THEORY-026: For a system driven without feedback through fixed kernels, the average work dissipated at each switch of the drive equals the memory that fails to predict the next input; that is nonnegative when the drive is Markov and, summed, bounds the protocol's dissipation from below when each kernel keeps its equilibrium stationary, so there only nonpredictive memory is wasteful; with feedback, in one binary channel no predictive agent is efficient

Source

• Still, Sivak, Bell & Crooks (2012), LIT-327, Eqs. (14)–(21) and footnote [24], as read in NOTE-295.
• Fiderer et al. (2025), LIT-041, Thms 4–5 (Thms 9–10 in the supplement) and Lemma 10, as read in NOTE-032.

What was actually shown

The identity (LIT-327). A system with state s_t is driven by a signal x_t through a fixed Markov kernel. Work is done only when the signal changes, and there is no feedback from system to signal. Then the average dissipated work in the step x_t → x_{t+1} is k_BT times I[s_t;x_t] − I[s_t;x_{t+1}] (Eq. 14). NOTE-295 re-derived this from footnote [24]; given the definitions it is an identity. Summed over a protocol, the “nostalgia” lower-bounds dissipation and excess work (Eq. 18). The heat released is then at least k_BT·(I_e + I_mem − I_pred) (Eq. 21). Here I_e is the Landauer term for entropy removed. The account prices only the part of memory that does not carry over to the next input. Predictive memory costs nothing in it.

The feedback case (LIT-041). Fiderer et al. model agent and environment as coupled hidden-Markov channels. They bound work per round by ⟨H(A_t|M_t) − H(S_t|M_t)⟩_t. Without feedback (a unifilar product environment), the efficient agents are exactly those that are maximally predictive and randomise-and-forget their actions (Thm 4). This is Boyd, Mandal and Crutchfield’s tape-setting principle, “randomize-and-forget your actions, predict your percepts”, extended beyond stationarity, and its prediction half is the LIT-327 slogan. With feedback, one binary memoryless channel makes the predictive, max-entropy-action and efficient agents three nonempty, pairwise-disjoint sets (Thm 5). Every predictive agent there extracts at most zero. NOTE-032 checked the numbers: the optimum is ≈ 0.272 bits per round, and uniform-random agents extract ≈ 0.189.

Read together, the two papers bound the claim. Predictive memory is what thermodynamic efficiency asks for only when the system does not act on its source. That reading is the record’s. LIT-041 presents itself as breaking the tape-setting principle, and NOTE-295 says to read the two as baseline and break.

What this does not say

• That the per-step nostalgia is a nonnegative cost under any drive. NOTE-295 found that nonnegativity needs a Markov drive, which the paper does not state. The note below proves it under that condition; that it fails without it is not shown (QUESTION-191). Under a non-Markov drive, a state that remembers x_{t−1} can predict x_{t+1} better than it tracks x_t, and then the step’s “dissipation” as defined is negative. The relaxation step ⟨ΔF^relax_neq⟩ ≤ 0 also needs an unstated stationarity condition.
• Anything per operation. Every quantity is an average over paths and protocols. The unit is nats, not bits, and neither paper states a “k_BT ln 2 per bit” cost.
• Anything about learning systems. Both papers hold the kernel fixed. LIT-327’s Discussion raises adaptation but proves nothing about it. LIT-041’s agent has no learning. Its “prediction and energy efficiency may be at odds in active learning systems” is a hedged extrapolation (NOTE-032, C14).
• That efficiency under feedback is a trade-off between prediction and forgetting. In the only proved example the efficient agent is memoryless (NOTE-032, C13). No interior optimum is exhibited.
• Anything about real hardware. Neither paper models a device.

