Gap Closure Status

Date: 2026-03-22 Agent: Arcane Sapience Commits: 51 (local, unpushed)

Gap 1: K_nm Validated Against Physical Systems

Status: PARTIALLY CLOSED — moderate correlations across 5 domains.

Five physical systems compared with SCPN K_nm (exponential-decay, all-to-all):

System	Modules	Topology ρ	Verdict
FMO photosynthesis (7 chromophores)	applications/fmo_benchmark.py	0.304	MODERATE
IEEE 5-bus power grid	applications/power_grid.py	0.190	WEAK
Josephson junction array (transmon)	applications/josephson_array.py	0.990	HIGH
EEG alpha-band (8 channels)	applications/eeg_benchmark.py	0.916	HIGH
ITER MHD modes (8 modes)	applications/iter_benchmark.py	−0.022	WEAK

Cross-domain summary: applications/cross_domain.py Measured: 2026-04-06 by Arcane Sapience using build_knm_paper27() and OMEGA_N_16.

What IS proven: Two systems show strong topology correlation (ρ > 0.5):

1. Josephson junction array (ρ=0.990): transmon coupling with all-to-all topology matches SCPN K_nm exponential decay almost perfectly. This is expected — both use distance-dependent coupling on a complete graph.
2. EEG alpha-band (ρ=0.916): neural oscillator PLV coupling structure matches SCPN hierarchy. Electrode distance → coupling decay is the physical mechanism.

FMO shows moderate correlation (ρ=0.304) — dipole-dipole coupling has distance dependence but different functional form (1/r³ vs exponential).

IEEE 5-bus (ρ=0.190) and ITER MHD (ρ=−0.022) show weak/no correlation — these systems have sparse, topology-specific coupling that does not match the all-to-all exponential-decay structure.

Honest assessment: The strong correlations in Josephson and EEG are real but expected — any system with distance-dependent coupling on a complete graph will correlate with exponential decay. This does NOT prove that the SPECIFIC K_nm values from Paper 27 are universal constants. It proves that the K_nm pattern is a reasonable model for systems where coupling decays with distance.

What is NOT proven: That the SPECIFIC K_nm values (K[1,2]=0.302 etc.) match any physical system's coupling constants. The topology shape matches; the magnitudes differ by 1-2 orders of magnitude.

What would close it fully: One system where K_nm values (not just pattern) match measured coupling to within experimental error.

Gap 1 status: PARTIALLY CLOSED (topology match in 2/5 systems).

Gap 2: Quantum Result Beyond Classical

Status: partially closed (narrow reading — 2026-04-18); broad reading remains OPEN.

Two separate readings of "beyond classical" are now distinguished (see docs/classical_irreproducibility.md):

• Narrow — signature irreproducibility: No classical simulator of the idealised Kuramoto-XY Hamiltonian can reproduce the observed +10.8 % mean / +17.5 % peak asymmetry, because $[H, P] = 0$ (every Hamiltonian term commutes with the total-parity operator) — proved symbolically in tests/test_classical_irreproducibility.py (28 tests at n=3, 4, 6). The observed non-zero asymmetry is therefore a hardware-noise signature by construction, not a property of the Hamiltonian we specified.
• Broad — quantum computational advantage: No efficient classical algorithm can simulate the hardware evolution at the deployed system size. Still open at n=4. Pathway unchanged from the table below.

Evidence	Module	Finding
DLA analysis	analysis/dynamical_lie_algebra.py	126/255 at N=4 — not trivially simulable but N=4 is classically trivial regardless
QFI	analysis/qfi.py	≈ 0 at default params — ground state is product-like
MPS baseline	benchmarks/mps_baseline.py	Small systems are MPS-tractable
QSVT resources	phase/qsvt_evolution.py	Optimal algorithm identified
Circuit cutting	hardware/circuit_cutting.py	32-64 osc scaling path
Koopman	analysis/koopman.py	BQP argument via Babbush et al.

What IS proven: The coupled oscillator problem IS BQP-complete (Babbush et al. 2023). The SCPN Hamiltonian has non-trivial DLA. Optimal simulation algorithms (QSVT) are identified.

What is NOT proven: That 16 qubits produce a result that classical computers cannot reproduce. At N=16, classical simulation is exact in seconds.

What would close it: Scale to N where classical simulation fails (estimated N30-40 for MPS, N50+ for exact diagonalisation). Circuit cutting enables this path.

Gap 3: Keep p_h1 = 0.72 Open Until Reproduced

Status: OPEN QUESTION — Monte Carlo falsified the square-lattice coincidence; public code and preprint framing now label it as an empirical/theoretical parameter rather than a derivation.

The initial finding A_HP(square) × sqrt(2/π) = 0.717 ≈ 0.72 was a coincidence.

Monte Carlo verification on the actual K_nm graph keeps the threshold open (2026-03-23):

A_HP (square lattice) = 0.8983 → p_h1 = 0.717 (0.5% from 0.72)
A_HP (K_nm graph, n=16) = 1.214  → p_h1 = 0.969 (35% from 0.72)

The Hasenbusch-Pinn amplitude is NOT universal across graph topologies. The complete graph with exponential-decay coupling has a different A_HP than the square lattice. The 0.5% match was topology-dependent, not a universal constant.

Module	Finding
analysis/bkt_analysis.py	T_BKT from Fiedler eigenvalue, bound-pair p_h1 = 0.813
analysis/bkt_universals.py	10 candidate expressions, best = A_HP(sq) × sqrt(2/π) = 0.717
analysis/monte_carlo_xy.py	A_HP(K_nm) = 1.21, p_h1(K_nm) = 0.97 — FALSIFIED
analysis/p_h1_derivation.py	Derivation chain valid for square lattice only

What IS proven: The BKT framework correctly describes the K_nm graph (T_BKT, vortex density, helicity modulus all self-consistent). The XY model physics works. But A_HP is graph-dependent.

What is NOT proven: p_h1 = 0.72 from first principles. It remains an open empirical/theoretical parameter. The Monte Carlo shows it is NOT a simple function of BKT universal constants on the K_nm graph.

What remains open: Why 0.72? Possible avenues:

1. Finite-size scaling: A_HP(n) → A_HP(∞) may converge differently
2. Noise/disorder effects not captured by clean MC
3. The threshold may relate to a different universality class
4. It may genuinely be empirical (no derivation exists)