Harvest record — the anyon-statistics chase (2026-09-27, one sitting)

Filed as a record so the chase survives, not only its outputs. The seed was meant for the generators and never reached them; it was chased sighted instead, and the chase is the content.

How it started, and why the seed did not go where it was meant to

The depth trial wanted a seed the human would write, to be fed blind to the generator bank one turn at a time. The human said he does not always start with an object in mind, and offered, as the one question he did have: is there a conserved quantity, in Noether's sense, behind the fact that swapping two bosons leaves the wavefunction unchanged, two fermions flips its sign, and anyons do something in between? That is a good question and a bad blind seed — it names the answer's neighbourhood (Noether) in the asking, and the trial's rule is object and tension only. So it was answered sighted, as the framework's other mode, and every step after was the human saying "pursue" to something the answer had exposed. The generators never saw it. What follows is the record of where the sighted chase went, what it filed, and where it was wrong.

The chase, in order

1. Exchange and Noether (N-016, rotation half added). Exchange is discrete; Noether needs a continuous symmetry, so there is no exchange current. What is conserved is the representation of the exchange group carried by the state, and which group that is depends on the dimension of the configuration space: the symmetric group in 3D and above (two values, ±1), the braid group in 2D (a circle of values, and more). The statistics is a topological label, not a charge, and the rotation-by-2π half of the story (spin–statistics) is the same π₁ read on the rotation group. Filed as the missing half of N-016.

2. Fusion channels, genus, Shannon. The genesis of anyons as superselection sectors; fusion as the algebra those sectors close under; the torus ground-state degeneracy as the number of sectors and its growth with genus; and the human's question whether information stored "nowhere" breaks Shannon — it does not, the information is in correlations that no local measurement reads, which is Shannon with a non-local alphabet. Noether's second theorem came up as the gauge-redundancy side (CASE-15). Conversation, not filed as its own object.

3. Klein bottle, computed not recalled (N-025). The human asked about non-orientable surfaces and was told the count depends on the theory; he said check. Computed for the quantum double D(G): torus |G|², Klein bottle |G|·|G[2]|, RP² |G[2]|, where G[2] is the 2-torsion — so odd-order groups are blind to non-orientability (Klein = 1 = RP² degeneracy... no: Klein = |G| for odd G, and the torus count is its square, as the human guessed for the odd case), and chiral orders cannot live on a non-orientable surface at all. Two errors caught in the computing: a sphere built as a a⁻¹ that was S¹ ∨ S², caught because the table showed the sphere degenerate; and a "count the self-conjugate anyons" heuristic that gives 1 for Z₃ against the actual 3. Filed as N-025: non-orientability sees 2-torsion.

4. Loop-closing as quantisation (N-025, second half; P-024 instance 7). The human said the loop-closing kept reminding him of the hydrogen atom. Held with conviction: wherever a physical quantity is quantised there is a loop that has to close — the allowed values are the representations of π₁ (or classes in H²), and 2D grants continuous statistics because its loops braid. Instances gathered: Bohr–Sommerfeld, flux quantisation, Dirac's monopole, the photoelectric effect read as the loop in the field mode, tunnelling as the loop closing in the analytic continuation (which reopened whether complex numbers are necessary or convenient — held open), and plektons, whose quantum dimension is quantised to 2cos(π/n) by the Jones index — quantised to irrational values. Cooper pairs, the Dirac sea and quaternions were raised and not pursued.

5. Codimension as a coordinate (P-024, settled). Is codimension properly in the framework? A defect of codimension c is classified by π_{c−1}(S^{c−1}); it is a selector like v5 §5, not a bridge that produces values. Settled as a coordinate on type. The falsifiable core kept: every localised source of quantised loop-holonomy is codimension 2, given flatness. The human's "why is three special" (Poincaré, Ricci flow) sits beside it, unpursued.

