run 035Falsified
../lab/runs/035-kuramoto-host-sync.mdPredictionf ∈ [0, 0.01] — every start synchronises: the Schreier path carries no winding, and every cycle of the graph is a relator, short relative to its embedding, so no twisted state is stable
Falsifierf ≥ 0.05 — the host has stable non-synchronised equilibria, U(1) sectors that a lattice has and an expander lacks
VerdictFALSIFIED — f = 1.000: all 200 random starts on Cay(G/St(4)) end in non-synchronised equilibria (199 converged to max|θ̇| < 10⁻⁶, one still moving at t = 2000), with r from 0.02 to 0.9 (histogram over [0,.2,.4,.6,.8,.9,.99,.999]: 61, 94, 34, 10, 0, 1, 0, 0). Controls: 64×64 torus 200/200 non-synchronised (validation passed), random 4-regular graph on 4096 vertices 0/200, and the host's own ⟨b,c,d⟩-coset graph (N = 1024) 2/200. Anatomy of twelve host equilibria: smooth states — no edge with |Δθ| > π/2 on any generator — 230–360 J above synchrony, carrying integer windings on the group's relators: ±1 around 0.4–2.3% of the (ad)⁴ octagons, ±1 around (ac)⁸ sixteen-cycles, ±1 and ±2 around (ab)¹⁶ thirty-two-cycles; never around the bcd triangles. The sealed reasoning had the sign backwards: "every cycle is a relator, short relative to its embedding" is exactly why they are stable — a winding-1 twisted state on an 8-cycle has phase step π/4, well inside stability, and the host's path-like spectrum (gap ~1/log²N) supplies no mixing to unwind it. The coset graph, which contracts d-steps and turns the octagon into a 4-cycle (step π/2, marginal), synchronises. The host has both the loops and the slowness that N-025's new clause requires, so it has U(1) sectors — local ones, on relators, a vortex glass rather than a global winding — like a torus and unlike an expander.
Sealed before any flow was integrated (2026-09-27, codifier). Between 0.01 and 0.05 is unanticipated.
Validation and controls. The 64×64 torus (N = 4096) must end non-synchronised from at least 20% of random starts — twisted states and pinned vortex pairs are stable there — or the integration is not finding the metastable states and the run is void. A random 4-regular graph on 4096 vertices (the theorem's regime in spirit, not in its constants at degree 4) and the level-4 coset graph (N = 1024) are reported. Runs that end with r < 0.999 while still moving (max|θ̇| > 10⁻⁶ at t = 2000) are reported separately: on the host they would make the fraction unsettled rather than scored.
What it tests. The symmetry-group axis proposed for P-025's grid: on an expander the Z₂ landscape (minimum bisection) is a glass and the U(1) landscape has one basin. Where does the intermediate host sit? Its Z₂ landscape is rugged (run 034's T = 0 barriers). If its U(1) landscape has one basin, it is expander-like for continuous symmetry and glassy for discrete — unlike a torus on both counts.
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| graph | N | non-synchronised of 200 | still moving | r min | t to converge |
|---|---|---|---|---|---|
| 64×64 torus | 4096 | 200 | 12 | 0.02 | > 2000 |
| Cay(G/St(4)) | 4096 | 200 | 1 | 0.02 | > 2000 |
| ⟨b,c,d⟩-coset graph | 1024 | 2 (r = 0.95, 0.98) | 0 | 0.95 | 160 |
| random 4-regular | 4096 | 0 | 0 | 1.000 | 80 |
What the host's equilibria are. Twelve further runs to convergence (max|θ̇| ≈ 10⁻⁸), phases saved. Every equilibrium is smooth: on all four generators the fraction of edges with |Δθ| > π/2 is 0.00–0.01. Energies above synchrony 88–358 J out of 8192. Windings: around the bcd triangles, none (a winding on a 3-cycle needs step 2π/3, unstable); around the (ad)⁴ octagons, ±1 on 0.4–2.3% of the 4096 octagons in every low-r run; around the (ac)⁸ sixteen-cycle through the identity, ±1 in five of twelve runs; around the (ab)¹⁶ thirty-two-cycle, ±1 or ±2 in six of twelve. The r = 0.89 run has no octagon windings and one (ab)¹⁶ winding of −2. So the states are vortex glasses of local windings on the relators, dozens at a time, each cheap (≈ 8(1 − cos π/4) ≈ 2.3 J for an octagon) and each stable because nothing mixes it away.
Reading. The symmetry axis proposed in P-025 now has the host on it: U(1) sectors present and local (torus-like), Z₂ landscape rugged (run 034, more than a random graph's). Neither is what a memory wants — the U(1) sectors are local defects, not global labels — and both are the same underlying fact: a spectrally slow graph keeps whatever its cycles can hold. The clause added to N-025 today (a sector needs the loop and the slowness) has its positive instance here and its negative in the coset graph one contraction away.
generated 2026-10-25the lab index