Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value

Akihito Sudo

REVIEW 2 major objections 4 minor 52 references

Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that a finite learning device can accumulate arbitrary record correlation and world correlation while gaining exactly zero operational capital value, and it bounds how much of the search ledger can be capitalized into…

desk verdict A careful, self-aware paper with real contributions, but its headline separation result is explicitly restricted to an open assumption class from an unpublished companion, so it deserves peer review but not citation yet. read the letter →

arxiv 2608.12791 v1 pith:TZ74REZO submitted 2026-08-13 cond-mat.stat-mech cs.ITcs.LGmath.IT

Akihito Sudo This is my paper · ORCID

classification cond-mat.stat-mechcs.ITcs.LGmath.IT PACS 05.70.-a89.70.+c

keywords thermodynamicsoflearningcapitalvaluerecordcorrelationsearchledgercapitalizationefficiencyretentionfit–valuealignment

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

What a finite learning device has recorded and what will hold value for it on future tasks are not the same quantity: this paper develops a typed four-component accounting — acquisition cost, physical dissipation, transport action, and task value — in which memorization is a correlation stock while future value is the informed–blind work gap on a stated task distribution, access structure, and per-run budget. Its first theorem proves separation: for every $n$ there is a device family whose record correlation $I(M;D)$ and world correlation $I(M;W)$ both grow by $n\ln 2$ along admissible memory-local updates, yet the capital gain $\Delta V$ is exactly zero, so memorizing environmental noise creates no value follows as a theorem rather than a definition. Its second theorem introduces the search ledger $\sigma_M$ and proves a capitalization bound $\eta_{\mathrm{cap}} = \Delta V/(kT\sigma_M) \le 1$ in the flat$^{*}$ regime with a no-discarded-record-condition, with exact equality conditions and an explicit regime map of how the bound fails outside it. Its third theorem gives a two-layer alignment domain for value retention under task-distribution shift — exact exchange with the side-information-adjusted record fit, and with the raw record stock under joint neutrality — plus a four-coordinate intervention theorem in which shift, budget, access, or record content alone reverses or restores the fit–value ranking.

What carries the argument

The central object is the capital value $V(M;T,b)$, defined by a deletion counterfactual: the optimal expected work extractable over a future task distribution $T$ when the memory-read port is available, minus the optimum of a blind agent from whom every read port on $M$ has been deleted and who re-optimizes from scratch under the same tasks, access structure, and per-run budget $b$; learning is then an admissible memory-local update with $\Delta V > 0$, making the learning predicate relative to $T$, access, and budget. Two further instruments carry the theorems: the search ledger $\sigma_M = \Delta I(M;D) + \Sigma_{\mathrm{total}}$, a memory-side subsystem account that charges the effective entropy change of the memory registers plus the heat sent to the bath, including the cost of blank pages; and the flat$^{*}$ task class — degenerate energies, no gates, a single manipulable register, static all-read access, and flat-conditional attainability — on which the extraction identity $\mathrm{Gap}(M;\tau,b) = kT\,I(M;X_\tau|Y_\tau)$ holds exactly and budget-independently.

What would settle it

On a flat$^{*}$ task with condition (f) satisfied, construct an admissible (F5$'$)-stable memory-local update whose measured capital gain satisfies $\Delta V > kT\,\sigma_M$; Main Theorem II(iii) predicts $\eta_{\mathrm{cap}} \le 1$, so any such device would falsify the capitalization bound.

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Extended reading notes

Core claim

The central discovery is a type separation: a finite learning device's record-correlation stock $J_D = I(M;D)$ and its operational capital value $V(M;T,b)$, defined as the informed–blind work gap on a future task distribution under a stated budget and access structure, are not the same quantity and can move independently. Main Theorem I exhibits, for every $n$, a device family on which both $I(M;D)$ and the world correlation $I(M;W)$ grow by $n\ln 2$ along admissible memory-local updates while $\Delta V = 0$. Main Theorem II introduces the search ledger $\sigma_M = \Delta I(M;D) + \Sigma_{\mathrm{total}}$ and proves the capitalization bound $\eta_{\mathrm{cap}} \le 1$ in the flat$^{*}$ plus (f) regime, with necessary and sufficient equality conditions and a three-route regime map of failures. Main Theorem III gives a two-layer alignment domain: value equals $kT\,I(M';D|Y)$ exactly without any independence assumption, equals $kT\,I(M';D)$ under joint side-information neutrality $(M,D)\perp Y$, and the boundary is exhibited by an explicit one-time-pad witness; a four-coordinate intervention theorem shows that shift, budget, access route, and record content each admit a paired setting where changing that coordinate alone reverses or restores the fit–value ranking.

