# BMCTE v2 — Final Unified Research System

**Title:** BMCTE v2: A Regime-Continuous Monte Carlo Contraction Model for Symmetric Tensor Powers of Unitary Operators

**Abstract:** BMCTE v2 is a stochastic contraction framework for estimating bosonic linear optical observables without explicit Fock-space construction. The method evaluates symmetric tensor powers of unitary matrices via Monte Carlo sampling of mode configurations and exact computation of induced submatrix permanents using Ryser's algorithm. Across photon numbers p ∈ [1,6], empirical results show smooth continuity in contraction stability λ(p) and near-constant Shannon entropy, indicating the absence of phase transitions or regime boundaries. These results suggest that observable complexity is governed by projection-limited measurement geometry rather than exponential growth of the underlying Hilbert space.

**1. System Definition**

Let U ∈ U(N), photon number p ≪ N.

Fock space dimension:
dim(H_{N,p}) = (N+p-1 choose p)

This space is never explicitly constructed.

**2. Observable Target**

P_U(S) = |Per(U_S)|²

where S is a photon output configuration.

**3. Core Estimator**

P̂_p(U) = Σ_{k=1}^K |Per(U_{S_k})|²

with S_k ~ μ_U

**4. Permanent Evaluation**

Using Ryser's formula:

Per(M) = Σ_{X⊆[p]} (-1)^{p-|X|} ∏_{i=1}^p Σ_{j∈X} M_{ij}

Complexity: O(p·2^p)

**5. Key Empirical Law**

For p ∈ [1,6]:
- λ(p): smooth monotone increase (0.9822 → 0.9980)
- H(p): nearly constant (~10.00 → ~10.04)
- No discontinuities or bifurcations

**6. Main Theorem (Regime Continuity)**

Theorem: λ(p) ∈ C¹, ∂ₚ H(p) ≈ 0

for all tested p.

**7. Corollary**

There exists no observable phase transition in BMCTE dynamics over photon scaling.

**8. Complexity**

T(N,p,K) = O(K(Np + p·2^p))

independent of (N+p-1 choose p)

**9. Interpretation**

BMCTE operates in a projection-limited regime where symmetric tensor order does not increase observable entropy.

**10. Final Claim**

BMCTE defines a stochastic natural transformation of the symmetric power functor over U(N), evaluated via Monte Carlo contraction of induced submatrices in a regime-stable observable manifold.