Refinement monotonicity of normalized Witten intersection numbers, and the nesting conjecture at genus 14–18

The compute-first gate: all fourteen exact rational values printed in Table 1 (p. 4) of arXiv:2603.15233v1 reproduced from the DVV recursion, together with two exact-rational instances of the dilaton relation Eq. (32), C(1,4) = C(4) = 35/144 and C(1,1,2,2,2) = C(2,2,2) = 175/648.

The four exact values and the four strict comparisons printed in the numbered Remarks (all on p. 5) of arXiv:2603.15233v1: C(0⁶,10) = 1616615/6718464 < C(4) and C(2³) < C(0,0,6) = 5005/15552 (the nesting fails once dⱼ = 0 is allowed), C(2⁵,8) = 727759375/2448880128 < 419588015525/1410554953728 = C(3⁴,5) (weak monotonicity in the length fails), and C(4,4) < C(3,5) (the [GNYZ] lexicographic monotonicity fails).

Engine controls at every genus 2 <= g <= 18, inside the same kernel check as the results: the one-point formula Eq. (19) in the form W(3g-2) = (6g-3)!!; Zograf's independent explicit two-point formula Eq. (44) in the integer form W(d₁, 3g-1-d₁) * g = zografS(g,d₁) at all 3g two-point cells of each genus; the dilaton relation Eq. (32) in the form W(1,d) = 3(2g-2+n) W(d) at every one of the 600,492 primitive tuples; and that the enumeration really is the primitive locus (each tuple has all parts >= 2, degree 3g-3+n and genus g).

The lower half of the source's Conjecture 1, C(3g-2) <= C(d), verified over the COMPLETE primitive locus of every genus 2 <= g <= 18 (600,492 tuples, p(3g-3) of them at genus g), with the strengthening that (3g-2) is the UNIQUE minimiser at each genus.

The OPEN half of the source's Conjecture 1, C(d) <= C(2³ᵍ⁻³), over the complete primitive locus of every genus 2 <= g <= 13 (39,783 tuples), with 2³ᵍ⁻³ the unique maximiser at each genus.

Genus 14, past the source's reported range: C(d) <= C(2³⁹) for each of the p(39) = 31,185 primitive tuples of genus 14, with 2³⁹ the unique maximiser. By the source's Eq. (32) this is equivalent to the full dⱼ >= 1 statement of its Conjecture 1 at this genus.

Genus 15, past the source's reported range: C(d) <= C(2⁴²) for each of the p(42) = 53,174 primitive tuples of genus 15, with 2⁴² the unique maximiser.

Genus 16, past the source's reported range: C(d) <= C(2⁴⁵) for each of the p(45) = 89,134 primitive tuples of genus 16, with 2⁴⁵ the unique maximiser.

Genus 17, past the source's reported range: C(d) <= C(2⁴⁸) for each of the p(48) = 147,273 primitive tuples of genus 17, with 2⁴⁸ the unique maximiser.

Genus 18, past the source's reported range: C(d) <= C(2⁵¹) for each of the p(51) = 239,943 primitive tuples of genus 18, with 2⁵¹ the unique maximiser. Together with WN6-WN9 this extends the source's verification from genus 13 to genus 18, i.e. by 560,709 further tuples.

REFINEMENT MONOTONICITY, genus 2 to 13: indexing the primitive tuples of genus g by the partition lambda = d - 1 of 3g-3, splitting one part of lambda into two positive parts STRICTLY increases C. Verified at every genus 2 <= g <= 13 over 515,715 single-split steps (2, 21, 90, 328, 959, 2588, 6311, 14579, 31703, 66315, 133182, 259637 per genus), zero violations. Since (3g-3) is the unique refinement-minimum and 1³ᵍ⁻³ the unique refinement-maximum, this IMPLIES the source's Conjecture 1 at those genera, with strict inequalities off the two extremes.

REFINEMENT MONOTONICITY at genus 14, 15, 16, 17, 18: 491,295 + 908,008 + 1,639,978 + 2,905,593 + 5,053,485 = 10,998,359 single-split steps, zero violations. With WN11 that is 11,514,074 split steps over genus 2 to 18, and it implies the open half of the source's Conjecture 1 at each of those genera.

Negative control, a too-large claim refuted: NEITHER half of Conjecture 1 survives dropping the hypothesis dⱼ >= 1. At g = 2 these are the source's own Remark (C(0⁶,10) < C(4) and C(2³) < C(0,0,6)); the genus-14 pair C(0⁶,46) < C(40) and C(2³⁹) < C(0,0,42) is computed here and shows the hypothesis is not an artefact of small genus.

Negative control, a too-small claim refuted: neither bound of Conjecture 1 can be narrowed. Replacing C(3g-2) by the next value up, or C(2³ᵍ⁻³) by the next value down, breaks the statement already at g = 3 – C(7) < C(2,6) and C(2,2,2,2,3) < C(2⁶). The uniqueness of the two extremisers, asserted at every genus in WN4 and WN5-WN10, is the same statement in general.

Negative control, a natural strengthening refuted: C is monotone in NEITHER direction of the dominance order on lambda = d - 1. lambda = (4,2) dominates (3,3) yet C(3,5) > C(4,4), while the global minimum sits at the dominance-largest partition (3g-3) and the global maximum at the dominance-smallest 1³ᵍ⁻³ (C(7) < C(2⁶) at g = 3). So dominance is not the order behind Conjecture 1; refinement, a proper suborder of reverse dominance, is.

The dilaton reduction: because C(1,d) = C(d) exactly (the source's Eq. (32)) and a part dⱼ = 1 contributes 0 to sumⱼ (dⱼ - 1), deleting all the 1s preserves the genus and, for g >= 2, leaves a non-empty tuple with all parts >= 2. Hence Conjecture 1 for dⱼ >= 1 is EQUIVALENT, genus by genus, to Conjecture 1 for primitive d – so the source's primitive checks at g <= 13, and rows WN5-WN10 here, already carry the full dⱼ >= 1 statement at those genera.