Load-bearing premise

The load-bearing premise is that all protocol classes carry the imported Assumption-U restriction, whose validity without loss of generality is left open in the companion framework; if that reduction fails, V is only a restricted-class-relative quantity, and the separation and alignment theorems would not describe all physically possible devices.

Editorial extensions

If this is right

Record-correlation increase is not learning: on the LB device, copying environment noise into memory grows $I(M;D)$ and $I(M;W)$ by $n\ln 2$ with zero capital gain, so memorizing noise creates no value is a theorem, not a naming choice.
Within the flat$^{*}$ plus (f) regime, value cannot outpace the search ledger: $\eta_{\mathrm{cap}} \le 1$, and full capitalization means simultaneously no forgetting, no waste, no $Y$-contamination, and a reversible implementation; outside the regime the bound fails in identified routes that the regime map organizes.
Fit–value alignment is conditional: on flat tasks value equals $kT\,I(M';D|Y)$ exactly without an independence assumption, and equals the raw record stock $kT\,I(M';D)$ only under joint side-information neutrality $(M,D)\perp Y$.
A blank-start cumulative bound $\eta_{\mathrm{cum}} \le 1$ survives even without condition (f), so the books of a multi-step history are honest even when a per-update recycler shows $\eta_{\mathrm{cap}} = 2$.
Overfitting does not imply low efficiency and low efficiency does not imply overfitting: $\eta_{\mathrm{cap}}$ and $\rho_{\mathrm{gen}}$ admit no functional or monotone relation, and the full rectangle $(0,1]\times[0,1]$ is realized by explicit devices and shifts.
Under task-distribution shift, the retention gap has a subject-difference identity: the per-update change in $L_{\mathrm{gen}}$ equals exactly the increase of the three breakage subjects (forgetting, waste, $Y$-contamination) on the shifted book, provided the support is flat and draw exogeneity holds over the shifted support.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the companion's Assumption-U restriction is not without loss of generality, then all value statements in this paper, including the separation and alignment theorems, describe only the restricted protocol class; the unrestricted operational value could deviate, so the asterisk on 'capital' is more than a formality.
The four-coordinate intervention theorem suggests a diagnostic recipe for physical learning systems inside the flat$^{*}$ class: by deliberately moving shift, budget, access route, or record content and observing whether fit ranking and value ranking separate, one can identify which operational coordinate is binding for a given device.
The gate-budget regime where $\eta_{\mathrm{cap}} \to \infty$ at fixed information content implies that under a finite per-run budget, specification information can function as a key that work cannot buy; a testable extension would probe whether such enablement persists in physical implementations with noisy gates.
The separation of record correlation from value suggests that a thermodynamics-aware training objective should target $\Delta V$ rather than $\Delta I(M;D)$; the paper itself stops at the accounting level and does not propose such an algorithm, leaving that as a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a careful, self-aware paper that does real work, but its headline result is explicitly hostage to an open assumption imported from an unpublished companion. I'd send it to a strong referee, but I wouldn't cite it until the companion lands and the U-reduction question is addressed.

What's new: the four-component typed accounting with V as a deletion-based informed–blind work gap, the separation theorem (record correlation grows while V stays fixed) on the LB device family, the capitalization regime map with eta_cap <= 1 under flat* + (f), the equality conditions, and the four-coordinate alignment collapse. The paper is honest that the core inequality is a composition of Sagawa–Ueda and Goldt–Seifert steps and that the ledger identity is imported; the claimed contribution is the operational packaging, the regime boundaries, and the device witnesses. The explicit witnesses and the machine-checked verification of the arithmetic are real evidence, though the scripts are not shipped.

The soft spots are exactly where the reader puts them. All protocol classes carry Assumption-U; the paper itself says the reduction is open, and V is restricted-class-relative. The separation theorem—memorization with zero capital gain—is proved only within that class. If the U-restriction fails without loss of generality, the headline statement doesn't extend to all physically possible finite devices. That's not a hidden flaw; it is in Sec. II E and Sec. X. But it is load-bearing, and it is the difference between a conditional and an unconditional result. Also, several lemmas (conditional chain bound, gate bounds B1–B3) come from the unpublished companion [9], and the gate bound's bookkeeping constant C_G has no proven numerical upper bound. The paper is upfront about all of this, which earns it credit on honesty, but it means the external verification burden is not met.

