Σ_G — The Grounding Ledger

What this document is — and is not

It is: the composition of three quantitative results already recorded (with evidence classes and run pointers) in the Collapse Certificate — the price of fresh verified truth (cert §8.4/§8.4.1), the schedule on which grounding must arrive (cert §3.1), and the budget that grounding imposes on safe self-improvement (cert §8.4 third road) — stated as one frame with a shared falsification path. It is written by Alex Place with AI coding agents (disclosed; agent sessions author and revise under operator review; the dates are real project dates — this project runs in 2026).

It is not: a peer-reviewed paper, a certificate, or a claim of mechanism novelty. The qualitative rule it rests on — ungrounded self-reference degrades; external verification is the escape — is established prior art (Shumailov 2024; Feng et al. 2024; Huang et al. 2024, all cited in the parent). Every load-bearing number below is imported from the parent certificate's audit table, not re-derived here.

Novelty: thin, named, and bounded. A 2026-07-17 corner audit (recorded in the parent's Appendix M) found the three quantitative corners composed here have close prior art in adjacent fields and open lanes in the LLM literature:

The composed frame — one ledger with all three columns priced from the same system's own measured quantities — is this document's only claimed contribution, and it is a framing contribution until the §5 experiments run.

Glossary — internal names → standard terms

Plain-language summary

Price. Fresh verified truth is the only currency that can inform selection — genuinely rank which model, answer, or update is better. Deterministic self-checks (perplexity, confidence — measured twice, same number) buy nothing in that role: their errors stick to the champion exactly like a lucky test score does. But the E-P falsification (2026-07-17) found a twist the original law missed: stochastic self-checks — and even plain added noise carrying zero information — recover a real fraction of the budget by shaking the champion's lucky seat (de-ratcheting), the same trick the Thresholdout referee performs deliberately. So the price column has three tiers now: deterministic self-checks are worthless for selection, fresh randomness is half a substitute (it unsticks, it doesn't rank), and only fresh truth does both. Internal signals detect; fresh randomness de-ratchets; only fresh truth informs.

Schedule. A loop that feeds on its own output commits to its trajectory at a measurable rate, and the outside evidence needed to un-commit it grows at that same rate, then saturates. So grounding is not an event you trigger when an alarm rings — by the time drift alarms are reliable, the cheap part of the rescue window is gone. Grounding is a cadence: inject external evidence with a period set by the loop's measured commitment half-life.

Budget. Because only fresh truth selects (price), and because reusing one test set lets lucky checkpoints win, the rate at which a self-improving system can safely accept updates is capped by the rate at which it can source fresh verified tasks. Recycling helps — retired test sets, arbitrated by a noise-adding referee, measurably extend the budget without corrupting the scoreboard — but recycling stretches the budget; it never removes the cap.

The one-line ledger: what grounding buys (price), when it must arrive (schedule), and how fast the whole system may improve (budget) are three computable functions of the same measured quantities — and none of the three can be bought with internal signals.

How to audit this document

Every number below maps to an artifact already in this repo. Nothing in this document adds a new experiment; §5 names the experiments that would.

If a command above fails on a fresh clone, this ledger has drifted from the code and should not be trusted until reconciled — the parent's intended failure mode, inherited.

1. The Price — what a unit of fresh truth buys, and why nothing else can buy it

Status: MEASURED (simulation) for the selection half; MEASURED (one model) for the detection half. The law is a statement of mechanism, not a theorem.

Let a selection process choose among candidates (checkpoints, answers, updates) using a scoring signal. Distinguish two roles:

• Selection — ranking candidates to promote a champion. Adaptive: the process feeds back.
• Detection — flagging a candidate as grossly broken. Non-adaptive, or nearly so.

The measured price schedule (cert §8.4, §8.4.1, §8.6-5; all run pointers in the audit table):

The mechanism (the freshness law — as re-stated after its own kill test fired, 2026-07-17). A deterministic intrinsic signal's error sticks to the champion exactly the way a fixed holdout's lucky draw does, so it cannot re-anchor the selection ratchet. But the E-P falsification (#2692, pre-registered) found the original one-line law too strong: re-drawn measurement noise — though it carries zero information — rescues a large fraction of the budget by de-ratcheting the champion's seat (a zero-information dither reproduces the entire rescue; it is the same mechanism Thresholdout buys deliberately with Laplace noise). Selection error decomposes into a stuck part (sets the ratchet) and a fresh part (breaks it); only re-drawn external truth both breaks the ratchet and ranks candidates, which is why fresh truth still strictly dominates every internal arm. In three lines now: internal signals detect; fresh randomness de-ratchets; only fresh truth informs.

