The Grounding Deadline: a scheduling consequence of the collapse certificate
Date: 2026-07-05 · Evidence class: PROVEN core (synthetic, V-metric) + MEASURED inputs + PROJECTION (Ouro) · Loop stage: Verify/Converge Artifacts: experiments/sigma0_grounding_deadline.py, data/sigma0/grounding_deadline_report.json, certificate §3.1
Question
Can the certificate's own objects (γ, c, P) + grounding + math produce a new answer to collapse — not another detector? Run postulate → kill-search → derive → machine-check.
Kill-search first (what was already taken)
- "Early intervention beats late; models commit" — published, with per-token commitment counts
(arXiv:2604.23235). Not ours.
- Contraction metrics determine basins; bounded-disturbance divergence bounded by rate and
condition number — classical control (e.g. contraction-metric literature). Not ours.
- Not found: a certified deadline formula for grounding computed from a collapse
certificate's own quantities, with a conditioning condition for when it binds. That composition survived the search.
The result (stated at its honest size)
Inside a certified basin {V ≤ c} with V∘F ≤ γV: escape with anchor budget B requires B > B*(n) = (√c − γ^{n/2}√V₀)/√λ_max(P) — rises geometrically at the certified rate, saturates at B*_∞ = √(c/λ_max(P)). Any B < B*_∞ has a computable deadline n*(B). Math: corollary-grade (P-norm triangle inequality + geometric decay) — the value is the composition and the design consequence, not the mathematics.
Machine-checked by construction (exact trust-region max of V(x+a), not our own bound):
| regime | prediction | measurement |
|---|---|---|
| normal basin (cond 1.8) | deadlines 1.6 / 2.9 / 6.0 | last escapes//— bites, conservative |
| sliver (cond 6·10⁵) | ceiling B*_∞ ≈ 0.002 → no practical bite | escapes at every depth |
| Ouro (ρ=0.88 measured, strongly non-normal) | sliver regime → weak deadline | retro-dicts the measured anchoring null |
Design consequences (the actionable part)
- Grounding is a schedule, not only an event — period below the commitment half-life
ln2/ln(1/ρ) (~5.4 steps at Ouro's measured rate; PROJECTION).
- Canary demoted to audit — critical slowing down is detectable only after deep
contraction, exactly when the escape budget has saturated; canary-triggered Σ₀⁻¹ fires at the expensive end of the window in well-conditioned basins. Consistent with (not proven by) the measured rank≠route split: early-curve logprob routed +, late-curve canary routed −.
- cond(P) decides the regime — hard deadline (well-conditioned) vs soft (sliver); cheap to
measure alongside any certified rate.
Honest limits
Additive-anchor model of grounding (real grounding enters as tokens via attention); the Ouro story is a retrodiction of our own null, not a prospective test; a prospective test needs a well-conditioned loop (e.g. STARS-trained, arXiv:2605.26733) where the deadline should bite. Landed as a certificate design note (§3.1), deliberately not a theorem-style section.