267 records · 122 runs · 49 probes · 40 notes · 33 cases · 9 panels · 14 exemplars · index density · read at a354234
One list, every id, state glyph, one line — ordered by id within each kind, retired sinking to the bottom of the group and excluded from its count. An id that is a slug rather than a number sorts after the numbered ids and is listed, not folded into a range.
runs110 · +12 retired
017Is the walk-run transition Froude number the kinematic ceiling, or a biological constant?run
018Does a dissipative chaotic PDE have discrete Ruelle resonances?run
019What survives of the lattice hypothesis when the lattice is replaced by a quasicrystal?run
020Is a number that recurs across unrelated systems a count or a well-measured magnitude?run
021Can this framework compute a positivity bound rather than cite one?run
022Is the Ulam eigenvalue independent of where the cell boundaries fall?run
023Does the Ruelle resonance survive as the attractor dimension grows?run
024Does any non-reversible feature kill product form, as the Markov audit claims?run
025Does positivity give only bounds, or does it sometimes give values?run
026Does the triple-correlation bias in the Liouville sums survive to 10⁹, and does per-scale information stay flat?run
027Does the swirl component grow without bound in the axisymmetric run, or saturate?run
028Does "completion and algebraic closure don't commute at p" have a rate?run
029Is the quantum Mixmaster (Artin) billiard spectrum Poisson, despite the classical dynamics being chaotic?run
030On the intermediate-growth host Cay(G/St(4)) (n = 4096), is the domain-wall free-energy barrier controlled by the host's local, low-dimensional structure — or does the entropy of near-minimal cuts eat it?run
031Does the minimum bisection of Cay(G/St(n)) track the Schreier-fibre count |G_n|/2^n — a separator of order n/log n whose exponent climbs toward 1 — or does the exponent 0.76 measured between n = 128 and n = 4096 (run 030) hold?run
032Run 030 found two metastable families of near-minimal cuts on Cay(G/St(4)) (≤ 224 edges from METIS, 270 from disorder).run
033The Ihara primes of the host are its relators.run
035Expanders are globally synchronising (the Kuramoto flow has no stable equilibrium but synchrony — no twisted states, no U(1) loop-closing sectors).run
036Expander codes (Sipser–Spielman, Tanner) need unique-neighbour expansion of small sets; amenability kills expansion only asymptotically.run
037Trained to grokking on multiplication mod p, which characters does a small network learn — the additive characters of Z/p, or the multiplicative (Dirichlet) characters, i.e. Fourier modes in discrete-log coordinates on (Z/p)^x?run
038At level 4 the sign of the 176-cut's projection onto the λ₂ irreducible (the sibling-odd representation, dimension 8) is a balanced cut of exactly 192 = (3/4)·256, and the bands up to dimension 72 reproduce 176, so 11/16 = 3/4 − 1/16.run
039At level 4 the 176 minimum bisection of the ⟨b,c,d⟩-coset graph is exactly the sign of its projection onto the lowest 73 Laplacian eigenvectors — the constant, the λ₂ = 4−4cos(π/8) irrep (multiplicity 8, share 0.697), two bands it is orthogonal to (4−4cos(π/6), multiplicity 8, and 2−√2, multiplicity 16; together +0.0002), and the 4−4cos(π/5) band (multiplicity 40, +0.189) — with agreement 1.000 and the cut 176 on the nose.run
040Run 034's instrument (Franz–Parisi at T = 1) failed its control and was withdrawn, but its second look stood unsealed — at T = 0, greedy pair-swap paths between minimum bisections of the level-4 ⟨b,c,d⟩-coset graph climb ~50 edges above the endpoints (28% of the 176 cut), against ~2 on a random 4-regular graph of the same size and ~0 on the level-3 coset graph.run
041Run 036 measured edge expansion of the ⟨b,c,d⟩-coset graph at 0.78 of a random 4-regular graph's at s = N/8, and P-026 read the intermediate cell's code ceiling from it.run
042Two "two-flavor" splits have appeared on the intermediate-growth host.run
043Five self-similar groups on the binary tree — odometer (nucleus 3, polynomial growth), Grigorchuk (5, intermediate), IMG(z² + i) (4, intermediate), Basilica (7, exponential, amenable), Aleshin (not contracting, free, non-amenable) — measured by the instruments of runs 033–042 at matched vertex count.run
044The quantum reading of the intermediate cell (P-045) — the boundary as a quasicrystal, the bulk as a flat-band localizer — is either a transport signature or a decoration.run
045Run 044 sealed the graph Laplacian as the walk Hamiltonian, and on the Schreier graph the Laplacian cancels every loop — which is exactly where the graph's aperiodicity lives (which of b, c, d fixes a leaf follows the substitution).run
046Runs 044 and 045 walked the Grigorchuk Schreier graph under isotropic operators (the Laplacian, then the equal-weight Hecke operator) and found a ballistic path both times — every leaf carries exactly one loop, so equal weights make the loop potential constant and the operator periodic.run
047Run 046 found the Grigorchuk boundary walks like a quasicrystal under the anisotropic Hecke operator (β = 0.800) and run 044 found a random expander's walk saturates by t ≈ 2 and a path's is ballistic.run
048Run 040 produced seventeen level-5 minimum bisections at exactly 90,112 and run 042 sorted them into ten classes (twins at |q| ≈ 0.953 are one class — a balanced random half of the free cosets flipped), with the overlaps between classes on a ladder 0, 0.33, 0.48, 0.64, 0.80.run
