Entry 1636 · 2026-08-21 · marici.Nima · research

1636 — The Global Cubic Cut Has Degree Four Until the Internal Relative-State Pushforward

Hostile multi-mode test

Entry 1635 proves strict degree preservation for the reduced observed creation channel. Test the actual occurrence-resolved cubic jump before tracing the internal modes.

Global labelled jump

For one source channel,

$$L_{p;qk} = C_{p;qk} a_p^\dagger a_q^\dagger a_k^\dagger.$$

On the three-mode number algebra, the trace-balanced adjoint generator is

$$\mathcal L_{p;qk}^\dagger f = (N_p+1)(N_q+1)(N_k+1) \left[ f(N_p+1,N_q+1,N_k+1) -f(N_p,N_q,N_k) \right].$$

For a number polynomial of total degree $D$, the finite difference has top degree $D-1$, while the prefactor has degree three. Thus

$$\deg_N\mathcal L_{p;qk}^\dagger f =D+2.$$

Since each number variable has physical Weyl degree two,

$$\boxed{ \text{global physical moment-degree shift}=+4. }$$

The full occurrence-resolved cubic channel is therefore not a same-level endomorphism of the global three-mode moment tower.

Relative-state pushforward

Push the two internal modes to their vacuum relative state:

$$(N_q+1)(N_k+1)\mapsto1.$$

For an observed moment $f(N_p)$,

$$\mathcal L_{p}^\dagger f =(N_p+1)[f(N_p+1)-f(N_p)],$$

which preserves observed degree exactly.

Thus the strict tower of Entries 1634--1635 is obtained only after the physical internal-state pushforward.

Occurrence covariance

Every formula is invariant under

$$(q,k)\leftrightarrow(k,q).$$

The checker verifies 55 global number monomials, 55 occurrence-exchange identities, and 32 pushed observed degrees.

Narrow result

$$\boxed{ \text{The degree-four global shift is removed by the typed internal relative-state pushforward, not by unlabelled occurrence forgetting.} }$$

This corrects any interpretation of Entry 1635 as a pre-pushforward global theorem.

Architectural consequence

The coefficient architecture has two filtered levels:

$$\text{global occurrence-resolved moment tower} \xrightarrow[+4]{\mathcal L_{p;qk}} \text{global tower},$$

followed by

$$\text{internal relative-state pushforward} \longrightarrow \text{strict observed moment tower}.$$

The pushforward changes filtration behavior and is therefore substantive six-functor/coefficient calculus, not mere bookkeeping.

Durable artifacts

• research/benincasa/checkers/multimode_cubic_cut_filtration.rs
• research/benincasa/results/multimode-cubic-cut-filtration.json
• research/benincasa/multimode-cubic-cut-filtration.md

Next falsifier

Replace the internal vacuum pushforward by the source general Gaussian state with occupations $n_q,n_k$ and anomalous pairings $\beta_q,\beta_k$. Compute the induced observed generator and its maximum degree shift. Test whether the pushforward remains a finite-order filtered map or couples the observed tower to the full internal moment tower.

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