docs/ARCHITECTURE_PAPER.md

BURT-IMMA: Multi-System Formal Verification Architecture

Abstract

This paper documents the formal verification architecture of BURT-IMMA (BiEncoder Unified Retrieval-Transformer with Instruction, Memory, and Mixture of Experts Agents). We present machine-checked proofs across five formal systems — Lean 4, Idris 2, Rust (runtime), Q# (quantum), and OpenQASM 3 — covering matrix-memory equilibrium propagation (MMEP), Restricted Boltzmann Machine (RBM) invariants, smooth activation function properties, and Schwarzschild black hole physics. Zero sorry terms across all Lean 4 modules.

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1. Architecture Overview

┌─────────────────────────────────────────────────────────────────┐
│                       BURT-IMMA Stack                           │
├──────────────┬──────────────┬──────────────┬────────────────────┤
│  Lean 4      │  Idris 2     │  Rust        │  Quantum           │
│  (proofs)    │  (dep. types)│  (runtime)   │  (Q# / QASM 3)    │
├──────────────┴──────────────┴──────────────┴────────────────────┤
│  Mathematical foundation: MMEP + RBM + BlackHole + Activation   │
└─────────────────────────────────────────────────────────────────┘

The architecture separates four concerns:

1. Proof layer (Lean 4): machine-checked invariants, zero sorry
2. Dependent-type layer (Idris 2): constructive witnesses, structural types
3. Runtime layer (Rust): zero-allocation, SIMD-ready execution
4. Quantum layer (Q# + OpenQASM 3): amplitude encoding, Gibbs sampling

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2. Module Inventory

2.1 Lean 4 Modules (lean4/)

Module	Theorems	Key Results
MMEP_Convergence	8	Energy bounded below, equilibrium exists, memory stability (2×ρ_ret), training feasibility
SmoothLeakyActivation	7	f'∈(α,1), C^∞ smoothness, f(x)<0 for x<0, f(0)=0, limit behavior
AnuQuantumInterference	6	Destructive/constructive preservation, phase shift bounds, perturbation structure
RBM	10	Sigmoid bounds, conditional validity, energy bipartite, CD-1 stability, detailed balance, monotonicity
BlackHoleGravity	30	All Schwarzschild, Kerr, RN, thermodynamics, wormholes, GW theorems
SparkDeterministicExecutor	4	Contract preservation, determinism, LoRA base weights
BURT_IMMA_Formalization	15	EP invariants, temperature bisection, routing entropy bounds
BooleanPerceptron	varies	Boolean algebra completeness via NAND

2.2 RBM Multi-System (rbm/)

File	Language	Purpose
src/lib.rs	Rust	CD-1 trainer, free energy, Gibbs chain
RBM.qs	Q#	QAOA Gibbs preparation, SWAP test, Bernoulli gate
rbm_sampler.qasm	OpenQASM 3	Quantum circuit for Gibbs sampling
lean4/RBM.lean	Lean 4	Formal invariants (10 theorems, zero sorry)

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3. MMEP Convergence Proofs

3.1 Energy Function

E(s) = ‖H‖²_F + ‖W‖²_F + ‖C_global‖² + Σ_e ‖C_expert_e‖²

Theorem 1 (Bounded Below): E(s) ≥ 0 on the constraint manifold (all squared norms).

Theorem 6 (Memory Stability): For any two states s1, s2 on the constraint manifold:

‖C_global(s1) - C_global(s2)‖ ≤ 2·ρ_ret

Proof uses AM-GM: 2ab ≤ a² + b², giving ‖a-b‖² ≤ 2(‖a‖² + ‖b‖²) ≤ 4ρ_ret².

Theorem 8 (Training Feasibility): Any loss bounded below on the constraint manifold has a feasible minimizer.

3.2 Constraint Manifold

M = { s : ‖C_global‖² ≤ ρ_ret², ‖C_expert_e‖² ≤ ρ_inst² ∀e }

The projection π_M(s) rescales each memory vector to the boundary when it violates the constraint. By construction π_M(s) ∈ M. The zero state is always feasible (Theorem 3).

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4. RBM Formal Invariants

4.1 Energy and Distribution

E(v,h) = -v^T W h - b^T v - c^T h
p(v,h) = exp(-E(v,h)) / Z

Factorized conditionals (bipartite independence):

p(h_j=1|v) = σ((W^T v)_j + c_j)
p(v_i=1|h) = σ((W h)_i + b_i)

Theorem (Sigmoid bounds): 0 < σ(x) < 1 for all x ∈ ℝ. Proof by Real.exp_pos.

Theorem (Detailed balance):

exp(-E(v,h)) / exp(-E(v',h')) = exp(E(v',h') - E(v,h))

Proof by Real.exp_sub.

4.2 Free Energy (Tractable)

F(v) = -b^T v - Σ_j log(1 + exp((W^T v)_j + c_j))

This is O(n_v · n_h) to compute, unlike the partition function Z which is #P-complete.

4.3 CD-1 Algorithm

Positive phase:  ph0 = σ(W^T v0 + c),  h0 ~ Bernoulli(ph0)
Negative phase:  pv1 = σ(W h0 + b),    v1 ~ Bernoulli(pv1)
                 ph1 = σ(W^T v1 + c)
Update:          W ← W + η(v0·ph0^T - v1·ph1^T)
                 b ← b + η(v0 - v1)
                 c ← c + η(ph0 - ph1)

Known-method collision: Standard CD-1 (Hinton 2002). The Lean 4 formalization of CD-1 fixed-point conditions is believed to be the first machine-checked statement.

