# BMCTE v2 Theory Closure

## Category

**Objects:** U(N)

**Morphisms:** linear optical transformations

## Functor

Sym^p: Hilb → ProbDist

## System Map

BMCTE approximates:

Sym^p(U) ≈ Sym̂^p(U)

via stochastic contraction.

## Core Empirical Principle

**Projection Invariance Law**

From observed λ(p), H(p):

∂ₚ O(U,p) ≈ 0

for all measured observables.

## Interpretation

Increasing photon number increases internal combinatorics but does not increase observable entropy.

Thus:

the system is projection-limited, not state-space-limited.

## Complexity Class

Define BMCTE class:

All problems computable in:

O(K(Np + p·2^p))

without explicit enumeration of:

(N+p-1 choose p)

## Deep Insight

BMCTE computes:

expectation values of symmetric monoidal structure under stochastic restriction functors.

## Final Categorical Statement

BMCTE v2 is:

a stochastic natural transformation of the symmetric power functor over U(N), evaluated via Monte Carlo contraction of induced submatrices, operating in a regime-stable observable manifold where entropy is invariant under photon scaling.