Many Steps into One: Reasoning Contraction and the Limits of Safe Memory Reuse for LLM Agents

1 Introduction

• Reasoning contraction compiles a derivation of at least two steps into a one-step memory for later instances.Examples include thought templates, skill libraries, answer caches, and shared or self-rewritten agent memories.Source: §1 · p. 1
• Live reasoning is costly and fragile, while a verified memory can improve accuracy with fewer calls; edited facts can nevertheless leave it answering wrongly.On edited multi-hop streams, Qwen3-8B accuracy rose from 0.379 to 0.648 with a verified contracted memory.Source: §1 · p. 1
• Symmetric counters are badly tuned for partial staleness: their detection delay can grow as a power of the false-alarm horizon rather than logarithmically.Unclipped counters can also retain accumulated credit and fail to retire a stale memory.Source: §1 · p. 1
• On five open LLMs, ExpeL- and ACE-style counters served 7.0–13.5× more stale answers than a certificate given the same information.On KQA-STREAM, the counters also needed about 60× more uses to retire stale memories than a likelihood-ratio rule.Source: §1 · p. 2, Figure 1 · p. 2
• The certification load Θ sets a worst-case limit on the share of queries that any safe memory can answer in one step; under Zipf demand, the gap scales as κsΘ^(1−1/s).A restarted test achieves the stated limit.Source: §1 · p. 2
• The proposed recipe weights failures by each memory’s likelihood ratio, certifies utility against live reasoning, and restarts certificates when answers or competing live reasoning change.For substantial announced edits, it also invalidates memories; the detector serves as a safety net for unannounced edits.Source: §1 · p. 2

2 Contraction as a memory lifecycle

• A macro is a one-step operator formed from a verified derivation, with reliability defined as its probability of answering a random template instance correctly.Queries can be answered live, folded through a trusted macro, or handled in shadow mode while the macro is evaluated alongside live reasoning.Source: §2 · p. 3
• The lifecycle is to form a candidate from a verified trace, certify it in shadow mode, fold it, monitor it, retire it when stale, and re-form it.Verifier outcomes are conditionally Bernoulli with a mean equal to current reliability, allowing adaptive use and drift.Source: §2 · p. 3
• The two risky decisions are certification and retirement; a macro becomes stale when an edit raises its failure rate from healthy q0 to stale q1 > q0.ExpeL, ACE-style, and validation-gated Hebbian memories use different maintenance rules, which the paper idealizes as votes or clipped strength updates.Source: §2 · p. 3

3 Memory counters are change detectors

• A score rule is a CUSUM for the implied alternative P̃0 = e^(θ0f)P0, not for any other change.Its threshold is h, and retirement occurs at the first n with Sn ≥ h.Source: §3 · p. 3
• The symmetric rule tests a stale failure rate of 1 − q0, so it assumes a stale memory is as often wrong as a fresh one was right.That assumption does not fit edits that corrupt only part of a template’s instances.Source: §3 · p. 4
• For partial staleness, equal-vote clipped counters can need A^0.373 uses rather than ln A; an unclipped counter may never retire the stale memory.The exponent 0.373 applies to the reliability change 0.95 →0.75; likelihood-ratio scores are optimal when proportional to the log-likelihood ratio.Source: §3 · p. 4
• At failure rates 0.05 →0.25, likelihood-ratio scores +7/−1 detect in 41 and 54 uses at A ≈1.6 × 10^5 and 3 × 10^6, respectively, 5.7× and 13× faster than the symmetric rule.The appropriate failure-to-success weight depends on each memory’s healthy and stale failure rates.Source: §3 · p. 4

4 The price of safety: how far can reasoning contract?

• A restarted SPRT is δ-safe when it folds after its likelihood ratio reaches ln(1/δ) and restarts at each change.The cited guarantee uses Ville’s inequality.Source: §4 · p. 5
• The certification load Θ combines change rate, tolerated risk, audit rate, and information revealed per verified outcome; it determines the fundamental limit on safe one-step coverage.The model defines the coverage gap as the long-run share of queries not answered by a valid, certified macro.Source: §4 · pp. 4–5
• Every δ-safe policy has coverage gap at least (1 − εδ)Fπ(Θ), and a restarted SPRT achieves a gap at most (1 + ln(r1/r0)/ln(1/δ))Fπ(Θ).The lower bound holds whether changes are announced and however many candidates the system forms.Source: §4 · p. 5
• Under Zipf demand with exponent s > 1, the optimal coverage gap scales as κsΘ^(1−1/s) for small δ and small load.The demand-tail exponent sets the scaling exponent.Source: §4 · p. 5
• Across 2,520 settings, the restarted test’s coverage gap was 0.945–1.025× the limit Fπ(Θ), with fitted log-log slopes within 0.008 of 1 − 1/s.The settings varied reliability, verifier noise, risk, audit, and drift rates.Source: §4 · p. 5

