Lab

PredictionBlocking, finite buffers and state-dependent service each destroy product form

Falsifiera non-reversible feature that leaves the product-form error at baseline

VerdictHALF WRONG - state-dependent service kills it, blocking does not

Confidence0.75

Predictionthe level-spacing ratio ⟨r⟩ ≈ 0.39 (Poisson) for odd Maass eigenvalues with R in [9, 150]

Falsifier⟨r⟩ near 0.53 (GOE) — the arithmetic-quantum-chaos anomaly does not hold on this object, and the spectrum is that of a generic chaotic billiard after all

VerdictFALSIFIED — ⟨r⟩ = 0.527 ± 0.033 on the 72 odd level-1 levels in R ∈ [9, 49.98]: 4.3σ from Poisson 0.38629 and within 1σ of GOE 0.5307 (large-N) and 0.53590 (3×3 surmise). The falsifier fired. Completeness 72/72 against LMFDB, none missed or mislocated, Weyl predicting 72.7. Not an artefact of the method: LMFDB's own levels over the same window give the same 0.527, and the estimator reproduces both ensemble controls. SCOPE, not a hedge on the token: the arithmetic Poisson anomaly is an asymptotic claim, and the sealed prediction names R ∈ [9, 150] while this scores R ∈ [9, 49.98] — the bottom third of the sealed range, ~72 of ~818 odd levels (8.9%), where the Weyl log term is 43% of the count against 15% at R = 150. A GOE-like bottom window does not settle the anomaly at large R.

PredictionL_min ≥ 0.60 and Δpos ≤ 1 — both families are crosswise cuts, localised at the same place on the Schreier path, and differ in how they partition the few fibres they thread

FalsifierL_min ≤ 0.35 (a family is spread along the path — lengthwise — and the fibre picture is wrong) or Δpos ≥ 3 (the families sit at different places on the path — positional metastability, not fibre-internal)

VerdictFALSIFIED on both clauses — L_min = 0.128, Δpos = 10. Neither family is crosswise. All six run-030 partitions (176–270 edges) cut a-edges only — zero b/c/d edges, zero within-fibre edges — spread evenly over all eight a-steps of the Schreier path (18–46 per step), and every one of the 16 fibres is split 47–53% / 53–47%. The two families agree on only 55–67% of vertices per fibre: uncorrelated partitions. The cut is lengthwise, orthogonal to the path, invisible to the Schreier projection. What it is: a balanced ±1 labelling constant on the ⟨b,c,d⟩-cosets (each a K₄ of four elements) that minimises the a-edges it frustrates — i.e. a minimum bisection of the coset graph (N/4 vertices, 4-regular via a-edges). Bisecting that graph directly with METIS gives 176 at level 4, below every cut found on the full graph (run 030's 198 becomes ≤ 176; c₄ = 0.69 of a fibre). The coset graph has normalised gap 0.076 against 0.0128 for the full graph and bisects at 8.6% of its edges against 15% for a random 4-regular graph of the same size: a sparse, frustrated-under-balance graph. That is what explains the glassy walls, and it is a known landscape — minimum bisection of a sparse graph is a spin glass (Fu–Anderson 1986): exponentially many near-optimal bisections, mutually uncorrelated, separated by extensive barriers. Run 030's families 25% apart in cut, uncorrelated, unreachable from each other at T = 1, with the wall's energy drifting for 10⁵ sweeps, are that landscape seen through the Ising model. The fibre-threading guess is withdrawn.

Predictionf ∈ [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.

Prediction(verbatim from N-028, dated 2026-09-27, before the level-5 eigenvectors were computed) the sign-rounding of the 90,112 partition's projection onto its 16-dimensional λ₂ irrep is a balanced cut of 98,304 = (3/4)·131,072, agreeing with the minimum on ≈ 99% of cosets, and 90,112 = 98,304 − 8,192 is the same 3/4 − 1/16

Falsifiernot written separately in N-028 — the prediction is an exact constant, so any other cut size falsifies it. The "≈ 99%" clause is an open constant and would not pass the seal validator; it is recorded as an observation, not scored.

VerdictFALSIFIED as an identity. The λ₂ eigenspace has dimension 16 with eigenvalue 0.0768589 (= 4 − 4cos(π/16)) to seven digits, and the next band is 0.1362967 = 4 − 4cos(π/12), the second irrep on the same ladder — the representation-theoretic picture from N-028 §1 holds exactly at level 5. But the sign-rounded cut is 98,288 (98,284 at tolerance 1e-6), sixteen edges short of 98,304; the 66 cosets whose projection lies within 1e-5 of zero are the only ambiguity, and flipping all of them together in either direction gives 98,302 — two edges short, and only by resolving every tie the same way. The rounding is balanced exactly (Σ sign = 0 without rebalancing, at both tolerances), which is a structural hit. Agreement with the minimum is 97.3%, not ≈ 99%. So 3/4 is the leading term (0.74988 of the unit) and the bands take 0.06238 rather than 1/16 = 0.0625; the two-term identity 11/16 = 3/4 − 1/16 is a level-4 accident. The constant 11/16 itself is untouched — it is exact at both levels because it is measured. What the host has now done three times in one day: hand over a clean identity at the small level (N/log N, the fibre count of run 031, 3/4 − 1/16) that fails by a part in 10³–10⁴ at the next.