6. The trade-off: non-locality is conserved (P-025). "Anyons protect information by storing it nowhere — did that come at a cost, something global made local?" Yes, and it is a conservation law with a literature: Bravyi–Haah–Verstraete, Bravyi–Poulin–Terhal (k·d^{2/(D−1)} ≤ cn), Baspin–Krishna. Measurement-based preparation relocates the non-locality to classical feed-forward; good qLDPC codes relocate it to connectivity; the 4D toric code relocates it to dimension. The single measure is the distance d, a positive integer. Falsified in session: "every resource ≥ d" fell to the 4D toric code (zero active correction), replaced by a frontier claim; the distance exponent α_max(D) = (D−1)/D. Counterexamples formulated in correction time and dimension. The human's "does the non-locality have to be somewhere big, without breaking no-communication?" — capped by BPT, and no-communication is untouched because none of it is signalling.

7. Haah, sphere packing, Leech, selection effect (N-021 instance). The association from fracton codes to sphere packing to the Leech lattice was followed one step: perfect codes filed as a sighted instance of the selection-effect probe. Fractons were named and left.

8. "Why exactly three?" — the count (P-025 correction, OPS method). Three open corners had been named. The human asked why three; then how to get an honest count rather than "three because those are what happen to be known". Answer: find the classification the instances are drawn from (Gromov's growth trichotomy), cross it with the binary that split the instances (manifold or not), read the no-go theorems' hypotheses as the closed cells. Six cells. Error found by the method: the "open corner" self-correcting + constant rate + finite D was closed by BPT all along — constant rate forces d = O(1). One cell structurally empty (no compact manifold has intermediate-growth π₁), which is evidence the grid is the right grid. Filed to OPS as a method: when a count feels like what happens to be known, it is. The human's standing rule, stated then: no basing things on empirical counts when no theorem says the count is the count.

9. The cell nobody has: intermediate growth (N-026, R22). "Can we examine the one nobody has?" The Grigorchuk group's finite quotients built and checked against the closed-form orders (128, 4096, 4,194,304). Growth exponent 0.68; effective dimension unbounded, D_eff = α ln n. Amenable, so d = o(n) — a third regime between Euclidean and expander. Three literature searches: empty. First code on it: homological, k large, d_X stalls at the systole, which is 2^n against log₂|G_n| ≈ (5/8)2^n — logarithmic in the qubits. Closed by the contracting property. Layer-code route named; embedding stretch computed (behaves like a 2–3D lattice at 4096). Two scripts OOM-killed and rerun with caps.

10. Close the partial cell: 3D (P-025 update). Literature as of October 2025: layer codes (Williamson–Baspin) have a linear energy barrier in 3D, saturating Bravyi–Terhal and BPT — the log L ceiling was on translation invariance, not dimension. But Baspin's free-energy barrier paper proves layer codes have constant f̄ (paths over the wall grow exponentially in its height) and poses the general 3D case as open — in so many words, the question this chase had arrived at. Cored product codes on the pinwheel tiling are the self-similar candidate, numerics only.

11. Scale-dependent dimension as a fourth mechanism (run 030, R23). The speculation: a host low-dimensional at small scales and high-dimensional at large could hold walls that are rigid (local) and large (global). "Lets pursue!" — and "does it control path entropy" — "yess follow". Sealed (ρ = TΔS/ΔE ∈ [0.05, 0.30], falsifier ≥ 0.50), then measured on the Ising memory against three controls. Outcome: unanticipated (ρ = 0.29 ± 0.03, on the band's edge; the 2D control at 0.34 outside the band, because the sealed reference 0.136 was a miscalculation of Onsager — recorded). The finding: entropy is local and the host's is 2D's; the barrier is the minimum cut; rigidity is a membrane property, high local dimension, so the speculation had the sign backwards. Not a fourth mechanism. New: the wall is glassy — metastable cut families 25% apart, which no lattice has. N-026 corrected twice (the Fiedler separator was ≥ 2× loose; T_c = 2.05 below 2D's 2.27). The method named at seal (window sampling in M) failed validation and was replaced, on the record.