050Run 048 found the level-5 minimum bisections are not one small orbit — 16 seeds, 16 orbits, ≥ 4,096 states each, no stabiliser within reach — and that the generator d carries the seam to its twin at zero barrier, so part of the free-spin layer is symmetry.run
051Run 049 was withdrawn on its control (the sealed p-grid could not resolve the square lattice at L = 2187).run
052Run 051 found the all-singleton carpet U′ in a different percolation class from carpets H and U at the same dimension (1/ν = 0.276 against 0.547 and 0.524), but its exponent sits beside the gasket's (0.263), and the gasket is finitely ramified.run
053Runs 051 and 052 found three carpets of identical Hausdorff dimension whose percolation exponents 1/ν (0.547, 0.524, 0.276) sit at a near-constant fraction of their left–right cut exponents x (0.631, 0.631, 0.316): 1/ν ÷ x = 0.867, 0.830, 0.873.run
054Run 053 found carpets V and H sharing a cut exponent (0.631) and a bond count at every size, yet differing in 1/ν by 0.070 (0.477 against 0.547), with U between them (0.524).run
055Run 054 found that the minimum-cut entropy per cut bond, s = ln N / C, orders carpets H, U and V (one cut exponent, 0.631) exactly as their percolation exponents do, s 0.693 < 1.058 < 1.153 against 1/ν 0.547 > 0.524 > 0.477.run
056The square lattice, measured with run 051's estimator at the same sizes, gave 1/ν = 0.7516 ± 0.0049 in run 053 and 0.7372 ± 0.0049 in run 055 — a shift of 0.014, about twice the standard error of a difference between two independent runs.run
057Birch and Swinnerton-Dyer read the rank of an elliptic curve as a growth exponent — the slope of log ∏_{p≤X} N_p/p against log log X. That is the same instrument runs 051–056 use on carpets: a global exponent fitted from local counts across a few decades of scale, where pair slopes drift.run
058Runs 054–055 found that the number of distinct minimum cuts orders the carpets' percolation exponents.run
059Run 058's one-block polynomial ordered all eight lattices' exponents but not their values (y reads carpets high by 0.009–0.065, the square low by 0.020), and put the thresholds p* 0.03–0.06 above the measured ones.run
060Run 058's crossing polynomials R(p) are exact.run
061Run 059's corrected exponent failed because it read the 81×81 lattice's slope at the one-block fixed point p*, where that lattice is already on the shoulder of its crossing curve.run
062On all seven carpets counted so far, the number of minimum left–right cuts obeys an exact recursion across generations, N(g+1) = c·N(g)^C₁, with C₁ the one-step cut size and c a constant (H 1, U 3, V 4, T 8, U′ 4, A 36, B 81/2500), checked as exact integers at g = 1→2→3 and in logarithm to g = 4.run
063Run 062's pool contains cut twins — distinct D4-symmetric 9×9 generators with identical minimum-cut counts N(g) at every generation (entries 5 and 12 twin X; 10 twins Y; 11 twins Z).run
064Probe P-048 argues that block renormalization fails to converge on the Sierpiński-type carpets because they are infinitely ramified (blocks share whole edges).run
065Run 064 found that the ⟨b,c,d⟩-coset tower is not an exact renormalization under φ(μ) = μ(8 − μ)/2 (18 of 24 new level-4 eigenvalues orphans).run
066The block exponents y_g = ln P_g′(p_g*)/ln 9^g rose from g = 1 to g = 2 on every lattice (run 061).run
067A carpet may use a different 9×9 pattern at each level.run
068The construction (a 9×9 pattern keeping 64 cells, outer ring kept, symmetric under the eight symmetries of the square, centre removed) admits exactly 63 patterns.run
069Feigenbaum's universality lives in a function, not only in δ: the fixed point of the doubling operator.run
070Run 067 found that stacking two carpets of the same cut class (X over Y, both cut 4) averages their exponents, while stacking a cut-2 pattern with a cut-4 one (X over U′) lands 3.5σ below the average.run
071Run 068 measured 14 of the 63 patterns and left a hierarchy, post hoc: the minimum cut C₁ sets a band of exponents (cut 2: 0.260–0.349; cut 3: 0.372–0.427; cut 4: 0.429–0.551; cut 5: 0.588–0.613), and within a band the number of minimum cuts N₁ sets a sub-band, more minimum cuts lower (cut 2: N₁ = 4 at 0.303–0.349, N₁ = 16 at 0.260–0.276; cut 3 monotone in N₁).run
072Run 069 found the width-standardized crossing curve indistinguishable across carpets by a Kolmogorov–Smirnov distance, but reported a moment that is not flat: the skewness of the standardized threshold distribution at L = 6561 is positive on every carpet (+0.11 to +0.55) and near zero on the square (+0.011), and in the three twin pairs whose exponents separate, the lower-exponent member has the lower skewness.run
073Run 072 found the skewness g₁ of the threshold distribution reproducible across reseeds but still moving between L = 729 and 6561 on four carpets (g₁(6561) − g₁(729): T −5.2 combined errors, X +3.2, Z +2.8, X1 −2.5).run
074Run 073 read the skewness g₁ of the threshold distribution at three sizes a factor of 9 apart and found it built between 81 and 729 on thirteen of fourteen carpets, then turning back, levelling off or still climbing.run
075Each 9×9 pattern has an exact one-block map, the crossing polynomial f(p) of run 058 (rg058.poly).run
076Stacking two carpet patterns A and B gives, at the one-block level, the 2-cycle of their crossing polynomials, with y_AB = ln(R_A′(r)·R_B′(q))/ln 81 at q = R_A(r), r = R_B(q).run
