ERRAND: Budgeted Maintenance of Agent Memory

Beining Wu; Zihao Ding; Jun Huang

cs.AISep 2, 2026

ERRAND: Budgeted Maintenance of Agent Memory

Authors: Beining Wu, Zihao Ding, Jun Huang

Organizations: Department of Electrical Engineering and Computer Science South Dakota State University, Brookings, SD 57007, USA

Abstract

Deployed agents run on handed-over knowledge: a frozen policy consults a briefing of consolidated items written before the stream begins. The world then moves while the store stands still: paths close, flags change, price bands move; every item was true at handover, and the failure is staleness, not ignorance. We introduce ERRAND, which treats revalidation as a priced errand: a recheck competes with the task it protects for the same scarce actions, funded only when the value per action of resolving a doubt clears a running wage. The errand index is single-peaked, vanishing at both ends of belief, so certainty in either direction costs nothing; free en-route receipts maintain on-path knowledge, and repair writes a version, never a deletion. Under equal action budgets in two drifting tool-use worlds, ERRAND clears every non-oracle policy on the preregistered calibers, primary in every setting and conditional at every binding budget, leading eager revalidation by 10.0pp at the base cap. Restraint wins: given no cap, ERRAND stops on its own, spending 11.0% of steps, while uncapped eager revalidation spends 70.7% and still finishes 4.5pp behind capped ERRAND. The margin sits where the briefing's coverage is thinnest, the shadow price of long-tail knowledge: a small budget, well priced, beats a bigger store that never rechecks.

Figures & tables

Figure 1: Priced revalidation beats spending the same budget without prices, everywhere we deploy it. Left: success across nine deployment axes; badges: mean gain per axis family. Right: stale-use share (

\downarrow

) across six settings, Errand lowest in all. One protocol: frozen agent, drifting world, hard action caps; oracle arms are priced in Table 1 .

Figure 2: The budget buys trajectory changes, never attention. Beliefs over grounded atoms drift toward the prior and reset on free en-route receipts; the single-peaked index prices each off-path item; the deadband gate funds an errand only when value per action clears the running wage $\hat{\nu}$ ; repair chooses the cheapest sufficient grade and writes a version, never a deletion. Abeyance closes use while the comparator keeps watching.

Panel of Figure 3 — Figure 3: Priced errands buy the frontier, and the scheduler stops spending on its own. (a) Success vs. measured spend across caps; the dotted line is where uncapped Errand settles. (b) Every Errand cap converges to one pace. Bands/whiskers: seed s.d.

	Budget axis (mid tier, 8B)						High gap		4B floor		Average		$\Delta$ (ours $-$ row)
	$b{=}6$		$b{=}12$		$b{=}24$		$b{=}12$		$b{=}12$
Method	ITT $\uparrow$	Cond. $\uparrow$	ITT $\uparrow$	Cond. $\uparrow$	ITT $\uparrow$	Cond. $\uparrow$	ITT $\uparrow$	Cond. $\uparrow$	ITT $\uparrow$	Cond. $\uparrow$	ITT	Cond.	ITT	Cond.
No maintenance ( $\nu{=}\infty$ )	51.1	48.9	51.1	48.9	51.1	48.9	45.4	45.5	43.8	43.6	48.5	47.1	$\uparrow$ 16.8	$\uparrow$ 8.5
Bigger memory, no maint.	51.2	51.9	51.2	51.9	51.2	51.9	46.5	45.8	43.6	45.0	48.7	49.3	$\uparrow$ 16.6	$\uparrow$ 6.4
Fixed-cadence recheck	58.1	53.5	61.4	56.3	64.9	57.4	56.5	55.6	46.7	43.9	57.5	53.3	$\uparrow$ 7.9	$\uparrow$ 2.4
Equal-budget random	55.1	52.5	60.8	55.4	66.2	56.6	54.4	52.6	46.4	44.7	56.6	52.4	$\uparrow$ 8.8	$\uparrow$ 3.3

Table 1: Under equal action budgets, priced revalidation clears every non-oracle policy on ITT in every setting and on the conditional caliber at every binding budget. ITT: success over all errand-relevant steps; Cond.: success on judged-domain uses (both %,

\uparrow

). Frontier arms: five seeds (eleven on the high-gap pair); baselines: three; dispersion and paired intervals: Appendix E . Zero-spend rows are cap-invariant, so one run repeats across budget columns. Underline: best non-oracle baseline per column;

\Delta

: average margin of Errand over the row, shaded by size. † Oracle reads staleness labels, wins Cond. but loses ITT; excluded from

\Delta

Figure 4: Priced revalidation wins on both calibers, and the ITT margin survives every budget cap. Success on (a) all steps and (b) knowledge steps under both coverage-gap regimes; boxes: best baseline (BB) and its cross-regime drop. (c) Errand vs. eager revalidation across caps. Error bars: seed s.d.