12. What is left, sealed (run 031). Whether the cut's exponent climbs toward 1. Reasoning after the fact from the Schreier path: the bisection tracks |G_n|/2^n, which is n/log n — so the prediction is b₅ ∈ [65,000, 140,000], falsifier ≤ 40,000. If it holds, the host beats a 3D lattice as a classical memory from n ~ 10⁶, with 2D walls. Handed to Claude Code.

13. Same evening, three more (runs 032, 033; P-026). "N/log N gives me prime counting vibes" → the function-field prime number theorem as orbit counting, and the Hausdorff dimension 5/8 of the group in Aut(T) as the separator's constant. "Mirror image → Langlands dual?" → the duality present is the trace formula's (Ihara = Selberg on graphs: relators ↔ eigenvalues), not a dual group. "Likely explains → does explain": run 032 sealed the fibre-threading guess and falsified it — the cuts are lengthwise, a-edges only, bisections of the ⟨b,c,d⟩-coset graph, and the glassy walls are the spin-glass landscape of sparse-graph bisection (Fu–Anderson); level-4 minimum now ≤ 176. "Push!" on the graph PNT: run 033 confirmed (κ = 0.665) that the host's primes — its relators — reach the free density sub-exponentially, the amenable signature; P-026 files the growth trichotomy as a primes-upstairs / zeros-downstairs coordinate (Kesten–Grigorchuk–Ihara). Run 031 handed to Claude Code, its mechanism paragraph corrected under the seal.

14. Later the same evening (runs 034–036, N-028). "Do test" on the glass: run 034 sealed the Franz–Parisi barrier at T = 1 and withdrew by its own rule (the positive control showed none; bulk entropy dominates at that temperature), but its T = 0 interpolation barriers stand: ~50 extra cut edges between every pair of the coset graph's minima at N = 1024, against a median of 2 on a random regular graph and 4 on a torus, up from ~0 at N = 32. Three Quanta results on expanders (synchronisation, Hamiltonicity, explicit randomness) read onto the grid: N-025 gains the clause "a sector needs the loop and the slowness"; P-025 gains a symmetry axis and the expander-code reading of the cell's ceiling; P-008 gains the reason the Helfgott–Radziwiłł route stops at the logarithmic average (the Haar weight). "N/log N gives me prime-counting vibes" → chased and broken twice in N-028 (depth against width in the same tree; an RH-scale correction against a constant, glassy one). Run 035 (Kuramoto on the host) falsified: every random start ends non-synchronised, in vortex glasses of windings on the relators (ad)⁴, (ac)⁸, (ab)¹⁶ — the host has U(1) sectors because it is spectrally slow, and its coset graph, one contraction away, does not. Run 036 confirmed (ratio 0.78): the coset graph expands small sets like a random graph up to 16 vertices and loses expansion gradually after — the Sipser–Spielman guarantee survives at N = 1024 with a constant-factor loss. Run 031 (Claude Code, in progress): b₅ = 90,144 = 0.688 fibres so far, in the sealed band, c stable across levels 4 → 5 and the level-4→5 exponent 0.90.

15. Run 031 landed (Claude Code): b₅ = 90,112 = 11/16 of the fibre count, exactly, confirmed, the exact value predicted under the seal an hour before. Five of six dated level-5 predictions in N-028 hit (λ₂ = 4 − 4cos(π/16), multiplicity 16, projection 0.68, Rayleigh 11/32); the sixth — the cosine profile — was one of three degenerate shapes, not the minimum's. The N/log N sentence now reads: an exact rational constant, a spectral skeleton with a vanishing share of the cost, a glassy landscape above. The two explainers rewritten in plain language and linked from the site.

16. The three chases (N-028 §1–2, same night). Why the λ₂-eigenspace has dimension 2ⁿ⁻¹: it is the sibling-odd irreducible representation of the quotient, character matched on 16 words. Why the 4 − 4cos(π/12) band: the second irrep on the same ladder, not a period of 12. Why 11/16: the sign of the 176-cut's λ₂ projection is exactly 192 = ¾·256, so 11/16 = ¾ − 1/16 at level 4 — sealed as a dated level-5 prediction, 98,304. A recalled "fact" that 90,112 is a McKay–Thompson coefficient was computed and is false (T₂ₐ's q² coefficient is 96,256).

