The Ihara primes of the host are its relators. Kesten–Grigorchuk say their density is the free density (cogrowth maximal) iff the group is amenable, with the deficit a sub-exponential "error term". Is that error term visibly sub-exponential at word length ≤ 48 on the Grigorchuk group?

Sealed before the count was run (2026-09-27, codifier). Below 0.45 is unanticipated.

Why this is the push on the graph prime number theorem. In a (q+1)-regular graph the primitive backtrackless cycles of length m number ~ qᵐ/m — N/log N once more — with the error term set by the Ihara poles, i.e. by whether the graph's Riemann hypothesis holds. For the infinite Cayley graph of a group the primes are the relators, and their count per vertex is R(m) up to the tail correction. Kesten (spectral radius 1 iff amenable) with Grigorchuk's cogrowth theorem (cogrowth rate maximal iff amenable) says: the graph PNT holds at full density qᵐ upstairs iff the group is amenable — and those are exactly the groups whose finite quotients fail the graph RH. Expanders are the mirror: the universal cover is a tree with no primes at all, and the quotients can be Ramanujan. Primes dense upstairs, zeros off the line downstairs; primes absent upstairs, zeros on the line. This run measures the "error term" on the intermediate-growth host, where the density is full but approached sub-exponentially.

Controls, reported not scored. Two Coxeter groups on four involutions in the integral Tits representation, same reduced-word normalisation: Ã₃ (4-cycle, m = 3 on neighbours, 2 on opposites; virtually ℤ³, amenable, polynomial growth — expect κ ≈ 0.6 from p ~ m^{−3/2}) and the same graph with one opposite pair set to m = 3 (non-affine, contains F₂, non-amenable — expect κ closer to 1, but a polynomial prefactor keeps it below 1 at this scale, which is why the control is not the seal).

Also reported. The tailless counts C(m) (cyclically reduced relators — the Ihara primes proper), the local exponent γ(m) = log₃ R(m) / m, and the shortest relators found by length.

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Measured, same evening

Per-step decay of ln p over the windows 8→16, 16→24, 24→32, 32→40, 40→48: Grigorchuk 0.119, 0.077, 0.064, 0.057, 0.051; Ã₃ 0.122, 0.070, 0.051, 0.040, 0.033; H 0.181, 0.157, 0.140 (to 32). κ = 0.665 for the Grigorchuk group (band [0.45, 0.85]); 0.47 for Ã₃; 0.77 for H on the shorter matched windows, where the Grigorchuk value is 0.54.

Shortest relators: Grigorchuk R(3..8) = 6, 6, 12, 36, 78, 272 (bcd and its rotations at length 3; the group has odd relators); Ã₃ has relators at even lengths only, R(4) = 4 (the two commuting pairs), R(6) = 16; H has R(4) = 2 (one commuting pair), R(6) = 14.

Reading. At every accessible length the three groups have relator-density exponents within 0.1 of each other (0.77–0.88): the level does not separate amenable from not, as the seal's prose said it would not. The trend does: the two amenable groups' decay rates fall steadily toward zero — the polynomial one faster than the intermediate one, as p ~ m^{−3/2} against exp(−c m^β) requires — while the non-amenable control's rate is two to three times larger at the same lengths and, by Kesten, must level off at −ln ρ > 0. So the host's primes obey the graph prime number theorem at full density with a sub-exponential error term, which is the amenable signature, and the same amenability is why the quotients' Ihara zeta fails its Riemann hypothesis (run 030 correction, N-026 correction 3: the gap is a path's). That is the pushed analogy, measured on one side.

What would sharpen it. Fit β: log(−d ln p/dm) against log m over m ∈ [16, 48] gives the stretched exponent directly (the five windows give slope −0.5 to −0.65, i.e. β ≈ 0.35–0.5, from a coarse fit — not sealed; for Ã₃ the same fit gives the 1.5/m of a ℤ³ walk exactly: 0.034 predicted, 0.033 measured at m = 44). And the finite quotients' side: the Ihara zeta of Cay(G/St(4)) from its adjacency spectrum, its non-trivial poles against the Ramanujan circle, against the relator count R(m) for m below the systole 16, where the quotient's primes are the group's own.