Clean Sheets and Checkpoints: Evidence Complexity for Certifying Sequential Agent Risk

1. Introduction

• With common milestone probability p, completion risk scales as approximately Bp^k under full reset and (Bp)^k/k! under full progress retention in their respective rare-risk limits.Checkpoint resets lie between these cases.Source: §1 · p. 1
• For four milestones and ten zero responses each, the target risk of at most 1/100 at 97.5% confidence is certified up to 169 attempts analytically, 185 with computer assistance under full reset, but only 10 under full retention.The examples use the assumptions of Section 2.Source: §1 · p. 1
• The paper asks how many zero audit responses certify completion probability at most ϵ within B attempts, and how to allocate those responses across milestones.Cost is measured in the zero world, where all milestone success probabilities are zero.Source: §1 · pp. 1–2
• For segments chosen before the audit, completion-window counting yields finite certificates and rare-risk evidence costs, with optimal response allocation determined by whether segments are active or free.The introduction also identifies a distinct reduced allocation at a single active-to-free crossover.Source: §1 · p. 2
• The contribution connects reset geometry, finite campaign-risk certification, phase-dependent evidence complexity, and necessary test allocation.The paper characterizes these connections using classical ingredients including confidence limits, Wald’s identity, log-concavity, and change of measure.Source: §1 · p. 2

2.1. Campaigns, milestones and reset semantics

• A campaign is a chain of k mandatory milestones; failure at a milestone returns the state to the start of its segment, preserving completed segments and erasing progress within that segment.Full reset is one segment, while full progress retention uses segments of length one.Source: §2.1 · p. 2

2.2. Deployment processes

• Deployment processes may adapt, randomize, retain memory, and receive user guidance, but must follow the chain transitions and cap success probability by p_i after every complete history.Under these conditions, the independent chain upper-bounds completion probability within B attempts.Source: §2.2 · p. 2

2.3. Audit observations and validity

• The transport assumption requires prepared-state audit probabilities to cap deployment success probabilities at every history; the audit does not test this link.The audit oracle permits adaptive milestone queries, and reaching or constructing milestone states is not charged.Source: §2.3 · p. 3

2.4. The certification geometry

• In certification geometry, each zero response contributes log-likelihood ratio u_i = -log(1-p_i), and the optimal information allocation determines the minimum zero-world cost up to an additive k.The campaign-risk boundary need not be convex, and the paper does not assume convexity.Source: §2.4 · p. 3

3. Finite certification under different reset rules

• For any nonempty selected segment set A, counting possible end positions of uninterrupted successful runs bounds the chance of completing the chain.The window count uses the chain-transition and history-wise domination conditions, not independence.Source: §3 · pp. 3–4
• When A is fixed before responses, querying each selected milestone equally often and certifying only on an all-zero record gives a valid certificate with fewer than c_α/τ_A + K_A responses.Milestones outside the selected set are not queried, and their parameters are unrestricted.Source: §3 · p. 4
• The certificate applies to adaptive deployment processes satisfying the model and arbitrary unqueried milestones; selecting A to minimize the count uses only the model and the target budget and risk parameters.The resulting rule remains valid because the segment choice is prospective.Source: §3 · p. 4
• The selected-segment window coefficient is specific to the chain, and the union bound may be loose when selected parameters or Bβ_j are not small.Choosing segments or the acceptance rule after observing responses requires separate calibration.Source: §3 · p. 4
• For counts fixed in advance, a plausibility rule can certify multiple budget-risk pairs at once while bounding the probability of certifying any unsafe pair by α.This rule certifies only when the responses are all zero.Source: §3 · p. 4

4.1. Finite information bounds

• The information optimum bounds the minimum zero-world cost between cα/I∗ and fewer than k responses above it, while the diagonal threshold gives simpler bounds but no leading constant or allocation.The lower bound applies to every valid evaluator; a fixed-count all-zero design gives the matching upper bound within fewer than k responses.Source: §4.1 · p. 4