Figure 5: The prices are calibrated, and they track the world, not the calendar. (a) All twelve points clear the diagonal. (b) Five orderings at matched spend; dots: seeds. (c) Errand holds flat; past

\kappa^{\ast}

(dotted) the half-life policy collapses (circled). (d) By coverage-gap severity band, Errand holds while eager falls away; arrows: the conditional gap in points at equal budget. Bands: seed s.d.

Variant	ITT $\uparrow$	Cond. $\uparrow$	Spend	$\Delta$ ITT
Full Errand	71.3	61.3	10.3%	–
w/o deadband gate	67.8	58.3	10.8%	↓3.5
coverage sum $\to$ max	68.1	58.5	10.7%	↓3.2
index: belief only	66.0	57.1	11.0%	↓5.3
index: age only	64.7	57.0	11.0%	↓6.6
$\mu_{a}{:=}0$ (no regime return)	66.3	57.9	11.0%	↓5.0

Table 2: Every component pays its way; belief calibration pays most. Single-component ablations at

b{=}12

: three seeds, every variant paired with the full arm of its own wave.

\Delta

ITT: drop versus full Errand , shaded by size.

Policy	Tokens/step		LLM calls	Sched.	Tokens	$\Delta$ calls
	prompt	gen.	/step	ms/step	/errand
Errand ( $\hat{\nu}_{0}$ )	431	64	1.10	40	361	–
Eager ( $\nu{\to}0$ )	455	72	1.40	33	334	↑0.30
Fixed-cadence	417	60	1.53	30	355	↑0.44
Equal-random	426	62	1.44	29	361	↑0.35

Table 3: Scheduling never spends a model call. Per-step accounting at

b{=}12

. The scheduler is closed-form arithmetic (

40

ms/step, no LLM invocation);

\Delta

calls: extra LLM calls per step versus Errand , shaded by size, whose targeted errands replace repeated failure recovery.

Figure 6: One item, two endings. The same briefing item flips mid-stream. Eager revalidation, budget spread thin, serves the stale value and fails; Errand holds inside the deadband until a co-located errand returns the new spec for free, and succeeds.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Role	Default	Source
$T$	horizon (steps)	$2{,}000$ ; $500$ -step prefix re-read	instrument manifest
$W$	warm-up excluded from every metric	$300$ steps	instrument manifest
$b$	errand-action cap per hundred steps	$\{6,12,24\}$ ; dense $\{2,3\}$	preregistered axis
$c$	errand cost (actions)	$d+1$ for site distance $d$	audited per atom
$(\lambda_{a},\mu_{a})$	flip-out and flip-back intensity, group A	$(0.006,\ 0.020)$	drift schedule
	groups B, C	$(0.002,\ 0.0025)$	drift schedule

Appendix

Table A.1 : Protocol constants with preregistered defaults.

Symbol	Meaning	Symbol	Meaning
$\lambda_{a},\mu_{a}$	flip-out and flip-back intensity	$g_{i}$	forgone gain of a suspended item
$\kappa_{a}$	total flip intensity $\lambda_{a}+\mu_{a}$	$\ell_{i}$	misuse loss of a served stale item
$q_{a}$	staleness belief	$N_{i}$	usage receipts of item $i$
$q_{a}^{\infty}$	stationary suspicion $\lambda_{a}/\kappa_{a}$	$\hat{u}_{i}$	usage rate
$q_{i}^{\ast}$	disposition threshold	$\tau_{i}$	durability horizon
$\hat{\nu}$	wage (shadow price of the budget)	$\xi(p,i)$	locality discount, in $(0,1]$

Appendix

Table B.1: Notation used in this appendix.

Evidence channels	$\kappa{=}0.18$		$\kappa{=}0.4$		$\kappa{=}1.04$
	$c{=}1$	$c{=}5$	$c{=}1$	$c{=}5$	$c{=}1$	$c{=}5$
no receipts at all	0.060	0.060	0.060	0.060	0.060	0.060
use receipts only	0.258	0.258	0.163	0.163	0.103	0.103
use $+$ en-route receipts	0.407	0.358	0.259	0.194	0.140	0.061

Appendix

Table B.2: Stationary value of three evidence policies (two-state value iteration).

System	Staleness model	Return ( $\mu_{a}$ )?	Repair
GLOVE ( Yin and Du, 2026 )	reactive mismatch on re-encounter; no temporal model	no	delete and replace
STALE ( Chao et al., 2026 )	one-way LLM-adjudicated invalidation	no	archive; read-time block
AdaMEM ( Zhang et al., 2026a )	store frozen after construction	n.a.	regenerate a transient strategy
SkillGuard ( Fan et al., 2026 )	one-shot contract check at ingestion	no	one-shot rewrite
Feedback loops ( Cui et al., 2026 )	outcome bookkeeping; hand-set threshold	edge only ∗	deprecate; no reinstate
MemTX ( Li et al., 2026 )	declared validity interval, never re-evaluated	edge only ∗	typed cascade; supersede

Appendix

Table B.3 : How prior models treat flip-back. ∗ A revival edge is stated but carries no trigger or price.