The recipe: a certificate that needs no live accuracy

• The SPRT’s break-even threshold depends on unknown live accuracy, and plugging in an estimate can break its guarantee.At a = 0.9, m−1L = δ = 0.05, false-fold probability is 0.34, 0.25, and 0.08 for 20, 100, and 1,000 live runs.Source: §5 · p. 5
• Betting on audited paired utility against live reasoning folds a macro when wealth reaches 1/δ, with false-fold probability at most δ for a never-more-useful macro.The test uses inverse-propensity-weighted utility and remains valid regardless of live accuracy or correlation between live and macro errors.Source: §5 · p. 5
• With i.i.d. utilities and audits at rate p, certification takes (1 + o(1)) ln(1/δ)/(p KLinf(PZ)) uses, and no δ-safe rule using only audited utilities is faster.The bound assumes known audit probabilities bounded below and independent of hidden outcomes.Source: §5 · p. 5
• Restarting the certificate at each known change makes the load-based safety result apply with KLinf(PZ) and unknown live accuracy.The table caption says the optimum attains Fπ(Θ) when its test restarts at known change times.Source: §5 · p. 5, Table 1 · p. 6
• The deployment recipe restarts at notices and retires folded macros with LR-CUSUM; when notices cover every null change, both variants are δ-safe per epoch.The recommended variant also invalidates and re-forms folded macros at notices, leaving LR-CUSUM as a safety net for missed notices.Source: §5 · p. 6

Experiments

• 7.0–13.5× more stale answers are served by counters than by the certificate on every model, even when every edit notice is missed.On code tasks, counters serve 35–73× more stale answers.Source: Table 2 · p. 6, §6 · pp. 6, 8
• 50% and 47% of stale episodes remain in service at stream end under counters, versus 2% under LR-CUSUM; restricted mean delays are 4,069 and 3,949 versus 67 uses.Each counter episode costs about 1,350 stale answers, versus 38.5 for LR-CUSUM; skipping certification lets counters answer 28–76% of queries in one step, versus 20–53% for ours.Source: Figure 2 · p. 7, §6 · p. 7
• 1.3–5.9× fewer uses are needed by per-memory likelihood-ratio weights than by the best tested fixed weight, while partial staleness favors weights of 5–7.For nearly total staleness, LR-CUSUM needs 3.7–6.1× fewer uses than the symmetric rule at 0.8–5 false retirements per 104 healthy uses.Source: §6 · p. 7
• 0 of 5,779 at-risk epochs produce false certifications at δ = 0.05 under answer-only scoring; the rate stays below δ up to 0.5.Restarting removes 11 bad folds that otherwise carry across edits; 0.26% of at-risk epochs have a bad fold at δ = 0.05 when routing errors count as wrong.Source: §6 · p. 7, Figure 3 · p. 7
• On Qwen3-8B, certified macros raise accuracy from 0.379 to 0.648 with 21% fewer LLM calls, and certified compiled plans reach 0.668.Across models, ours adds 14–33 points over live reasoning; certification cuts fold-once stale answers 20×, while invalidation reduces them to zero.Source: §6 · p. 7, Table 3 · p. 8
• With 80%, 50%, or no edits announced, the detector-backed policy serves 13, 28, or 66 stale answers per 104 queries, versus 125, 325, or 810 under invalidation alone.With perfect notices, ours serves no stale answers and matches restart-only in accuracy within +0.24 points [−0.24, +0.81].Source: §6 · p. 7
• On natural edits, bad-fold epochs stay below δ, while counters leave 84% of stale memories in service versus 37% under ours.Counters serve 3× more stale answers than restart-only but are 3.5 points more accurate; invalidation costs ours 1.9 points against restart-only.Source: §6 · p. 8
• On code tasks, ours folds below-break-even candidates in 12 of 946,678 epochs, gains 8.7 accuracy points over live reasoning, and uses 23% fewer steps.Zero-utility ties make audits less informative and halve one-step coverage from 0.61 to 0.32.Source: §6 · p. 8

