External verification can correct individual outputs while leaving a self-reinforcing population in the basin of a wrong consensus. We study how the timing and addressing of a fixed verification budget affect recovery in an asynchronous binary register. For a general nonlinear response, we derive the minimum fuel required to cross a basin boundary under a peak verification constraint. For a finite population, an exact birth--death calculation gives the probability of subsequent wrong consensus after a pulse. Our main asymptotic result identifies the critical budget window: a leading term Nlog(x0/b) and a correction of order N, with separate variance contributions from repeated verification targets and autonomous amplification after verification stops. The distinction is substantial: with 16 majority-updated slots and 14 initially wrong, 9 random checks cross the mean-field budget threshold, whereas 23 are required for 95% eventual recovery in the exact model. A prospectively specified experiment records 13,392 language-model responses, including calibration and 108 held-out trajectories. Calibration produces different fitted response regimes, but all four adjusted schedule-comparison intervals include zero. A distributional audit also finds that modest mean-prediction error can conceal a large underestimate of terminal consensus occupancy. The results support risk-calibrated reset scheduling under a specified update contract, while explicitly separating it from distinct-target checking and unrestricted evidence broadcast.
Figures & tables
Figure 1: Deterministic majority response. (A) Stable equilibrium branches are solid and the unstable branch is dashed. (B) Directed verification changes the drift. (C) All schedules spend B=0.75 over eight update sweeps from x0=0.875 . Early and late pulses use u=1 for duration B ; spread verification uses u=B/8 . These are solutions of the specified ODE, not measured LLM trajectories.
Figure 2: Control boundaries. (A) Minimum fuel under a peak cap from x0=0.875 to b=0.5 for majority-of-three. Portions outside the plotted range include infeasible caps; the required fuel diverges as a barrier becomes impassable. (B) The sign of the exact commutator ( 12 ) changes, ruling out a universal early-is-better claim for terminal loss.
Figure 3: Exact finite-state calculations for g3 and i0=7N/8 . (A) Rescaling the check budget by its N window approaches Theorem 4 . (B) Exact 95% recovery budgets exceed the mean-field threshold; the two-term asymptotic formula remains an approximation. (C) Avoiding repeated targets changes the budget, illustrating why addressing must be part of the system specification.
N
KMF
Risk at KMF
Exact K0.05
Two-term K0.05
16
9
0.476
23
18.93
64
36
0.484
60
55.76
256
144
0.483
187
183.15
1024
574
0.490
657
652.83
4096
2293
0.495
2456
2451.75
Table 1: Perfect random checks followed by autonomous majority updates. KMF=⌈Nlog(7/4)⌉ . Risk is eventual wrong consensus, not terminal error at a fixed horizon.
Model
0 wrong reports
1 wrong report
2 wrong reports
3 wrong reports
GPT-4.1 mini
0
2
95
96
GPT-4.1 nano
8
32
60
90
Table 2: Calibration wrong-output counts out of 96 valid responses per cell. Each row uses the same 96 task records across the four report compositions.
Figure 4: Measured response curves constructed from four conditional probabilities. Bands are pointwise 95% task-bootstrap intervals; they are not simultaneous guarantees for all x . Filled and open markers denote stable and unstable fitted equilibria. The pooled calibration curves yield different fitted response regimes.
Model
Schedule
Final wrong
Integrated wrong
All correct
All wrong
Mini
Early
0.337
0.403
11
6
Mini
Spread
0.597
0.766
3
1
Mini
Late
0.455
0.946
0
0
Nano
Early
0.514
0.504
8
7
Nano
Spread
0.424
0.585
6
1
Nano
Late
0.306
0.726
4
0
Table 3: Held-out trajectories: 18 episodes per row, each with 116 model calls and 12 deterministic checks. Endpoint counts are out of 18.
Model
Contrast
Mean difference
Lower
Upper
Mini
early minus spread
-0.260
-0.543
0.035
Mini
late minus spread
-0.142
-0.332
0.075
Nano
early minus spread
0.090
-0.078
0.229
Nano
late minus spread
-0.118
-0.295
0.066
Table 4: Primary paired terminal-error differences. Negative values favor the pulse. Intervals use the declared four-comparison adjustment.
Figure 5: Live held-out results. Top: mean observed wrong fractions (solid) with pointwise 95% task-bootstrap bands and independently calibrated finite-state predictions (dashed). Bottom: individual paired task differences and means with adjusted primary intervals. The late pulse acts just before measurement; low terminal error alone is not evidence of stable autonomous recovery.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Phase
Model
Attempts
Terminal
Input tokens
Output tokens
Calibration
Mini
385
384
67,840
1,920
Calibration
Nano
384
384
67,840
1,920
Pilot
Mini
49
48
8,480
240
Pilot
Nano
48
48
8,480
240
Trajectories
Mini
6,268
6,264
1,106,640
31,320
Trajectories
Nano
6,265
6,264
1,106,640
31,320
Appendix
Table 5: Recorded API usage. Attempts include transport retries; terminal requests count the single retained outcome per request identifier. Input and output tokens are provider-reported totals over successful responses.
Model
Schedule
Predicted final
Observed final
Trajectory MAE
Mini
Early
0.316
0.337
0.039
Mini
Spread
0.687
0.597
0.025
Mini
Late
0.458
0.455
0.008
Nano
Early
0.434
0.514
0.066
Nano
Spread
0.357
0.424
0.040
Nano
Late
0.283
0.306
0.028
Appendix
Table 6: Independently calibrated finite-state predictions and held-out means. MAE is averaged over the 129 recorded states.
All correct
All wrong
Model
Schedule
Predicted
Observed
Predicted
Observed
Mini
Early
0.617
0.611
0.258
0.333
Mini
Spread
0.085
0.167
0.050
0.056
Mini
Late
0.004
0.000
0.000
0.000
Nano
Early
0.040
0.444
0.024
0.389
Nano
Spread
0.046
0.333
0.001
0.056
Appendix
Table 7: Post-analysis terminal-distribution audit. Entries are endpoint probabilities, not eventual absorption probabilities. Observed values are proportions of 18 episodes.