Estimand	Reads	Carrier	Preregistered band	Outcome	Verdict
StaleCost	stale collapse on the frozen arm	R1 ′	$\geq 40$ pp	$+79.5$ ${}_{\pm\text{\scriptsize 0.6}}$ pp	in band
FrontierGap	ITT, Errand vs. eager @ $b{=}12$	W1-3	CI $>0$ and $\geq{+}3$ pp	$+10.0$ pp [+8.6, +11.4]	in band
SpendSave	spend saved at the same cap	W1-3	$\geq 5\%$ , cond. within $1.5$ pp	$10.3$ ${}_{\pm\text{\scriptsize 1.9}}$ % ( $\Delta$ cond $+1.8$ pp [-0.8, +4.5])	in band
DominanceCell	caps where Errand dominates eager	W1-3	$\geq 2/3$	$1/3$	1/3: single-cap claim
CondMatched	matched conditional gap, high tier	W1-5 $+$ W3-5	CI $>0$ at half-width $\leq 5$ pp	$+4.9$ pp [+3.3, +6.6]	in band
GapPrice	ordering premium, high minus mid tier	W1-5 $+$ W1-3	CI $>0$	$+3.8$ pp [+1.8, +5.8]	in band

Appendix

Table C.1 : Preregistered interpretation bands and outcomes. Bands were frozen before the full waves; verdicts are mechanical.

Tier	Seeds	Pairs (forecast)	Half-width (pp)	Gate at 480
mid	5	451	5.27	below 480 upgrade
mid	8	714	4.19	meets
high	5	462	5.21	below 480 upgrade
high	8	739	4.12	meets
high	11	1016	3.51	meets W3 - 5
realized	11	1016	1.66	met

Appendix

Table C.2: Design-stage pair-yield forecast and the realized cohort.

Panel of Figure 16 — Figure C.1 : The design study, realized. (a) The forecast is one $c/\sqrt{n}$ law, so both tiers share a curve; the realized cohorts halve its constant. Dotted, the $480$ -pair gate; dashed, the $5$ pp criterion. (b) Measured spend against the allowance: both arms empty the purse until the cap stops binding for Errand alone. (c) The two ladder segments, paired by seed; whiskers, $95\%$ paired $t$ . (d) Forecast pair yield against the gate (dashed); dots, realized cohorts.

Group	Atoms	$c$ range	Orders/run	Uses/run	Receipts/run	Stale share
A	8	1–2	20.0	43.5	58.7	13.8%
B	8	2–3	16.0	98.5	182.7	15.3%
C	3	3–3	22.0	17.1	28.2	15.9%
D	2	4–5	14.0	10.5	25.8	18.2%
F	4	5–7	8.0	0.2	15.7	0.0%

Appendix

Table D.1: Roster by atom group, per run on the Errand arm at the base cap.

Tier	Gate	B1	B2	B3	B4a	B4b ′	B5	Verdict
	threshold	$\geq 12$	$\geq 30$	$\geq 150$	$\geq 75\%$	$\geq 2$	$\geq 8$
high	measured	23.3 ${}_{\pm\text{\scriptsize 0.6}}$	69.7 ${}_{\pm\text{\scriptsize 13.5}}$	199 ${}_{\pm\text{\scriptsize 4}}$	85.4 ${}_{\pm\text{\scriptsize 6.2}}$	4.7 ${}_{\pm\text{\scriptsize 0.6}}$	16.0 ${}_{\pm\text{\scriptsize 7.2}}$	PASS (3 seeds)
mid	measured	22.2 ${}_{\pm\text{\scriptsize 1.5}}$	77.0 ${}_{\pm\text{\scriptsize 7.6}}$	202 ${}_{\pm\text{\scriptsize 13}}$	85.6 ${}_{\pm\text{\scriptsize 3.3}}$	5.4 ${}_{\pm\text{\scriptsize 0.9}}$	19.8 ${}_{\pm\text{\scriptsize 3.8}}$	PASS (5 seeds)

Appendix

Table D.2: Substrate gates at calibration, against their thresholds.

