The length spectrum of q-discrete Painlevé I over a finite field: an exact form of the bin conjecture at prime q, and verification past the published range

Compute-first gate: every number the source prints reproduced – the q=2, q=3 and q=4 orbits, the full q=5 frequency lists L₁, L₂, L₄, and the q=499 bin frequencies 113831/54446/13551/19140/6935/42097 summing to 250000

M_q and the bins Bₘ of Conjecture 1.2.B in exact integer arithmetic (no square roots), reproducing the source's printed q=499 bins [456,544], [228,272], [152,181], [114,136], [92,108], [1,90]

The maximum reduced orbit length is exactly q + 1 + maxd: d² < 4q, the largest integer strictly below the Hasse bound, at every prime power q <= 131 except q = 128 – and at q = 503 and q = 1009

Negative controls: the 2 sqrt q term cannot be dropped (reduced length 7 > q+1 at q = 3); the sharpness statement is not a theorem for all q (at q = 128 the maximum is 150, not 151); the bins are strictly stronger than the Hasse bound (78 passes the bound at q = 131 but lies in no bin); and the sweep is not vacuous

q-Painleve I over F₅03: Conjectures 1.2.A and 1.2.B hold for all 3 534 638 orbits, over every s in F₅03^* and every coset, with maximum reduced orbit length 548 and minimum 1

The kernel band: Conjectures 1.2.A and 1.2.B for every orbit over F₂, F₃, F₅, F₇ with no native_decide axiom, plus a cross-check that the array engine and the array-free kernel engine return the same maxima

What the integer checks mean, and two facts at every q: hasseOK is equivalent to #(gamma)/r <= q + 2 sqrt q + 1, inBin is exactly the integer test (mL-(q+1))² <= 4q and that test is membership in Bₘ, consecutive bins below M_q are disjoint, and ord(s) divides the period of any state with nonzero time over an arbitrary field

The source's Magma verification re-run from scratch: Conjectures 1.2.A and 1.2.B for every orbit at all 45 prime powers q <= 131 (44 in one sweep, q = 128 separately), 3 327 974 orbits, on an independent implementation

q-Painleve I over F₁009, more than twice the source's range: Conjectures 1.2.A and 1.2.B hold for all 5 756 256 orbits with s among the twelve smallest elements of F₁009^*, M_q = 8, maximum reduced orbit length 1073 = hasseStrictMax 1009

Conjecture 1.2.B is an equality for prime q: the set of realised reduced orbit lengths is exactly the union of the M_q bins with 2 and 3 removed, at every prime 3 <= q <= 131; and this fails at proper prime powers (4, 9, 25, 128)