Sharp axial geometry for the full birth-time law: necessity beyond the union bound

[LightChainr]

Result delivered, not a new enumeration assignment

For honest axial square-site tori with 2 <= w_n <= m_n and w_n*m_n -> infinity, under the explicitly cited critical square-site RSW, site sharpness and matching-duality inputs,

all fixed interior Q_n(u) -> p_c
    iff convergence is uniform on compact u-intervals
    iff F_n tends to the threshold step
    iff log(m_n)/w_n -> 0.

The new part is necessity, including logarithmic-width sequences. An upper union bound failing to vanish was never a proof of necessity. Here a seam-closed ring of crossing events has probability at least exp(-eta*w) at a FIXED p_eta < p_c; independent vertex-disjoint transverse bands amplify it. This forces P0 -> 0 and keeps each fixed lower quantile away from p_c on a witnessing subsequence. The median/balance root is deliberately NOT included in that obstruction.

This complements #718/#735: on m=ceil(exp(d*w)), w -> infinity still suffices for the balance root, but not for the full law. No new numerical p_c, exponent, or continuum identification. No necessity theorem for arbitrary oblique periods. The missing uniform oblique-corridor lemma is stated precisely as the next question.

Files and validation

• notes/axial-quantile-geometry-criterion-20260913.md: proof, quantifier order, source reading boundaries and prior-art caveat.
• scripts/axial_ring_gluing.py: free-rectangle crossing BFS versus an independent lifted-graph winding detector.
• tests/test_axial_ring_gluing.py: 3 mathematical tests, passed locally.
• results/research-control-20260913/axial-ring-controls.json: 21,760 exhaustive configurations, 2,365 deterministic uneven-cell controls, zero implication failures; exact Fraction FKG checks at 1/3, 1/2, 2/3.

At 4x2 the sufficient ring event has 17 configurations, while winding has 35: the construction is explicitly not an event identity. Controls do not prove RSW or the all-size theorem.

Python compilation passed. Full repository CI not run by this delivery. No existing code, production data, frozen design or old result is replaced. Independent all-size proof review and systematic novelty comparison remain publication work. This proof is supplied as a consequence of named inputs, not as a claim to have invented RSW/strip gluing.

Owner-delegated research redirection on 2026-09-13. Advances #613; #276 stays completed. Related #716/#718/#735 and #650.