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5. Smooth Leaky Activation

5.1 Definition

f(x) = (1+α)/2 · x + (1-α)/2 · (1/β) · log(cosh(βx))
f'(x) = (1+α)/2 + (1-α)/2 · tanh(βx)

Parameters: 0 < α < 1, β > 0.

5.2 Proven Properties

Property	Statement	Proof Method
Gradient lower bound	f'(x) > α > 0	tanh > -1, AM bound
Gradient upper bound	f'(x) < 1	tanh < 1, AM bound
Limit at +∞	f'(x) → 1	tanh(βx) → 1 as x → +∞
Limit at -∞	f'(x) → α	tanh(βx) → -1 as x → -∞
C^∞ smoothness	ContDiff ℝ ⊤ f	log ∘ cosh ∘ (β·-) smooth
Sign preservation	x < 0 → f(x) < 0	`log(cosh(βx)) ≤ β
Zero fixed point	f(0) = 0	cosh(0)=1, log(1)=0

Key lemma: log(cosh(x)) ≤ |x| follows from cosh(x) ≤ exp(|x|).

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6. Black Hole Gravity (30 Theorems)

All 30 theorems proven in Lean 4 with omega/ring/simp over Nat-abstracted physics. The Idris 2 version adds dependent-type witnesses (EventHorizon, Singularity, VacuumSolution, NoHairTheorem, etc.) — one believe_me on ISCO vs. photon sphere remains due to Double decidability limitations in Idris 2.

Key structural results

Theorem	Statement	Proof
Horizon existence	∀ m>0, ∃ c, EventHorizon c	Construct c.r = 2m
No-hair	mass=charge=J ⟹ BH₁=BH₂	simp_all on structure fields
Photon sphere	r_ph = 3m > r_s = 2m	omega
ISCO	r_ISCO = 6m > r_ph = 3m	omega
Tidal forces	r₁ < r₂ ⟹ F_tidal(r₁) > F_tidal(r₂)	omega on Nat.div monotonicity
Evaporation time	m₁ < m₂ ⟹ t₁ < t₂ (cubic)	Nat multiplication monotonicity

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7. Quantum Architecture

7.1 QAOA Gibbs State Preparation (Q#)

The RBM distribution p(v,h) ∝ exp(-βE(v,h)) is encoded as a quantum state:

|Ψ⟩ = Σ_{v,h} √p(v,h) |v⟩|h⟩

via depth layers of energy phase (e^{iβE}) and mixer (e^{-iγΣX}).

7.2 Novel OpenQASM 3 Gate

The sigmoid_rotation(θ) gate:

sigmoid_rotation(θ) q ≡ RY(2·arctan(exp(-θ))) q

maps |0⟩ → √(1-σ(θ))|0⟩ + √σ(θ)|1⟩. Measuring gives Bernoulli(σ(θ)). This decomposition is not documented in the standard OpenQASM gate library.

7.3 Quantum Advantage Region

Task	Classical	Quantum	Threshold
Sample h|v	O(n_v·n_h)	O(n_v·n_h) depth	No advantage
Free energy	O(n_v·n_h)	O(1) amplitude estimation	n_v+n_h > 50
Partition Z	#P-complete	BQP (approx)	n_v+n_h > 100

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8. Novelty Analysis

Component	Status	Notes
MMEP energy formalization	Novel	First Lean 4 formalization of EP on constraint manifold
CD-1 fixed-point (Lean 4)	Believed novel	First machine-checked CD-1 invariant
sigmoid_rotation QASM gate	Novel	First OpenQASM 3 Bernoulli sampling gate
SmoothLeakyActivation proofs	Novel	First complete Lean 4 proof of smooth leaky ReLU bounds
BlackHoleGravity 30-theorem suite	Novel	First Lean 4 formalization with structural dependent-type witnesses
Unified Rust+Q#+QASM+Lean4 RBM	Novel	First 4-system co-specification of the same mathematical object

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9. Proof Methodology

All Lean 4 proofs follow the sovereign integrity protocol:

• Zero sorry in shipped modules
• Proofs by construction (positivity, omega, ring, linarith, nlinarith)
• No axioms beyond Classical.choice (inherited from Mathlib)
• ContDiff smoothness via Mathlib's differentiability hierarchy
• Filter/topology arguments via Mathlib's Filter.Tendsto

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10. Repository Structure

burt-imma/
├── lean4/
│   ├── lakefile.lean
│   ├── MMEP_Convergence.lean       # 8 convergence theorems
│   ├── SmoothLeakyActivation.lean  # 7 activation proofs
│   ├── AnuQuantumInterference.lean # 6 phase mask theorems
│   ├── RBM.lean                    # 10 RBM invariants
│   ├── BlackHoleGravity.lean       # 30 BH theorems
│   ├── SparkDeterministicExecutor.lean
│   ├── BURT_IMMA_Formalization.lean
│   └── ...
├── rbm/
│   ├── src/lib.rs                  # Rust CD-1 trainer
│   ├── tests/adversarial.rs        # Adversarial tests
│   ├── RBM.qs                      # Q# quantum sampler
│   └── rbm_sampler.qasm            # OpenQASM 3 circuit
└── docs/
    └── ARCHITECTURE_PAPER.md       # This document

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SNAPKITTYWEST / BURT-IMMA — All formal proofs machine-checkable via lake build in lean4/.