Minimal Recurrent Behavioral Memory for Imitation under Partial Observability

Xianyao Li; Fang Xu; Rui Min; Ruitong Tian; Jing Du

cs.LGSep 22, 2026

Minimal Recurrent Behavioral Memory for Imitation under Partial Observability

Authors: Xianyao Li, Fang Xu, Rui Min, Ruitong Tian, Jing Du

Abstract

What is the least recurrent memory needed to reproduce a specified expert under partial observability? The instantaneous requirement is the conditional entropy of the expert's behavioral quotient, but recurrence must also preserve distinctions that future observations will not restore before use. We characterize this minimal recurrent behavioral memory by a compatibility relation: under transitivity its classes attain the exact minimum, while the general case is an entropy minimization over closed compatible state assignments, with exact certificates on finite instances. A sole-carrier measurement protocol separates behavioral sufficiency, excess code rate, and information carried by observations or other memory paths; experimental bit requirements refer to the induced symbolic behavioral model under the stated occupancy. Across manipulation tasks, learned code rates remain near zero- and two-bit requirements as hidden modes grow to $512$ , and anticipatory memory follows a $2\to1\to0$ requirement despite zero instantaneous demand during waiting. Learning this representation remains difficult: event-agnostic future-behavior supervision yields $36/40$ sufficient seeds with one frozen configuration and improves the longest-horizon pixel setting from $0/8$ to $6/8$ sufficient held-out seeds (closed-loop success from $0.08$ to $0.57$ ). On unmodified community benchmarks, the protocol certifies delay-independent requirements, which sufficient codes match at mid-delay. The supervision aids commitment but can induce predictive surplus; annealing it lets imitation and rate training reduce that surplus, separating the information-theoretic target from the ability to learn it.

Figures & tables

training target	gap 6	gap 10	gap 20	$M=16$	$M=32$
teacher internal state	$2/8$	$0/8$	$0/8$	$0/8$	$1/8$
task-informed future behavior	$8/8$	$8/8$	$8/8$	$8/8$	$5/8$
event-agnostic future behavior	$8/8$	$7/8$	$8/8$	$6/8$	$7/8$
event-agnostic: closed-loop success	$.99$	$.87$	$.90$	$.79$	$.93$

Table 1: Learning near the certified two-bit requirement. First-gap rates are reported below; counts give full-gate sufficient seeds out of eight, and the last row gives all-seed closed-loop success. The event-agnostic configuration is frozen; all

40

verdicts reproduce on independent recordings. The task-informed reference uses event structure.

learner	first $\to$ second gap (bits)	sufficient	closed-loop success
plain	$1.99\to 1.52$	$1/16$	$0.14$
event-agnostic, annealed	$2.00\to 1.13$	$13/16$	$0.65$
task-informed reference	$2.00\to 1.00$	$15/16$	$0.61$

Table 2: Pixel code rates versus the certified requirement. A ′ at gap 20 requires

2\to 1

bits. Sixteen seeds per row; rates average sufficient seeds, success all seeds. Inputs retain probe and phase channels. The

60\%

schedule was selected on seeds 0–7 and tested unchanged on seeds 8–15.

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

	finite toys	signpost corridor	A ′ (distortion-constrained)	A ′ (attainability)	Task A
domain	finite POMDPs	grid navigation	Franka, kinematic attach	Franka, kinematic attach	Franka, physics-simulated $2$ kg grasp
regime (solver)	exact; (A2), (A4), transitive	exact at $W{=}1$ ; certified at $W{=}3,5,\dots,13$	exact	exact	exact: $\sim_{t}$ transitive, (A4) fails at one step
learner	raw VQ, lr $3\!\times\!10^{-4}$	raw VQ, tanh-bounded state	plain $-$ R	scaffold-RF	plain $-$ R
$\beta$	$0.03$	$0.01$	$\leq 10^{-3}$	$2\!\times\!10^{-3}$	$10^{-3}$
data, steps	6k	6k	$N{=}1152/3584$ ; 10k	$N\approx 4$ k	$N{=}448$ ; 10k
role	exact validation	cross-domain replication	distortion-constrained	attainability only	matched performance

Appendix

Table 3: Benchmark families, regimes and frozen primary configurations (

N

= training episodes;

K=16

codes throughout; the Franka runs use batch 512).