Honest scope. The selection half is simulation-shaped (32 seeds, synthetic hill-climb; shape, not constants). The detection half is one model (Ouro-1.4B),truth tasks, and a corruption ladder whose "subtle badness" cell came back empty by construction. The nearest external work has since occupied the detection lane (Model-Centric Diagnostics, arXiv:2601.16874; statistical degradation detection, arXiv:2602.10144) — the incremental ΔAUC-over-a-tiny-external-gate design may still be unique; the lane is not.

2. The Schedule — when grounding must arrive

Status: the inequality is PROVEN and machine-checked on synthetic maps (cert §3.1); the mapping onto real token-level grounding is CONJECTURED; the one real-model measurement to date is a retrodicted null, honestly recorded below.

Inside a certified basin {V ≤ c} with per-step contraction V(F(x)) ≤ γV(x), γ < 1 (V = xᵀPx), an external anchor of budget ‖a‖ ≤ B can escape only if

$$B \;>\; B^*(n) \;=\; \frac{\sqrt{c} - \gamma^{n/2}\sqrt{V_0}}{\sqrt{\lambda_{\max}(P)}} \;\xrightarrow{\;n\to\infty\;}\; B^*_\infty,$$

so any budget B < B*_∞ stops working after a computable deadline n*(B), and the loop's commitment half-life is lnlnlnlnlnln 2 / ln(1/γ)/ ln(1/γ)/ ln(1/γ)/ ln(1/γ)/ ln(1/γ)/ ln(1/γ) steps. The design consequences (cert §3.1):

1. Grounding is a cadence, not only an event — inject external evidence with a period below

the commitment half-life.

1. Reactive triggers are structurally late — critical-slowing-down alarms become reliable

only after contraction is deep, which is when B*(n) has saturated. (Scoped claim: proven in the synthetic setting; consistent with, not proven by, the measured rank≠route split.)

1. Conditioning decides the regime — cond(P) of the local basin decides whether the

deadline is a hard constraint (well-conditioned) or toothless (sliver). Measuring it is cheap and should accompany any certified rate.

Positioning against prior art (added with the 2026-07-17 audit). "Compute the next intervention time from a certified Lyapunov decay rate" is self-triggered control — a mature field (Heemels, Johansson & Tabuada, CDC 2012; Lyapunov event-triggered stabilization with a known convergence rate, arXiv:1803.08980). This section is that field's recipe applied to a new plant. What the LLM literature has instead is the event-triggered side — FLARE (arXiv:2305.06983) and DRAGIN (arXiv:2403.10081) trigger retrieval reactively on uncertainty — and naive fixed-interval retrieval with hand-picked periods. The open lane is exactly the self-triggered counterpart: a retrieval cadence derived from the loop's measured dynamics. Citing the control literature imports its guarantees; it does not diminish the lane.

Phaseexperiments/sigma0_scheduled_grounding.pymeasured (2026-07-17, #2690). The scheduled-vs-reactive race (experiments/sigma0_scheduled_grounding.py,seeds, timing the only variable, canary swept over all healthy-quantile thresholds): in a well-conditioned basin the charitably-calibrated alarm (FPR 1.5%) lands at median stepvs the cadence tick at step 3, and the alarm premium is 2.25× — scheduled grounding reaches ≥90% escape at 0.40·B*∞ where reactive needs 0.90·B*∞. In the sliver basin a forced fire-time sweep is fully timing-indifferent — the Ouro-regime null, produced rather than retrodicted. Synthetic maps; single-shot policies; one detector family.

Phase 0.5 — the prospective shot-call held (2026-07-17, #2690). The first prospective test ran on a real learned system: the trained §6 reservoir. Every quantity was computed from the linearization before any sweep — ρ(J)=0.9836, cond(P)≈4.8·10⁴, pre-registered regime call sliver / timing-indifferent — and the true nonlinear dynamics then confirmed it: the measured per-turn grounding kick of the real conversation stream is 14,000× the escape ceiling, and escape succeeds at every depth even at a hundredth of that kick (experiments/sigma0_reservoir_deadline.py,tests). The theory called its shot on a system it wasn't tuned for, with the instrument chosen by the call, and the call held.

The accumulating pattern (and the honest null). Both real learned loops measured so far land outside the well-conditioned regime: Ouro-1.4B is expansive and strongly non-normal (ρ(J) ≈ 8–11, cert §6 [#2029] correction; its anchoring null measured by experiments/ouro_canary_vs_logprob.py is retrodicted by the theory), and the reservoir is near-critical sliver — despite a design spectral radius of 0.9. The schedule has therefore still never been tested where it should bite, and the pattern now suggests that regime may only exist in deliberately stabilized loops (JSRR/STARS, ρ <— the gate the project already adopted, cert §1.2.3). That is experiment E-S (§5), Phases 1–2 — motivated by measurement now, not assumption.