4.2. Cycle lengths

• For each segment, a cycle ends at failure or completion; its mean length is κj, and independent segment attempt counts determine the chain’s completion probability within budget.Wald’s identity gives expected segment attempts κj/βj when βj > 0.Source: §4.2 · p. 4

4.3. Phase interiors

• In phase interiors, active segments remain rarely completed within budget while free segments are nearly always completed; the active set consists of segments at least as long as a threshold length.On the diagonal, segment j is active when Bp^ℓj is small and free when it is large.Source: §4.3 · p. 5
• The sharp evidence complexity has exponent a/KA, and the Theorem 3.1 certificate attains its leading constant; the exponent falls as fewer, longer segments remain active.The rounding condition cαn/tn → ∞ is needed for the stated constant.Source: §4.3 · p. 5
• Every asymptotically optimal evaluator allocates equal expected response shares to active milestones and vanishing shares elsewhere, although finite-budget optimal allocations need not be uniform.Under full retention, the limiting allocations are uniform.Source: §4.3 · p. 5

4.4. A single crossover

• When one segment crosses from active to free, its milestones receive smaller limiting shares than active milestones, and the crossover-aware certificate attains the optimal leading constant.For segments (3, 1) at λ = 2, the length-one segment receives about half the limiting expected share of each active milestone.Source: §4.4 · pp. 5–6
• A crossover segment consumes a random fraction of the budget, adding a factor ha(λ) to risk; this factor increases from 0 to 1.The asymptotic crossover results do not identify the finite-budget optimum or cover crossovers of several equal-length segments.Source: §4.4 · p. 6

5. Cycle-aware certification

• For a two-transition segment with probabilities (1, q), counting cycles gives VB ∼ Bq/2 when Bq → 0, recovering the half-window coefficient.Each cycle lasts two attempts; the window count alone does not capture this factor.Source: §5 · p. 6
• The cycle-aware certificate for unqueried prefixes is valid, certifies with probability one when queried probabilities are zero, and uses fewer than cα/τ′ + K′ responses.The design selects segments and queried milestones before responses; its bound can benefit from unqueried prefixes.Source: §5 · pp. 6–7
• When only a specified set of milestones is queryable, the cycle-aware certificate is asymptotically optimal within that restricted class, with queryable milestone counts determining active/free classification.Its leading constant includes a factor 1/σj for each active segment, and limiting allocation is uniform over queryable milestones in active segments.Source: §5 · p. 7
• For the six-transition family, the cycle-aware certificate costs asymptotically cαn(Bn/(2ϵn))1/3, while the window certificate costs about 1.26 times the optimum.This comparison assumes the stated sure transitions and phase-interior and rounding conditions.Source: §5 · p. 7

6. Numerical examples and a conditional historical reanalysis

• The numerical values illustrate the theorems and are not used in their proofs.The examples use exact chain recursions, binary64 or exact rational arithmetic, or integer-arithmetic branch and bound.Source: §6 · p. 7

6.1. Evidence across budget regimes

• At B = 10^6, the smallest displayed certificate uses about 1.2 × 10^3 responses under full reset and 1.6 × 10^7 under full retention.Under full retention, the lower benchmark is within k responses of the minimum; the certificate is about 4% above that benchmark.Source: §6.1 · p. 7

6.2. Cycle-aware accounting

• 63 versus 78 responses at B = 100, and 78 versus 96 at B = 185, make the cycle-aware certificate about 19% smaller than the window certificate.The cycle-aware counts are 5.8% and 5.1% above the lower benchmark at those budgets; the window counts are 31% and 29% above it.Source: §6.2 · p. 7