choice	premise	evidence
$C_{t}$ is the only carrier across time	Definition 2	Appendix B.2 : bypass under-counts in three domains
persistence (residual proposal)	recurrence	non-persistent code: $0/8$ sufficient (trivial)
discrete codebook	directly measurable accounting	continuous: KL upper bound $6.9$ bit, not comparable (Table 17 )
prior conditioned on $O_{t}$	matches the conditional-rate objective	no empirical advantage over an unconditional prior in these families
forecaster used at training only	sole carrier at evaluation	forecaster removed before every reported rate and rollout; $C_{t}$ is the only cross-time variable (§ 5 )

Appendix

Table 4: Design choices, premises and evidence (§ 4 ).

phase	$\beta_{1}$ (0.50)	$\beta_{2}$ (0.50)	$\theta$ (0.03)	mass bin (0.25)	distractor $z$ (0.25)
identification (scan)	1.00	1.00	0.10	0.26	—
gap 1	0.53	0.51	0.04	0.23	—
grasp	0.99	0.47	0.05	0.24	—
gap 2	0.51	0.46	0.03	0.18	0.99
place	0.49	1.00	0.04	0.26	—

Appendix

Table 5: Leak probe on A ′ (tier 4, gap 6): held-out accuracy (

n=128

episodes) of a classifier on raw observation windows, per phase and hidden variable; chance in parentheses. Bold entries are the steps at which the visibility convention marks the class visible.

gap	architecture	$\beta$	sufficient	first-gap rate	post-use rate	behav. error	closed-loop success
6	hierarchical	0	8/8	$2.03\pm 0.08$	1.00	0.000	1.00
6	hierarchical	$10^{-3}$	8/8	$2.09\pm 0.17$	1.00	0.000	1.00
6	intent	0	8/8	$2.02\pm 0.05$	1.09	–	0.96
6	intent	$10^{-3}$	8/8	$2.03\pm 0.08$	1.00	–	0.94
20	hierarchical	0	8/8	$2.04\pm 0.10$	1.00	0.027	0.96
20	hierarchical	$10^{-3}$	8/8	$2.03\pm 0.08$	1.00	0.000	1.00

Appendix

Table 6: Strict read-side architectures on A ′ with behavioral-forecast supervision (8 seeds per cell; theory

2.00\to 1.00

; forecaster removed at evaluation). Without the forecast target the same architectures are sufficient on

0/8

seeds in every cell.

$\delta$ (mm)	0	0.5	1	2	4	8
closed-loop success	$0.59$	$0.52$	$0.61$	$0.83$	$0.84$	$0.96$
slot accuracy	$0.68$	$0.68$	$0.79$	$0.93$	$0.95$	$0.98$
held-out action error (MSE)	$0.107$	$0.116$	$0.053$	$0.015$	$0.022$	$0.007$
behaviorally correct seeds	$5$	$5$	$7$	$15$	$14$	$16$
seeds passing the sufficiency gate	$5$	$4$	$7$	$4$	$6$	$3$
flag A: correct, rate $<0.9$ bit	$0/5$	$0/5$	$0/7$	$0/15$	$1/14$	$0/16$

Appendix

Table 7: Injected-leak audit. Top:

16

seeds per magnitude

\delta

. Correct means held-out class error below

0.05

at the first place step. Flag A, fixed before the runs, pairs correctness with second-gap rate below

0.9

bit. The exploratory flags pair correctness with failure of the full gate (B) or second-gap memory alone (B ′ ). Bottom: false alarms on separate clean recordings, conditioned on correctness.

	CPU, batch 1 (ms)			GPU, batch 64 (ms)
$T$	Transformer	GRU	DIACRITIC	Transformer	GRU	DIACRITIC
33	1.13	0.55	0.54	1.77	1.12	1.18
128	1.78	0.55	0.55	1.83	1.18	1.17
256	3.51	0.56	0.53	2.12	1.15	1.18
512	9.03	0.55	0.54	5.45	1.10	1.13
1024	29.3	0.56	0.55	16.8	1.13	1.17

Appendix

Table 8: Per-step inference latency (median of 200 steps) as a function of the history length

T

, with the widths of the paper’s models. The Transformer re-encodes its token history at every step (no key–value cache), so these timings describe that implementation; its carried history is

64(T-1)

floats (8.2 kB at

T=33

, 262 kB at

T=1024

), compared with 128 bytes for the GRU and 129 bytes for DIACRITIC (state plus an 8-bit code index).