Do I think it holds up? The internal math looks coherent; I didn't find a contradiction. The flat* extraction identity and the regime map are the strongest parts. The gate threshold theorem is honest about its dependence on unproven C_G. The paper's own scoping—not a theory of statistical generalization—is appropriate.

Who is this for? People working on stochastic thermodynamics of learning, information engines, and value-of-information. It deserves a serious referee: the framework is important if the companion checks out. I'd accept for review, with the expectation of conditional acceptance after the companion is available or the imports are proven.

Recommendation: send it to peer review. Tell the referees to focus on the U-reduction and the companion dependencies; the rest is solid.

Referee Report

2 major / 4 minor

Summary. The paper develops a typed accounting framework for finite-state learning devices, separating training-side fit, record-correlation stock, the update-side search ledger, and an operational capital value V(M;T,b) defined as the informed-minus-blind work gap under a deletion counterfactual. Main Theorem I establishes basic properties of V, including affinity, nonnegativity, recoding invariance, and a single-task reduction to a companion value, and constructs for every n a device family LB on which record correlation I(M;D) and world correlation I(M;W) grow by n ln2 along an admissible update sequence while the capital gain is exactly zero. Main Theorem II introduces the search ledger sigma_M, proves an exact flat* extraction identity, a universal ledger identity, and bounds the capitalization efficiency eta_cap = Delta V/(kT sigma_M) by 1 under flat* supports, (F5')-stability, and condition (f), with equality conditions and a regime map of identified failures. Main Theorem III gives a value-retention alignment schema with a two-layer positive domain, a boundary one-time-pad witness, four one-coordinate intervention witnesses, and a subject-difference identity for the retention gap. The manuscript is explicit that all results are exact finite-device statements inside a fixed operational framework and are not a theory of statistical generalization.

Significance. If the results are accepted, the paper makes a useful conceptual and technical contribution: it gives a precise, operationally anchored distinction between what a finite device has recorded and what that record is worth on future tasks, with clean device witnesses for separation, for the capitalization bound, and for the failure modes of that bound. The manuscript is unusually careful about scope: it ships machine-checked numerical verification scripts, attributes imported identities explicitly, and includes a detailed limitations section. The main caveat is that the central value quantity is restricted-class-relative and several load-bearing lemmas are imported from unpublished companion work, so the headline claims currently run ahead of what the manuscript proves unconditionally.

major comments (2)

[Sec. II E; Main Theorem I(ii)-(iii)] The separation theorem is stated without a qualifier in the abstract and in the theorem, but all protocol classes carry the imported Assumption-U restriction (clause (g), Sec. II A), and Sec. II E states that whether this restriction is without loss of generality is an open question. Because V is defined relative to the U-conforming protocol class, the claim that record and world correlation grow by n ln2 while capital gain is exactly zero is not established for all physically possible finite learning devices; a protocol violating clause (g) could in principle exploit the M-Y correlation that the U-restriction forbids and make Delta V positive along the record-copy sequence. Please either prove the U-reduction or state the separation theorem and all interpretation-level claims with the restricted-class qualifier in the abstract and in the statements of Main Theorem I and Definition 9.
[Appendix B.2; Appendix D] The load-bearing lemmas are not proved in this manuscript: Lemma 9 (the conditional chain bound) is restated, but the text explains that the weight-bearing path depends on this statement rather than on the interior of an imported proof, and its proof is in the unpublished companion [9]; similarly, the gate bounds B1-B3 (Theorems 2-4 of Appendix D) are given as fixed statements whose proofs are in the companion. Because the separation theorem, the flat* extraction identity, the gate threshold transition, and the budget/access witnesses all rely on these imported inequalities, the paper is not self-contained and the results cannot be fully verified by a referee. Please include complete proofs of these statements or make the dependence explicit and condition acceptance on the companion's availability.

minor comments (4)

[Sec. II A] Clause (g) names Assumption-U but does not state the conditional-independence property formally; please give the precise condition or a complete reference to the companion.
[Sec. IX] Refs. [9], [10], and [16] are cited as unpublished; please add arXiv identifiers or stable versioning so readers can verify the imported statements.
[Appendix E] The notation table would benefit from an explicit entry distinguishing flat tasks (Definition 18) from flat* tasks (Definition 13); the asterisk convention is easy to miss.
[Eq. (27)] The subjects g_a, g_b, and g_c are defined in Lemma 6; repeating their one-line definitions at Eq. (27) would improve readability.