Predictioncontrol passes as stated (m = 40, s ∈ [0.885, 0.887], c = 176, a = 1.000). At level 5 — m ∈ [64, 96]; the bands strictly between λ₂ and the π/10 band add at most 0.01 to the share; s at the π/10 boundary ∈ [0.84, 0.92]; c ∈ [90,112, 90,600] and a ∈ [0.995, 0.99999] — two bands still determine the minimum to leading order, and the level-4 exactness fails at level 5 the way ¾ failed in run 038

Falsifierany of — no eigenvalue within 1e-4 of 0.381966 (the π/10 band does not exist at level 5); s at the π/10 boundary ≤ 0.78 (the minimum's weight is spread beyond two bands); c ≥ 91,000 or a ≤ 0.99 (the two-band skeleton does not determine the minimum even to leading order). Exactness — c = 90,112 and a = 1.000 — misses the prediction's last clause and is scored unanticipated, not falsified — it would mean the level-4 identity survives and 038's failure is specific to the first band alone. Control failing withdraws the run.

VerdictFALSIFIED as written — the falsifier's clause "a ≤ 0.99" fires (agreement 0.97715) — with four of five prediction clauses hit and the cut exact. Control passed (level 4 reproduced). Level 5, 208 eigenvectors through the π/10 band: the band exists at 4 − 4cos(π/10) = 0.195774 with multiplicity 79 (+1 straggler at 0.1996; 80 by the doubling), inside [64, 96]; the two bands between λ₂ and π/10 — 4 − 4cos(π/12), mult 16, and 0.149230, mult 96 — add 0.0000 to the share; the share at the π/10 boundary is 0.8575 ∈ [0.84, 0.92]; and the sign of the projection is a balanced cut of exactly 90,112, the minimum, at the bottom of [90,112, 90,600]. The fired clause measured agreement with the one representative χ; 2a − 1 = 0.954 is the twin overlap of run 042 (0.953), so the rounding landed on a twin of χ — the same minimum cut, a different balanced half of the ~45,500 free spins — and the seal, written before runs 040 and 042 showed the minimum is a degenerate valley, took agreement with χ for determination of the minimum. Scored as written; the codifier's error is recorded. What the run found is stronger than what it predicted: the minimum bisection of the level-5 coset graph is the sign of its own projection onto two irreducible bands — the sibling-odd irrep (16) and the π/10 band (80) — exactly, as it was at level 4 from 73 eigenvectors. The first identity on this host to survive a level-up, and it survived in the form the level-up rule allows: exact in the cut, exact in the partition up to the free spins.

PredictionRelator-driven — u(N/8)/u_random ≥ 0.5 for IMG, Basilica and Aleshin alike: no group but Grigorchuk has the square obstruction, because no other has a Klein-four subgroup to contract, and (σβ)⁴ = 1 in IMG yields 8-cycles, not squares. Growth-driven, on the Cayley graphs — expansion ratio ρ_Φ: IMG ∈ [0.25, 0.60], Basilica ∈ [0.40, 0.90], Aleshin ∈ [0.85, 1.05], ordered IMG < Basilica < Aleshin with Grigorchuk's 0.374 at or below Basilica; cut-fraction ratio: IMG ≤ 0.50, Basilica ∈ [0.40, 0.90], Aleshin ≥ 0.85; barrier as % of cut: IMG ≥ 20%, Basilica ∈ [8%, 25%], Aleshin within a factor of 1.5 of its control's; Kuramoto f: IMG ≥ 0.90, Basilica ∈ [0.20, 0.95], Aleshin ≤ 0.10. The two intermediate groups agree in class — both at least a factor of two from their controls on ρ_Φ and the cut fraction, both above 20% on the barrier — while their numbers differ; the two exponential groups separate by amenability, Basilica between the intermediate pair and Aleshin on every growth-driven quantity.

Falsifierany of — Aleshin's ρ_Φ ≤ 0.60 (the free group's level-5 quotient is not expander-like at N = 1024, and the growth axis is not visible at this size); IMG's barrier ≤ 10% of its cut (intermediate growth without the glass — the barrier is a Grigorchuk fact, not a growth fact); Basilica's ρ_Φ ≤ 0.374 (exponential growth with worse expansion than the intermediate host — the axis reversed); Aleshin's f ≥ 0.50 (an expander that does not synchronise; either the graph is not an expander or Abdalla et al. do not apply at this size); or u/u_random ≤ 0.10 for any of IMG, Basilica, Aleshin (the square obstruction is not specific to the Klein-four contraction). Between prediction and falsifier is unanticipated. Control: the Grigorchuk row's reproduction at 432c51e is the control and passed before sealing; the random d-regular controls must be connected or the row is withdrawn.

VerdictFALSIFIED — the relator clause's falsifier fires: Basilica's unique-neighbour minimum at N/8 is 1 of its control's 101 at best-of-20 and 0 of 94 at best-of-40, u/u_random ≤ 0.10, so the square obstruction of N-034 is not specific to the Klein-four contraction; a torsion-free exponential-growth group with no squares has zero unique-neighbour expansion by another route. IMG (0.38) and Aleshin (0.19) also miss the ≥ 0.5 prediction without firing it. Growth-driven clauses: ρ_Φ inside every band — IMG 0.508, Basilica 0.474, Aleshin 0.853 — but not in the predicted order: Grigorchuk 0.374 < Basilica 0.474 < IMG 0.508 < Aleshin 0.853, the three contracting groups clustered and the free group apart, which is the nucleus coordinate ordering (amenable vs not), not the growth class; cut-fraction ratios 0.506 / 0.46 / 0.83 (IMG and Aleshin miss their edges by 0.006 and 0.02); barriers 53% / 40% / 24% of the cut against controls of 42% (random 3-regular is itself glassy) / 8.0% / 8.4% — IMG's clause hit, Basilica's and Aleshin's missed high, and every Cayley graph including the free group's has walls three to five times its control's, so barriers separate Cayley graphs from random graphs rather than growth classes; Kuramoto — IMG 1.000 (hit), Basilica 0.020 (miss: it synchronises like an expander), Aleshin invalid because kuramoto.py's fixed dt = 0.2 is unstable at degree 6 (dt·λ_max = 2.4 > 2), so that falsifier clause is unmeasured. Controls connected; calibration passed. Basilica is amenable-like in geometry (expansion, cut, barrier) and expander-like in dynamics (synchronisation) — the split the run did not predict and the one worth a probe. IMG's cogrowth κ = 0.797 against Grigorchuk's 0.665, shortest prime (ac)⁴ at length 8, confirming the relator.