	parameters	training time	at deployment: carried state / per-step latency
forecaster, 3000 steps (default)	$0.49$ – $0.51$ M	$11$ – $12$ s	removed
random-offset forecaster, 100 steps (timing only)	$0.51$ M	$\approx 1$ s	removed
compact student, 10 000 steps, with or without the auxiliary head	$0.19$ M	$\approx 800$ s	$129$ bytes / $0.55$ ms
full-history Transformer baseline, 10 000 steps	$0.49$ M	$\approx 650$ s	$64(T-1)$ floats / $1.1$ – $29$ ms for $T=33$ – $1024$

Appendix

Table 9: Cost bookkeeping on A ′ gap 20 (one B200 GPU; medians over runs). The forecaster and the auxiliary head exist only during training.

sufficient seeds / closed-loop success
	$M=4$	$M=8$	$M=16$	$M=32$
ours, $K=16$	8/8 / 0.91	8/8 / 0.96	8/8 / 0.98	8/8 / 0.98
ours, $K=64$	8/8 / 1.00	8/8 / 1.00	8/8 / 0.99	8/8 / 1.00
sys-ID, $K=16$	1/8 / 0.11	0/8 / 0.08	0/8 / 0.06	0/8 / 0.07
sys-ID, $K=64$	8/8 / 0.71	3/8 / 0.18	0/8 / 0.08	0/8 / 0.09
sys-ID, $K=128$	8/8 / 0.79	8/8 / 0.34	0/8 / 0.12	0/8 / 0.16

Appendix

Table 10: Task A controls, 8 seeds per cell. Top block: seeds sufficient at the place step / closed-loop success (128 episodes per seed). Bottom block: code rate during grasp (bits) /

I(C;\mathrm{mass}\mid\bar{O})

(bits).

$M$	objective	sufficient (place)	grasp rate (bits)	$I(C;\mathrm{mass}\mid\bar{O})$	success
128	ours, $K=16$ , $\beta=10^{-3}$	8/8	0.03	0.01	0.99
128	ours, $K=16$ , $\beta=0$	8/8	0.18	0.02	0.99
128	sys-ID, $K=16$	0/8	3.73	0.62	0.04
128	sys-ID, $K=256$	3/8	6.84	1.40	0.17
512	ours, $K=16$ , $\beta=10^{-3}$	8/8	0.03	0.01	0.99
512	ours, $K=16$ , $\beta=0$	8/8	0.10	0.02	0.98

Appendix

Table 11: Task A at

M=128

and

512

(8 seeds per row; 128 closed-loop episodes per seed).

task	learner	sufficient	first-gap rate	transport rate	closed loop
readout-2	$-$ R	8/8, 7/8, 8/8	2.00, 2.02, 2.03 [2.00]	1.10, 1.23, 1.06 [1.00]	0.94, 0.85, 0.87
readout-2	forecast	8/8, 8/8, 8/8	2.02, 2.08, 2.03 [2.00]	1.11, 1.27, 1.09 [1.00]	0.95, 0.90, 0.96
readout-2	sys-ID, largest $K$	4/8, 6/8, 1/8	4.82, 5.56, 7.49	4.49, 4.74, 7.93	0.25, 0.26, 0.19
readout-3	forecast	6/8, 5/8, 4/8	3.03, 3.09, 3.17 [3.00]	3.06, 3.01, 3.09 [3.00]	0.25, 0.28, 0.15
readout-3	forecast, $K=8$	0/8, 0/8, 0/8	2.73, 2.92, 2.87	2.49, 2.46, 2.76	0.05, 0.05, 0.05
readout-3	sys-ID, largest $K$	6/8, 0/8, 0/8	4.83, 5.78, 7.94	4.78, 5.78, 8.31	0.32, 0.07, 0.03

Appendix

Table 12: Readout task, 8 seeds per cell,

K=16

unless noted (sys-ID largest

K

: 256, 256, 1024); each cell lists

M=32

128

512

. Rates are all-seed means except for the

K=16

forecast row on readout-3, which averages sufficient seeds; closed-loop success is always averaged over all seeds. Behavioral-learner verdicts replicate on the fresh recordings; the largest-

K

readout-2 sys-ID count at

M=512

changes from

1/8

0/8

. Theory is in brackets.