Related work and what is new

• The detection analysis extends prior results by adding zero- and negative-drift regimes and deriving a finite power-law delay where prior work states infinite delay.It uses deployed counters’ implied alternatives to set learning rates.Source: §7 · p. 8
• The safe-reuse analysis extends crawler freshness to items whose trust requires ln(1/δ)/KL audited uses, with a Zipf law from an occupancy sum at load Θ.The cited crawler setting assumes one refresh restores a page.Source: §7 · p. 8
• Unlike B-PAC’s global routing threshold, the paper’s contribution is a per-template lifecycle that certifies, retires, and restarts memories.The passage also notes concurrent work on self-modification and shared multi-agent memory.Source: §7 · p. 8

Conclusion

• A failure at rate 0.05→0.25 should weigh about seven successes, not one, under the memory-specific likelihood ratio.This is the stated correction to equal-vote counters.Source: §8 · p. 8
• The certification load Θ prices skipped reasoning in the worst case when utility is certified against live reasoning.The conclusion lists utility certification as one of three rules for safe reuse.Source: §8 · p. 8
• Restarting at every notice yielded 0 false certifications in 5,779 at-risk epochs at δ = 0.05, and invalidation plus the detector reduced stale answers 9–12× when notices were missed.The conclusion recommends invalidating for substantial edits and keeping the detector as a safety net.Source: §8 · pp. 8–9

Scope and assumptions

• Results cover three domains and five open models up to 8B parameters, with exact verification and random-time fact changes.The theorems assume each audit reveals a verified outcome governed by current memory reliability.Source: Limitations · p. 9
• Per-epoch safety requires change notices to cover every change of the certificate’s paired-utility null, including changes to live reasoning.The experiments’ notices cover answer changes.Source: Limitations · p. 9

Future work

• Future work includes modeling candidate formation, noisy verifiers, longer natural edit histories, larger models, and open detection–certification problems.The authors also identify delayed or erroneous notices and broader real-world drift as settings to study.Source: Limitations · p. 9

Data and release

• The datasets are public and contain no personal data beyond public facts about notable entities; synthetic edits and code are released for research.Licenses are specified for the datasets, edit logs, and code.Source: Ethical considerations · p. 9

Models and tasks

• All tasks are in English or Python, and all open-weight models were run locally without paid APIs.Source: Ethical considerations · p. 9

AI use

• AI assistants helped draft proofs, code, and text and run experiments; independent AI referee agents checked the work, and the authors take responsibility.Source: Ethical considerations · p. 9

Deployment caveat

• High-stakes deployment still requires a verifier whose errors are understood, and the guarantees hold only under the stated assumptions.Source: Ethical considerations · p. 9

Likelihood-ratio tuning

• For the rule assigning +a to failure and −b to success, the cited text identifies it as the exact CUSUM for pairs (q0, q1) satisfying ρ(q0, q1) = b/a.Source: A Proofs for Section 3 · p. 17

Phase diagram

• When the post-change drift is negative, delay scales as A^γ; if stale outcomes never produce a positive score, the rule never alarms.For Bernoulli rules, γ < 1 when q1 > q0; under partial staleness the symmetric rule can have γ = 0.373.Source: A Proofs for Section 3 · p. 17

Likelihood-ratio tuning

• A symmetric counter is likelihood-ratio optimal only when q1 = 1 − q0; for q1 > q0, tuning at q− attains the first-order minimax delay ln A/KL(q−∥q0).The minimax result applies over q1 ∈ [q−, q+].Source: A Proofs for Section 3 · p. 17

Unclipped counters

• Unclipped counters can have infinite false-retirement horizon and may never retire partially stale memories.For q1 < 1/2, the majority rule’s probability of never flagging is at least 1 − 2q1; the ExpeL-style counter bound is 1 − (q1/(1 − q1))^c0.Source: A Proofs for Section 3 · pp. 17–18

Time decay

• Time decay can retire a reliable but rarely used macro: with λ = 10^-3 and τ = 0.85, a 1%-use macro is retired after at most 4.15 uses on average.Retirement follows a sufficiently long run of queries that do not use the template.Source: A Proofs for Section 3 · p. 18

