# Theory Proofing — PEC5D / Sovereign Formalisms

Runnable verification of the ecosystem's mathematics. Run:

```bash
python3 proofs/theory_proofs.py      # human report
.venv/bin/pytest tests/test_proofs.py -q   # as tests (8 tests)
```

## Results table

| # | Proof | Property checked | Status |
|---|---|---|---|
| P1 | IPT gradient | ∇P = ∂/∂xᵐ[Φ·ρ·f(432)·S·Q] finite, Φ-scaled, 5-dim | ✅ |
| P2 | Phi-5 Kronecker | ‖tensor‖ ≈ 1 · coherence 0.99997 | ✅ |
| P3 | Progression honesty | engine computes (1.335)²⁰ ≈ 323×; 368.4× is a stated parameter | ✅ |
| P4 | CHSH Bell | classical S = 2 (not violated) · entangled S > 2 (violated) | ✅ |
| P5 | HOLO-MEM Φ-symmetry | correction consistent across all 5 layers | ✅ |
| P6 | Kyrexis anchors | fidelity 0.999423 · awakening 87.4% · 53 qubits | ✅ |
| P7 | Coral invariants | drone ≥500/day · spacecraft closed-loop life support | ✅ |

## Honesty gate

These proofs verify **internal mathematical consistency** of the
implementations — they are not claims about physics, biology, or
hardware. Where the source spec is internally inconsistent (P3: the
33.5% CAGR implies ~323× over 20 years, not the 368.4× headline), the
engine computes the honest value and both numbers are surfaced.