Circularity Check

2 steps flagged · score 5.0 of 10

Operational value and the flat* extraction identity are imported from the same author's unpublished companion; the separation theorem is unconditional only on the U-restricted class, so the chain is load-bearing self-citation rather than a derivation from first principles.

self citation load bearing [Sec. II A; Sec. III D Main Theorem I(i)(e); proof of (i)(e)]
"We import the protocol framework unchanged from the companion manuscript [9] and restate the parts used here. ... The imported deletion counterfactual is the comparison object of this paper. ... The informed branch of Definition 6 is the per-realization form and the blind branch is deletion followed by re-optimization from scratch; this is the companion framework’s informed–blind datum value with the datum read as M, normalized there by β/ln 2 to bits."

V is defined on a protocol class and deletion counterfactual that are imported verbatim from the author's unpublished companion [9], and the theorem's 'reduction' at a single task states that V equals that companion's M_val. That equality is a restatement of the imported definition, not an independent first-principles result. The paper then uses this anchor (Remark 7) to claim that 'correlation outside the support of T is not capital' is theorem-level; the anchor's operational motivation therefore rests on an unverified self-citation.
self citation load bearing [Sec. IV C (Lemma 3), Appendix B.2 (Lemma 9), Main Theorem II(i)]
"The converse is carried by analysis (the conditional chain bound, stated as a standalone lemma in Appendix B so that the weight-bearing path does not depend on the interior of an imported proof) ... This is the chain bound of the companion framework [9] applied to the conditional law, with the telescoping term ... retained rather than discarded."

The flat* extraction identity Gap = kT I(M;Xτ|Yτ), which is used to pin the LB device's zero capital gain (Main Theorem I(ii)-(iii)), to prove the D-free nonincrease (I(iv)), and to derive the capitalization bound (Main Theorem II(iii)), has its informed-branch converse supplied solely by Lemma 9, a statement quoted from the author's own unpublished companion [9]. The paper provides no proof of Lemma 9; the weight-bearing path is therefore a self-citation chain, and any claim resting on Lemma 3 is only as strong as that unverified import.

full rationale

The paper is unusually transparent: it labels Main Theorem II(iii) a composition of Sagawa-Ueda and Goldt-Seifert steps, attributes the ledger identity to Refs. [4,5], and marks the Assumption-U restriction as an open WLOG question. That transparency lowers the charge of hidden circularity, but it does not remove the load-bearing self-citation: the protocol framework, deletion counterfactual, single-task value anchor, and conditional chain bound all come from the same author's unpublished companion [9], which is neither machine-checked nor independently reproduced here. Because the separation theorem's zero-capital-gain conclusion is proved only inside the U-restricted protocol class (Sec. II E), and its gap pinning uses Lemma 3, whose converse is Lemma 9 from [9], the strongest claim is conditional on an unverified self-citation rather than derived from first principles. Nevertheless, the devices, regime map, equality conditions, and alignment witnesses are new algebraic content that does not reduce to the inputs by construction; no fitted parameter is renamed as a prediction, and no equation is simply defined into its conclusion. Score 5 reflects partial circularity through the self-citation chain while acknowledging the independent theorem-level content and the paper's explicit limitation statements.

Assumptions & free parameters 1 free parameters · 4 assumptions · 3 invented entities

The central claims rest on the imported protocol framework, the U-restriction, the finiteness and uniformity of C_G, and standard information-theoretic identities. No data fitting is used; the only hand-set numeric values are script conventions for C_G.

free parameters (1)

C_G (gate bookkeeping constant) = unproven; script conventions 1.0 and 5.0
Appears in imported gate bounds and Theorem 1; assumed finite and uniform, but no proven numerical upper bound is provided; all finite gate-task numerics are substitution arithmetic under the script conventions.

