Primes upstairs, zeros downstairs — the growth trichotomy as a prime-number-theorem / Riemann-hypothesis dichotomy on the host graph

Claim. For a group G with a finite symmetric generating set, the Ihara "primes" of the infinite Cayley graph are its relators (primitive closed non-backtracking cycles), and two theorems make their density and the finite quotients' spectra one fact seen from two sides:

• Upstairs (Kesten 1959; Grigorchuk 1980, Cohen 1982). The relator count R(m) grows at the free rate qᵐ — the graph prime number theorem at full density — iff G is amenable. Non-amenable G has cogrowth rate strictly below q: its primes are exponentially thin.
• Downstairs (Ihara; Alon–Boppana). A finite (q+1)-regular graph's Ihara zeta satisfies its Riemann hypothesis iff the graph is Ramanujan, |λ| ≤ 2√q. A sequence of finite quotients of an amenable group has spectral gap → 0 and fails it maximally; quotients of a free group can satisfy it (LPS, via Eichler–Deligne's Ramanujan–Petersson — the same conjecture run 029's Maass forms leave open on the real-analytic side).

So along the growth trichotomy of P-025's grid: polynomial and intermediate growth (amenable) = primes dense upstairs, zeros off the line downstairs; exponential/non-amenable = primes thin upstairs, and the quotients can put every zero on the line. The expander cell that carries the good qLDPC codes is the prime-free one (universal cover a tree). The intermediate cell's host has primes at full density with a sub-exponential deficit, exp(−c·m^β) with β ≈ 0.35–0.5 at accessible lengths (run 033, confirmed: κ = 0.665), and its quotients' gap is a path's, 1/log²N (run 030, N-026 correction 3).

Why it is filed as a probe and not a note. The two theorems are old; the claim is that they are one coordinate of the grid — that "how the graph RH fails" and "how dense the relators are" are the same axis as the growth type, and that the self-correction question of P-025 (rigid walls without a free-energy collapse) lives on that axis. The instance where this earns something: a host whose quotients fail the RH logarithmically (gap 1/log²N, cut N/log N) rather than polynomially — the intermediate cell — sits between the lattice and the expander on every one of the three readings at once. Whether a code can use that intermediate failure is N-026's open question.

What would falsify the reading as a coordinate. A finitely generated amenable group whose finite quotients are Ramanujan-like along a sequence (gap bounded below) — impossible by Kesten if the quotients converge to the group, so the falsifier is really: a non-amenable group whose Cayley graph's relator count reaches the free rate along a subsequence of lengths. Kesten–Grigorchuk forbid it. The claim is therefore a theorem restated as a map coordinate, and the only open part is the rate: the exponent β of the deficit on the intermediate host, and whether the quotients' pole distribution (Ihara zeta of Cay(G/St(n))) can be predicted from it.

Not claimed. Anything about the Riemann hypothesis over ℤ. The classical 1/log x is Mertens' product over sieving primes; nothing here touches it. The N/log N of the host's separator (run 031, sealed) is orbit–stabiliser counting, the function-field shape, and the "mirror image" the human noticed between orbit counting and stabiliser counting is the geometric/spectral duality of the trace formula (Ihara = Selberg on graphs), not a dual group.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

Runs 043 and 047 found the ordering follows amenability, not the growth trichotomy; Basilica (exponential growth, amenable) is the twin that separates them (N-038 §2).

• Run 029 — falsified. lab/runs/029-artin-billiard-maass.md: FALSIFIED — ⟨r⟩ = 0.527 ± 0.033 on the 72 odd level-1 levels in R ∈ [9, 49.98]: 4.3σ from Poisson 0.38629 and within 1σ of GOE 0.5307 (large-N) and 0.53590 (3×3 surmise).
• Run 033 — confirmed (2026-09-27, measured). lab/runs/033-host-primes-cogrowth.md: CONFIRMED — κ = 0.665.
• Run 041 — unanticipated (2026-09-28, measured). lab/runs/041-unique-neighbour-expansion.md: Clause 1 withdrawn (control band miscalibrated;
• Run 043 — falsified (2026-09-28, measured), via the builds_on thread. lab/runs/043-growth-axis-on-one-tree.md: FALSIFIED — the relator clause's falsifier fires: Basilica's unique-neighbour minimum at N/8 is 1 of its control's 101 at best-of-20 and 0 of 94 at best-of-40, u/u_random ≤ 0.10, so the square obstruction of N-034 is not specific to the Klein-four contraction;