17. Run 038: the identity falls. Level-5 rounding gives 98,288, not 98,304 — sixteen edges, two beyond the tie band. ¾ is the leading term, not a term. Third small-level identity of the day to fail one level up (after N/log N and the fibre count); the constant 11/16 itself is untouched.

18. Run 039 sealed: the two-band skeleton at level 5. At level 4 the minimum is exactly the sign of its projection onto the lowest 73 eigenvectors — two weighted bands, two empty ones between. Sealed leading-order-yes, exactness-no; control passed; under the seal, dated before data, the observation that the level-5 band list is the level-4 list pulled back by θ ↦ θ/2, so bands that carry weight at level 4 land between λ₂ and π/10. Running on the M5.

19. Run 040: the T = 0 barrier, sealed on its own terms. Clause 1 confirmed — 47 edges between 176-minima (27% of the cut), 0 at level 3, 12 on the random control (4%; run 034's "~2" was a lucky seed set). Clause 2 on the M5: nine 90,112 minima by the full-graph route, a C interpolator whose level-5-only indexing bug the level-4 control could not see and the first level-5 step did, two duplicate minima caught by the distinctness check.

20. Run 041 → N-034: there is no expander code on the host, at any level, and it is the relators. A Tanner code at degree 4 is degenerate (local distance 2, girth 4), so the sealable question was the theorem's hypothesis, unique-neighbour expansion. The annealer found zero at every size; the reason is a theorem — the coset graph's edges are disjoint (ad)⁴ squares, left vertices chain into cycles of length 2ⁿ⁻¹, and their left vertices have every neighbour twice. Withdrawn as sealed (the codifier's control band was miscalibrated); the finding stands by construction. The same squares are the vortex traps of run 035 and the shortest primitive cycles run 033 counts.

21. P-036: the code question moves to the squared graph — a [12, 8, 3] local code on the length-2-walk multigraph, decoding radius and minimum-distance estimate against the random control, BernoullisBook's job; sealed as run 042 once the control's numbers exist. Also this stretch: the two explainers rewritten with hindsight and given a reMarkable Paper Pro edition through a Design chat and a Code chat (five silent print failures caught by a gate, the whole bundle moved to one page size, a diagram that had never shipped restored, two glossary drifts fixed); four attributions made from memory verified and pinned; the manifest health check fixed.

22. Run 042: the valley/wall split is the local/global split — unanticipated. The twin clause confirmed on all eight pairs: a twin difference of ~24,700 cosets is ~22,000 connected components, all zero-modes, a balanced random half of the ~45,500 free cosets flipped. The valley clause missed its line on 12 of 36 pairs. Free spins are the local flavour; the fronts are the global one.

23. The growth axis on one tree (P-043 → run 043, Lab_Tree_Model) — falsified, and the falsification is the finding. Five self-similar groups on the binary tree — odometer, Grigorchuk, IMG(z² + i), Basilica, Aleshin — at matched N with the instruments of runs 033–042. The relator clause fired: Basilica, torsion-free, no squares, has zero unique-neighbour expansion by another route — unions of cosets of ⟨g²⟩ — and Aleshin by the same route by construction (the annealer's 8 was the annealer). N-036: two routes to zero, both g². Expansion orders by the nucleus (contracting 0.37–0.51, free 0.85), not by growth class; barriers separate Cayley graphs from random graphs (24–53 % against 8 %); Basilica amenable in geometry and expander-like in dynamics. Alongside: N-035, the sequential-circuit reading (latch = Hensel, register = nucleus, carry = sections, activity × growth as a 3 × 3 grid); P-044, the seam orbit as a register, waiting on the level-5 element list.