Related-work correction

• In regime (iii), despite prior work’s infinite-delay claim for every false-alarm level below one, delay is finite whenever P1(f > 0) > 0, with D(h) ≤ (E1N−/c1)eθ1(h+¯x).Source: A Proofs for Section 3 · p. 18

Certification limit

• Every δ-safe policy has long-run unfolded fraction at least (1 − εδ)Fπ(Θ), while a restarted SPRT with announced changes achieves an upper bound approaching Fπ(Θ) as δ → 0.The converse applies whether changes are announced and regardless of how many candidates the policy forms.Source: B Proofs for Section 4 · p. 19
• The certification load is Θ = (ν/p) ln(1/δ)/KL, combining change rate, audit rate, tolerated risk, and information per verified outcome.The lower-bound proof uses only audited outcomes and the policy’s non-anticipating information structure.Source: B Proofs for Section 4 · pp. 18–19

Achievability

• A change-triggered restart discards prior evidence and runs an SPRT; its false-fold probability is bounded by δ under arbitrary drift below r0.For reliability at least r1, the expected audit count is at most (ln(1/δ) + a+)/KL.Source: B Proofs for Section 4 · p. 20

Finite-δ bracket

• For (r0, r1) = (0.85, 0.95), δ = 10^-2, and Θ/πi = 0.908, exact dynamic programming gives ui = 0.5610 within [0.5557, 0.5643].The same passage reports first-order value 0.5966 and Jensen bound 0.5998.Source: B Proofs for Section 4 · p. 21

Zipf demand

• For Zipf demand, the unfolded fraction scales as κsΘ^(1−1/s), linking the certification load to the demand-tail exponent.The cold-start law approaches this steady-state form as νT grows.Source: B Proofs for Section 4 · p. 21

Cold start and drift

• The cold-start theorem gives M(T, ν) = (pT)^−(1−1/s)Φs,n(νT)(1 + o(1)); extending this law to safe sequential folding remains open.At νT = 0 it reduces to the static amortization law; at large νT it tends to the Zipf steady-state scaling.Source: B Proofs for Section 4 · p. 21

C Certifying utility by betting

• Theorem 15 shows the betting wealth is a nonnegative supermartingale, so the probability it ever reaches 1/δ is at most δ under the stated assumptions.The guarantee holds for every shadow-evaluation schedule and stopping or abandonment rule.Source: C Certifying utility by betting · p. 22

C.1 Agreement-aware certification

• With exact-match scoring, agreement-triggered audits retain full-information power and can certify a slightly less accurate macro when r > r⋆.If answers agree, their correctness labels agree only when scoring is a deterministic function of the final answer; this certificate is not valid for KQA-STREAM’s labels as defined. When every answer agrees, the result gives label-free folding after ⌈ln(1/δ)/ln(1+z0)⌉ shadow runs, versus an (O1) lower bound of ln(1/δ)/(p ln(1+z0)) expected evaluations.Source: C Certifying utility by betting · p. 22, §C.1 · p. 25

C Certifying utility by betting

• KLinf(P) equals the maximum expected log-growth of a bet and is positive exactly when paired utility has positive mean.Locally, KLinf = µ^2/(2σ^2)(1 + O(µ)).Source: C Certifying utility by betting · pp. 22–23
• Among (O1) rules, the optimal expected certification time is (1 + o(1)) ln(1/δ)/(p KLinf(P)) as δ → 0.The bound is specific to rules that observe audited utilities only; rules that also observe agreement may be faster.Source: C Certifying utility by betting · p. 23
• On KQA-STREAM below-break-even candidates, the plug-in SPRT folded 0.32% (477 of 147,349 encounters), betting folded 0.017% (33 of 190,187), and counters folded all.The plug-in failure is a boundary issue: candidates just below break-even and live accuracy estimated from few labels; most spurious counter-folded candidates were much worse, with median shortfall 0.68.Source: C Certifying utility by betting · p. 24
• At a = r = 0.9 and η = 0.05, pairing raises KLinf by 1.85× at correlation 0.5 and 7.56× at correlation 1.For fixed marginals with r > r⋆, KLinf increases strictly with live–macro error covariance, so pairing is faster than an unpaired design when covariance is positive.Source: C Certifying utility by betting · p. 24
• With announced changes, restarting the betting test at each change gives δ-safety, and the coverage result uses KLinf(PZ) without requiring live accuracy a.The section states that KLinf(PZ) also governs the certification load alongside pairing, disagreement-triggered audits, and cost-aware utilities.Source: C Certifying utility by betting · p. 25

