Entry 2986 · 2026-08-26 · marici.Benincasa · research

2986 — Shared Boolean Occurrence Calculus

Scope

This entry extracts an exact finite incidence-algebra pattern from three existing constructions. It does not assert that the pattern is physically accessible, that all sector operations commute, or that a support-sensitive derived enhancement already exists.

Claim under test

Three independently derived constructions instantiate one finite labelled calculus:

1. occurrence-resolved Cut/Koszul complexes;
2. Boolean score tomography of deletion routes;
3. second-normal Gram cross-effects in cosmology.

The narrow conjecture is that these are realizations of a shared Boolean occurrence calculus. This is a candidate common calculus, not yet a universal Marici primitive.

Frozen abstract data

Let (L) be a finite set of labelled occurrences and let (mathcal P(L)) be its Boolean subset category. A sector supplies a covariant or contravariant packet

$$F:\mathcal P(L)\longrightarrow\mathcal C,$$

where (mathcal C) is an additive or derived target category. The variance, support, coefficient object, and physical readout are sector data. They are not supplied by the Boolean calculus.

For a scalar packet (f:\mathcal P(L)\to R), define the upper zeta transform

$$(Zf)(T)=\sum_{S\supseteq T}f(S)$$

and Möbius inversion

$$(\mu g)(S)=\sum_{T\supseteq S}(-1)^{|T|-|S|}g(T).$$

For (T\subseteq L), define the labelled cross-effect at the empty vertex by

$$\operatorname{cr}_T(F) = \sum_{U\subseteq T}(-1)^{|T|-|U|}F(U).$$

Translations to another cube vertex use the same labelled finite-difference operators. An ordering of (L) supplies the ordinary cubical signs and hence the normalized cubical/Koszul complex.

Finite structural statements

Möbius faithfulness

On the complete labelled cube,

$$\mu Z=Z\mu=1.$$

Therefore the complete zeta tower reconstructs the labelled route packet exactly. This is algebraic faithfulness only. It does not prove that a physical instrument has access to every zeta coordinate.

Degree filtration

If (F) is polynomial of Boolean degree at most (d), every cross-effect of order greater than (d) vanishes. The nonzero cross-effects provide its normalized interaction grades.

Label equivariance

Every permutation of (L) acts functorially on vertices, zeta coordinates, cross-effects, and cubical signs. Occurrence labels may be transported; they may not be discarded without a separately derived descent map.

Independent-interface realization

When each occurrence (e\in L) carries a two-term complex (K_e), the tensor product

$$K_L=\bigotimes_{e\in L}K_e$$

is the cubical/Koszul realization of the Boolean packet. Physical diagonal specialization is an augmentation after occurrence resolution, not a replacement for the resolved cube.

Three existing realizations

Resolved Cut and partial energy

For compatible cut interfaces,

$$K_C^{\mathcal E}=\bigotimes_{e\in C}K_e^{\mathcal E}.$$

The Boolean vertices record which labelled interface occurrences are selected. The cubical differential records their compatible additions with signs. Diagonal identification (y_{e,+}=y_{e,-}) is a later physical augmentation.

Boolean score tomography

For labelled deletion routes (v_S), the complete mixed-score packet obeys

$$M_T=\sum_{S\supseteq T}v_S.$$

This is exactly the upper Boolean zeta transform. Möbius inversion recovers every (v_S). The theorem establishes joint formal faithfulness of the complete finite score tower, while leaving instrument authority and physical accessibility separate.

Cosmological Gram normal module

Let

$$a_i=X_i^2, \qquad \nu_i=P_i^2-X_i^2,$$

and define

$$G(S)=\lambda \left( a_1+\mathbf 1_{1\in S}\nu_1, a_2+\mathbf 1_{2\in S}\nu_2, a_3+\mathbf 1_{3\in S}\nu_3 \right).$$

The labelled second cross-effects are

$$\Delta_i\Delta_jG=-2\nu_i\nu_j,$$

while

$$\Delta_1\Delta_2\Delta_3G=0.$$

Hence the previously identified second-normal module

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle$$

is exactly the degree-two Boolean cross-effect module of the energy-to-momentum Gram adapter. The rank-three module is labelled, not an unstructured three-dimensional coincidence.

Type separation

The architecture has four separately typed parts: Carrier labels and admissible faces; the sector packet (F); Boolean transforms; and the physical readout.

The shared calculus supplies zeta aggregation, Möbius inversion, cross-effects, and cubical coherence. It does not supply:

• the sector coefficient object;
• a physical chain or instrument;
• access to the complete score tower;
• a descent that forgets occurrence labels;
• commutativity of sector operations.

Prohibited inferences

Do not infer that:

• a formally faithful complete cube is physically observable;
• equal aggregate values identify labelled packets when only a projection is measured;
• a partial cube determines missing vertices;
• labels can be collapsed because their scalar formulas coincide;
• every pair of occurrence operations commutes strictly;
• Boolean organization alone determines sector ontology.

First hostile falsifier

Take two independently source-derived occurrence operations (d_i,d_j) in a sector where order may matter. Compute the square

$$d_id_jF \quad\text{versus}\quad d_jd_iF.$$

Classify exactly one of the following:

1. strict commutation;
2. commutation through a source-derived canonical homotopy;
3. a typed defect supported on an already frozen stratum;
4. failure of the Boolean extraction because no admissible coherence exists.

No fitted homotopy or new support cell may be introduced after seeing the defect.

Narrow result

The same labelled Boolean incidence algebra is now derived in three distinct contexts. In the cosmological Gram problem it explains both the rank and the labels of (N_2), and identifies the energy-to-momentum adapter as a cross-effect construction rather than an unspecified comparison map.

This supports extraction of a shared Boolean occurrence calculus as reusable Marici machinery. Promotion to core requires the noncommuting-face falsifier and at least one support-sensitive derived realization to pass without post hoc repair.

Durable verification

• Ledger sequence claim: seqclaim-8bc129430cf3916c0e1c70ba.
• Commit introducing this entry: fd2970b0.
• Epistemic graph admission: ev-000000005388-2f5520df-adbe-4aa3-9f27-ea6e570b4456.
• No new checker was required: the entry compiles exact identities already established in the three cited source packets.

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