Arm	Cap	Conditional (%)			ITT (%)	Spend		Errands	Judged	Cap hit
		judged	full	literal	all steps	% steps	cost/errand	per run	per run	runs
Errand ( $\hat{\nu}_{0}$ )	$b{=}2$	54.2 ${}_{\pm\text{\scriptsize 2.5}}$	54.1 ${}_{\pm\text{\scriptsize 2.6}}$	88.8 ${}_{\pm\text{\scriptsize 0.9}}$	58.4 ${}_{\pm\text{\scriptsize 1.0}}$	1.9 ${}_{\pm\text{\scriptsize 0.1}}$	1.03	32	–	3/3
	$b{=}3$	55.0 ${}_{\pm\text{\scriptsize 2.3}}$	54.8 ${}_{\pm\text{\scriptsize 2.2}}$	89.6 ${}_{\pm\text{\scriptsize 1.3}}$	60.7 ${}_{\pm\text{\scriptsize 2.3}}$	2.8 ${}_{\pm\text{\scriptsize 0.1}}$	0.99	50	–	3/3
	$b{=}6$	56.0 ${}_{\pm\text{\scriptsize 3.3}}$	55.8 ${}_{\pm\text{\scriptsize 3.4}}$	87.7 ${}_{\pm\text{\scriptsize 1.7}}$	65.8 ${}_{\pm\text{\scriptsize 1.5}}$	6.2 ${}_{\pm\text{\scriptsize 0.2}}$	1.04	102	131	5/5
	$b{=}12$	57.8 ${}_{\pm\text{\scriptsize 4.6}}$	57.7 ${}_{\pm\text{\scriptsize 4.6}}$	89.6 ${}_{\pm\text{\scriptsize 1.8}}$	71.9 ${}_{\pm\text{\scriptsize 2.3}}$	10.9 ${}_{\pm\text{\scriptsize 0.2}}$	1.09	170	132	5/5
	$b{=}24$	57.9 ${}_{\pm\text{\scriptsize 4.2}}$	57.8 ${}_{\pm\text{\scriptsize 4.0}}$	89.0 ${}_{\pm\text{\scriptsize 3.3}}$	71.5 ${}_{\pm\text{\scriptsize 2.7}}$	10.9 ${}_{\pm\text{\scriptsize 0.3}}$	1.09	170	120	0/5

Appendix

Table E.1 : Frontier grid: both spending arms at every cap, with the frozen arm as floor. Cap hit: runs that ever exhaust the trailing budget. Five seeds at

b\in\{6,12,24\}

and on the floor; three at the deep caps and uncapped.

Arm	Cap	Share of no-spend steps (%)
		no candidate	index at zero	gate below wage	cap reached	in flight
Errand ( $\hat{\nu}_{0}$ )	$b{=}2$	10.9	20.8	17.9	48.3	2.2
	$b{=}3$	12.1	21.6	18.2	45.0	3.1
	$b{=}6$	11.1	21.7	18.3	46.2	2.7
	$b{=}12$	12.2	39.0	33.5	12.3	2.9
	$b{=}24$	14.5	43.7	37.5	1.0	3.3

Appendix

Table E.2: No-spend steps by clause, share of each run’s no-spend total.

Method	$b{=}6$		$b{=}12$		$b{=}24$		high gap		4B floor
	ITT	Spend	ITT	Spend	ITT	Spend	ITT	Spend	ITT	Spend
No maintenance ( $\nu{=}\infty$ )	51.1	0.0	51.1	0.0	51.1	0.0	45.4	0.0	43.8	0.0
Bigger memory, no maint.	51.2	0.0	51.2	0.0	51.2	0.0	46.5	0.0	43.6	0.0
Fixed-cadence recheck	58.1	5.9	61.4	10.8	64.9	11.3	56.5	10.9	46.7	10.6
Equal-budget random	55.1	5.8	60.8	10.6	66.2	10.8	54.4	10.5	46.4	10.5
Eager revalidation ( $\nu{\to}0$ )	57.8	6.2	61.9	12.1	66.2	24.1	57.2	12.2	48.5	–

Appendix

Table E.3 : ITT and measured spend for all seven policies in five settings; dashes, spend not instrumented in that wave.

Matched conditional pairs by item class (1016 pairs, 11 seeds; seed-clustered):
Arm (high tier)	Cond. judged	Cond. full	ITT	Spend	$\Delta_{\mathrm{ord}}$ (pp)
Errand ( $\hat{\nu}_{0}$ )	59.5 ${}_{\pm\text{\scriptsize 2.4}}$	59.4 ${}_{\pm\text{\scriptsize 2.5}}$	67.1 ${}_{\pm\text{\scriptsize 1.3}}$	10.7 ${}_{\pm\text{\scriptsize 0.2}}$	7.3 ${}_{\pm\text{\scriptsize 1.0}}$
Eager revalidation ( $\nu{\to}0$ )	54.6 ${}_{\pm\text{\scriptsize 1.5}}$	54.5 ${}_{\pm\text{\scriptsize 1.4}}$	57.2 ${}_{\pm\text{\scriptsize 0.6}}$	12.2 ${}_{\pm\text{\scriptsize 0.4}}$	3.3 ${}_{\pm\text{\scriptsize 0.7}}$
A_onpath	261 pairs		$\Delta$ cond $+8.4$ pp [+6.1, +11.1]
B_onpath	248 pairs		$\Delta$ cond $+4.6$ pp [+1.9, +7.2]
C_fetch	256 pairs		$\Delta$ cond $+3.3$ pp [-1.2, +7.8]

Appendix

Table E.4: High-tier arms and the matched pairs, by item class.