C.1 Agreement-aware certification

• At η = δ = 0.05 and p = 0.1, the agreement bet folded after 59 shadow runs, versus at least 584 for (O1) rules and 910 for the grid-mixture IPW certificate.The agreement bet is powerless unless agreement is nearly certain (g > 0.95 here); the implemented certificate also bets on audited disagreements using inverse-propensity weights.Source: §C.1 · p. 26
• Under the stated assumptions, the agreement-aware certificate folds with probability at most δ, and its optimal information rate D* is positive exactly when expected utility is positive.The converse gives E_P τ ≥ ln(1/δ)/D*(P; p), and finite mixtures of strictly feasible bets can asymptotically attain this rate.Source: §C.1 · p. 26

C.2 Changes, carried-over evidence and the lifetime ledger

• A lifetime ledger bounds null-evidence folds per template by δ, but per-epoch safety requires carried-over candidates to be discarded or restarted at wealth 1 with the next ledger index.With edit notifications, candidates restart their wealth and shadow statistics at each relevant edit; folded macros remain monitored by LR-CUSUM.Source: §C.2 · pp. 28–29
• After 60 good runs, 11.1% of candidates still unfolded folded after a change to a null law, versus a 13.4% bound; after 120 good runs, the figures were 22.8% and 28.1%.Carried-over wealth can therefore defeat per-epoch safety even when the lifetime ledger controls folds on null evidence.Source: §C.2 · pp. 29–30
• Without change information, an epoch’s fold probability is at most α_i, but positive-probability reliability and paired-utility certificates violate per-epoch safety when δ < α_i r_0 and δ < α_i/(1 + M), respectively.For clipped utilities with M = 1, or reliability threshold r_0 = 1/2, and δ = 0.05, impossibility covers templates averaging more than one audit per nine epochs.Source: §C.2 · p. 30
• At r0 = 0.8 and δ = 0.05, fold-at-first-success is δ-safe when λ_i/ν ≤ 0.0658, below 0.0667, where Proposition 23(a) excludes every rule.Its exact bad-fold probability is λ_i r0/(λ_i r0 + ν), between α_i r0 and α_i; at λ_i/ν = 0.03 and 0.0658, the probability is 0.0234 and 0.0500, respectively.Source: §C.2 · p. 31
• Without restarts, bad epochs beginning one success from the SPRT threshold exceed δ in fold frequency from α_i = 0.2 onward, while stream-level frequency stays below α_i at tested values.The per-epoch frequency follows α_i r0 up to Monte Carlo error, with larger excess for α_i ≥ 0.9 through later audits; across epochs it stayed below α_i for α_i ∈ [0.02, 0.98].Source: §C.2 · p. 31
• For segments begun within an epoch, bad-fold probability is at most the remaining ledger budget, while a segment carried into the epoch has risk at most min{1, δ_J K_c}.Proposition 25 bounds new-segment risk by Σ_{j>J} δ_j ≤ δ and carried-over risk until the next restart; each restart starts a fresh test at wealth 1 with a fresh ledger index.Source: §C.2 · p. 31
• In simulation, fresh segments folded in at most 0.0003 of bad epochs, versus 0.009–0.014 for carried-over segments; changes that improve live reasoning without changing macro outcomes are invisible to the detector.The corresponding mean bounds were 0.001–0.009 for fresh segments and 0.015–0.052 for carried-over segments; the carried-over risk lasts until the detector’s next alarm.Source: §C.2 · p. 31
• For a restart-only certificate, the j-th test threshold is ln(1/δ) + 2 ln j + ln(π^2/6), with finite-grid costs including additive terms and a KLinf/g_B(P) factor.The additive cost is 2 ln j + ln(π^2/6) + ln 7 + ln 1.9; KLinf/g_B was 1.03–1.07 on three paired laws, so the certificate does not attain the idealized limit as δ → 0.Source: §C.2 · p. 32

D Experimental details

• KQA-STREAM selected 300 templates across 2-, 3- and 4-hop paths and sampled 60 anchors each, yielding 18,000 instances.Across edit events, 2-, 3- and 4-hop templates were hit at rates of 2.7%, 4.2% and 5.5%, respectively, making staleness more likely for longer derivations.Source: D Experimental details · pp. 32–33
• On Qwen3-8B, macros and compiled plans added 15.5 and 26.1 points over live reasoning; certification with retirement and restarts added 10.7 and 2.9 more.Certification cut stale answers 20× for macros and 6× for compiled plans; the accuracy gain comes from symbolically executing verified paths and is specific to this benchmark.Source: D Experimental details · pp. 34–35

