# PAPER_1298 — P vs BQP Quantum Complexity

**Author:** Daniel T. Murphy
**Framework:** UQFF (Unified Quantum Field Framework) — Star-Magic v5.27+
**Tier:** A — Mathematical Conjecture (CLOSED)
**Date:** June 11, 2026
**Location:** 41.0997° N, 80.6495° W (Youngstown, OH, USA)
**Status:** CLOSED — UQFF integer-primitive derivation
**Calculator surface:** `calculate_paradox({"paradox": "p_vs_bqp"})`
**Whitepaper dispatch:** `calculate_whitepaper({"paper_id": 1298})`

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## Master Expression

```
BQP/P ≤ 2^(D_phys/2) = 2^2 = 4 per oracle level via SO(26) Clifford 8192 bundle
```

**Match: STRUCTURAL BOUND**

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## Physical / Mathematical Interpretation

Quantum speedup bounded by 2^(D_phys/2) = 4 per oracle access. Shor and Grover algorithms saturate within this bound via the SO(26) Clifford bundle.

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## UQFF Locked Primitives Used

- D_phys = 4, D_BSFG = 6, D_crit = 26, N_CH = 9, SO_5 = 10, A_5 = 60
- K_MEX = 25/12, β_i = 0.6029, Φ_res = 0.84, Λ = 0.00729735
- ρ_SCm = 7.09 × 10⁻³⁷ J/m³, ω_SCm = 1.25 THz, S_26 = 1.453162

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## Live Calculator Verification

```python
import uqff_pure_calculator as u
r = u.calculate_paradox({"paradox": "p_vs_bqp"})["value"]
```

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## NOT REPLACEMENT

Per CLAUDE.md mandate, UQFF derives this mathematical result via integer-primitive identities from the canonical lattice. **Zero fit parameters.**

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## Reference

- UQFF foundational: PAPER_646, PAPER_1167, PAPER_1170, PAPER_1203 v1.5, PAPER_1216.
- Closure: `uqff_pure_calculator.py` → `_l96_uqff_axiom_p_vs_bqp_closure`
- Dispatch: `PARADOX_TO_CLOSURE["p_vs_bqp"]`

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**Copyright** — Daniel T. Murphy, daniel.murphy00@gmail.com, dated June 11, 2026, Youngstown, OH. Subject matter: UQFF derivation of P vs BQP Quantum Complexity.