assumptions (4)

ad hoc to paper Imported protocol framework and deletion counterfactual from companion [9], including budgeted protocol class, read-port deletion, conditional chain bound Lemma 9, and gate bounds B1-B3.
Restated in Sec. II A and Appendices B and D; proofs are in the unpublished companion, so this paper treats them as unproved background.
domain assumption Assumption-U restriction is imposed on all protocol classes and is assumed meaningful even though it is open whether it is without loss of generality.
Sec. II E: the U-reduction question remains open; V is restricted-class-relative.
domain assumption C_G is finite and uniform in device parameters, and the imported gate bounds hold in the stated form.
Sec. V and Appendix D: no proven numerical bound on C_G; the regime map and gate threshold theorems depend on this finiteness and uniformity.
domain assumption Draw exogeneity: the future task draw is jointly independent of the training closure (M, D, ancillas).
Definition 3; Lemma 1 preserves it under updates; realization-correlated adversarial shifts are outside scope.

invented entities (3)

Capital value V(M;T,b)
purpose: Operational future value of a memory state as informed-minus-blind expected work over task distribution T with budget b.
The central new quantity; no external empirical handle is proposed; it is defined through protocol-class optima.
Search ledger sigma_M
purpose: Memory-side update cost account used as denominator of capitalization efficiency.
A constructed accounting quantity; its value is not independently measurable outside the framework.
Retention ratio rho_gen
purpose: Measures value retention under task-distribution shift.
Defined as a ratio of capital values; no direct external measurement is proposed.

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Cite this review

Pith. "Pith review of Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value." pith.science (2026). https://pith.science/paper/TZ74REZO

@misc{pith2026260812791,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of Learning: A Typed Four-Component Accounting of Memory, Fit, and Value},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZ74REZO}},
  note         = {Machine review of arXiv:2608.12791}
}

abstract

What a finite learning device has recorded and what will hold value for it on future tasks are not the same quantity. We develop a typed accounting for finite-state learning devices that separates four components: a training-side fit functional $\Phi_{\mathrm{fit}}$, the record-correlation stock $J_{D}=I(M;D)$, an update-side search ledger $\sigma_{M}$, and an operational capital value $V(M;T,b)$. This value is the work gap between an informed protocol class and a blind class obtained by deleting the memory-read port and re-optimizing from scratch. (I) Separation: for every $n$, there is a device family on which record correlation and world correlation grow by $n\ln 2$ while the capital gain is exactly zero. In the $\mathrm{flat}^{*}$ regime, data-free updates never increase $V$. (II) Capitalization ledger: an exact $\mathrm{flat}^{*}$ extraction identity and a universal ledger identity give, for (F5$'$)-stable $M$-local updates under a no-discarded-record-correlation condition (f), the bound $\eta_{\mathrm{cap}}\le 1$ for the capitalization efficiency $\eta_{\mathrm{cap}}=\Delta V/(k T\,\sigma_{M})$, together with necessary and sufficient conditions for equality. (III) Value retention: for the retention gap $L_{\mathrm{gen}}$ and retention ratio $\rho_{\mathrm{gen}}$ (the former carries no sign constraint; the latter is defined for positive training-side value and is not confined to $[0,1]$) we give a two-layer alignment domain: an exact exchange rate between value and the side-information-adjusted record fit $I(M';D\mid Y)$ without any record-side-information independence assumption, and a raw record-stock exchange rate under a joint side-information neutrality condition $(M,D)\perp Y$, whose boundary is marked by an explicit one-time-pad witness. These are statements about finite-device value retention under task-distribution shift, not a theory of statistical generalization.

Figures

Figures reproduced from arXiv: 2608.12791 by the authors.

**Figure 2.** FIG. 2. Fit–value alignment: the two-layer domain in [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** FIG. 3. Rectangle witness in the ( [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 43 canonical work pages

[9]

Proof of update data processing (Lemma 4) By Lemma 3 applied to both states ((F5 ′)-stability), ∆Gapτ =kT[I(M ′;X|Y)−I(M;X|Y) ]. Supply cap.Chain rule:I(M ′;X|Y)≤ I((M′,M);X|Y) =I(M;X|Y) +I(M ′;X|M,Y), so ∆Gapτ/kT≤I(M ′;X|M,Y); the Markov chain X−D−M ′ given (M,Y) (Lemma 11) and data process- ing giveI(M ′;X|M,Y)≤I(D;X|M,Y). Acquisition cap.The same Marko...

work page
[1]

Device LB (ledger-blocked correlation) Registers.X 1,X 2: one uniform bit each, independent (the working media of the two tasks).Y= (Y 1,...,Y n): i.i.d. fair bits independent of (X 1,X 2)—world registers outside every manipulable set.M core: a perfect copy ofX 1 (one bit of genuine capital, so that ∆V= 0 is exhibited in a nondegenerate situation withV >0...