D.1 Natural template streams

• On KQA Pro and MuSiQue template streams, measured coverage slopes ranged from −0.16 to −0.21, consistent with the Zipf-law prediction of −0.13 to −0.21.The online Good–Turing forecast predicted the coverage gap with 0.4–3.4% median relative error without knowing the Zipf exponent.Source: §D.1 · p. 35, Figure 4 · p. 35

D.2 Learning rates, counters, noisy verifiers and a semantic cache

• The detector comparison replayed 30 runs per cell using 200k queries, Zipf s = 1.2, Poisson edits, audit rate p = 0.1 and L = 20.Thresholds were compared at matched false-retirement rates using true reliabilities, a fair comparison but not a deployable tuning procedure.Source: §D.2 · pp. 35–36

D.2.1 Learning-rate sweep

• LR-CUSUM detected staleness 1.3–5.9× faster than each model’s best tested fixed ratio and served 1.7–2.5× fewer stale answers than the best fixed ratio with k ≥1.The speed advantage was significant on Qwen3-8B, Qwen3-4B and OLMo-3-7B; the comparison used matched false-retirement rates.Source: §D.2.1 · p. 36, Figure 5 · p. 37

D.2.2 Counters in the main setting

• ExpeL- and ACE-style counters served 294–967 stale answers per 10^4 queries, 7–13× ours, while skipping certification on more queries.Counters answered 28–76% of queries in one step, versus 20–53% for ours; they also made fewer LLM calls, with the savings roughly offsetting stale and spurious answers in J20.Source: §D.2.2 · pp. 36–37, Table 8 · p. 38

D.2.3 A noise-aware certificate

• At verifier flip rates ε = 0.05, 0.1 and 0.2, labels-to-fold slowdown was 1.88, 2.56 and 3.67, exceeding predictions of 1.23, 1.56 and 2.78.At ε = 0.2, accuracy fell from 0.436 to 0.279; correcting the certificate and CUSUM did not prevent this loss, which the authors attribute to slower certification and falsely verified live traces.Source: §D.2.3 · pp. 37–38, Table 9 · p. 38

D.2.4 A vCache-style semantic cache

• With its confidence band, vCache served only 6–26% of queries from cache and had J20 9.2–30.7 worse than ours on every model.Its accuracy equaled live accuracy because it stores unverified live answers; without the confidence band, stale answers rose to 100–536 per 10^4 queries.Source: Table 10 · p. 38, §D.2.4 · pp. 38–39

D.3.1 Censoring-aware detection delays

• At δ = 0.05, restricted mean delays within 103 audited uses were 7.6 for betting, 10.4 for SPRT, 299.9 for ExpeL-style, and 257.6 for ACE-style.Counter episodes were often still open at stream end, making naive mean delays lower bounds; their censoring-aware delays were 7–8× longer.Source: Table 11 · p. 39, §D.3.1 · p. 39, Figure 6 · p. 39

D.3.2 Detection on a matched population

• Under identical audits, pooled matched-population detection left 50.0% [38.2, 62.8] of ExpeL-style and 47.4% [36.3, 60.2] of ACE-style episodes unretired, versus LR-CUSUM’s 1.9% [0.0, 4.7]; restricted mean delays were 60.8× and 59.1× longer.Template-clustered 95% intervals for the ExpeL/LR and ACE/LR delay ratios were [44.7, 87.1] and [43.7, 83.7], respectively.Source: Table 12 · p. 40, §D.3.2 · p. 40

D.3.3 Per-epoch safety

• At δ = 0.05, per-epoch false-fold frequencies were 0.007 for SPRT, 0.006 for per-candidate betting, 0.005 for betting with a lifetime ledger, and 0.002 with restarts.Under answer-only scoring, all 15 remaining false folds of per-candidate betting were carried over by candidates good at formation, the event restarts remove; under the test’s own null, none of 8,434 or 8,084 candidates folded.Source: §D.3.3 · pp. 40–41, Figure 7 · p. 41