$M=32$ , $N=448$ (every mass in training)
configuration	$H(C\mid\bar{O})$	$I(C;\mathrm{mass}\mid\bar{O})$	$H(C^{\rm id}\mid\bar{O})$	$I(C^{\rm id};\theta\mid\bar{O})$	success
DIACRITIC ( $K{=}16$ )	0.34	0.01	–	–	0.99, 0.88
shared, weight 1 ( $K{=}16$ )	2.83	0.73	–	–	0.07, 0.05
shared, weight 0.1 ( $K{=}256$ )	4.52	0.80	–	–	0.38, 0.23
dual carrier ( $K^{\rm id}{=}256$ )	0.46	0.03	4.12	2.73	0.90, 0.84
dual, stop-gradient	0.29	0.00	4.35	2.45	0.93, 0.88

Appendix

Table 13: Task A, sys-ID controls (8 seeds): transport rates of the behavioural code

C

and of the identification code

C^{\rm id}

(bits), mass information carried by

C

\theta

information carried by

C^{\rm id}

, and closed-loop success (mean, min). “shared” = the

\theta

head reads

C

itself.

sufficient seeds	gap 6	gap 10	gap 20	$M=16$	$M=32$
EA, anneal $60\%$ (frozen)	$8/8$ ( $8/8$ )	$7/8$ ( $7/8$ )	$8/8$ ( $8/8$ )	$6/8$ ( $6/8$ )	$7/8$ ( $7/8$ )
EA, anneal $40\%$	—	—	$7/8$ ( $7/8$ )	—	$7/8$ ( $7/8$ )
EA, not annealed	$6/8$ ( $6/8$ )	$8/8$ ( $7/8$ )	$7/8$ ( $7/8$ )	$8/8$ ( $8/8$ )	$6/8$ ( $6/8$ )
TI	$8/8$ (—)	$8/8$ ( $8/8$ )	$8/8$ ( $8/8$ )	$8/8$ ( $8/8$ )	$5/8$ ( $5/8$ )
teacher state	$2/8$	$0/8$	$0/8$	$0/8$	$1/8$
rate / closed loop	gap 6	gap 10	gap 20	$M=16$	$M=32$

Appendix

Table 14: Forecast supervision on A ′ (state observations), 8 seeds per cell. Top: sufficient seeds under the full-trajectory gate (in parentheses: the same models re-evaluated on an independent recording). Bottom: first-gap

\to

second-gap rate of the sufficient seeds (bits; theory

2.00\to 1.00

) / closed-loop success over all seeds (128 episodes per policy). EA = event-agnostic random-offset forecast; TI = task-informed forecast.

horizon	learner	sufficient	rate (bits)	success	slot accuracy
gap 6	plain $-$ R	$2/8$	$1.99\to 1.00$ ( $n{=}2$ )	$0.17$	$0.60$
gap 6	task-informed forecast	$7/8$	$1.99\to 1.12$	$0.25$	$0.94$
gap 20	plain $-$ R	$1/16$	$1.99\to 1.52$ ( $n{=}1$ )	$0.14$	$0.56$
gap 20	event-agnostic, not annealed	$7/8$	$2.15\to 2.01$	$0.40$	$0.80$
gap 20	event-agnostic, annealed from $40\%$	$12/16$	$2.01\to 1.13$	$0.51$	$0.82$
gap 20	event-agnostic, annealed from $60\%$	$13/16$	$2.00\to 1.13$	$0.65$	$0.89$

Appendix

Table 15: Pixel A ′ : sufficient seeds (full-trajectory gate), rates of the sufficient seeds (theory

2.00\to 1.00

), and closed loop over all seeds (128 episodes per policy).

target	gap 6	gap 10	gap 20	$M=16$	$M=32$
task-informed forecast of pending behavior	8/8	8/8	8/8	8/8	5/8
teacher’s internal state	2/8	0/8	0/8	0/8	1/8
unsupervised (best variant)	3/8	2/8	1/8	0/8	0/8
forecast: rate before use (theory 2.00)	2.01	2.05	2.09	2.08	2.13
forecast: rate after use (theory 1.00)	1.07	1.06	1.10	1.09	1.11
forecast: closed-loop success	0.95	0.89	0.92	0.84	0.73

Appendix

Table 16: Distillation controls on A ′ and P-I (8 seeds per cell; full-trajectory gate of § 4.1 ; the event-agnostic objective is in Table 14 ). Top block: seeds sufficient. Bottom block: rates and closed-loop success of the sufficient seeds under the forecast target.