077Feigenbaum's universality sorts one-hump maps by the order z of their maximum; the amplitude of the maximum drops out.run
078Runs 076–077 found that, at fixed fixed-point separation δ, the one-block composition deviation |D| depends on cut class C₁ (partial +0.60 over all pairs; +0.72 over unequal-class pairs with the amplitude (ln N₁)/C₁ also held) and not on the amplitude.run
079Run 074 read H, the standard 3×3 carpet, at five sizes a factor of 3 apart (L = 81, 243, 729, 2187, 6561; generations 4–8) and found, post hoc, that the skewness steps alternate — small, large, small, large — with the large steps ending at even generations.run
08043 census patterns have a measured 1/ν (14 measured by run 068, 15 by run 071, 14 carried as record values from earlier runs without errors).run
081Run 073 found the skewness g₁ of the crossing threshold falling between L = 729 and 6561 on two carpets, T (−0.191 ± 0.038) and X1 (−0.116 ± 0.045), while the exact one-block recursion rises at every depth on all 63 patterns.run
083PN-005's House named the route the one-block recursion cannot represent: a lattice crossing that passes through a top-level block without crossing it left to right.run
084Run 082 was withdrawn on a mis-written control, but what its lab saw before stopping reshapes the ladder: rung 1 is wall-bound at the carpet's sizes (the threshold within 2.8 widths of 1 at L = 729), its slopes sit below both the carpet and the one-bit map and in the wrong order (X below T), and the one-bit map sits above the carpet.run
085Run 083 found, on two carpets, that the route the one-block recursion cannot represent is the majority route at every size, that the lag Δ = t − t_coarse between the lattice crossing and the coarse block crossing is about −0.8 of the threshold width on both patterns at both sizes, and that the share of exact ties (Δ = 0) is the one per-sample quantity that separates T from X (10% against 5%).run
086The census has 63 patterns and 49 crossing polynomials; the 14 extra patterns share a polynomial with another, in eight groups of two to four, and the completed census measured one member of each.run
087P-050 read the census's local slopes and found the cut-4 exponents reading as a fan around y₉ — the spread of the local slope wider over 729→6561 than over 81→729 — carried by five patterns whose stored drift reads positive while thirty of thirty-five fall, only one of them beyond two drift errors: #60 (+0.043), #63 (+0.010), #10 (+0.012), #3 (+0.007), #33 (+0.001).run
088Run 044's bulk clause was confirmed — on the Grigorchuk level-4 coset graph the quantum walker stays near its origin at long times (PR/N = 0.096, return floor 76/N) against a random 4-regular graph's PR/N = 0.99 — and P-045 reads it as a transport signature of the intermediate-growth cell.run
089The one-block map f = R_P carries the depth-k threshold law to the depth-(k+1) law by one fixed function, h = f⁻¹: Q_{k+1} = h ∘ Q_k, so the step map Q_{k+1} ∘ Q_k⁻¹ is h at every depth and, by Koenigs, the map is linear in a coordinate φ with φ ∘ h = λ⁻¹·φ.run
091Each sample's threshold is a minimax — the lowest level at which some crossing path is fully open — and its two tails are set by two different optimizations: crossing far below threshold needs the cheapest path, the ring, of length ∝ L; failing to cross far above it needs the cheapest blocking cut, of size ∝ C₁^g = L^(ln C₁/ln 9).run
092The calibration note on run 081 found the bootstrap error of the skewness g₁ right within a file (pooled block-to-bootstrap ratio 1.00 over ten files) while files of one graph differ beyond their block errors at L = 6561 — T4: 0.172 ± 0.026 on samples053/T4.txt (the file run 073 read), run 070's 0.282 ± 0.028, run 081's 0.248 ± 0.020 (one-way F over blocks, p = 0.008); X1's F4: 0.216 against 0.348 (Welch on blocks, p = 0.005) — and not at 729.run
093The crossing polynomials of the 49 census fibres carry squares and a composition that nothing in the record explains.run
094Run 084 scored rung 3 of the ladder on S₃ = {1, 4, 7}: X's local slope over 729→6561 is below T's by 0.131 ± 0.007 where the carpet has it above by 0.18.run
095P-054's look found that on T and X at L = 729 the sizes of invasion bursts — the runs of accepted links from one forced link to the next, growth from the whole left side — follow a power law with a ripple about it whose period is a factor of about 55–64 in size, T's twice X's and none on the square.run
096Scaffold S2 (single-exponent scaling, w ∝ L^(−1/ν)) has its first data against it in a reading, not a result: run 079's five ln w values for H (#17, the Sierpinski carpet's second generation; the only census pattern with sizes at every power of 3), 48,000 samples each at L = 81, 243, 729, 2187, 6561, lie on a straight line in ln L with χ² = 8.28 on 3 degrees of freedom; the weighted quadratic coefficient is +0.00319 ± 0.00112 (+2.85 errors); the pair slopes fall 0.553, 0.550, 0.541, 0.533; in the census's unit the drift is −0.0147 ± 0.0058.run
097Run 083 found that the lattice crosses before the graph of its 64 top-level blocks does in 85% of samples on T and X at L = 729, a block counting as open once it is crossed left to right on its own bonds; run 085 that the mean lead is 0.46 to 1.92 widths over the census.run