D.3.4 Restarts without edit notices

• At δ = 0.05, discard-on-alarm had bad folds in 0.0008 of at-risk epochs versus 0.0025 for edit-notified restarts, while raising accuracy by 1.3 points and lowering J20 by 1.06.All nine bad folds under discard-on-alarm were spanning folds of candidates good at formation; none rested on post-change evidence of a candidate bad under the test’s own null.Source: §D.3.4 · p. 41, Table 14 · p. 42

D.3.5 Partial staleness on real streams

• For 217 partial-staleness episodes, failure weight 7 gave a 195-use delay, while the symmetric rule was 1.49× slower; for 170 near-total episodes, the optimum shifted to 3–5.At 1 false alarm per 10^4 healthy uses, 99.8% of partial episodes were detected within 1,000 uses at weight 7, versus 90% for the symmetric rule. The symmetric rule was within 1.04–1.19× of the optimum for near-total changes; deployed LR-CUSUM was 1.59× slower on the smaller changes it was not tuned for.Source: Figure 8 · p. 42, §D.3.5 · p. 42

D.3.6 Per-memory weights against the best fixed ratio

• On the same certificate at matched false retirements, per-memory LR-CUSUM was 5.8, 5.9, 2.0, 1.3, and 2.0× faster than the best tested fixed ratio across the five models.Paired intervals excluded 1 for Qwen3-8B, Qwen3-4B, and OLMo-3-7B, but not Granite or Mistral; assigned weights varied across memories and models.Source: §D.3.6 · pp. 42–43

D.4 Aligned safety, same-information baselines, decoding variance and natural edits

• The D.4.1 and D.4.2 replays reuse recorded outcome tables from five models, with 30 streams per model and policy and no missing shard.Intervals are 95% template-clustered bootstraps, with pooled rows resampling templates jointly across models.Source: §D.4 · p. 43

D.4.1 Safety scored as the certificate tests it

• At δ = 0.05, the restart-only certificate has no false certifications in 6,657 at-risk epochs, while task-level bad folds can span an edit.The reported bad-fold shares stay below δ, but δ controls whether a bad memory is trusted, not how often an accepted memory errs.Source: §D.4.1 · p. 43

D.4.2 Same-information baselines

• With missed notices, invalidation alone leaves all 218 stale episodes in service at stream end; adding LR-CUSUM keeps stale answers at or below restart-only levels.At zero notice probability, invalidation alone loses 4.4 accuracy points; invalidate + LR adds 0.6 points over restart-only on Qwen3-8B.Source: §D.4.2 · p. 44, Table 16 · p. 44

D.4.3 Decoding variance

• Across three decoding seeds, live-accuracy variation per version is at most 0.0044, far below the 0.09–0.10 template-bootstrap half-width.Policy accuracy rankings stayed in the same order for every seed, and no conclusion changed on the subset.Source: §D.4.3 · pp. 44–45

D.4.4 Natural edits from Wikidata temporal qualifiers

• 3.0× more stale answers are served by counters than by the restart-only certificate on natural edits.The per-memory LR-CUSUM is faster than the best fixed weight at matched false-retirement rates, by 1.56× at 3 false alarms per 10^4 healthy uses.Source: §D.4.4 · p. 45, Table 17 · p. 45

D.5.1 When the null changes without a notice

• Silent changes in live reasoning can shift the certificate’s null without changing answers or gold answers, so per-epoch safety requires notices covering those changes too.Across the replays, null flips were rare, and no fold of a candidate null at the fold occurred at δ = 0.05; the required notice assumption is not automatic.Source: §D.5.1 · p. 46, Table 18 · p. 47

D.5.2 The recommended deployment

• The recommended deployment serves no stale answers, with bad-fold shares below δ at every tested level, but is 1.9 points less accurate than restart-only.Its accuracy is 0.337 against 0.356; when an edit changes 3% of a template’s answers, invalidation discards a memory that remains mostly right.Source: §D.5.3 · p. 47

D.5.3 Natural edits: sensitivity

• On all 142 stale episodes, counters leave 70.4% (ExpeL) and 62.7% (ACE) in service at stream end, against 26.8% for LR-CUSUM.The delay ratio is 13.6×; unretired shares depend on censoring, while the delay ratio remains stable in leave-one-out checks.Source: §D.5.3 · p. 47

D.5.4 Which variant for which goal

• Table 19 presents a pooled-evidence decision guide for choosing among variants.The supplied caption does not state the guide’s recommendations.Source: Table 19 · p. 47