variant	sufficient	post-use rate (bits)	rate measurement	success
DIACRITIC ( $-$ R, $\beta=10^{-3}$ )	✓	$1.38$ (near)	✓	$0.97$
scaffold $\to$ DIACRITIC ( $N\approx 4$ k)	✓	$0.99$ – $1.05$ on $5/6$ (strongest)	✓ ( $\alpha=0$ at evaluation)	$0.87$ – $1.00$
$\beta=0$	✓	$1.53$ (redundant)	✓	$0.96$
continuous bottleneck	possible	KL bound $6.9$ ; nuisance $0.32$	upper bound only	—
GRU bypass	✓	$1.06$ (apparent under-rate)	$\times$ (not the sole carrier)	$0.97$ – $1.00$
system identification	budget-dep.	$2.2$ – $3.2$ (learned rate)	✓	$0.06$ – $0.79$

Appendix

Table 17: Sufficiency and minimality on A ′ (gap 6,

N=1152

unless noted). Minimality is the post-use rate against the

1.00

-bit boundary. The variational KL bound of the continuous model is not comparable to hard-code entropy and is therefore not read as a minimality result. Appendices D.4 and C.3 report additional variants.

policies	$n$	$S_{\Gamma}$ (gap 2 )	slot accuracy	success	action error (cm)
teacher-forced sufficient ( $-$ R, seeds 0, 6)	2	$1.00$ , $0.95$	$0.92$ , $0.94$	$0.66$ , $0.36$	$3.1$ , $5.7$
insufficient, $-$ R	6	$\leq 0.08$	$0.42$ – $0.66$	$0.09$ – $0.27$	$8$ – $12$
insufficient, scaffold-R	8	$\leq 0.07$	$0.41$ – $0.54$	$0.10$ – $0.42$	$7$ – $10$

Appendix

Table 18: Closed-loop evaluation of the 16 pixel policies (128 episodes each;

S_{\Gamma}

under the policy’s own occupancy).

cue bits, variant	$L=5$	$L=10$	$L=20$	$L=40$	$L=80$
1 bit, $-$ R	8/8 / 1.25	8/8 / 1.19	7/8 / 1.00	3/8 / 1.00	5/8 / 1.00
1 bit, $-$ RF	8/8 / 1.12	8/8 / 1.50	8/8 / 1.38	5/8 / 1.10	6/8 / 1.17
1 bit, scaffold-RF	8/8 / 1.17	8/8 / 1.00	8/8 / 1.06	8/8 / 1.19	6/8 / 1.08
2 bits, $-$ R	6/8 / 2.00	4/8 / 2.00	1/8 / 2.00	0/8 / —	0/8 / —

Appendix

Table 19: T-maze: seeds sufficient / corridor rate of sufficient seeds (bits; theory

\log_{2}R

	gap 6		gap 20
learner	sufficient	closed loop	sufficient	closed loop
plain $-$ R	$1/8$ / $1/8$	$0.37$ / $0.51$	$0/8$ / $0/8$	$0.37$ / $0.24$
task-informed forecast	$3/8$ / $3/8$	$0.21$ / $0.51$	$2/8$ / $2/8$	$0.21$ / $0.35$
task-informed forecast from step 0	$8/8$ / $8/8$	$0.10$ / $0.30$	$7/8$ / $7/8$	$0.07$ / $0.32$
event-agnostic, not annealed	$2/8$ / —	$0.19$ / $0.51$	$0/8$ / —	$0.13$ / $0.35$
system identification	$8/8$ / —	$0.08$ / $0.38$	$7/8$ / —	$0.06$ / $0.40$

Appendix

Table 20: Weighing task, 8 seeds per cell: sufficient seeds on the training recording / on the independent recording, and closed-loop success / grasp-side accuracy over all seeds (chance

0.25

learner	$L=10$	$L=20$	$L=50$	$L=100$
GRU policy, continuous state	8 (1.00)	8 (1.00)	8 (1.00)	7 (0.93)
compact carrier, plain imitation	1 (0.53)	1 (0.53)	0 (0.46)	0 (0.46)
compact carrier, event-agnostic forecast	8 (1.00)	7 (0.94)	5 (0.80)	0 (0.48)
compact carrier, $-$ RF (decoder sees future observations)	—	1 (0.53)	0 (0.46)	0 (0.46)
compact carrier, annealed continuous scaffold	—	0 (0.46)	0 (0.47)	0 (0.47)
GRU policy, 8-dimensional state	—	0 (0.47)	0 (0.41)	0 (0.35)