3. The Budget — how fast the system may safely improve

Status: MEASURED (simulation) on an imported PROVEN backbone (Dwork/Thresholdout). The compounding design is HEURISTIC.

Because only fresh truth selects (§1), and because a reused promotion set lets lucky checkpoints ratchet in (cert §8.4), the safe update rate is capped:

rate(safe promotions) ≤ rate(fresh verified task sourcing), stretched — not uncapped — by controlled reuse.

The measured decomposition (cert §8.4 third road; tests/test_sigma_theta_thresholdout.py): wrapping the fixed holdout in a Thresholdout referee and ablating the burned pool separates two effects that naive accounting conflates:

• Validity is bought by the mechanism. The champion's reported-vs-true gap stays below both

other arms at every n ≤(n=50: fixed 0.482 vs Thresholdout 0.187 vs fresh 0.194) — fresh per-query noise breaks the sticky-luck ratchet.

• Extraction is bought by the burned pool. With no accumulated pool, extraction collapses

to the naive-fixed arm; with a realistic accumulated pool (4n), the managed arm beats even the same-n fresh flow for n ≥(21.8 vs 18.0 at n=100) while staying more honest than fresh.

The compounding design (HEURISTIC): retire each promotion set into the burned exploration pool after use and let the referee arbitrate pool-versus-holdout feedback. Retired truth is not dead — the tiers form a cycle, not a conveyor. The rotating-tier structure is cert §8.4.1's four-tier discipline, unchanged.

Protocol layer implemented and validated (2026-07-17, #2691). The promotion-evidence protocol itself — Fixed / Fixed+dither (the E-P knob promoted to an arm) / Fresh-flow / Thresholdout+pool, with provenance ledgers and budget accounting — is now implemented model-agnostically (experiments/sigma_theta_abc/holdout_protocol.py) and validated at task-level Bernoulli-stuck fidelity (outcomes deterministic on re-evaluation — the real stuck-luck mechanism, an upgrade over the Gaussian sims): 7/7 checks,seeds. Orderings hold at the higher fidelity, and the dither arm extracts 0.856 vs fixed 0.617 at identical fresh-task cost — the cheapest intervention measured. Planted teeth are rejected by the real seven-condition gate with arm-appropriate reasons (fixed catches a contaminated hack only via the provenance ledger; fresh rejects it on merit). What remains of E-B is exactly one thing: running this protocol around real training on L4.

Positioning against prior art (added with the 2026-07-17 audit). The problem shape — adaptive promotion against a reused evaluation with limited feedback — is the Ladder (Blum & Hardt, arXiv:1502.04585), built for competition leaderboards; the mechanism imported here is Thresholdout (Dwork et al., arXiv:1506.02629), which has almost no applied literature. The validity/extraction decomposition and the burned-pool cycle appear unpublished. The standing counterpoint: Roelofs et al. (NeurIPS 2019) meta-analyzed ~100 Kaggle competitions and found little adaptive overfitting in practice — so the severe small-n staleness measured here may describe an adversary real promotion pipelines don't resemble. Whether the ratchet materializes at practical scale is exactly what experiment E-B (§5) exists to answer; until it runs, the budget cap is a simulation-shaped claim.

4. The composed ledger

Status: TARGET — a design frame, exactly as the parent's Part III is. Nothing below is proven, and the three columns rest on different evidence classes (see §1–§3).

One system, three computable columns from its own measured quantities:

The joint statement — the ledger's only new sentence:

A self-improving loop's safe operating point is set by three couplable constraints: it must ground at least on the schedule's cadence (or re-commitment outruns the anchor), it may promote at most at the budget's rate (or selection outruns fresh truth), and it must price signals by re-drawability when deciding what counts as grounding at all. All three constraints are computable from measured quantities of the same object — none requires an oracle, and none can be relaxed by internal signals.

In plain words: the loop has a metabolism. It must eat outside evidence at a floor rate, it can grow at a ceiling rate, and no amount of introspection substitutes for food.

5. Falsification path (before trusting any of this)

Three experiments, ordered by cost, each tracked as a repo issue. Each names its kill condition — what result would delete which column.

