Block systems for the SL(2,ℤ)-action on orbits of primitive origamis, and maximal Veech groups in H(2) and H(1,1)

H(2): complete enumerations at n = 3..7 – 3, 9, 27, 36, 90 primitive origamis in 1, 1, 2, 1, 2 orbits; and the source's own Figure 2.1 graph G₃

H(2): the Hubert-Lelievre A/B cardinalities at n = 5, 7, 9, 11 – 18/9, 54/36, 108/81, 225/180 – reproduced from the definitions and matched against the formula

H(1,1) below Zmiaikou's threshold: empty at n = 2, 3, one orbit at n = 4, 5, and two orbits of sizes 24 and 24 at n = 6 whose monodromy groups are Alt(6) and a primitive group of order 120 – not Sym(6)

Zmiaikou's Conjecture 1 at n = 7, in full: exactly 160 primitive 7-square origamis in H(1,1), falling into exactly two SL(2,Z)-orbits of cardinalities 144 and 16; the parity invariant identifies which orbit is which

H(1,1) orbit cardinalities at n = 8, 9, 10: 96/144, 72/432, 240/432, matching zmA and zmS

H(1,1) at n = 12 and n = 14, the even half arXiv:2602.21984 calls open: orbit cardinalities 480/960 and 1008/2160, matching Zmiaikou's even-n formulas exactly

H(1,1) at n = 11, 13, 15: orbit cardinalities 240/1200, 560/2520, 960/4032

H(4): complete enumerations at n = 5, 6, 7 – 40, 225, 775 primitive origamis in 4, 7, 6 orbits

H(4) orbit cardinalities at n = 9, 10, 11 (25, 90, 135 / 10, 135, 450 / 180, 200, 420); externally, the orbit *counts* are 7, 7, 7, matching Delecroix-Lelievre plus Lanneau-Nguyen's Prym count

The soundness layer, kernel-clean for an arbitrary finite type: the two moves commute with simultaneous conjugation; the commutator hvh⁻¹v⁻¹ is fixed on the nose by both; the monodromy subgroup and the <h,v> <= Alt(n) condition are invariants

The orbit-certificate machinery: breadth-first saturation returns only reachable origamis, and a closed list containing the seed contains the whole orbit

Negative controls: canon is conjugation-invariant over all 120 relabellings of all 24 five-square origamis and so are the moves; a non-transitive pair with the right commutator is excluded; primitivity is not vacuous (4 -> 10 and 48 -> 88 without it); a mis-transcribed move leaves the stratum; one generator alone gives cusps of width 7 inside orbits of size 16 and 144; neither n = 7 orbit is the whole stratum and the two together are

CORRECTION to Zmiaikou's thesis Table B.1 (p.129): the seventh orbit length for 6-square origamis in H(4) prints 60; the correct value is 120 (total 225, not 165)

The H(2) block dichotomy, 3 <= n <= 21: for every SL(2,Z)-orbit of primitive n-square origamis in H(2) the only block systems of the SL(2,Z)-action are the two trivial ones together with, when the monodromy is Sym(n), the parity partition B1,B2,B3 of arXiv:2602.21984 Prop 3.3, whose block at the seed has exactly |O|/3 elements. In particular the Alt(n)-monodromy orbit Gₙ^B is primitive at every odd n in range: the Veech group of such an origami is a maximal subgroup of SL(2,Z), hence (since -I lies in it) of PSL(2,Z). By Hubert-Lelievre/McMullen/Lelievre-Royer these are all the orbits at each n, and the certified orbits are verified to exhaust the primitive locus at every n of the range (h2ₒrbitsₑxhaust).

The H(1,1) block dichotomy, odd 5 <= n <= 15: the same statement in the other hyperelliptic stratum. The Alt(n)-monodromy orbits (Zmiaikou's Aₙ: sizes 16, 72, 240, 560, 960) are primitive; the Sym(n) orbits (Sₙ: sizes 24, 144, 432, 1200, 2520, 4032) have the parity partition as their unique non-trivial block system, with block at the seed of size |O|/3. These are all the orbits at each odd n in range: for odd n every primitive origami of H(1,1) carries one of two HLK-invariants (arXiv:2602.21984 Section 6, from Duryev Thm 3.1), the invariant is SL(2,Z)-invariant so the two classes partition the locus, Kappes-Moller give the two class sizes and they are exactly zmA n and zmS n – so an orbit of that size is the whole class.