6.3. A conditional reanalysis of a published clean sheet

• The reanalysis treats the forty zero responses as fixed in advance and assumes the deployment process satisfies the model’s domination condition.The published Bayesian credible bound is not a fixed-parameter confidence guarantee or a campaign guarantee, and the paper does not compare it with these certificates.Source: §6.3 · p. 8
• For the four-milestone record, the conditional calculation certifies up to 169 attempts analytically and 185 with a computer-assisted bound under full reset, but only 10 under full retention.At B = 186, the symmetric point already has risk about 0.0100093, so 185 is the largest budget the stated plausibility rule can certify.Source: §6.3 · p. 8
• A rational asymmetric probability vector has greater risk at B = 185 than the symmetric point, while the cycle-length explanation for the 169-to-185 increase is explicitly heuristic.The reported risks are approximately 0.0099791 and 0.0099549, respectively.Source: §6.3 · p. 8
• The calculation inherits the model’s transport assumptions: prepared milestone states may differ from deployment states, and attempts may share a latent cause.The four milestones may also omit alternative routes to the goal.Source: §6.3 · p. 8

7. Related work, limitations and conclusion

• Milestone-based rare-event estimation is related to multilevel splitting, but this paper certifies campaign-level probability rather than only estimating a rare probability.The cited multilevel-splitting idea estimates a rare probability as a product of conditional probabilities from intermediate states.Source: §7 · p. 8
• The certificate’s multiplicative gain over black-box testing requires condition (D3), which excludes arbitrary dependence among component failures.Bishop and Povyakalo’s conservative bound for positive Nmin is min{1, log(1/α)/Nmin}; passing all milestones in one traversal has the same logical structure as a system failing only when every component fails.Source: §7 · p. 9
• Language-model risk work differs in what is sampled and who owns the budget: this paper audits prepared-state attempts while deployment budgets and resets belong to the agent.The related work includes sampled conversations, censored generation streams, and repeated-run benchmark scores.Source: §7 · p. 9
• The transport assumption is untested and may fail when prepared states differ from deployment states, attempts share a latent cause, or the agent improves across attempts.Condition (D1) excludes alternative routes and skipped milestones, and reaching milestone states is not costed.Source: §7 · p. 9
• The leading-constant and allocation results are asymptotic and limited to phase interiors and single crossovers with fixed λ; the B = 185 certificate is computer-assisted, not formally verified.Crossovers of several segments, fixed risk, finite-budget designs, and nonzero safe parameters remain open.Source: §7 · p. 9
• In rare-risk phase interiors, the certificate attains the optimal leading zero-world cost, and asymptotically optimal audits allocate equally across active milestones and vanishingly little to free ones.A crossover-aware certificate attains the optimal leading constant at a single crossover.Source: §7 · p. 9

Impact Statement

• The certificates may inform deployment decisions, but they hold only under stated modeling and transport assumptions that audits do not test.The paper reports its reanalysis as a conditional calculation, not a claim about the evaluated systems.Source: Impact Statement · p. 9
• A certified attempt budget can change substantially when the modeling assumptions change.The paper states that it reports assumptions next to results to guard against misuse as unconditional safety claims.Source: Impact Statement · p. 9

• V_R(i) is nondecreasing in the horizon, state, and each success probability, and is strictly increasing along a probability ray when B≥k and V_B(p)<1.Source: B. Proofs for Section 2 · p. 12
• Every process satisfying the history-conditional caps completes within B attempts with probability at most V_B(p), and the independent chain attains that bound.Source: B. Proofs for Section 2 · p. 12
• Average milestone success rates do not suffice: in the two-milestone example, completion probability is r rather than V_2(r,r)=r^2.After reaching state 1, the next attempt succeeds with probability one, violating the conditional cap.Source: B. Proofs for Section 2 · p. 13
• Under full reset with k=5 and B=9, the unsafe sets are nonconvex in both logit and u coordinates at the stated thresholds.The proof uses positive tangent curvature at specified points, with exact signs checked by rational computation.Source: B. Proofs for Section 2 · p. 13
• The nonconvexity result does not undermine the proposed certificates, which use log-concave bounds and do not assume the unsafe set is convex.The result does not settle whether I*=t generally or address the separate computation in Section 6.3.Source: B. Proofs for Section 2 · p. 14

