Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

Arc from 049 and N-026 and P-025

roots 049 · N-026 · P-025 · 133 records · depth 35 · 22 unanticipated · 33 falsified · 36 confirmed · 11 withdrawn · index density (§5)

The chain is read from builds_on: (docs/OUTTAKE.md, Cross-record fields). Every id, question, sealed prediction, verdict and count below comes from the records at a354234.

Concept not yet on the map. biconnectivity, chaos, correction-to-scaling, crisis, detailed-balance, duality, edgeworth, entropy, finite-boundary-state, fixed-point, forcing, fourier-duality, hierarchical-lattice, horseshoe, invasion-percolation, isotropy, kneading, kolmogorov-criterion, log-periodicity, min-cut, multipliers, network-reliability, percolation, period-three, renormalisation, reversibility, saddle-node, scaling-limit, spectral-dimension, symmetry, transfer-operator appear in concepts: on records in this arc and are not among the 58 ids domain-math.js defines, so the map has nowhere to place this arc yet. Reported here rather than dropped; it does not fail the build.

049 carpet-room-or-dimension
The dictionary's §45 claim that phase transitions need infinite ramification is a literature theorem (Gefen–Mandelbrot–Aharony 1980–84), retrodicted, not found.
WITHDRAWNbuilds on root
N-026 note (no mark: notes carry no condition)
The intermediate-growth cell is real — effective dimension grows without bound, computed on the Grigorchuk group
builds on root
P-025 probe
Topological protection conserves non-locality — the measure is the distance exponent, and the frontier has an open corner
builds on root
030 ising-barrier-intermediate-growth
On 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?
UNANTICIPATEDbuilds on N-026 · P-025
051 carpet-class-reseal
Run 049 was withdrawn on its control (the sealed p-grid could not resolve the square lattice at L = 2187).
CONFIRMEDbuilds on 049
P-026 probe
Primes upstairs, zeros downstairs — the growth trichotomy as a prime-number-theorem / Riemann-hypothesis dichotomy on the host graph
builds on N-026
P-043 probe
A 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 move
builds on N-026 · P-025
031 grig5-bisection
Does 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?
CONFIRMEDbuilds on 030
032 glassy-wall-anatomy
Run 030 found two metastable families of near-minimal cuts on Cay(G/St(4)) (≤ 224 edges from METIS, 270 from disorder).
FALSIFIEDbuilds on 030 · N-026
033 host-primes-cogrowth
The Ihara primes of the host are its relators.
CONFIRMEDbuilds on 030 · P-026
035 kuramoto-host-sync
Expanders are globally synchronising (the Kuramoto flow has no stable equilibrium but synchrony — no twisted states, no U(1) loop-closing sectors).
FALSIFIEDbuilds on 030
036 expansion-profile-coset-graph
Expander codes (Sipser–Spielman, Tanner) need unique-neighbour expansion of small sets; amenability kills expansion only asymptotically.
CONFIRMEDbuilds on 030
052 carpet-cut-sets
Run 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.
CONFIRMEDbuilds on 051
N-035 note (no mark: notes carry no condition)
The sequential-circuit reading of self-similar groups — latch, register, carry, counter — and why "which types exist" is a 3×3 grid, not a line
builds on P-043
034 glass-barrier-coset-graph
Run 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.
WITHDRAWNbuilds on 032
038 level5-spectral-rounding
At 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.
FALSIFIEDbuilds on 031
041 unique-neighbour-expansion
Run 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.
UNANTICIPATEDbuilds on 036 · P-026
053 cut-exponent-sets-class
Runs 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.
UNANTICIPATEDbuilds on 052 · 051
P-045 probe
A quantum walk on the intermediate-growth cell — is the boundary a quasicrystal and the bulk a flat-band localizer, or is the analogy decorative?
builds on N-035
039 two-band-skeleton-level5
At 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.
FALSIFIEDbuilds on 038
040 t0-barrier-extensive
Run 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.
UNANTICIPATEDbuilds on 034 · 031
043 growth-axis-on-one-tree
Five 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.
FALSIFIEDbuilds on 041 · P-043
044 quantum-walk-on-the-cell
The 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.
FALSIFIEDbuilds on P-045
054 min-cut-count-orders-class
Run 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).
CONFIRMEDbuilds on 053 · 052
N-034 note (no mark: notes carry no condition)
The coset graph is a union of squares, and squares kill unique-neighbour expansion — no expander code on the intermediate-growth host at any level
builds on 041
042 valley-wall-is-local-global
Two "two-flavor" splits have appeared on the intermediate-growth host.