098Within a group of patterns that share one core, how much does a pendant part lower the lattice threshold at L = 729?run
100On H's two stored sets, with the mean, ln S and g₁ fitted jointly under PG-03's three one-term forms, tying the correction exponent ω to 1/ν costs Δχ² = 11.28 (each file's covariance; 8.54 with diagonal errors) on all five sizes and 1.47 (0.59) with L = 81 left out.run
101On the sixteen stored minimum bisections of the level-5 ⟨b,c,d⟩-coset graph (cut 90,112), read on 2026-10-04, exactly 40,960 free cosets are independently flippable on every one, and they lie over the map to level 4 the same way on every one.run
102The free one-term form (ω its own parameter; the forms are run 100's), fitted on run 100's six files at L = 81 (H4a, H4b), 243 (H5a, H5b), 729 (H6) and 6561 (H8), predicts H's three moments at L = 2187 — ln S = −5.4449911 ± 0.0012293, mean = 0.6843628 ± 0.0000088, g₁ = 0.3634693 ± 0.0042900 (profile errors, χ² ≤ min + 1 in all nine parameters).run
105The free one-term form (ω its own parameter; the forms are run 100's), fitted on run 100's seven new files at L = 81 (H4a, H4b), 243 (H5a, H5b), 729 (H6), 2187 (H7) and 6561 (H8), predicts H's three moments at L = 19,683 — ln S = −6.6218285 ± 0.0032965, mean = 0.6833995 ± 0.0000125, g₁ = +0.4230883 ± 0.0074911 (profile errors, χ² ≤ min + 1 in all nine parameters; the fit is χ² = 12.406 on 12, 1/ν = 0.5260, ω = 0.3009).run
106Bond percolation on the b = 2 diamond hierarchical lattice (each bond replaced at every level by two parallel paths of two bonds; level n has 4^n bonds and "size" L = 2^n).run
107Does the orbit through which the cut-2 wall enters the second ring order the lattice exponent?run
108Does the algebraic connectivity λ₂ of a generator's kept-cell graph (the second-smallest eigenvalue of its graph Laplacian at generation 1) order the lattice exponent 1/ν across the census as well as the cut readings do?run
109For each stored sample, the second crossing t₂ — the fraction of bonds open when left and right are first joined after the single bond that completed the first crossing is deleted and the same bond order continues — and the gap (t₂ − t)/w in widths of the first crossing's pile.run
110On the diamond hierarchical lattices (b, s) with recursion F_{n+1} = 1 − (1 − F_n^s)^b, does the sign of the lean g₁ of the threshold distribution at level n = 24 follow the rule "positive iff b < s" on the three rungs the House named?run
111On the diamond hierarchical lattice with recursion R_{b,s}(p) = 1 − (1 − p^s)^b, is λ(b, s) = R′_{b,s}(p*) equal to λ(s, b) on the pairs (2,5)/(5,2), (3,5)/(5,3), (2,6)/(6,2)?run
112Does the object-side predicate "G owns an object in the 24-component" reproduce the symmetry-side split of the 26 sporadic groups into the happy family (20) and the pariahs (6) exactly, group for group?run
113Does the inherited predicate — "G is the stabiliser, in Co₀ or the Monster, of a named substructure of a 24-object, or sits in a chain of such stabilisers" — reproduce the happy/pariah split of the 26 sporadic groups exactly, on the readers' reports of run 112, with no new reader?run
114For a hierarchical lattice whose rule is a Boolean function f on k independent children, the RG map is R(p) = Σ_j c_j p^j (1 − p)^{k−j} with c_j the number of accepted inputs of weight j, 0 ≤ c_j ≤ C(k, j).run
116Under four kinds — W (the whole Golay code, Leech lattice or V♮), V (a piece of Λ or the Golay code), R (a coordinatisation of Λ), F (the fixed set of a non-identity element or subgroup of M on V♮) — does every happy-family group have at least one kind, with the roots exactly {M₂₄, Co₁, M}?run
117On the exclusive hierarchical rules of run 114 (not-all-equal and the band rules [a, b], k = 2 … 12), does the distance to period three O(R) increase with k along NAE, and is the stretch |λ| at the interior fixed point above 1.8 on every rule that has a period-3 orbit?run
118For which conjugacy classes pX of elements of prime order p in the Monster is the centraliser C_M(pX) a direct product p × S with S a sporadic simple group, and is p = 11 the only such prime whose S is a Mathieu group?run
119On the 19 cut-2 census patterns, does flipping one cell of the 15-cell wedge break boundary isotropy visibly — the left–right and top–bottom transfer operators acquiring different singular values — for every flip that removes the pattern's D4 symmetry, and does the size of the break order by the flipped cell's symmetry class?run
122On census generators run 119 did not build, does a single-cell flip change the block transfer operator T exactly when the cell lies on a simple path between two distinct cells of the left and right columns?run
123Off the entropy plateaus, does the Li–Yorke slack O₂ rank the period-3 band rules at each k as their topological entropy does?run
124Is the Li–Yorke slack O₂ a reading of the period-3 saddle-node's unfolding, ranking the period-3 band rules at each k as the split between their two 3-cycles' multipliers does?run
125On the continuous band family, is the stable-born 3-cycle's multiplier at or below −8 at every point where the map reaches the log 2 ceiling, the crisis R²(c) = q?run
127On the census generators other than generator 1, does the number of one-way pairs of the block transfer operator equal |U|·(511 − |U|), with U fixed by the right-column holes alone?run