C. Proofs for Section 3

• For a fixed selected set of segments, the certificate counts possible completion-window patterns and bounds each pattern using the history-conditional success caps.A union bound over the weakly increasing window offsets yields the finite-budget bound.Source: C. Proofs for Section 3 · p. 14

D.1. The cycle-length renewal lemma

• For an active set A, renewal bounds express the completion probability through segment completion probabilities and cycle means; when all rates vanish, the cycle means tend to one.Source: §D.1 · p. 15

D.2. Phase interiors

• In phase interiors, active segments form a length threshold set; in the last phase, segments of maximal length are active together.Source: §D.2 · p. 16

D.3. Finite budgets, full retention and boundary cases

• At B=3 and p*=0.1, the example has optimal w1*=0.4865 and uniform information 0.105325 below I*=0.105361.Source: §D.3 · p. 17
• With full progress retention, uniform allocation is exact at every budget; the rare-risk limits are also uniform under full reset and full retention.Source: §D.3 · p. 17
• With p1=1/4 fixed in a length-two segment, VB/(Bp1p2) tends to 4/5 because the cycle mean is κ=5/4.Thus the cycle-free window coefficient need not be accurate when rates do not vanish.Source: §D.3 · p. 17
• For one milestone at fixed risk, the evidence scale is cαB/(-log(1−ϵ)), not the rare-risk approximation cαB/ϵ.Source: §D.3 · p. 17

D.4. A single crossover

• The crossover profile e_a(λ) decreases from 1 to 0 as λ grows, interpolating between the crossover segment's active and free behavior.For a = 1, the passage gives explicit expressions for h_1(λ) and e_1(λ).Source: §D.4 · p. 18
• At a single crossover, every asymptotically optimal evaluator allocates limiting weight according to ω(λ): equally across active milestones, less across the crossover segment, and zero elsewhere.The crossover-aware certificate attains the leading constant.Source: §D.4 · p. 19
• The crossover-aware certificate is asymptotically optimal, while fixed allocations other than ω(λ), including uniform allocations on active milestones, fall short.The two window-certificate choices that include the active segments, with or without the crossover segment, both have limiting efficiency below one.Source: §D.4 · p. 20
• For segments (3, 1), the best window-certificate bound has a minimum factor of 0.851 near λ ≈ 2.35, where its two terms are equal.Across the listed geometries, the worst-case factor ranges from 1.11 to 1.26.Source: §D.4 · p. 20
• The asymptotic optimality result excludes simultaneous crossovers and fixed ε, while the crossover-aware certificate remains valid at every budget.The asymptotic result also excludes varying declared constants and efficiency at nonzero safe parameters.Source: §D.4 · p. 20

E.1. Parameter classes

• In restricted parameter classes, fixed or capped unqueryable probabilities can be set to their declared caps when evaluating risk, without changing audit observations.The restricted classes therefore yield the same certification problem when they share the same queryable set and declared constants.Source: §E.1 · p. 21
• When the target risk is at least the maximum risk in a restricted class, the class has no unsafe parameter and certification before any response is valid.The maximum is attained when all queryable coordinates are set to one and unqueryable coordinates take their declared constants.Source: §E.1 · p. 21
• For targets below that maximum, the restricted diagonal has a unique threshold, and every valid evaluator has a zero-world response-cost lower bound.The bound is expressed using the restricted information quantity I_Q* in the cited result.Source: §E.1 · p. 21