PredictionPath — β ∈ [0.95, 1.05], PR/N ≥ 0.5, P̄_∞ ≤ 5/N. Fibonacci chain — β ∈ [0.30, 0.95] (anomalous, the positive control). Grigorchuk Schreier graph — β ∈ [0.30, 0.90], strictly anomalous like the Fibonacci chain and unlike the path, with γ ∈ [0.20, 0.95]. Random 4-regular — saturated (t_c < 4), PR/N ≥ 0.6, P̄_∞ ≤ 5/N. Coset graph — PR/N ≤ 0.30 and P̄_∞ ≥ 0.02: the flat bands hold the walker near its origin at long times, against an expander's PR/N ≥ 0.6.

FalsifierSchreier graph β ≥ 0.95 (the boundary walk is ballistic — a decorated path, and the quasicrystal reading is decorative) or β ≤ 0.10 (localized outright, which the Cantor spectrum does not predict either); coset graph PR/N ≥ 0.60 (the flat bands do not localize the walk and the bulk transport is expander-like); or the positive control failing — Fibonacci β outside [0.30, 0.95] — which withdraws the run, since the instrument then cannot see anomalous spreading at N = 1024. Between prediction and falsifier is unanticipated.

VerdictFALSIFIED on the boundary clause — the Grigorchuk level-10 Schreier graph walk is ballistic under the sealed Laplacian, β = 0.973 ≥ 0.95, with the return floor (1.5/N), decay (γ = 0.82) and participation (0.999) of the path control. Controls all pass — path β = 1.000, PR/N = 1.000, floor 1.5/N; random 4-regular saturated at t_c = 1.7, PR/N = 0.992, floor 2.9/N; Fibonacci chain β = 0.763, the positive control for anomalous spreading, inside [0.30, 0.95]. The bulk clause is CONFIRMED and strong: on the coset graph the long-time participation ratio is PR/N = 0.096 (≤ 0.30) and the exact return floor 0.0746 = 76.4/N (≥ 0.02) — the walker occupies a tenth of the graph where the expander's occupies all of it; the flat bands localize. The reason the boundary clause failed is the codifier's operator, not the host: the seal named the graph Laplacian, and in the Laplacian every loop cancels — a loop adds to the diagonal and to the degree equally — while the Schreier graph's aperiodicity lives entirely in its loops (which of b, c, d fixes a leaf follows the substitution). Bartholdi–Grigorchuk's Hecke operator and Grigorchuk–Lenz–Nagnibeda's Schrödinger operator are the adjacency operator with loops on the diagonal. The seal measured a path with alternating hoppings 1 and 2 and found a path. Scored as written; the corrected operator is sealed as run 045 with the same predictions, before any Hecke walk was run.

PredictionPath — β ∈ [0.95, 1.05], PR/N ≥ 0.5, P̄_∞ ≤ 5/N. Fibonacci chain — β ∈ [0.30, 0.95] (anomalous, the positive control). Grigorchuk Schreier graph — β ∈ [0.30, 0.90], strictly anomalous like the Fibonacci chain and unlike the path, with γ ∈ [0.20, 0.95]. Random 4-regular — saturated (t_c < 4), PR/N ≥ 0.6, P̄_∞ ≤ 5/N. Coset graph — PR/N ≤ 0.30 and P̄_∞ ≥ 0.02: the flat bands hold the walker near its origin at long times, against an expander's PR/N ≥ 0.6.

FalsifierSchreier graph β ≥ 0.95 (the boundary walk is ballistic — a decorated path, and the quasicrystal reading is decorative) or β ≤ 0.10 (localized outright, which the Cantor spectrum does not predict either); coset graph PR/N ≥ 0.60 (the flat bands do not localize the walk and the bulk transport is expander-like); or the positive control failing — Fibonacci β outside [0.30, 0.95] — which withdraws the run, since the instrument then cannot see anomalous spreading at N = 1024. Between prediction and falsifier is unanticipated.

VerdictFALSIFIED on the boundary clause, identically to run 044 — β = 0.973 under the equal-weight Hecke operator, floor 1.5/N, γ = 0.816, PR/N = 0.999, the path's numbers — and the reason is structural, found by counting: every one of the 1,024 leaves carries exactly one loop (1,022 with one, the two path ends with three), so with equal weights the loop potential is a constant and the Hecke operator is the loop-free adjacency plus the identity, a period-2 chain. The aperiodicity of the Schreier graph is which generator loops at each leaf, invisible to any isotropic operator. Grigorchuk–Lenz–Nagnibeda (arXiv:1412.6822, abstract) prove the Cantor spectrum for the anisotropic Laplacians — unequal generator weights — which is the operator run 046 seals. Controls unchanged: path β = 1.000; Fibonacci β = 0.688 (positive control inside [0.30, 0.95]; adjacency instead of Laplacian shifts it from 0.763); random saturated, PR/N 0.992; coset graph PR/N 0.096, floor 76/N (the bulk clause, not re-tested, identical since the coset graph is regular and loop-free). Scored as written.