CORRECTION to arXiv:2602.21984 Section 4.1. Its remark ends "Notice that (ST)² does not seem to appear in the Veech group of any primitive origami in H(2)". False: (ST)² fixes 2 origamis in G₅^B, 4 in G₇^B and 4 in G₁1^B. Explicit witness X₇ = ((1 2)(3 4)(5 6 7), (1 3 5 7 2 4 6)), a primitive 7-square origami in H(2) with both permutations even (so in G₇^B), whose Veech group contains (ST)² but neither ST nor (TS)². Every other entry of that remark is reproduced exactly for n <= 13, with one omission: ST² and S²T each also fix one origami of G₃.

Uniform algebraic seeds, proved for an arbitrary finite type and every n at once: for any h: Perm a and any point a, [h, swap a (h a)] = swap (h a) (h² a) * swap (h a) a, a 3-cycle when a, h a, h² a are distinct (the stratum H(2)); and [h, swap a (h² a)] = swap (h a) (h³ a) * swap a (h² a), of cycle type (2,2) when a, h a, h² a, h³ a are distinct (the stratum H(1,1)). With Mathlib's cycle-and-swap closure theorems the monodromy is Sym(n) (for the second, when n is odd), so the seeds are primitive origamis. The matching List-level seeds make every orbit in the family reachable from a formula.

Negative controls for the block machinery: the dichotomy is false one stratum over – the 120-element Alt-monodromy H(4) orbit from ((0 1... 6),(0 1 3)) is imprimitive, carrying the source's X, -I.X system of 60 blocks of size 2, and dichCert returns false on it; -I = (T S⁻¹ T)² fixes every origami of the H(2) and H(1,1) n = 7 orbits and none of that H(4) one; the partition of G₇^A by sgn h alone is not a block system while the parity pair is (three classes of 18 in 54); both trivial partitions are block systems; a non-canonical seed and a wrong stratum are rejected. Both directions of the decidable/propositional bridge are proved, so the false results are genuine refutations. Also bound to this row and previously unnamed in the label: parity_dynamics (the parity partition's dynamics under the generators) and stsqₘovesₑvery_block ((ST)² moves every block of the parity system), both @[dossier "origami-orbits" "origami-orbits-18"] theorems in BlockControls.lean and both cited by the paper's Proposition 6.3. [LABEL WIDENED 2026-08-30 (post-publication referee pass 2026-08-30, commit 7aea71ae) to match the row's BINDING SET, which is the compiler-checked truth: the tags are what familyₛtatus reads out of the environment, so a label that names fewer theorems than the row is bound to understates what the row certifies. Twelve declarations carry this row's tag; the label now covers them.]

The block-system certificate machinery, kernel-clean: the least congruence generated by one pair refines every congruence containing it (gen_cong); a replayed derivation log produces only related pairs whatever the log contains (runIns_gen), so the union-find search that writes the log carries no proof obligation; the index model is faithful to the four moves and transports back to the orbit, giving the trichotomy for every partition of the orbit invariant under T⁺⁻¹, S⁺⁻¹ (block_dichotomy, blockₚrimitive, parityᵢs_block).

The certified H(2) orbits exhaust the primitive locus, 3 <= n <= 21. With hlTot n = (3/8)(n-2)n² prod_(p|n)(1-p⁻2) – Hubert-Lelievre's count of all primitive n-square origamis in H(2), certified here to equal hlA n + hlB n at every odd n of the range – the orbit lengths reported by the family's 28 block certificates satisfy: |O| = hlTot n at n = 3 and at every even n, and |O_A| + |O_B| = hlTot n at every odd n >= 5. Combined with the classification quoted in arXiv:2602.21984 Section 2 (a single orbit for n = 3 and even n >= 4, two for odd n >= 5) this upgrades "the seed reaches an orbit" to "the certified orbits are all of them" – the step that was previously unchecked at even n, where the source states the classification but no cardinality. Negative controls: at odd n neither orbit alone reaches hlTot n, and the two nearest wrong closed forms (3/8)(n-1)n² prod and (3/8)(n-3)n² prod disagree with hlTot at every n of the range.