128Past k = 30, does the band family's closest approach to −8 at the crisis keep approaching from below at the rate k⁻², while the other cycle approaches +8 at the slower rate k⁻¹?run
129At k = 72 to 128, do the closest approach's four scaled quantities extrapolate to the derived limits, d₋k² → 27.2532, d₊k → 8.28684, (b − a)/√k + k^{−1/2} → 0.931582 and (a + b)/(2k) → 0.683325?run
130A host that keeps the dilations by 3 (H) may carry comb teeth only at the frequencies those dilations allow, the even teeth of the base-64 comb in ln(burst size).run
131On sixteen fresh 9 × 9 blocks, none in the census and half with a kept column-7 cell cut off from the left, does the number of one-way pairs of the block transfer operator equal |U′|·(511 − |U′|), with U′ from the corrected flood-fill rule?run
132On five census blocks no operator run has built (20, 35, 36, 39, 44), does the graph-only core rule say exactly which single-cell toggles give the same block operator?run
133On fresh draws of H at L = 2187, if the comb's even-tooth period P is fitted instead of fixed at 8, do the odd teeth vanish at the fitted period?run
134Does the forbidden k = 3 tooth on H come from the full-side source?run
135On the nine blocks with no one-way pairs, is the block operator reversible in the detailed-balance sense in some diagonal metric?run
136On one row of nine blocks (a 9 × 81 strip) of each of H, U, U′, T and X, how far does chaining the block's left-to-right operator T fall short of the strip's exact crossing?run
138(T) On fresh full-side draws of H8 at L = 6561, does the 3/2 tooth (k = 3) survive every member of the declared set of backgrounds when both steps are modelled?run
034Run 032 identified the glassy walls of run 030 with the minimum-bisection landscape of the ⟨b,c,d⟩-coset graph, a spin glass by Fu–Anderson.run
049The dictionary's §45 claim that phase transitions need infinite ramification is a literature theorem (Gefen–Mandelbrot–Aharony 1980–84), retrodicted, not found.run
082Probe P-049 builds, for any census pattern, the rung-k lattice F_k: the carpet with its block-crossing bonds thinned to k per shared edge at every level, so that adjacent blocks of any level are joined by exactly k bonds.run
090Each sample's threshold is a minimax — the lowest level at which some crossing path is fully open — and its two tails are set by two different optimizations: crossing far below threshold needs the cheapest path, the ring, of length ∝ L; failing to cross far above it needs the cheapest blocking cut, of size ∝ C₁^g = L^(ln C₁/ln 9).run
099On H's two stored sets, with the mean, ln S and g₁ fitted jointly under PG-03's three one-term forms, tying the correction exponent ω to 1/ν costs Δχ² = 11.28 (each file's covariance; 8.54 with diagonal errors) on all five sizes and 1.47 (0.59) with L = 81 left out.run
103For each census pattern with stored widths at 81, 729 and 6561, the stored drift is d = s(81, 729) − s(729, 6561), where s(L₁, L₂) = −[ln w(L₂) − ln w(L₁)] / ln(L₂/L₁) is the local slope of the 20th–80th percentile width w of the per-sample crossing threshold.run
104For each census pattern with stored widths at 81, 729 and 6561, the stored drift is d = s(81, 729) − s(729, 6561), where s(L₁, L₂) = −[ln w(L₂) − ln w(L₁)] / ln(L₂/L₁) is the local slope of the 20th–80th percentile width w of the per-sample crossing threshold.run
115On the twenty happy-family sporadic groups, do the three kinds of substructure they stabilise — V (a piece of the Leech lattice or Golay code), R (a ring or coordinate structure on the Leech lattice), F (the fixed points of an element or small subgroup of the Monster on V♮) — nest as R ⊂ V ⊂ F, group for group?run
120With the distance to period three measured on the map's core in both orientations (O₂), is O₂'s sign exactly the period-3 test on run 114's 286 exclusive rules, and does O₂ increase along the not-all-equal rules?run
121Does O₂'s sign — the Li–Yorke slack on a hump map's core, in either orientation — separate period three exactly on hierarchical rules no run has computed: the NAE and band rules at k = 13 … 16, and random rules from the full box at k = 7 and 8?run
126On the 63 census generators, does the number of one-way pairs of the block transfer operator equal |U|·(511 − |U|), with U fixed by the right-column holes alone?run
137(T) On fresh full-side draws of H8 at L = 6561, does the 3/2 tooth (k = 3) survive every member of the declared set of backgrounds?run
probes49
P-001Chain decay of Ω-class correlations in blocksprobe
P-002Ω-parity organisation of block fluctuations vs. the sieve parity problemprobe
P-003Does the index theorem straddle two licence mechanisms, or reduce to one?probe
P-004Positivity's two roles: does it produce data or remove a hypothesis?probe
P-005Do the two singular limits commute — c → ∞ and ν → 0?probe
P-006Does the chromatic/anonymous discriminator turn the false-friends list into a decision procedure?probe
P-007Does the Differential roster have a seat for the friction itself, or only for its explanation?probe
P-008Is there no exceptional scale for dyadic Chowla sums?probe
P-009Are the two live twin-prime bets two halves of one statement?probe
P-010Determined but uncomputable — a fifth magnitude slot, or a different axis?probe
P-011Is rediscovery count correlated with namelessness?probe
P-012Interferometric access gives shape without magnitudeprobe
P-013Does the T/G/P/M family split have a stated domain?probe