Prediction(1) One orbit — every one of the ten classes has maximum |q| ≥ 0.95 against the first seed's orbit, and the seeds fall into one orbit up to twin flips (the exact-match count may be less than ten; twins are free-spin choices, not symmetry images). (2) |O| ∈ [2⁴, 2⁸] — the stabiliser has order between 2¹⁴ and 2¹⁸ — and the seam is aligned with the tree: k* ≤ 4, the stabiliser contains everything that fixes the first four levels. (3) The orbit's overlap spectrum is exactly the ladder — the distinct values are 0, 0.33, 0.48, 0.64, 0.80 each within ±0.02, and nothing else below 0.95; 0.953 does not appear as an orbit overlap (twins are not images). (4) The generator image with the largest |q| is at |q| = 0.80 ± 0.02 and its barrier is 39,235 within 10 %; barriers are non-increasing in |q| down the ladder.

Falsifierany of — the ten classes span two or more orbits with some class at maximum |q| ≤ 0.85 against the first seed's orbit (the minima are not one crystal and "register" is the wrong word); |O| > 4,096 (the cap: reported as "> 2¹²"; the seam has no tree-aligned stabiliser); an orbit overlap value in (0.85, 0.95) or above 0.95 other than 1 (symmetry images at twin distance — the ladder and the free spins are not separate structures); the barrier to the largest-|q| generator image below 20,000 or above 60,000 (the symmetry path is not the quench path's cost — the write costs half, or more than 1.5 times, what run 040 measured). Any control failing in (i) withdraws the run. Between prediction and falsifier is unanticipated.

VerdictFALSIFIED on every clause, controls all passing (L_h label-preserving involutions; orders of ab, ac, ad = 16, 8, 4 by both constructions; every π_h an automorphism of Q₅; every image recounts to 90,112). (1) Not one orbit: the 16 seeds are 16 distinct orbits within the cap; four classes (s31, s34, s39, s43) reach only max |q| = 0.80 against the first seed's orbit. (2) |O| ≥ 4,096 (cap hit); the BFS depth profile is the Grigorchuk sphere series through radius 12, so no two elements of length ≤ 12 give the same state, and all 9,655 Schreier words are the identity — no stabiliser within reach, k* undetermined. (3) Twin-level overlaps appear inside the orbit (0.952–0.955, 446 pairs) and the spectrum has twelve clusters, two wider than ±0.02. (4) π_d(seed) is the seed's twin — |q| = 0.954, barrier 0 — and the generator images at 0.80 (π_b, π_c) cost 5,720, a seventh of run 040's 39,235. What holds: the barrier falls monotonically with |q| down the ladder (38,038 at q ≈ 0 to 5,720 at 0.80 to 0 at 0.954).

Prediction(1) d is exactly the free-spin symmetry: π_d(core(S)) = core(S) for all 16 seeds, while π_a, π_b, π_c each change ≥ 5 % of the core. (2) The quotient does not rescue the crystal: |O_c| still hits the 4,096 cap for the first seed, and the 16 seeds give ≥ 8 distinct core orbits — the seam's symmetry breaking is in the core, not in the free layer. (3) The core stabiliser within the cap is non-trivial (it contains at least d), with leaf-group order ≥ 2 and m₁ = 1 (it fixes the first level — d acts trivially on the first letter).

Falsifierany of — π_d changes more than 1 % of the core on any seed (the twin map is not a free-spin symmetry and run 048's zero barrier is a coincidence of the interpolator); |O_c| ≤ 256 for the first seed (the crystal picture holds on the core — P-044 was right about the core and wrong only about the free layer); the core stabiliser within the cap is trivial (then d does not fix the core and clause 1 is already false). Controls as in run 048 (all must pass, or withdrawn). Between prediction and falsifier is unanticipated.

VerdictFALSIFIED on clauses 1 and 3, clause 2 confirmed; controls and equivariance pass (64/64). d is not exactly the free-spin symmetry: π_d changes 0.31–0.36 % of the core on 14 seeds and 18 % on the twin pair s24/s17 (falsifier: > 1 % on any seed). The core stabiliser within the cap is trivial — all 9,655 Schreier words are the identity, m_k = 1 at every level (falsifier: trivial stabiliser). Clause 2 holds: every seed's core orbit hits the 4,096 cap with the Grigorchuk sphere series, and the 16 seeds give 16 distinct core orbits — the symmetry breaking is in the core. Six of ten classes sit at core overlap 0.9987 against the first seed's core orbit; the other four at 0.849.

Prediction(A1) D₂(p*) tracks the threshold gap: Spearman ρ(D₂(p*), p* − p∞) ≥ 0.7 over the eight lattices. (A2) It does not track the exponent offset: |ρ(D₂(p*), y − measured 1/ν)| < 0.5. (A3) The first-order correction moves the exponent toward the measurement: |y_c − measured| < |y − measured| on at least six of eight. (B) The square's (243, 729, 2187) fit is closer to 0.75 than its (81, 243, 729) fit, and within 0.008 of 0.75.

Falsifierany of — ρ(D₂(p*), p* − p∞) ≤ 0.3 (the zeroth-order commutator is not what shifts the threshold); y_c closer than y on three or fewer of eight (the first-order commutator does not carry the exponent's error); the square's (243, 729, 2187) fit more than 0.0175 from 0.75 (larger sizes make the bias no better, 2S beyond 056's gap). A failed Monte Carlo sanity check (P₂ at p = 0.5 on the square within 0.01 of 0.5) withdraws the run. Clause A2 has no falsifier. Between prediction and falsifier on any clause is unanticipated.