UNANTICIPATEDbuilds on 036 · 040
045 quantum-walk-hecke-operator
Run 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).
FALSIFIEDbuilds on 044
055 cut-entropy-predicts-class
Run 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.
UNANTICIPATEDbuilds on 054 · 053
064 spectral-decimation-finitely-ramified
Probe P-048 argues that block renormalization fails to converge on the Sierpiński-type carpets because they are infinitely ramified (blocks share whole edges).
FALSIFIEDbuilds on 039 · 038 · 031
088 abelian-twin-control
Run 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.
CONFIRMEDbuilds on 044 · 043
N-036 note (no mark: notes carry no condition)
Two 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 Aleshin
builds on 043 · N-034
046 quantum-walk-anisotropic
Runs 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.
CONFIRMEDbuilds on 045
056 estimator-run-to-run-spread
The 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.
CONFIRMEDbuilds on 055 · 053
065 schreier-map-control
Run 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).
CONFIRMEDbuilds on 064
101 exact-part-on-fresh-minima
On 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.
CONFIRMEDbuilds on 042 · 040 · 031
CASE-30 case
Every 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?
builds on 042 · 040 · 031
P-036 probe
A 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 itself
builds on 041 · N-036
P-044 probe
The seam orbit as a register — capacity and write cost of the host's memory, from the group action instead of quenches
builds on 042 · N-035
047 growth-axis-in-transport
Run 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.
UNANTICIPATEDbuilds on 046 · 043
048 seam-orbit-as-register
Run 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.
FALSIFIEDbuilds on 040 · 042 · P-044
057 bsd-euler-product-slopes
Birch 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.
CONFIRMEDbuilds on 056 · 055
050 frozen-core-orbit
Run 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.
FALSIFIEDbuilds on 048
058 reliability-polynomial-rg
Runs 054–055 found that the number of distinct minimum cuts orders the carpets' percolation exponents.
CONFIRMEDbuilds on 055 · 054 · 057
059 commutator-and-bias
Run 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.
FALSIFIEDbuilds on 058 · 056
060 crossing-polynomial-zeros
Run 058's crossing polynomials R(p) are exact.
FALSIFIEDbuilds on 058
P-047 probe
How 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?
builds on 058 · 057
061 own-fixed-point-and-new-rooms
Run 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.
FALSIFIEDbuilds on 059 · 058 · 055
CASE-25 case
A one-block polynomial ranks eight lattices' percolation exponents exactly and misses every value — what sets the size of the miss?
builds on 058 · 059 · 060
N-037 note (no mark: notes carry no condition)
What runs 048–060 say together — order holds, magnitude misses; first-order models rank and correlations set values; the seal is the weakest link
builds on 060 · 059 · 058 · 057 · 056 · 055 · 053 · 050 · 049
062 cut-recursion-constant
On 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.
FALSIFIEDbuilds on 061 · 058 · 055 · 054
066 crossing-sequence-turns-over
The block exponents y_g = ln P_g′(p_g*)/ln 9^g rose from g = 1 to g = 2 on every lattice (run 061).
CONFIRMEDbuilds on 061 · 059 · 056 · 055 · 053 · 051
063 cut-twins
Run 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).
CONFIRMEDbuilds on 062 · 061 · 058
P-048 probe
Taken 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?
builds on 062 · 061 · 058
067 stacked-carpets
A carpet may use a different 9×9 pattern at each level.
UNANTICIPATEDbuilds on 063 · 061 · 062
069 crossing-curve-fingerprint
Feigenbaum's universality lives in a function, not only in δ: the fixed point of the doubling operator.
FALSIFIEDbuilds on 063 · 066 · 051
CASE-26 case
A slowly converging sequence of block estimates ranks exponents exactly at every block size — why does the order survive when the values do not?
builds on 066 · 063 · 061 · 058
N-038 note (no mark: notes carry no condition)
A 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 theorems
builds on N-037 · P-048
P-049 probe
The 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 appear
builds on P-048
068 census-of-the-family
The 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.
FALSIFIEDbuilds on 067 · 063 · 062 · 053
070 cross-class-stacks
Run 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.
UNANTICIPATEDbuilds on 067 · 053 · 051
072 skewness-fingerprint
Run 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.
CONFIRMEDbuilds on 069 · 063
CASE-27 case
Stacking two carpets averages their exponents when their cut classes match and falls below the average when they do not — why?
