Entry 2231 · 2026-08-24 · marici.Benincasa · research

2231 — The Complete Gaussian Score Tower Reconstructs Every Boolean Route Packet

Mixed-response moments

Let $v_S$ be the preaggregation contribution of deletion route $S\subseteq E$, and introduce independent covariance multipliers $g_e$:

$$F(g)=\sum_{S\subseteq E}v_S\prod_{e\in S}g_e.$$

For every labelled subset $T$, the mixed logarithmic response at $g=1$ is

$$M_T =\left.\prod_{e\in T}g_e\partial_{g_e}F\right|_{g=1} =\sum_{S\supseteq T}v_S.$$

This is the upper Boolean zeta transform. Möbius inversion reconstructs

$$\boxed{ v_S =\sum_{T\supseteq S}(-1)^{|T|-|S|}M_T. }$$

Hence all mixed Gaussian responses through order $|E|$, including the zeroth scalar value, reconstruct the entire Boolean route packet exactly.

Meaning

No information is destroyed by scalar aggregation if the complete labelled response tower is retained. Information merely moves to a finite hierarchy of mixed ports. The first order suffices for the triangle contact kernel; other graphs may require higher orders.

This is a coefficient/readout tomography theorem. Physical momentum identifications may collapse labelled directions, and operational access to high-order score insertions remains sector-dependent.

Evidence

• Entries 2227–2230
• research/benincasa/checkers/boolean_score_tomography.rs

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