VerdictFALSIFIED 2026-09-29, on clause A3. The sanity check passed (P₂(0.5) on the 81×81 square 0.5064 ± 0.0004). A3: y_c = ln P₂′(p*) / ln 81 is further from the measured 1/ν than y on all eight lattices (0 of 8 closer; y_c runs 0.05–0.63 against measured 0.28–0.74), so the falsifier (three or fewer) fired. A1 unanticipated: ρ(D₂(p*), p* − p∞) = 0.48, between the 0.7 prediction and the 0.3 falsifier (0.21 on the seven carpets alone). A2 held: ρ(D₂(p*), y − measured) = −0.12. B unanticipated: the (243, 729, 2187) fit is 0.7335 ± 0.0073, closer to 0.75 than the (81, 243, 729) fit's 0.7313 but 0.0165 away — outside the predicted 0.008, inside the 0.0175 falsifier. The body records that A3 failed on its own construction, and a post-hoc lead the data show.

Prediction(1) The nearest zeros dominate: f₄ ≥ 0.5 on at least six of the eight generators. (2) The disc holds: every zero of every generator satisfies |p_j − 1| ≤ 1.

Falsifierany of — f₄ ≤ 0.2 on at least six of eight (the exponent is a property of the whole zero set, not of the zeros near the fixed point); any zero with |p_j − 1| > 1.05 on any generator (the disc fails). A failed control withdraws the run. Between prediction and falsifier on either clause is unanticipated.

VerdictFALSIFIED 2026-09-29, on clause 2. Both controls passed on the second pass (the zeros rebuild the coefficients to 1e-257; λ from λ = ℓ + p*·Σ 1/(p* − p_j) equals run 058's to every printed digit); the first pass failed on the harness's precision, not the data's, and was discarded. Clause 2 falsified outright: every generator has zeros outside |p − 1| ≤ 1, from 15 of 28 (U′) to 48 of 120 (square), reaching |p_j − 1| = 1.41 (square) to 2.42 (B), far past the 1.05 falsifier. Clause 1 held as written — f₄ ≥ 0.5 on seven of eight (square 0.165; H 1.04, U 1.46, U′ 1.43–2.29, T 0.95, V 0.96, A 1.08, B 1.31) — but the body records that the picture it was meant to test does not: the nearest zeros overshoot the whole sum and farther zeros cancel them.

Prediction(A) |y₂ − measured| < |y − measured| on at least six of the eight lattices, and y₂ orders the eight as the measured 1/ν does on at least 26 of the 28 pairs. (B) Among the three new rooms, the offset y − measured 1/ν orders exactly as D₂′(p*) does (larger D₂′, larger offset), and over all eleven lattices the Spearman correlation of D₂′(p*) with y − measured is at least 0.7.

Falsifierany of — y₂ closer than y on three or fewer of eight (a bigger block does not improve the renormalization at this size); over all eleven lattices, Spearman ρ(D₂′(p*), y − measured) ≤ 0.3 (the post-hoc ranking of run 059 does not survive new rooms). A failed Monte Carlo sanity check (P₂(0.5) on the square within 0.01 of run 059's 0.5064) withdraws the run. Between prediction and falsifier on any clause is unanticipated.

Prediction(A) c₁ = c₂ = c₃ on all 19 generators (as exact rationals where the counts fit in 128 bits; to 10⁻⁹ in the logarithm only past overflow). (B) The misses order as c: miss(X) < miss(Y) < miss(Z); and over the eight cut-4 lattices (H, U, V, A, B, X, Y, Z) the Spearman correlation of ln c with the miss, average ranks for ties (Y ties U in c), is at least 0.85. Noise band: two misses closer than 0.008 (twice the estimator's run-to-run spread, √2 for a difference, rounded up) are undetermined in order, neither confirming nor falsifying.

Falsifier(A) any generator whose c_g differ as exact rationals, or, past overflow, by more than 10⁻⁹ in the logarithm; or whose minimum cut does not scale as C₁^g. A generator the counter cannot count is reported as not applicable. (B) miss(Z) < miss(X) − 0.008, or the eight-lattice Spearman correlation at most 0.5. Between prediction and falsifier on either clause is unanticipated. If run 061 withdraws its Part B, or if clause A fails on X, Y or Z (whose c values assume the recursion), clause B is withdrawn.

PredictionNo orphans: at level 4 every distinct eigenvalue is inherited or decimated from level 3, and at level 5 every converged distinct eigenvalue among those on disk is inherited or decimated from level 4. In each control, at least half of the non-inherited distinct eigenvalues are orphans.

Falsifiermore than 5% of the non-inherited distinct eigenvalues at level 4, or at level 5, are orphans — the spectrum is not a decimation of the level below, and "finitely ramified ⇒ exact renormalization" fails on this host. If a denominator is below 20, a single orphan counts as more than 5%; this is accepted and stated. If in either control 10% or fewer of the non-inherited distinct eigenvalues are orphans, the classification cannot tell a self-similar tower from that cover and the run is withdrawn (C2 passing the half-angle rule would mean the rule belongs to 2-power covers, not to self-similarity). Between prediction and falsifier (a few orphans, up to 5%) is unanticipated.

Prediction(A) The classes order by cut: among the measured members (a sample, so a statement about sample maxima; cut 4 already has 11 measured members, cut 2 will have 6, cut 3 and cut 5 all), the maximum measured 1/ν rises strictly with c, max(c = 2) < max(3) < max(4) < max(5). (B) The ceiling holds: no measured member exceeds its class ceiling by more than 0.012. Overall: confirmed if (A) and (B) are both confirmed.