Variant	ITT (%)	Cond. (%)	Spend (%)	$\Delta$ ITT [95% CI]	Predicted
Full Errand	71.3 ${}_{\pm\text{\scriptsize 2.0}}$	61.3 ${}_{\pm\text{\scriptsize 0.7}}$	10.3 ${}_{\pm\text{\scriptsize 0.4}}$	–
w/o deadband gate (pure ranking)	67.8 ${}_{\pm\text{\scriptsize 0.7}}$	58.3 ${}_{\pm\text{\scriptsize 1.7}}$	10.8 ${}_{\pm\text{\scriptsize 0.3}}$	$-3.5$ pp [-7.4, +0.4]	$-$ ✓
coverage sum $\to$ max	68.1 ${}_{\pm\text{\scriptsize 2.9}}$	58.5 ${}_{\pm\text{\scriptsize 1.1}}$	10.7 ${}_{\pm\text{\scriptsize 0.5}}$	$-3.2$ pp [-6.7, +0.4]	$-$ ✓
index: belief only	66.0 ${}_{\pm\text{\scriptsize 2.7}}$	57.1 ${}_{\pm\text{\scriptsize 2.0}}$	11.0 ${}_{\pm\text{\scriptsize 0.2}}$	$-5.3$ pp [-7.1, -3.5]	$-$ ✓
index: age only	64.7 ${}_{\pm\text{\scriptsize 4.3}}$	57.0 ${}_{\pm\text{\scriptsize 1.9}}$	11.0 ${}_{\pm\text{\scriptsize 0.2}}$	$-6.6$ pp [-13.3, +0.1]	$-$ ✓
$\mu_{a}{:=}0$ (no regime return)	66.3 ${}_{\pm\text{\scriptsize 3.0}}$	57.9 ${}_{\pm\text{\scriptsize 3.6}}$	11.0 ${}_{\pm\text{\scriptsize 0.2}}$	$-5.0$ pp [-8.6, -1.4]	$-$ ✓

Appendix

Table E.5: Single-component ablations at the base cap, paired with the full arm.

Allocator	$b{=}6$ (tight)		$b{=}24$ (loose)
	ITT (%)	$\Delta$ vs. numeric	ITT (%)	$\Delta$ vs. numeric
numeric index (receipts priced in)	67.1 ${}_{\pm\text{\scriptsize 1.4}}$	–	72.5 ${}_{\pm\text{\scriptsize 1.5}}$	–
Whittle closed form	65.0 ${}_{\pm\text{\scriptsize 0.8}}$	$-2.2$	70.6 ${}_{\pm\text{\scriptsize 1.0}}$	$-1.9$
myopic $\mathrm{VOI}\hat{u}/c$	62.3 ${}_{\pm\text{\scriptsize 0.8}}$	$-4.8$	70.7 ${}_{\pm\text{\scriptsize 1.3}}$	$-1.8$

Appendix

Table E.6: Allocator comparison at the tight and loose caps: three seeds, allocators paired within one wave.

Coverage $m$	Binding budget				Non-binding budget
	open channel		gated comparator		open channel		gated comparator
	light $c$	heavy $c$	light $c$	heavy $c$	light $c$	heavy $c$	light $c$	heavy $c$
$1$	1.7 ${}_{\pm\text{\scriptsize 1.2}}$	2.3 ${}_{\pm\text{\scriptsize 1.3}}$	1.6 ${}_{\pm\text{\scriptsize 1.0}}$	1.9 ${}_{\pm\text{\scriptsize 1.1}}$	-0.1 ${}_{\pm\text{\scriptsize 0.9}}$	-0.0 ${}_{\pm\text{\scriptsize 0.8}}$	-0.4 ${}_{\pm\text{\scriptsize 0.9}}$	-0.2 ${}_{\pm\text{\scriptsize 0.9}}$
$0.5$	5.6 ${}_{\pm\text{\scriptsize 1.0}}$	7.4 ${}_{\pm\text{\scriptsize 1.3}}$	4.8 ${}_{\pm\text{\scriptsize 0.8}}$	6.1 ${}_{\pm\text{\scriptsize 1.1}}$	-0.3 ${}_{\pm\text{\scriptsize 0.9}}$	0.1 ${}_{\pm\text{\scriptsize 0.9}}$	0.1 ${}_{\pm\text{\scriptsize 0.8}}$	0.1 ${}_{\pm\text{\scriptsize 1.0}}$
$0.25$	7.2 ${}_{\pm\text{\scriptsize 1.4}}$	10.5 ${}_{\pm\text{\scriptsize 1.4}}$	6.0 ${}_{\pm\text{\scriptsize 1.0}}$	8.6 ${}_{\pm\text{\scriptsize 0.9}}$	0.5 ${}_{\pm\text{\scriptsize 0.7}}$	0.4 ${}_{\pm\text{\scriptsize 1.1}}$	0.1 ${}_{\pm\text{\scriptsize 0.7}}$	0.2 ${}_{\pm\text{\scriptsize 0.7}}$
$0.1$	8.4 ${}_{\pm\text{\scriptsize 1.2}}$	11.8 ${}_{\pm\text{\scriptsize 1.1}}$	6.6 ${}_{\pm\text{\scriptsize 1.1}}$	9.8 ${}_{\pm\text{\scriptsize 1.0}}$	-0.1 ${}_{\pm\text{\scriptsize 1.1}}$	0.2 ${}_{\pm\text{\scriptsize 0.8}}$	0.0 ${}_{\pm\text{\scriptsize 1.0}}$	-0.1 ${}_{\pm\text{\scriptsize 0.7}}$

Appendix

Table E.7 : Ordering premium

\Delta_{\mathrm{ord}}

(pp) over coverage, cost tail, budget layer, and channel.