E.2. The general cycle-aware certificate

• For each selected segment, the cycle length measures attempts up to a prefix failure or the first queried milestone, including the terminating attempt.The mean truncated cycle length σ_j is between 1 and h_j + 1, and equals 1 when the segment begins with a queried milestone.Source: §E.2 · pp. 21–22
• The certificate bounds completion risk for parameters below declared caps and for every process in the stated transition class.If the target is at least the certificate's risk bound, certifying before any response is valid.Source: §E.2 · p. 22
• The cycle-aware certificate queries selected milestones equally often and certifies only when every response is zero, with validity over every class sharing the queryable set and declared constants.Its response count is less than c_α/τ′ + K′.Source: §E.2 · p. 22
• Counting failed cycles can sharpen the window bound when a selected segment has an unqueried prefix before its first queried milestone.When all selected segments begin with a queried milestone, the cycle term is never smaller than the window term.Source: §E.2 · p. 22

E.3. Restricted sharp complexity

• For the restricted sharp-complexity result, every evaluator attaining the asymptotic cost allocates limiting weight 1/K′_A to each queryable active milestone and zero to other queryable milestones.The certificate of Theorem E.3 attains the leading constant.Source: §E.3 · p. 23
• In the (2, 3) segment example, the minimum expected response count is asymptotic to c_α(B_n/(2ε_n))^1/2, attained by querying milestones 4 and 5 only.The limiting allocation on queryable milestones is (0, 1/2, 1/2).Source: §E.3 · p. 23
• Cycle lengths alter the restricted constant only when an active segment has fixed, capped, or unqueried milestones before its first queried milestone.Declared constants after the first queried milestone enter through the same factor as in the window bound.Source: §E.3 · p. 24
• The asymptotic result excludes crossover sequences, fixed ε, declared constants varying with n, and efficiency at nonzero safe parameters.These exclusions do not affect the validity of the finite certificates.Source: §E.3 · p. 24

Numerical checks

• The forward state recursion computes V_B(p); for large budgets, repeated squaring computes the transition matrix power, with reported relative differences of 1.3 × 10^-15 and 5 × 10^-14 in the stated test cases.Table 1 thresholds were bracketed to width 10^-12 and their sign changes verified with rational arithmetic.Source: F. Numerical methods and reproduction · p. 24

Numerical comparisons

• On the final plotted budget interval ending at B = 106, the lower-benchmark exponents are 0.2506 for full reset, 0.3340 for segments (3, 1), 0.5001 for segments (2, 2), and 1.0000 for full retention.Figure 2 displays the smallest prospective certificate across the specified Theorem 3.1 segment sets and Proposition 4.9 pairs.Source: F. Numerical methods and reproduction · pp. 24–25
• For segments (3, 1), combining Theorem 3.1 with Proposition 4.9 brings the certificate to at most 2.8% above cα/t for B ≥30, versus up to 18.1% for Theorem 3.1 alone.At smaller budgets, the combined certificate is at most 6.9% above cα/t, where rounding dominates.Source: F. Numerical methods and reproduction · p. 25

Numerical checks

• The Figure 3 comparison uses 121 log-spaced integer budgets from 10 to 105 and ϵ values 10^-2 and 10^-4; its rounded counts at B = 100 and 185 match Table 1.The caption identifies the plotted quantities as unrounded certificate cost relative to the lower benchmark for window and cycle-aware coefficients.Source: Figure 3 · p. 24, F. Numerical methods and reproduction · p. 25

Numerical comparisons

• Full reset certifies more attempts than full retention in every Table 3 case, by factors ranging from 2 to about 7,200.The factor grows with k, n, and ϵ; the displayed cases with fewer milestones, fewer responses, or stricter risk have factors of approximately 2–3.2.Source: F. Numerical methods and reproduction · p. 26, Table 3 · p. 26

Reproduction

• The six-transition-family diagnostics are numerical observations, not certificates; the supplement reports an independent recomputation of all 1,278,508 leaf bounds for the B = 185 certificate.The reported normalized-gradient weights yield 62 responses in a fixed-count design, but the passage explicitly says this observation proves nothing.Source: F. Numerical methods and reproduction · p. 26