Falsifier(A) is falsified if any adjacent pair of classes has the lower-cut maximum above the higher-cut maximum by more than 0.012; confirmed if every adjacent pair is strictly ordered; otherwise (an adjacent pair within 0.012 either way) unanticipated. (B) is falsified if any member exceeds its class ceiling by more than 0.016 (about 3.5 errors, so that about 30 members tested give an expected false break well under 0.1); confirmed if none exceeds it by more than 0.012; a member between 0.012 and 0.016 above makes (B) unanticipated. The run is falsified if either clause is, else unanticipated unless both are confirmed. Withdrawn as underpowered if the median bootstrap error on the new exponents exceeds 0.006, or if stage 1's pivotal control fails on more than 3 patterns.

PredictionThe shape is a fingerprint: for all 3 scored twin pairs, the scaled D_twin exceeds both members' scaled D_self and the largest scaled reseed D.

FalsifierAt most 1 of the 3 scored pairs separates — the curve shape does not separate twins beyond size and sampling noise. Exactly 2 separating pairs is unanticipated. Checked first: if the scaled D_self exceeds twice the largest scaled reseed D on more than 7 of the 14 carpets, shapes are still changing with size and the run is withdrawn as uninformative (nothing is read from shape at these sizes).

Prediction(A) Bands hold: every new cut-2 pattern measures below 0.3718 (the lowest cut-3 exponent) and every new cut-4 pattern above 0.4270 (the highest cut-3 exponent). (B) Cut-2 sub-bands hold: the lowest N₁ = 4 exponent (all) exceeds the highest N₁ = 16 exponent (all). (C) Within cut 4, the exponent falls with N₁: Spearman(N₁, 1/ν) over all measured cut-4 patterns is at most −0.5. Confirmed if A, B and C all hold.

Falsifier(A) any new cut-2 above 0.3718 + 0.012, or any new cut-4 below 0.4270 − 0.012; a member within 0.012 on the wrong side makes A unanticipated. (B) the highest N₁ = 16 exponent above the lowest N₁ = 4 exponent by more than 0.012; within 0.012 either way, unanticipated. (C) Spearman at or above 0; between −0.5 and 0, unanticipated. The run is falsified if any clause is falsified, confirmed if all three hold, and unanticipated otherwise. Withdrawn as underpowered if the median bootstrap error on the new exponents exceeds 0.006. Reported, not scored: Spearman(y₉, 1/ν) within each cell and each class; whether the cut-5 reversal (N₁ = 54 above N₁ = 36) has any counterpart in cut 4.

PredictionThe drift is a converging correction to scaling: at least 3 of T, X, Z, X1 have D > 0 (equivalently r ∈ (0, 1)), and at least 2 of those have the 5th percentile of D above 0. Illustratively, with run 072's values the boundary D = 0 is g₁(81) = 2g₁(729) − g₁(6561): T above 0.5544, X below 0.1211, Z below 0.2334, X1 above 0.4478; the scored value uses the recomputed k-statistic g₁.

Falsifier2 or more of the 4 have the 95th percentile of D below 0: the drift reverses direction or grows with L, and the approach to the limit is not a single shrinking correction. Between prediction and falsifier, unanticipated.

Prediction(A) First order. The interquartile range of s over the 49 distinct functions is at least 0.2 × the median |s|; and on pairs with 0 < |δ| ≤ 0.01 and |s_A − s_B| ≥ 0.1 × median |s| (count reported), the median of |D/D₁ − 1| is at most 0.1. (B) Both signs. D < 0 on at least 5% of the pairs with |δ| > 0.02 (976 pairs at sealing). (C) Cut class does not protect. Bin all pairs by |δ| into quintiles; within each bin take the median |D| of equal-C₁ pairs over the median |D| of unequal-C₁ pairs, dropping bins with fewer than 5 pairs on either side; the median of the bin ratios is at least 0.5. (D) Separation beats class. The partial Spearman correlation of |D| with |ΔC₁| controlling for |δ| (average ranks for ties) is below 0.15 in absolute value. Confirmed if all four hold.

Falsifier(A) IQR(s) below 0.05 × median |s| (s effectively constant, D₁ ≡ 0, and the House's second-order statement stands), or on the filtered set the median of |D/D₁ − 1| exceeds 0.5. (B) D < 0 on fewer than 1% of the pairs with |δ| > 0.02: the surplus is effectively forced. (C) the median bin ratio is below 0.2: at equal separation, equal cut class keeps the map close to averaging. (D) the partial Spearman exceeds 0.3 in absolute value. Each clause is scored on its own: the run is falsified if any clause is falsified, confirmed if all four hold, unanticipated otherwise. A clause whose set has fewer than 10 pairs (or, for C, fewer than 3 usable bins) is unscorable; with any clause unscorable the run cannot be confirmed, and with fewer than three scorable it is unanticipated unless a scored clause is falsified.

Prediction(A) The skewness alternation replicates. The fresh H contrast Δ⁴g₁ (samples079 only) is above +2 errors. Positive is the phase run 074 saw: g₁ high at even generations, so the steps ending at an even generation are the large ones. (B) The width alternates in the same phase. The fresh H contrast Δ⁴ln w is below −2 errors: ln w low (the width narrowed further) at even generations, so the pair slopes ending at an even generation are the large ones. Confirmed if both clauses are confirmed.