• E-P → #2692 (**done 2026-07-17 — the

kill condition fired; §1 re-priced**) · E-S → #2690 (Phase#2690+ Phase 0.5 done 2026-07-17 — the prospective shot-call held on the trained reservoir; Phases 1–2 need a JSRR-stabilized loop, now motivated by measurement) · E-B → #2691 (protocol layer done 2026-07-17; the L4 training run is the sole remaining gap — fully specified as a self-contained handoff in SIGMA0-EB-L4-RUNBOOK.md, with pre-registered verdicts and the Roelofs rule)

1. E-P (price — RUN 2026-07-17, kill condition FIRED). The prediction tested: a

stochastic internal signal (re-drawable noise, checkpoint-fixed bias) still cannot extend the selection budget. Measured outcome (experiments/sigma_update_stochastic_signal.py, pre-registered): H-P1 confirmed (bias→noise error shift raises extraction 0.81 → 9.36 at n=50), but H-P2 refuted — re-drawn noise at fixed bias rescued extraction 7.9×, and a post-hoc zero-information dither control reproduces the entire rescue (6.42 vs 6.43). The law was too strong; §1's price table and mechanism paragraph are re-priced accordingly (de-ratcheting is real and nearly free; information still requires fresh truth, which still strictly dominates). The optional real-model arm folds into E-B.

1. E-S (schedule — needs a well-conditioned loop). Prospective deadline test on a

JSRR/STARS-stabilized loop (ρ < 1, measured cond(P) moderate): predict B*(n) and the half-life from the measured γ and P before the run; then measure actual correction cost vs injection depth; then race scheduled grounding (period from the half-life) against FLARE-style reactive grounding on a factuality benchmark. Kill condition: correction cost flat in depth on a well-conditioned loop kills the §2 mapping (the additive-anchor model fails); reactive ≥ scheduled everywhere kills the design consequence.

1. E-B (budget — the cloud A/B/C run, the parent's standing empirical gap). The three arms

live around a real RLVR/distill step: fixed holdout vs fresh flow vs Thresholdout+pool, scored on truly hidden tasks. Kill condition: if the fixed-holdout ratchet does not materialize at realistic n and gate counts (the Roelofs counterpoint), the budget cap is a worst-case story and §3 must be re-scoped to adversarial promotion processes.

Sequencing note: E-P falls out of the existing harness nearly for free; E-S piggybacks on the already-merged JSRR gate (cert §1.2.3, PR #2237); E-B is the same cloud-L4 run the parent's §8.6 teeth 2–3 already require — this ledger adds no new infrastructure demands.

6. Relation to the parent certificate

The parent remains the source of record for every theorem, measurement, and evidence class; this ledger must never be cited as upgrading any of them. If the two documents disagree, the parent wins and this one is stale.

References (lineage; every arXiv ID verified on its stated date)

Verified 2026-07-17 (this document's corner audit):

• W. P. M. H. Heemels, K. H. Johansson & P. Tabuada, An Introduction to Event-triggered and Self-triggered Control, IEEE CDC— the schedule column's prior-art field.
• arXiv:1803.08980 — Lyapunov Event-triggered Stabilization with a Known Convergence Rate — intervention timing from a certified decay rate.
• arXiv:2305.06983 — Jiang et al., Active Retrieval Augmented Generation (FLARE) — the reactive/event-triggered side of LLM retrieval timing.
• arXiv:2403.10081 — DRAGIN: Dynamic Retrieval Augmented Generation based on the Real-time Information Needs of LLMs — token-level reactive triggering.
• arXiv:1502.04585 — Blum & Hardt, The Ladder: A Reliable Leaderboard for Machine Learning Competitions — adaptive promotion gating with limited feedback.
• R. Roelofs et al., A Meta-Analysis of Overfitting in Machine Learning, NeurIPS— the budget column's standing counterpoint (little adaptive overfitting across ~100 Kaggle competitions).
• arXiv:2601.16874 — Model-Centric Diagnostics: A Framework for Internal State Readouts; arXiv:2602.10144 — When LLMs get significantly worse: a statistical approach to detect model degradations — the detection lane, now occupied (see cert §8.6-5 correction).
• Abdelfattah et al., Zero-Cost Proxies for Lightweight NAS, ICLR 2021; NAS-Bench-Suite-Zero, NeurIPSD&B — internal proxies empirically failing to replace validation (the price column's empirical parallel in NAS).

Verified earlier, in the parent (cited here via the cert, not re-verified): Dwork et al. arXiv:1506.02629 (Thresholdout); STARS arXiv:2605.26733; per-token commitment arXiv:2604.23235; TRPO arXiv:1502.05477; Gao arXiv:2210.10760; Shumailov Nature 2024; Feng arXiv:2406.07515; Huang arXiv:2310.01798.

Per the parent's §References rule — an early draft of the cert once carried four fabricated arXiv IDs — no ID above is cited unverified, and each carries its verification date.