Panel of Figure 26 — Figure E.1 : Closing probes. (a) Ordering premium over the coverage gap: binding layers climb, slack layers pin to zero; band, twenty-seed s.d. (b) ITT across a tenfold drift range; the half-life prior collapses past the crossover while Errand tracks the known-rate reference. (c) Measured against predicted coverage gain by flip group; every group sits above the diagonal.

Policy	$\kappa{=}0.1$	$\kappa{=}0.25$	$\kappa{=}0.5$	$\kappa{=}0.75$	$\kappa{=}1$
Errand ( $\hat{\nu}_{0}$ )	69.7 ${}_{\pm\text{\scriptsize 1.3}}$	69.5 ${}_{\pm\text{\scriptsize 1.3}}$	68.7 ${}_{\pm\text{\scriptsize 0.6}}$	66.5 ${}_{\pm\text{\scriptsize 2.4}}$	67.5 ${}_{\pm\text{\scriptsize 1.6}}$
Known-rate reference	71.7 ${}_{\pm\text{\scriptsize 1.6}}$	72.0 ${}_{\pm\text{\scriptsize 0.9}}$	70.6 ${}_{\pm\text{\scriptsize 0.2}}$	68.0 ${}_{\pm\text{\scriptsize 1.3}}$	70.4 ${}_{\pm\text{\scriptsize 2.5}}$
Eager ( $\nu{\to}0$ )	66.0 ${}_{\pm\text{\scriptsize 0.5}}$	65.2 ${}_{\pm\text{\scriptsize 0.2}}$	63.5 ${}_{\pm\text{\scriptsize 0.3}}$	61.1 ${}_{\pm\text{\scriptsize 1.0}}$	62.5 ${}_{\pm\text{\scriptsize 2.6}}$
Literature half-life policy	68.3 ${}_{\pm\text{\scriptsize 0.8}}$	65.9 ${}_{\pm\text{\scriptsize 1.4}}$	58.2 ${}_{\pm\text{\scriptsize 0.3}}$	50.6 ${}_{\pm\text{\scriptsize 1.2}}$	46.7 ${}_{\pm\text{\scriptsize 2.5}}$

Appendix

Table E.8: ITT across drift intensity on the tool-API world.

Flip group	Predicted gain (pp)	Measured (pp)	Excess (pp)
flip A	3.10 ${}_{\pm\text{\scriptsize 0.00}}$	4.64 ${}_{\pm\text{\scriptsize 0.54}}$	1.54 ${}_{\pm\text{\scriptsize 0.54}}$
flip B	4.50 ${}_{\pm\text{\scriptsize 0.00}}$	6.07 ${}_{\pm\text{\scriptsize 1.04}}$	1.57 ${}_{\pm\text{\scriptsize 1.04}}$
flip C	5.20 ${}_{\pm\text{\scriptsize 0.00}}$	7.41 ${}_{\pm\text{\scriptsize 0.13}}$	2.21 ${}_{\pm\text{\scriptsize 0.13}}$
flip D	6.80 ${}_{\pm\text{\scriptsize 0.00}}$	9.24 ${}_{\pm\text{\scriptsize 1.16}}$	2.44 ${}_{\pm\text{\scriptsize 1.16}}$
pooled	4.90 ${}_{\pm\text{\scriptsize 1.39}}$	6.84 ${}_{\pm\text{\scriptsize 1.91}}$	1.94 ${}_{\pm\text{\scriptsize 0.81}}$

Appendix

Table E.9: Predicted versus measured coverage gain by flip group.

Arm	4B backbone		8B reference
	ITT (%)	Cond. (%)	ITT (%)	Cond. (%)
No maintenance ( $\nu{=}\infty$ )	43.8 ${}_{\pm\text{\scriptsize 2.2}}$	43.6 ${}_{\pm\text{\scriptsize 1.3}}$	51.1 ${}_{\pm\text{\scriptsize 2.3}}$	48.9 ${}_{\pm\text{\scriptsize 2.3}}$
Eager revalidation ( $\nu{\to}0$ )	48.5 ${}_{\pm\text{\scriptsize 1.5}}$	46.0 ${}_{\pm\text{\scriptsize 1.3}}$	61.9 ${}_{\pm\text{\scriptsize 2.0}}$	56.0 ${}_{\pm\text{\scriptsize 4.5}}$
Errand ( $\hat{\nu}_{0}$ )	50.6 ${}_{\pm\text{\scriptsize 1.3}}$	47.1 ${}_{\pm\text{\scriptsize 1.3}}$	71.9 ${}_{\pm\text{\scriptsize 2.3}}$	57.8 ${}_{\pm\text{\scriptsize 4.6}}$

Appendix

Table E.10: Capability floor: the headline arms at 4B beside their 8B reference.