Connections

• LIT-338 (IB) is the ancestor, through Still 2009, of the “maximise predictive power at fixed memory” objective that LIT-327’s Discussion recommends.
• LIT-308 bounds information acquired by entropy production, and LIT-327 bounds information kept but useless by dissipation. The Landauer term they share is the subject of THEORY-030.
• LIT-523 (Seifert’s review, read in NOTE-420) is the framework, not a source. For one protocol started in equilibrium, LIT-327’s dissipation equals ⟨W⟩ − ΔF − k_BT·D_KL(p_τ‖p_eq), and the review’s Eq. 150 says that is nonnegative. That is a protocol total for continuous Langevin dynamics; it does not settle the sign of the per-step nostalgia this account’s promote_when asks about.
• THEORY-431 (Crooks and Jarzynski’s fluctuation theorems) is the trajectory-level background. This account rests on an average-level identity for a discrete Markov chain and uses no fluctuation theorem.

Note of 2026-10-19: the identity and the bound, re-derived

Re-derived for the promotion audit (batch 045, ADR-047 v2). The setup is LIT-327’s. A system with state s is driven by x. Energies are E(s, x), and p_eq(s|x) ∝ exp(−βE(s, x)). At each step the drive switches x_t → x_{t+1} with s_t held, which is where work is done. Then s relaxes under a kernel K_x at fixed x_{t+1}. Nothing feeds back from s to x.

1. The identity, with no further condition. Write F_neq(x) = ⟨E(s, x)⟩ − kT·H[p(s|x)] for the free energy of the conditional distribution, averaged over x. At the switch, ⟨W_t⟩ = ⟨E(s_t, x_{t+1})⟩ − ⟨E(s_t, x_t)⟩. The averaged F_neq changes by the same energy difference minus kT(H[S_t|X_{t+1}] − H[S_t|X_t]). So β(⟨W_t⟩ − ΔF_neq) = H[S_t|X_{t+1}] − H[S_t|X_t] = I[s_t;x_t] − I[s_t;x_{t+1}].
2. Nonnegative under a Markov drive. With no feedback, s_t depends on x_0, …, x_t and on noise independent of the drive. A Markov drive makes x_{t+1} independent of the earlier inputs given x_t. So s_t – x_t – x_{t+1} is a Markov chain, and data processing gives I[s_t;x_t] ≥ I[s_t;x_{t+1}].
3. A lower bound when each kernel keeps its equilibrium stationary. Start the protocol in equilibrium and write F_neq = F_eq + kT·D_KL(p(s|x) ‖ p_eq(s|x)). Then W − ΔF_eq = Σ_t (W_t − ΔF_neq at the switch) − Σ_t (ΔF_neq in the relaxation) + kT·D_final. Relaxation does no work. Relative entropy to a distribution the kernel keeps stationary cannot grow under that kernel, so each relaxation term is nonnegative. D_final ≥ 0. Hence βW_diss ≥ Σ_t (I[s_t;x_t] − I[s_t;x_{t+1}]).
4. A numerical check. For a 4-state system driven by a 3-symbol Markov chain through Metropolis kernels, the identity in step 1 holds to 10⁻¹⁰, and over six steps the dissipation, 0.426 nats, exceeds the summed nostalgia, 0.076.
5. The feedback channel. In LIT-041’s Theorem 5 channel, action 0 gives percept 0 and action 1 a fair coin. A memoryless agent that plays 0 with probability q extracts at most h(q) − h((1+q)/2) bits per round. That is largest at q = 1/√2, about 0.272, and equals 1 − h(3/4) ≈ 0.189 at q = ½. A predictive agent there has H(A|M) = 0, so it extracts at most −H(S|M) ≤ 0. In this channel no predictive agent is efficient.

Steps 2 and 3 meet the promote_when’s request for a derivation under the two conditions. A system in which the bound fails without them is not built, and neither is a physical test; both are QUESTION-191.

Note of 2026-10-20: the wider claim, filed Proposed

Under ADR-047 v4 the narrowing-loss sweep (batch 072) filed THEORY-817 as Proposed, extending this entry. It carries the claim this entry’s title gave up when batch 045 narrowed it: that once actions feed back on the drive, maximal prediction is generally not what thermodynamic efficiency requires, which Fiderer et al. assert from their one proved binary channel. This entry is its evidence and is not changed by the filing.