P-014Is QR's two-sidedness a distributive law both ways, or a reciprocity phenomenon in disguise? — RESOLVED: edge, via Artin reciprocityprobe
P-015Are dimensionless constants special values of L-functions?probe
P-017Does Collatz need all primes (an adelic obstruction)?probe
P-018Is causal bracketing an instance of non-associativity, or an interpretive gloss?probe
P-019Does every computable magnitude come through a convex dual?probe
P-020Is the Brauer–Manin obstruction an instance of the anonymous fixed-point obstruction?probe
P-021R1's recorded prior: a sealed bet on a run never executedprobe
P-022Are the topological-existence and constructive-arithmetic halves one phenomenon or two tools?probe
P-023Gap-spacing prediction — 2/5, with the diagnosed fix never builtprobe
P-024Codimension c is classified by π_{c−1} — settled as a coordinate, held pending a novel absenceprobe
P-025Topological protection conserves non-locality — the measure is the distance exponent, and the frontier has an open cornerprobe
P-026Primes upstairs, zeros downstairs — the growth trichotomy as a prime-number-theorem / Riemann-hypothesis dichotomy on the host graphprobe
P-027Can the staircase of a deep linear network's loss be read backwards, as a measurement of a chosen spectrum?probe
P-028Is the period-doubling past the edge of stability Feigenbaum's cascade, and in what dimension does that stop holding?probe
P-029Is the double-descent spike at p = n quantitatively the Marchenko–Pastur edge?probe
P-030Do a primality classifier's residual errors fingerprint its algorithm — Fermat pseudoprimes for a Fermat-like test, no-small-factor composites for a sieve?probe
P-031Grokking multiplication on the units of Z/561 — which characters appear when the group is not cyclic and no discrete log exists?probe
P-032Do a learned Franklin involution's errors concentrate exactly on the pentagonal-number shapes where the map is undefined?probe
P-033Under weight decay, does a partition-recurrence readout become sparse exactly on the generalized pentagonal lags, with alternating-pair signs?probe
P-034Eta's coefficients indexed by n versus by m — is one coordinate trivial and the other near-impossible?probe
P-035Are learning-curve shapes exactly two kinds — spectral readouts and competitions between coexisting solutions?probe
P-036A Tanner code on the squared coset graph — the empirical code ceiling of the intermediate-growth host, now that N-034 rules out an expander code on the graph itselfprobe
P-043A growth axis on one tree — five self-similar groups on the binary tree, the nucleus as the coordinate, and the claim that nucleus size predicts which instruments moveprobe
P-044The seam orbit as a register — capacity and write cost of the host's memory, from the group action instead of quenchesprobe
P-045A quantum walk on the intermediate-growth cell — is the boundary a quasicrystal and the bulk a flat-band localizer, or is the analogy decorative?probe
P-046Is "accidental symmetries are the wildly ramified ones" more than a reading — does ramification theory predict an integrality relation between the charge an operator violates and its dimension?probe
P-047How far is each carpet from a pure composition — is the drift in its pair slopes, and the misplacement of carpet B, a measurement of the commutator between crossing a block and crossing the arrangement of blocks?probe
P-048Taken as true, "the cut constant c sets the one-block miss" contradicts itself across block sizes — so is the block renormalization of an infinitely ramified carpet converging to its percolation exponent at all?probe
P-049The ramification ladder — between the one-bit recursion and the carpet lie finitely ramified relatives on which the crossing recursion is exact; climb them and see where the carpet's turn, and its exponent, first appearprobe
P-050The census read without a question — the one-block fixed point overshoots every lattice threshold by a fixed amount set by the ring share of the pivotal set; the exponents fan around y₉ and are still falling everywhereprobe
P-051The ladder made continuous — give the links across block seams their own chance; the contact set between blocks then has a dimension between 0 (a rung) and 1 (the carpet), and ramification becomes a dialprobe
P-052The dial that varies across the sample — gradient percolation on a carpet; one configuration locates the threshold and its sharpness, and the front should carry the holes' imprint at every factor of nineprobe
P-053A dial for every block — each block has its own crossing point and its neighbours shift it; the shift is a small table with one entry per pair of sides, and how fast the table settles is the correction the one-exponent scaling lacksprobe
P-054Invasion — grow the crossing from the left side by always opening the easiest link; it finds each draw's crossing point without a dial, its forced links and bursts are a second exponent, and the burst sizes on a carpet carry a ripple with the period of the hierarchyprobe
P-055Bits left in the table — compress the census with the one-block polynomial as the codebook and count what stays; PG-01's "finite number of boundary bits" becomes a number read from stored data, and columns that compress together are one objectprobe
notes34 · +6 retired
N-002What the logarithmic derivative can and cannot extractnote
N-006The logarithmic weight is the Haar measure of the scaling flow, not an entropy-step artifactnote