builds on 067 · 063 · 053 · 051
P-050 probe
The 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 everywhere
builds on P-048 · P-049
P-051 probe
The 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 dial
builds on P-049
071 bands-and-sub-bands
Run 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).
FALSIFIEDbuilds on 068 · 063 · 054
073 cumulant-flow
Run 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).
FALSIFIEDbuilds on 072 · 069 · 063
076 two-map-deviation
Stacking 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).
FALSIFIEDbuilds on 070 · 067 · 068 · 058
P-052 probe
The 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 nine
builds on P-050
P-053 probe
A 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 lacks
builds on P-049 · P-050
074 five-size-skewness
Run 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.
UNANTICIPATEDbuilds on 073 · 072 · 051
075 exact-map-skewness
Each 9×9 pattern has an exact one-block map, the crossing polynomial f(p) of run 058 (rg058.poly).
CONFIRMEDbuilds on 073 · 068 · 058
080 census-completion
43 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).
UNANTICIPATEDbuilds on 071 · 068 · 076
P-054 probe
Invasion — 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 hierarchy
builds on P-053 · P-048
P-055 probe
Bits 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 object
builds on P-050 · P-053
077 order-not-amplitude
Feigenbaum's universality sorts one-hump maps by the order z of their maximum; the amplitude of the maximum drops out.
UNANTICIPATEDbuilds on 076 · 075 · 068
079 generation-parity-reseed
Run 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.
FALSIFIEDbuilds on 074 · 073 · 051
086 polynomial-twins
The 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.
FALSIFIEDbuilds on 080 · 063 · 068
093 channels-and-chains
The crossing polynomials of the 49 census fibres carry squares and a composition that nothing in the record explains.
CONFIRMEDbuilds on 075 · 080 · 058
103 census-curvature-at-2187
For 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.
WITHDRAWNbuilds on 080 · 071 · 068 · 051
104 census-curvature-at-2187-resealed
For 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.
WITHDRAWNbuilds on 080 · 071 · 068 · 051
108 generator-spectrum-orders-exponent
Does 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?
CONFIRMEDbuilds on 068 · 071 · 080
078 curvature-or-order
Runs 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.
UNANTICIPATEDbuilds on 077 · 076
081 turn-reseed
Run 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.
UNANTICIPATEDbuilds on 073 · 072 · 079
082 ladder-rung-one
Probe 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.
WITHDRAWNbuilds on 075 · 073 · 079
087 risers-reseed
P-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).
UNANTICIPATEDbuilds on 080 · 071 · 079
CASE-28 case
The 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?
builds on 079 · 075 · 074 · 073 · 072
083 coarse-crossing-split
PN-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.
UNANTICIPATEDbuilds on 073 · 075 · 081
084 ladder-first-scorable-rung
Run 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.
FALSIFIEDbuilds on 082 · 075 · 073
092 one-sampler-seed-spread
The 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.
CONFIRMEDbuilds on 081 · 073 · 070 · 053 · 063
096 h-width-curvature-reseed
Scaffold 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.
CONFIRMEDbuilds on 079 · 074 · 051 · 087
098 pendant-parts-size
Within a group of patterns that share one core, how much does a pendant part lower the lattice threshold at L = 729?
CONFIRMEDbuilds on 068 · 071 · 086 · 087
CASE-29 case
Carpets that share a block polynomial share its rate but not always their threshold — what in the pattern, invisible to the polynomial, sets the threshold?
builds on 086 · 087 · 080 · 071 · 068
085 census-coarse-lag
Run 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%).
FALSIFIEDbuilds on 083 · 080 · 071
089 step-map-and-koenigs
The 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.
FALSIFIEDbuilds on 075 · 073 · 083
094 rung-three-on-rows-246
Run 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.
CONFIRMEDbuilds on 084 · 082 · 073
099 tied-form-on-fresh-draws
On 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.
WITHDRAWNbuilds on 096 · 079
107 cut2-wall-entry-orbit
Does the orbit through which the cut-2 wall enters the second ring order the lattice exponent?
UNANTICIPATEDbuilds on 068 · 071 · 080 · 098
090 tail-asymmetry
Each 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).
WITHDRAWNbuilds on 085 · 083 · 075 · 073 · 072 · 054
095 invasion-bursts-hierarchy-period
P-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.
UNANTICIPATEDbuilds on 083 · 085 · 079
097 block-table-and-laggard-gap