Falsifier(A) Δ⁴g₁ below +1 error: the skewness alternation of run 074 does not replicate. (B) Δ⁴ln w above −1 error: the width does not alternate in that phase. Between +1 and +2 errors (A), or between −2 and −1 (B), a clause is neither confirmed nor falsified. The run is falsified if A is falsified, whatever B shows; a confirmed B beside a falsified A reopens only the width line and does not rescue the run. Δ⁴g₁ below −2 errors is falsified as written but reported as a reverse-phase or trend-curvature signal, not as noise. A confirmed with B falsified is unanticipated: the alternation lives in the shape of the threshold distribution, not in its scale. A confirmed with Δ⁴ln w above +2 errors (both alternating, opposite phase) is unanticipated. Confirmed if both are confirmed; otherwise unanticipated.

Prediction(A) The ladder recovers the carpet's order at the first clear rung, on the guarded sizes: s₃₄(X) − s₃₄(T), the local slope over 729→6561 by the recursion, exceeds +0.05 (the carpet's gap there is 0.18, 0.545 against 0.365; rung 1's three-size slopes were 0.30 against 0.33). (B) The turn is at the first clear rung: on T, g₁(level 4) − g₁(level 3) is below −0.03; on X it is at or above 0. Confirmed if both hold.

Falsifier(A) the difference at or below 0: the rung still has X at or below T, as rung 1 did. Between 0 and +0.05, unanticipated. (B) T's step at or above +0.03: the first clear rung rises like the one-bit map. Between −0.03 and +0.03, or X's step below −0.03, unanticipated. The run is falsified if either clause is falsified, confirmed if both hold, otherwise unanticipated. A falsified A with a confirmed B says the shape is recovered before the slope; a falsified B with a confirmed A says the slope is recovered before the shape; either is a statement about which the boundary state buys first.

Prediction(A) The lag in width units is lattice-generic. ρ = ⟨Δ⟩/w lies in [−1.0, −0.6] on at least 45 of the 49 members. (B) The tie share is a class quantity. Over the 49, Spearman(s₀, C₁) is at or below −0.5. Confirmed if both hold.

Falsifier(A) fewer than 40 members inside [−1.0, −0.6]: the lag is pattern-specific. 40 to 44 inside, unanticipated. (B) Spearman(s₀, C₁) above −0.2: the tie share is not a class quantity, falsified; from −0.2 down to but not including −0.5, unanticipated (boundaries inclusive on the unanticipated side). The run is falsified if either clause is falsified, confirmed if both hold, otherwise unanticipated. A lone surviving A says the lag is a property of the lattice at threshold and the tie share is the pattern's; a lone surviving B says the reverse — the ties sort by class while the lag does not, and the lag is then the quantity to carry to the ladder.

Prediction(A) The boundary datum is real and small. Over the two no-island pairs, Q ≥ 6 (the two differences are jointly inconsistent with zero at the 5% level), and each |1/ν(new) − 1/ν(twin)| is below 0.020. (B) Attached cells are live. On both island pairs the attached member crosses earlier: δ < 0 by more than 3 combined errors, at L = 729. Confirmed if both hold.

Falsifier(A) Q ≤ 2: the two pairs are consistent with equal exponents (the median of Q under equality is 1.4) — the polynomial, with its pivotal structure, fixes the lattice exponent at these sizes and the boundary datum is below the census's resolution; falsified. Either |difference| at or above 0.020: the datum is not small; falsified in the other direction, and the twin pairing is re-read before anything is concluded. Q between 2 and 6 with both differences below 0.020: unanticipated. (B) either island pair within 3 combined errors, or either with the attached member crossing later (δ > 0 by more than 3 errors): falsified — the attached cells are as dead on the lattice as on the block, or they hinder. Both the right sign, one of them between 2 and 3 errors: unanticipated. The run is falsified if either clause is falsified, confirmed if both hold, otherwise unanticipated. A falsified A (equal) with a confirmed B says the one-block structure fixes the exponent at L ≤ 6561 while the threshold still moves with cells the block cannot see — the datum is in the threshold, not the exponent; a confirmed A with a falsified B says the exponent moves inside a fibre by something other than attached cells, and the no-island rearrangement is where to look.

Prediction(A) X is an iterated map at these sizes: g₈₁ and g₇₂₉ agree on the 19 nodes — rms z within [0.7, 1.3] and no |z_j| above 3.5 — after the half-file null is checked to lie in the same band. Confirmed if it holds.

Falsifier(A) rms z above 2.0, or any |z_j| above 4.5: the function that carries X's law from 81 to 729 is not the function that carries it from 729 to 6561 — the lattice's recursion changes with level, and no one-dimensional coordinate, Koenigs or other, makes it linear; falsified. rms z in (1.3, 2.0] with no node above 4.5: unanticipated. If the half-file null itself falls outside [0.7, 1.3], the rms is read against the null's value instead (confirmed within 0.3 of it, falsified beyond twice it), and the substitution is stated. One clause; the run takes its outcome.

PredictionAt L = 729, on T and on X. (A) The table's rule tracks the lattice: −0.30 ≤ Λ_t ≤ +0.30 and S_t ≤ 0.50. (B) The corners do it: C ≥ 0.30. Confirmed if both hold on both patterns.

Falsifier(A) Λ_t ≤ −0.55 on either pattern: pairs of sides do not carry the coupling, and it sits in where on a side a route enters (the ladder's question) or in several blocks at once. Λ_t ≥ +0.55 on either: a side is not a terminal — joined sides do not line up across a seam — and the table over-passes. S_t ≥ 0.66 on T or ≥ 0.79 on X (run 083's S_s): the rule does no better than the site rule sample by sample, whatever its mean; a mean inside the band is then two opposite biases cancelling (free seams early, middle thirds late). Between a bound of A and its falsifier, unanticipated. (B) C ≤ 0.10 on either pattern: the corner events add nothing beyond what along and across force, and the table has one entry. Between, unanticipated. The run is falsified if either clause is falsified, confirmed if both hold on both patterns, otherwise unanticipated. Readings fixed now: A falsified above with B confirmed — the corner entry is real and sides need splitting; the next rule uses the thirds, on which the recorder's same-side events bear. Λ_a ≥ 0 on either pattern (reported): the free seams over-pass even without the corners, and the lead that R_s showed was the full sides', not the corners' — read as against P-053's lead 1 whatever C is.