N-007Accidental symmetries are wildly ramifiednote
N-008Two kinds of local-to-global impossibility: chromatic and anonymousnote
N-009Characteristic p is rational flux: a Hofstadter reading of geometric Langlandsnote
N-011Spectral rigidity of a mixing matrix, and its one-bit residue (CP)note
N-012Residues of group actions — resolved by performing them, not by computingnote
N-013Stokes in two signatures — a holonomy equals an enclosed areanote
N-014Resurgence — a perturbatively invisible effect and its instanton are one factnote
N-015BSD as reciprocity, written as a product with one term per rungnote
N-016The braid group is π₁ of configuration space — commutativity as a dimensionnote
N-017Spacetime 3, 4, 6, 10 from the division-algebra determinantnote
N-018Three different threes — the ℤ₃'s in the Standard Model are distinctnote
N-019Large partial quotients are rare with a clean tail -- P(a_n >= k) = log2(1 + 1/k)note
N-020A float continued fraction is reliable only while q_n < 2^26 -- pi diverges at term 13note
N-021The selection-effect probe -- cross-chat evidence ledger for pattern-vs-mechanismnote
N-022Sixth mining pass — the necklace/greedy chat's own rediscovery and correction inventorynote
N-025A non-orientable surface sees only the 2-torsion — and loop-closing is what "quantized" meansnote
N-026The intermediate-growth cell is real — effective dimension grows without bound, computed on the Grigorchuk groupnote
N-028N/log N three ways — one shape, two breaks.note
N-029The PCA section's orientation is arbitrary, and it moves |lam2| by thirteen times run 022's quoted floornote
N-030Evidence for N-029 — which section each filed basis was fitted on, and how fragile the svd_flip convention is on itnote
N-031MPS and CPU training diverge for the same seed in R26_training_dynamics — so a retrain meant to reproduce 037's W_E must run on the same device, and be checkednote
N-034The coset graph is a union of squares, and squares kill unique-neighbour expansion — no expander code on the intermediate-growth host at any levelnote
N-035The sequential-circuit reading of self-similar groups — latch, register, carry, counter — and why "which types exist" is a 3×3 grid, not a linenote
N-036Two routes to zero unique-neighbour expansion on a Cayley graph, both through g² — the Klein-four squares of N-034 and the cosets of ⟨g²⟩ found on Basilica and Aleshinnote
N-037What runs 048–060 say together — order holds, magnitude misses; first-order models rank and correlations set values; the seal is the weakest linknote
N-038A reductio pass over the whole record outside the carpet thread — stale back-links, controls that do not control, agreements built in by construction, two-level laws, and misquoted theoremsnote
N-039Loop-in from the grokking chat, 2026-10-03 — "define the geometry by what the shapes optimise" is a duality bridge run backwards plus the rigidity question, not a new term; candidate §67 entries; two candidate probes on merging grokked seeds; a speculative physics chain with every attribution unverifiednote
N-040Returns — what the lattice does that the block map does not, and in which directionsnote
N-027Harvest record — the anyon-statistics chase (2026-09-27, one sitting)note
cases33
CASE-01Is there a fractional quantum Hall liquid on a thin Sierpinski carpet?case
CASE-02Which "3" is the number of particle families?case
CASE-03Is the global form of the Standard Model gauge group a fact or a convention?case
CASE-04Why does depth fail to be preserved under the local Langlands correspondence for wildly ramified groups, and what is the right definition of depth?case
CASE-05Is "time is a property of the vacuum" defensible?case
CASE-06Why does the weak force act only on left-handed particles?case
CASE-07Completion and algebraic closure don't commute at p; is there a number that measures it?case
CASE-08What geometry supplies the polarization?case
CASE-09Is a spectral symmetry class literally a reality type?case
CASE-10Is a continuous Frobenius structure or artifact?case
CASE-11Does the Chowla–Selberg formula break the sum-rule criterion?case
CASE-12Do the short lists have one generator or two?case
CASE-13Is reversibility load-bearing, or incidental?case
CASE-14Is "every invariant here is linear" a real limitation or a definitional one?case
CASE-15If gauge symmetry is redundancy, why does it have physical consequences?case
CASE-16Is Mostow rigidity an instance of the trace identity or independent of it?case
CASE-17Is the walk-run transition number kinematic or biological?case
CASE-18Is there a space on which a turbulent transfer operator is quasi-compact?case
CASE-19Is a critical exponent a derived quantity or a measured one?case
CASE-20Is "compute one quantity two ways" a single theorem or several?case
CASE-21What sets the magnitude of the annular Reynolds stress relative to the collapse requirement?case
CASE-22What conserved quantity does the signature flip at |a| = 2√q saturate?case
CASE-23When does a spectral gap control mixing?case
CASE-24Does an operator's spectrum determine it, and what repairs the failure?case
CASE-25A one-block polynomial ranks eight lattices' percolation exponents exactly and misses every value — what sets the size of the miss?case
CASE-26A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not?case
CASE-27Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why?case