Bridge to the carpet thread, 2026-09-29 (codifier). The carpet runs (049 onward; probe P-048) grew from a separate root — the dictionary's §43/§45, not this cell — but most of what they measure has an exact code reading, and results on either side should be read on the other. The translation: minimum cut C₁^g ↔ code distance (weight of the shortest crossing logical); min-cut exponent ln C₁/ln n ↔ distance growth d ~ L^{ln C₁/ln n}; number of minimum cuts ↔ number of minimum-weight logicals (the sub-threshold prefactor); percolation threshold ↔ erasure threshold (exactly 1/2 on the self-dual plane); 1/ν ↔ the sharpening of the failure curve at threshold (the same (p − p_c)·L^{1/ν} collapse); finitely ramified (no transition) ↔ bounded cuts, hence bounded distance, hence no growing code. What the carpets have already said: run 063 (confirmed) — twins with identical cut counts at every generation (same distance growth, same minimum-logical counts) differ measurably in threshold and exponent, so distance and logical counts do not fix behaviour near threshold; runs 064 (falsified) and 065 (confirmed) — this cell's ⟨b,c,d⟩-coset tower is not an exact spectral decimation, while the Schreier graphs are finitely ramified and so cannot host a growing code; run 066 (confirmed) — the carpets' block estimates converge slowly, from above. Open, and cheap when this cell wakes: the erasure threshold of the coset graphs measured the carpet way, primal and dual separately (on a non-self-dual host they must differ), with the reseed floor and the ratio estimator of P-048. Run 067 (stacked carpets) is listed in related: so the back-link check flags this probe until its outcome is read here.

Carpet-thread result, 2026-09-29 (late). Run 067 unanticipated. In code terms: stacking two hosts whose distances grow alike gives a composite whose threshold exponent is the average; stacking a bottleneck host (bounded-width cuts, distance growing slowly) with a wide one pulls the composite below the average — the bottleneck levels count for more than their share. For the empty cell: a host built by mixing growth types level by level should not be expected to inherit the mean of its parents' behaviour near threshold.

Carpet-thread results 068–079, read in code terms, 2026-10-01 (codifier). The translation is the bridge note's: minimum cut C₁ ↔ code distance per level, number of minimum cuts N₁ ↔ number of minimum-weight logicals, threshold ↔ erasure threshold, 1/ν ↔ how fast the failure curve sharpens with size.

• Correction to the 067 reading (run 070, unanticipated). "A bottleneck level counts for more than its share" did not survive. Three cross-class stacks gave three answers: X∘U′ below its parents' mean, X∘T and U′∘T above. Mixing host types level by level has no single rule, not even in sign. At L ≤ 6561 the two build orders of one stack can differ by more than the effect (0.023 on U′,T). Comparisons are finite-size limited before they are physics.
• Distance does not fix the sharpening; minimum logicals matter jointly (runs 068, 071). At fixed distance growth, more minimum-weight logicals means slower sharpening (cut 4: Spearman(N₁, 1/ν) = −0.77). With enough of them, a higher-distance host behaves like a lower-distance one near threshold: pattern #20 (C₁ = 4, 324 minimum cuts) measures 0.408, inside the cut-3 band. The classes overlap. Within one (C₁, N₁) cell the exponent still spreads (0.303–0.349), so even distance plus minimum-logical count does not fix it. The exact one-block exponent y₉ carries most of the rest.
• The failure curve is skewed, and the bottleneck sets the skew (runs 072–075). The threshold distribution is right-skewed on the carpets and symmetric on the plane. In the exact one-block map, the sign of the limiting skewness is set by the minimum cut against the shortest crossing (63 of 63). Inkling, held with conviction: in code terms the shortest crossing is the other logical type's weight, so a host whose two logical types have unequal weight has an asymmetric failure-threshold distribution, skewed toward the lighter side. A self-dual host should have none; the square shows none. On the lattice the skewness turns back on some hosts past L = 729 while the map's does not. That is filed as CASE-28.
• Concatenating two levels (runs 076–078). For the one-block composite, the exponent's deviation from the parents' mean is first order in how far apart the two thresholds are, weighted by how differently the two failure curves bend there. Distance class matters mostly through that bending (curvature mismatch), with a small direct residue (+0.185). The amplitude per level (minimum-logical count per unit distance) drops out across classes. In code terms: when two codes are concatenated, near threshold the composite inherits the mismatch in their failure-curve shapes more than the mismatch in their distances.
• No hidden level-parity term (run 079, falsified). At six times the samples, the factor-3 host H shows no alternation with generation in either skewness or sharpening. A reading built on it does not stand.

The empty cell, as the carpets bound it. Cut exponents ln C₁/ln 9 fill the interval from 0 (the gasket: finitely ramified, bounded distance, no growing code) up toward 1 (the plane: distance ~ L). Removing cells only thins cuts, so every carpet sits on the sub-Euclidean side, and no carpet approaches the expander side (distance growing with volume). Conviction, with where it breaks: the cell between Euclidean and expander hosts is not reachable by removing material from a planar block. It needs adding connectivity across scales (hyperbolic or product constructions). Where it breaks: a carpet's threshold behaviour is set by more than its distance exponent (the bullets above). So "sub-Euclidean in distance" need not mean "sub-Euclidean near threshold", and that is exactly what this thread's next host should test: the erasure threshold of the coset graphs, measured the carpet way, primal and dual separately. Run 080 (census completion) is in related: until its outcome is read here.