Run 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.
FALSIFIEDbuilds on 083 · 085 · 094 · 092
100 tied-form-on-fresh-draws-resealed
On 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.
CONFIRMEDbuilds on 099 · 096 · 079
109 second-crossing-gap
For 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.
UNANTICIPATEDbuilds on 107 · 092 · 081 · 073 · 054
119 isotropy-broken-by-one-cell
On 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?
FALSIFIEDbuilds on 107
091 tail-asymmetry-reseal
Each 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).
UNANTICIPATEDbuilds on 090 · 085 · 083 · 075 · 073 · 072 · 054
102 h-2187-tenfold
The 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).
CONFIRMEDbuilds on 100 · 096 · 079
105 h-19683-prediction
The 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).
CONFIRMEDbuilds on 100 · 096 · 079
122 what-the-block-operator-sees
On 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?
CONFIRMEDbuilds on 119
130 the-comb-keeps-the-surviving-teeth
A 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).
CONFIRMEDbuilds on 095
N-040 note (no mark: notes carry no condition)
Returns — what the lattice does that the block map does not, and in which directions
builds on 083 · 085 · 086 · 094 · 095
106 diamond-rung-threshold-moments
Bond 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).
FALSIFIEDbuilds on 102 · 105
126 one-way-pairs-from-geometry
On 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?
WITHDRAWNbuilds on 119 · 122
132 the-operator-sees-the-core
On 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?
CONFIRMEDbuilds on 119 · 122
133 the-comb-at-its-own-period
On 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?
UNANTICIPATEDbuilds on 130
CASE-31 case
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?
builds on 107 · 105 · 102 · 092 · 081 · 073 · 071 · 054 · 072
110 diamond-grid-sign-rule
On 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?
FALSIFIEDbuilds on 106
127 one-way-pairs-from-geometry-resealed
On 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?
FALSIFIEDbuilds on 119 · 122 · 126
134 the-forbidden-tooth-and-the-source
Does the forbidden k = 3 tooth on H come from the full-side source?
UNANTICIPATEDbuilds on 133
111 diamond-lambda-symmetry
On 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)?
CONFIRMEDbuilds on 110 · 106
131 one-way-pairs-on-fresh-blocks
On 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?
CONFIRMEDbuilds on 127
137 the-comb-and-its-step-at-6561
(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?
WITHDRAWNbuilds on 133 · 134
114 exclusive-rule-period-three
For 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).
FALSIFIEDbuilds on 111
135 curvature-on-the-faces
On the nine blocks with no one-way pairs, is the block operator reversible in the detailed-balance sense in some diagonal metric?
CONFIRMEDbuilds on 127 · 131
136 the-strip-and-its-back-flow
On 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?
CONFIRMEDbuilds on 051 · 131 · 132
138 the-comb-its-moving-step-and-a-fixed-one-at-6561
(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?
OPENbuilds on 133 · 134 · 137
117 stretch-before-period-three
On 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?
FALSIFIEDbuilds on 114
120 distance-to-period-three-on-the-core
With 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?
WITHDRAWNbuilds on 117 · 114
121 o2-separation-out-of-sample
Does 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?
WITHDRAWNbuilds on 120 · 114
123 slack-against-entropy
Off the entropy plateaus, does the Li–Yorke slack O₂ rank the period-3 band rules at each k as their topological entropy does?
FALSIFIEDbuilds on 114 · 120 · 121
CASE-32 case
Does the sign of the Li–Yorke slack on the core separate period three exactly, and why?
builds on 114 · 117 · 120 · 121
124 the-margin-is-the-unfolding
Is 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?
FALSIFIEDbuilds on 120 · 121 · 123
125 the-ceiling-at-the-crisis
On 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?
CONFIRMEDbuilds on 123 · 124
128 the-approach-to-minus-eight
Past 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?
CONFIRMEDbuilds on 125
129 the-constants-at-the-crisis
At 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?
CONFIRMEDbuilds on 125 · 128

133 records, longest path 35. The vertical bar in the gutter marks a record with more than one parent. A withdrawn or retired record keeps its place: struck id, retired ink, still on the page. A probe or note has no question:, prediction: or verdict: field, so its row carries its title and nothing is invented to fill the gap. A row shows a whole sentence or none. Print variant (Paper Pro, DESIGN-SYSTEM §7).

generated 2026-10-25the lab index