Atom group	Verb class	Fetch orders off	Fetch orders on
C_obj_value	fetch	0.00 ${}_{\pm\text{\scriptsize 0.00}}$	4.26 ${}_{\pm\text{\scriptsize 0.51}}$
D_path	go	1.14 ${}_{\pm\text{\scriptsize 0.04}}$	5.62 ${}_{\pm\text{\scriptsize 0.35}}$
F_obj_value	fetch	0.00 ${}_{\pm\text{\scriptsize 0.00}}$	0.17 ${}_{\pm\text{\scriptsize 0.00}}$

Appendix

Table E.11: Instrumental closure: uses per order with fetch orders off and on.

Method	ITT@500	ITT@2000	Stale@500	Stale@2000
No maintenance ( $\nu{=}\infty$ )	60.1 ${}_{\pm\text{\scriptsize 0.9}}$	51.1 ${}_{\pm\text{\scriptsize 2.3}}$	28.7 ${}_{\pm\text{\scriptsize 1.2}}$	43.2 ${}_{\pm\text{\scriptsize 1.2}}$
Bigger memory, no maint.	60.6 ${}_{\pm\text{\scriptsize 1.1}}$	51.2 ${}_{\pm\text{\scriptsize 1.3}}$	30.6 ${}_{\pm\text{\scriptsize 0.6}}$	53.1 ${}_{\pm\text{\scriptsize 1.8}}$
Fixed-cadence recheck	66.5 ${}_{\pm\text{\scriptsize 0.6}}$	61.4 ${}_{\pm\text{\scriptsize 0.1}}$	10.5 ${}_{\pm\text{\scriptsize 0.8}}$	17.1 ${}_{\pm\text{\scriptsize 1.9}}$
Equal-budget random	65.4 ${}_{\pm\text{\scriptsize 1.5}}$	60.8 ${}_{\pm\text{\scriptsize 1.8}}$	10.8 ${}_{\pm\text{\scriptsize 1.7}}$	18.6 ${}_{\pm\text{\scriptsize 0.8}}$
Eager revalidation ( $\nu{\to}0$ )	67.2 ${}_{\pm\text{\scriptsize 0.5}}$	61.9 ${}_{\pm\text{\scriptsize 2.0}}$	13.2 ${}_{\pm\text{\scriptsize 1.4}}$	21.9 ${}_{\pm\text{\scriptsize 1.0}}$
Errand ( $\hat{\nu}_{0}$ )	73.4 ${}_{\pm\text{\scriptsize 1.3}}$	71.9 ${}_{\pm\text{\scriptsize 2.3}}$	7.2 ${}_{\pm\text{\scriptsize 0.3}}$	12.2 ${}_{\pm\text{\scriptsize 1.0}}$

Appendix

Table E.12: ITT and stale-use share at the

500

-step prefix and the full horizon.

Method	$b{=}6$	$b{=}12$	$b{=}24$	high	4B	$T{=}500$
No maintenance ( $\nu{=}\infty$ )	43.2 ${}_{\pm\text{\scriptsize 1.2}}$	43.2 ${}_{\pm\text{\scriptsize 1.2}}$	43.2 ${}_{\pm\text{\scriptsize 1.2}}$	54.8 ${}_{\pm\text{\scriptsize 0.3}}$	46.5 ${}_{\pm\text{\scriptsize 0.4}}$	28.7 ${}_{\pm\text{\scriptsize 1.2}}$
Bigger memory, no maint.	53.1 ${}_{\pm\text{\scriptsize 1.8}}$	53.1 ${}_{\pm\text{\scriptsize 1.8}}$	53.1 ${}_{\pm\text{\scriptsize 1.8}}$	55.6 ${}_{\pm\text{\scriptsize 0.4}}$	51.4 ${}_{\pm\text{\scriptsize 0.5}}$	30.6 ${}_{\pm\text{\scriptsize 0.6}}$
Fixed-cadence recheck	26.5 ${}_{\pm\text{\scriptsize 0.2}}$	17.1 ${}_{\pm\text{\scriptsize 1.9}}$	14.8 ${}_{\pm\text{\scriptsize 0.8}}$	21.5 ${}_{\pm\text{\scriptsize 1.7}}$	18.8 ${}_{\pm\text{\scriptsize 1.1}}$	10.5 ${}_{\pm\text{\scriptsize 0.8}}$
Equal-budget random	28.3 ${}_{\pm\text{\scriptsize 0.8}}$	18.6 ${}_{\pm\text{\scriptsize 0.8}}$	15.0 ${}_{\pm\text{\scriptsize 1.4}}$	23.3 ${}_{\pm\text{\scriptsize 0.9}}$	20.0 ${}_{\pm\text{\scriptsize 1.5}}$	10.8 ${}_{\pm\text{\scriptsize 1.7}}$
Eager revalidation ( $\nu{\to}0$ )	31.6 ${}_{\pm\text{\scriptsize 0.5}}$	21.9 ${}_{\pm\text{\scriptsize 1.0}}$	13.9 ${}_{\pm\text{\scriptsize 0.4}}$	26.1 ${}_{\pm\text{\scriptsize 0.6}}$	24.9 ${}_{\pm\text{\scriptsize 1.7}}$	13.2 ${}_{\pm\text{\scriptsize 1.4}}$
Oracle detect-and-filter †	3.4 ${}_{\pm\text{\scriptsize 1.3}}$	3.4 ${}_{\pm\text{\scriptsize 1.3}}$	3.4 ${}_{\pm\text{\scriptsize 1.3}}$	2.6 ${}_{\pm\text{\scriptsize 1.3}}$	4.3 ${}_{\pm\text{\scriptsize 0.7}}$	–