24. The transport column (P-045 → runs 044–047). Two falsifications by the codifier's operators — the Laplacian cancels the loops, and the isotropic Hecke operator is periodic because every leaf carries exactly one loop — then run 046 confirmed: under a + b/2 + c/4 + d/8 the Grigorchuk boundary walk is anomalous, β = 0.800, beside the Fibonacci chain. Run 047 on all five boundaries: odometer 1.000, Grigorchuk 0.800, Basilica 0.595, IMG 0.440, Aleshin saturated at diameter 6 — the classes confirmed, the internal order unanticipated; in transport the nucleus decides. Unsealed: β smooth and monotone in the anisotropy, the spectral measure's D₂ saturating near 0.3 while the eigenstates' D₂ falls 0.88 → 0.56 with β.

25. P-036 resolved before the build — the cell hosts no code with an expander guarantee. The five-line check N-036 asked for: the squared and cubed coset graphs have zero unique-neighbour expansion from s = 8, double edges or not (the random square: 21–31); every Cayley graph of G/St(4) on non-involution generators from s = 4, by absorbing Klein-four subgroups such as ⟨ab(ac)⁻¹, (ad)²⟩; and no generating set escapes, because the quotients are 2-groups and ⟨g²⟩ lies in the Frattini subgroup, index ≥ 8. One criterion contains N-034's squares and N-036's cosets: a subgroup that absorbs every generator through a short relation gh ∈ H. Zémor's spectral condition fails on host and control alike, so the decoder would have measured the decoder. The arc that began at the empty cell — is there a code on it? — ends as a theorem-shaped closure: three routes, one mechanism, and the answer is no at the level of the guarantee.

Third tally, 2026-09-28. Filed: runs 042 (unanticipated), 043 (falsified), 044 and 045 (falsified), 046 (confirmed), 047 (unanticipated); notes N-035, N-036 (with addendum); probes P-043 (sealed as a run), P-044 (open), P-045 (closed with its arc), P-036 (resolved); the R27 directory. Errors caught: the relator clause of 043 twice (Basilica's zero, Aleshin's annealer 8 against a true 0); P-043's claim of squares in IMG (8-cycles, no Klein-four); two recursions in the wrong convention; the operators of 044 and 045; Kuramoto's step at degree 6; the band grouping merging λ = 0 on cycles; the anneal missing a cycle's bisection; and the P-036 gate reading the wrong run file. Nine. Same pattern as the first two tallies — a control, a construction, or an instrument caught each; none was caught by re-reading.

What the chase cost, and what it caught

Filed: one note extended (N-016), three notes (N-025, N-026, this), two probes advanced (P-024 settled as a coordinate, P-025 falsified once and corrected twice), one instance (N-021), two OPS method notes, two probes (P-026 and the axis in P-025), a note (N-028), seven runs (030 unanticipated, 031 sealed and running, 032 falsified, 033 confirmed, 034 withdrawn, 035 falsified, 036 confirmed), three research directories. Errors caught in session, all by the tooling or by computing what had been recalled: twelve (§3 twice, §6, §8, run 030's band, its method, the separator in N-026, its spectral "dimension", the fibre-threading guess, run 034's instrument and its torus expectation, and run 035's sealed reasoning). Nothing the generators did; everything the valve did.

The seed's own question — is there a Noether quantity behind exchange — got its answer in the first hour (no current; a representation label of π₁ of configuration space) and turned out to be the door, not the room.

Second tally, late 2026-09-27. Filed: runs 038 (falsified), 039 (sealed), 040 (clause 1 confirmed), 041 (withdrawn, theorem); N-034; P-036; N-028 scored under its own prediction; N-025 pinned. Errors caught in this stretch: the moonshine recall; the ¾ − 1/16 identity; the codifier's own control band in 041; the interpolator's level-5 indexing; two duplicate minima; and, on the print side, a font gate passing for the wrong reason, a bash-3.2 array, nested @font-face, media="print" suppressing the fetch, a CORS preload, an injection order, a list-marker overhang. Thirteen. Same pattern: none by re-reading, all by an instrument, a control, or a construction.