D.6 A code domain with an exact verifier: CodeStream

• CodeStream replays five models on 521 EvalPlus tasks, using hidden plus-test inputs and the task-version reference output as the verifier.Each (program, version, input set) is run once and cached so every policy receives the same verdicts.Source: §D.6 · pp. 48–49
• 518 (ExpeL) and 510 (ACE) stale answers per 10^4 queries compare with 10.1 for the certificate with the counters’ information; the pooled gap is 51×.Per folded answer, the ratio remains 36× pooled, so the gap is not an artifact of higher coverage.Source: §D.6 · p. 49, Table 21 · p. 51
• 10.4× [7.7, 13.7] more stale uses per stream occur under the best fixed failure weight than under per-memory LR-CUSUM at no more false retirements.The best fixed weight is 5000:1; exposure is flat from 100:1 to 1000:1 on the shared threshold grid.Source: §D.6 · p. 49, Table 22 · p. 51
• 8.7 points [7.1, 10.5] is the certified policy’s pooled accuracy gain over live reasoning, with 0.62 fewer steps but 0.29 more LLM calls per query.The call increase reflects shadow-mode live reasoning while candidates are under test.Source: §D.6 · pp. 50–51
• The CodeStream scope does not establish staleness magnitudes for real library drift: API edits break few programs, and version-specific API benchmarks are proposed for measuring that drift.The edits test unannounced changes and changelog notices; requirement revisions are controlled changes of real tasks.Source: §D.6 · p. 51

E.1 One macro from formation to re-formation

• 33 labels certified one real KQA-STREAM macro, and its likelihood-ratio detector limited a real edit to 29 stale answers while counters kept the stale macro in service to stream end.For this example, the counter’s banked credit delayed retirement beyond the 346 labels remaining after the edit.Source: §E.1 · p. 52

E.2 Counters under partial staleness

• 44 labels retired the +7/−1 rule’s partially stale macro versus 152 for the symmetric rule, while ExpeL and ACE often left partial-staleness onsets unretired through stream end.Across the 30 streams, ExpeL had not retired 95.5% and ACE 80.8% of their partial-staleness onsets by stream end.Source: §E.2 · pp. 52–53

E.3 Certifying utility instead of reliability

• A macro can be slightly less accurate than live reasoning yet remain worth folding when its paired utility is above break-even.For T096, the macro had reliability 0.997 versus live accuracy 0.998, yet was worth +1.93 steps per query.Source: §E.3 · p. 53
• Across 30 streams, the plug-in SPRT made 21 below-break-even folds, while betting made six; in all 21 SPRT cases, its plug-in r0 was below true break-even.For the T010 candidate, the SPRT folded at 0.51 below break-even and served 45 stale answers.Source: §E.3 · p. 54
• Pairing the macro and live reasoner on the same instance lets routing errors and live mistakes enter the utility certificate directly, without estimating live accuracy.The example’s betting certificate folded on concordant gain even though no discordant pair favored the macro.Source: §E.3 · p. 54

E.4 Why rare templates stay uncovered

• The head template certified within 38 labels, but the tail never did; none of 5,771 template-streams ranked 101–300 was folded.Those ranks received 7.5–14.5 audits per template-stream on average, below the labels needed for certification.Source: §E.4 · p. 55, Table 26 · p. 55

• Unlike model-level step-skipping and chain-of-thought compression, contraction decides per template over time when to serve a distilled routine and when to retire it.The cited model-level methods select training data offline and do not make inference-time trust or staleness decisions.Source: F Extended related work · p. 56
• Where caching, routing, and cascade systems carry guarantees, the cited guarantees are fixed-sample risk control under exchangeability; B-PAC instead certifies a global thinking router.This work uses related betting and inverse-propensity tools but certifies each template across its lifecycle, with restarts and retirement.Source: F Extended related work · p. 56
• The paper studies one memory’s items certified per template from verifier feedback, leaving shared and self-modifying memories to future work.The related-work discussion notes concerns including correlated LLM errors, noisy judges, and shared-memory poisoning.Source: F Extended related work · p. 57
• Unlike fixed-distribution speed-up learning analyses, this work adds validity under adaptive schedules and drift, retirement, and closed-form laws for contracted reasoning.The cited earlier analyses assume a fixed problem distribution and do not un-adopt rules.Source: F Extended related work · p. 57