PredictionOn this rung the tied form is exact: ω = 1/ν for all three moments, with the ratio of successive corrections equal to 1/λ to within 10⁻⁶ by n = 16. One clause. Reported, not scored, and written down now: the limiting skewness g₁,∞ is negative (R is concave at p*, R″(p*) = 4 − 12p*² ≈ −0.58, so the map compresses the upper side of the threshold distribution more than the lower), with |g₁,∞| between 0.1 and 0.6; the limiting ln S_n + n ln λ converges to a constant with corrections in powers of λ^{−n} only; and no term in the moments' expansion is log-periodic in n (the moments are analytic functions of λ^{−n} through the Koenigs linearisation of R at p*), so the comb of 2026-10-15 has zero harmonics on this rung for moments-versus-size, and the place to look for it is the free-energy-like sums over levels, not the threshold moments.

Falsifierthe ratio of successive corrections for any moment differs from 1/λ by more than 10⁻³ at n = 16 with the fit converged (the tie is not exact even here), or the corrections are not a power series in λ^{−n} (a second exponent appears that is not a multiple of 1/ν). Unanticipated: the ratio approaches 1/λ but more slowly than λ^{−n} (a marginal operator). Withdrawn if the quadrature does not converge to 12 digits by n = 24 (control (i)).

Prediction(The House's, PN-009 TEST 1 and FORCED 4.) g₁,24 > +0.02 on (2,4) and on (3,4), and g₁,24 < −0.02 on (4,3). Three clauses; confirmed only if all three hold.

Falsifierany one of the three with the opposite sign and |g₁,24| > 0.02. Unanticipated: any |g₁,24| ≤ 0.02, or a sign that differs between n = 18 and n = 24 on a scored rung. Withdrawn if control (i) or (ii) fails.

Prediction(A) the smallest k with a period-3 orbit under NAE is 6. (B) some band rule other than NAE has a period-3 orbit at a k smaller than NAE's smallest k. (C) at k ≤ 4 no rule of any kind has a period-3 orbit (the count is 0 at k = 2, 3, 4).

Falsifier(A) the smallest k is not 6 (any value in 2 … 12, or none up to 12). (B) no band rule other than NAE has a period-3 orbit at any k below NAE's smallest. (C) a nonzero count at k ≤ 4. Each clause scored separately; the run is confirmed only if all three hold. Unanticipated: a NAE period-3 orbit detected by the grid scan that mpmath fails to confirm to 1e-30 (a tangency; recorded, recomputed at 80 digits before scoring). Withdrawn if control (i) fails.

Prediction(A) O(NAE_k) is strictly increasing in k for k = 2 … 12 and changes sign between k = 7 and k = 8 (O(NAE_7) < 0 ≤ O(NAE_8)). (B) every rule with a period-3 orbit in RESULTS114 has |λ|max > 1.8. (C) for each k = 7 … 12, Spearman's ρ between O and |λ|max over the band rules at that k is ≥ 0.7. Each scored separately; the run is confirmed if (A) and (B) hold; (C) is reported as held or not at each k.

Falsifier(A) O(NAE_k) not strictly increasing on k = 2 … 12, or O(NAE_7) ≥ 0, or O(NAE_8) < 0. (B) any period-3 rule with |λ|max ≤ 1.8. Unanticipated: a rule with a period-3 orbit in RESULTS114 for which O < 0 (Li–Yorke's ordering absent while period three is present — possible when the orbit runs high-to-low; recorded with its orbit, and the definition of O is the seal's fault). Withdrawn if control (i) or (ii) fails.

Prediction(A) every one of the 266 non-centre flips has g > 1e−12 (the broken symmetry is visible in the singular values). (B) the median g over off-axis flips exceeds the median over diagonal flips and the median over midline flips. Each scored separately; confirmed if (A) holds; (B) reported as held or not.

Falsifier(A) any non-centre flip with g ≤ 1e−12. Unanticipated: a class whose every flip has g ≤ 1e−12 (a residual symmetry the codifier did not foresee makes the two operators isospectral — recorded, and the class becomes the theme's instance of a hidden symmetry). Withdrawn if control (i) fails.

Prediction(A) ρ(O₂, h) ≥ 0.9 at every k with |P_k| ≥ 10. Confirmed if (A) holds.

Falsifier(A) any eligible k with ρ < 0.9. Unanticipated, recorded and not scored: a period-3 rule with h < log φ − 1e−10 (it would contradict Štefan's bound, so it is an instrument fault), or a rule with a critical-point count other than 1. Withdrawn if any control fails.

Prediction(A) ρ(O₂, Δμ) ≥ 0.95 at every eligible k. Confirmed if (A) holds.

Falsifier(A) any eligible k with ρ < 0.95. Unanticipated, recorded and not scored: a period-3 rule with fewer than two 3-cycles found (a detector miss, since a period-3 rule has at least one cycle and a saddle-node pair), or with more than two. Withdrawn if any control fails.

Prediction(A) the observed count equals the predicted count on all 62 scored generators: 0 on the 7 the rule clears, and the stated count on each of the other 55. Confirmed if (A) holds.

Falsifier(A) any generator whose count differs from the prediction. Unanticipated, recorded and not scored: a one-way pair in which neither subset is in U. Withdrawn if a control fails.