Appendix

Table 21: Passive T-maze: seeds (of 8) that solve the task in closed loop (success

\geq 0.99

); mean success in parentheses.

task	delay	certified	suff., plain	suff., event-agnostic	learned rate
MemoryChain, 1 bit	10	$1.00$	$7/8$	$8/8$	$1.00$
MemoryChain, 1 bit	30	$1.00$	$4/8$	$8/8$	$1.00$
MemoryChain, 1 bit	100	$1.00$	$0/8$	$2/8$	$1.00$
MemoryChain, 2 bits	10	$1.99$	$7/8$	$7/8$	$1.99$
MemoryChain, 2 bits	30	$1.99$	$2/8$	$3/8$	$1.99$
MemoryChain, 2 bits	100	$1.99$	$0/8$	$0/8$	—

Appendix

Table 22: Certified requirement and learned rate on community benchmarks (8 seeds per cell). The certified value is the empirical requirement at mid-delay under the recorded occupancy (

512

episodes, hence

1.99

and

2.99

); the learned rate is

\hat{H}(C_{t}\mid O_{t})

at the same step over the sufficient seeds of both learners.

method	target representation	F	S	E	B	rate of the target’s fixed-demonstrator form under our solver
bisimulation ( Zhang et al., 2021 ) ; $\pi$ -bisimulation ( Castro, 2020 )	value / dynamics equivalence (policy-specific for $\pi$ -bisim.)	policy	–	–	–	not implemented
$\pi^{*}$ -irrelevance ( Li et al., 2006 ) ; support sufficiency ( Walsh, 2026 )	optimal-action equivalence (single decision)	optimal act.	–	rate–regret	–	single-decision quotient $=G_{E,t}$ : 0 bit in the gaps (Fig. 7 )
causal states / PSR ( Shalizi and Crutchfield, 2001 ; Littman et al., 2001 )	sufficiency for predicting observations	–	–	stat. complexity	–	observation-predictive form: $1.75$ bit on the re-reveal toy vs. $1.00$ for $\Gamma$ (App. D.3 )
multi-step inverse / ACSD ( Mhammedi et al., 2023 ; Lamb et al., 2023 )	control-endogenous latent	–	–	–	–	$0.3$ – $0.7$ bit of drift on the corridor, never sufficient (App. C.4 )
approximate information state ( Subramanian et al., 2022 )	sufficiency for reward and next observation	–	–	–	–	not implemented
stable quotients ( Zhang et al., 2026 ) ; Nerode quotient ( Nixon, 2026 )	minimal Markov / controller-equivalent state	–	–	–	–	not implemented

Appendix

Table 23: Representation targets in related work (§ 6 ). Property columns: F the target reproduces a fixed demonstrator’s action distribution; S continuations are compared over stochastic reachable supports, so future observations act as decoder side information; E minimality is defined through occupancy-weighted conditional entropy; B the general case yields a non-transitive compatibility bracket. Individual properties appear in prior work; the contribution here is their combination.

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Poincaré Meets Bellman: Revisable Memory, Operational Quotients, and Evidence-Supported Learning in Changing Environments

Xin Li

Department of Computer Science University at Albany

Bellman Equation Bounded-Memory Hamiltonian Invariant Synthesis

Open paper AskPDF arXiv

Memory consolidation determines both what a learner can do now and which changes remain implementable later. We develop a finite-model synthesis of operational state abstraction and optimal control under the stability-evidence-revision (SER) framework. ``Poincaré meets Bellman'' names two complementary roles: qualitative dynamics identifies reusable action-response structure, and dynamic programming prices acquisition, retention, reuse, merging, and forgetting. Recurrence enters separately through the timing and value of future demands. We distinguish active quotient merging from historical information erasure, characterize exact repair by zero-error functional coding and causal migration, and derive a Bellman recursion over the joint law of hidden state and complete deployed memory. A first-return model yields an explicit retention rule. Conditional results show how factor sharing avoids enumerating combinations and how independent informative observations improve identification, while leaving some zero-error evidence budgets unchanged. Finite enumerations verify the coding and retention calculations. The synthesis gives an exact benchmark for specified finite models, without claiming universal recurrence, bounded-memory open-ended learning, or tractable global planning.

Abstract

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