Appendix

Table E.13: Stale-use share across settings and the

500

-step prefix.

Axis	Family	Errand ITT	Eager ITT	$\Delta$ (pp)
Budget $b{=}6$ , base gap	Budget	66.1 ${}_{\pm\text{\scriptsize 1.4}}$	57.7 ${}_{\pm\text{\scriptsize 1.7}}$	$+8.3$
Budget $b{=}12$ , base gap	Budget	71.0 ${}_{\pm\text{\scriptsize 1.4}}$	61.7 ${}_{\pm\text{\scriptsize 0.6}}$	$+9.4$
Budget $b{=}24$ , base gap	Budget	72.4 ${}_{\pm\text{\scriptsize 2.6}}$	66.5 ${}_{\pm\text{\scriptsize 1.8}}$	$+5.9$
Budget $b{=}6$ , high gap	Coverage	60.0 ${}_{\pm\text{\scriptsize 0.9}}$	53.5 ${}_{\pm\text{\scriptsize 1.0}}$	$+6.5$
Budget $b{=}12$ , high gap	Coverage	66.8 ${}_{\pm\text{\scriptsize 1.4}}$	57.5 ${}_{\pm\text{\scriptsize 0.6}}$	$+9.3$
Budget $b{=}24$ , high gap	Coverage	66.7 ${}_{\pm\text{\scriptsize 1.4}}$	61.9 ${}_{\pm\text{\scriptsize 1.7}}$	$+4.8$

Appendix

Table E.14: The nine deployment axes of Figure 1 , as numbers; three seeds per axis.

Policy	Prompt tok/step	Gen. tok/step	LLM calls/step	Sched. ms/step	Tok/errand
Errand ( $\hat{\nu}_{0}$ )	431 ${}_{\pm\text{\scriptsize 12}}$	64 ${}_{\pm\text{\scriptsize 2}}$	1.10 ${}_{\pm\text{\scriptsize 0.04}}$	39.8 ${}_{\pm\text{\scriptsize 1.1}}$	361 ${}_{\pm\text{\scriptsize 4}}$
Eager revalidation ( $\nu{\to}0$ )	455 ${}_{\pm\text{\scriptsize 3}}$	72 ${}_{\pm\text{\scriptsize 4}}$	1.40 ${}_{\pm\text{\scriptsize 0.11}}$	32.7 ${}_{\pm\text{\scriptsize 0.8}}$	334 ${}_{\pm\text{\scriptsize 16}}$
Fixed-cadence recheck	417 ${}_{\pm\text{\scriptsize 9}}$	60 ${}_{\pm\text{\scriptsize 2}}$	1.53 ${}_{\pm\text{\scriptsize 0.08}}$	30.4 ${}_{\pm\text{\scriptsize 0.4}}$	355 ${}_{\pm\text{\scriptsize 13}}$
Equal-budget random	426 ${}_{\pm\text{\scriptsize 10}}$	62 ${}_{\pm\text{\scriptsize 2}}$	1.44 ${}_{\pm\text{\scriptsize 0.06}}$	28.8 ${}_{\pm\text{\scriptsize 2.1}}$	361 ${}_{\pm\text{\scriptsize 11}}$

Appendix

Table E.15: Per-step overhead, all columns, by policy.

Panel of Figure 35 — Figure G.1 : The cases, as recorded. (a) The item of Figure 6 : the world flips at step $641$ ; Errand ’s belief follows at $690$ off a free receipt while eager’s never does; dotted vertical, the resolve-by date; bottom ticks, each arm’s recorded events. (b) A full base-cap ledger: the trailing account rides the cap early, overrunning only while an errand is in flight, then the wage, not the cap, sets the pace; bottom ticks, errand steps.

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