumulants of the form scale times delta_i delta_j. Such a rank-one family has a load-bearing nonzero closure law: nonzero correlations on pairs 01 and 12 force the correlation on pair 02 to be nonzero.

The explicit symmetric three-shift target instead has adjacent values K_01=K_12=1 and distance-two value K_02=0. It therefore has no representation by one displacement vector and one global scale.

A separate public control theorem represents the constant-one pair family, showing that the obstruction is caused by shift structure rather than an empty radial model class.

References

• Truth anchor: D5/S3/Analytic/GlobalBitCumulantObstruction.one_global_bit_cannot_encode_all_pair_cumulants