CASE-28The skewness of the crossing threshold turns back on some carpets between L = 729 and 6561, while the exact one-block recursion's keeps rising — where does the turn come from?case
CASE-29Carpets that share a block polynomial share its rate but not always their threshold — what in the pattern, invisible to the polynomial, sets the threshold?case
CASE-30Every lowest-cut bisection found on a self-similar graph carries exactly the same number of independently flippable vertices — what fixes that number, and is there a ground state without it?case
CASE-31The crossing threshold is the weakest link of the strongest wall — does the number of near-strongest walls set the lean of the threshold pile, and its turn with size?case
CASE-32Does the sign of the Li–Yorke slack on the core separate period three exactly, and why?case
CASE-33Why is M₁₂ the one happy-family group read both as a piece of the Golay object and as a centraliser factor in the Monster, and is that one fact or two?case
panels9
PN-001what geometry supplies the positivity for a Lefschetz-trace argument · run 2026-09-14, derivation roster (physics) · two rounds, one lab step · resolved: openpanel
PN-002A one-block polynomial ranks exponents exactly and misses every value: what sets the miss? · run 2026-09-29, lab roster (physics) · two rounds, one lab step · resolved: openpanel
PN-003A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not? · run 2026-09-29, lab roster (physics) + Reductio (one-off, round 2) · two rounds, one lab step · resolved: openpanel
PN-004Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why? · run 2026-09-30, lab roster (physics) + Reductio (one-off, round 2) · two rounds, one lab step · resolved: openpanel
PN-005The skewness of the crossing threshold turns back on some carpets between L = 729 and 6561, while the exact one-block recursion's keeps rising — where does the turn come from? · run 2026-10-01, lab roster (physics) · one round, closed after House 1, no lab step · resolved: openpanel
PN-006The skewness of the crossing threshold turns back on some carpets between L = 729 and 6561, while the exact one-block recursion's keeps rising — where does the turn come from? · run 2026-10-01, lab roster (physics) + Reductio (one-off, round 2) · two rounds, no lab step · resolved: openpanel
PN-007Capet Lattice Crosses · run 2026-10-04, lab roster (physics) · one round, closed after House 1, no lab step · resolved: localpanel
PN-008Every lowest-cut bisection found on a self-similar graph carries exactly the same number of independently flippable vertices — what fixes that number, and is there a ground state without it? · run 2026-10-05, lab roster (physics) · one round, closed after House 1, no lab step · resolved: openpanel
PN-009PN-nnn — The crossing threshold is the weakest link of the strongest wall — does the number of near-strongest walls set the lean of the threshold pile, and its turn with size? · run 2026-10-20, lab roster (physics) · one round, closed after House 1, no lab step · resolved: localpanel
exemplars14
EX-01a free / quadratic theory — every moment reducible to the two-point function; solvable because nothing interactsexemplar
EX-02the model compact group — U(1); Fourier analysis and Pontryagin duality at their self-dual, rank-one base caseexemplar
EX-03the model reciprocity law — local symbols at every place multiplying to oneexemplar
EX-04"the sign is a homomorphism" — one ℤ/2 invariant realized identically as permutation parity, a determinant sign, and a Legendre symbolexemplar
EX-05a quantity determined only up to the action of a group G — no calculation fixes it; you perform a G-element and observe, or find independent structure that breaks Gexemplar
EX-06a determined quantity certifiable from one side only — which end of an interval is open is fixed by which direction admits a finite certificate (Σ₁ vs Π₁)exemplar
EX-07a spectrum forgets the shape (isospectrality); minor spectra remember the eigenvector moduli (the eigenvalue–eigenvector identity) — rigidity recovered up to a residueexemplar
EX-08compute one quantity two ways — Σλ = ∫K(x,x) — a spectral sum equal to a geometric integral, because the trace is conjugation-invariantexemplar
EX-09the structure is unique, so any invariant at all determines it — a second-order "the moduli space is a point" that licenses a determination without performing oneexemplar
EX-10an independent axiom is a residue — undetermined by the rest, certifiable only by exhibiting a model, resolved not by proof but by choosing among consistent completions (Euclidean / hyperbolic / spherical = flat / negative / positive curvature, the three constant-curvature homogeneous geometries)exemplar
EX-11the abelian-to-nonabelian transition read through commutativity-as-flatness — an abelian reciprocity is a flat symmetry-action of a group of characters, and turning the symmetry side nonabelian (curved) is the general stepexemplar
EX-12a method that recovers the correct quantity but is blind to its place — the value is right, the location or global form unexhibitableexemplar
EX-13local-to-global reciprocity as one object: the invariants at all places, combined, cancel to the identityexemplar
EX-14the right coordinate makes a structure obvious — a change of chart that trivializes what looked hard and reveals the invariant that was always thereexemplar
generated 2026-10-25nothing on this page is maintained by hand