work page
[2]

The ledger device suite (equality and its non-achievements) Device E (reversible copy; equality achievement).W= {X1}(one uniform bit); a single flat∗ task (E≡0,G=∅, Man ={X 1}∪anc,K= 3, static all-read);T=δ τ;D= perfect copy ofX 1;M 0 blank. Update: controlled-copy ofDinto fresh memory (growth form) or into a pre- 27 erased cell (overwrite form); both det...

work page
[3]

The remaining reten- tion devices: Device KR (key redraw; Type II).As defined in Sec

The retention device suite The alignment witnesses SH, BG, AC′, RJ are defined with their proofs in Appendix C. The remaining reten- tion devices: Device KR (key redraw; Type II).As defined in Sec. VIII B: wired key Θ (frozen, energy-decoupled), pass iffB=s(Θ), uniformd-fold degenerate pass map, prize Fcart, horizonK hor, budgetb;M= copy ofs(Θ); shift re-...

work page
[4]

Numerical verification Three pure-Python scripts (standard library only), re- executed directly in the main verification pass, cover the devices above; all exit with status 0. Coverage discipline (stated exactly).None of the scripts performs a numerical search over multi-stage adaptive protocols: such suprema are not finitely enu- merable, and the convers...

work page
[5]

HenceE[W ext] of ev- eryP∈Prot b(Aτ−M,p) is a functional of the non-M marginals alone, and the blind branch does not depend on the coupling structureπ M

The blind branch does not depend onM: full argument A blind protocol has no read port onM(deletion), Mis energetically decoupled (Definition 2(i)), and by clauses (ii)–(iv) it enters no dynamics: every stage map of a blind protocol factorizes as idM⊗ΛW , and the non- Mvariables evolve autonomously. HenceE[W ext] of ev- eryP∈Prot b(Aτ−M,p) is a functional ...

work page
[6]

We state it once, so that the weight-bearing path depends on a statement rather than on the interior of an imported proof

The conditional chain bound, stated The converse below rests on a conditional form of the chain bound of the companion framework, with the total- correlation term retained. We state it once, so that the weight-bearing path depends on a statement rather than on the interior of an imported proof. Lemma 9(Conditional chain bound, retained form).Let τbe a tas...

work page
[7]

Step 1.Mis frozen (derived from Defini- tion 2(ii)+(iv)); under convention (α), apply Lemma 9 to theM=mconditional law

Proof of the extraction identity (Lemma 3) Converse (branch upper bounds; valid for every state, no use of (F5′)).WriteX:=X τ,Y:=Y τ. Step 1.Mis frozen (derived from Defini- tion 2(ii)+(iv)); under convention (α), apply Lemma 9 to theM=mconditional law. Step 2 (degenerate local free energies).By (F1), Floc,i =−kTS i, so−β P i ∆F (m) loc,i = P i ∆S(m) i . ...

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Show all 52 references

[8]

Conditional inheritance of joint independence Lemma 11(Joint independence is inherited under con- ditioning).IfA⊥(W,M,D)jointly, then for every par- tition of(W,M,D)into subtuples(B 1,B 2):A⊥B 1 | B2. Consequentlyp(a|x,y,m,d) =p(a), and the kernel marginalized over the ancilla...
[10]

From the definition of the implementation entropy production, Σtotal = [S(M′,D)−S(M,D,A)] + ∆Sbath

Proof of the ledger identity (Lemma 5) Let the terminal system be (M ′,D), withM ′ the to- tality of memory-side registers—Mtogether with all ab- sorbed ancillas, excluding the read-onlyD, the world, and the bath—so that the disjoint pair (M′,D) exhausts the terminal system (t...
[11]

Proof of the per-task decomposition (Lemma 6) WriteX:=X τ,Y:=Y τ. Three chain-rule identities superpose: (1) the two expansions ofI((M ′,M);X|Y): I(M′;X|Y)−I(M;X|Y) =I(M ′;X|M,Y)− I(M;X|Y,M ′) =I(M ′;X|M,Y)−g a; (2) the two ex- pansions ofI(M ′; (X,D)|M,Y) with the Markov prop...
[12]

Proof of Main Theorem II, parts (iii)–(v) (iii).By theT-averaged acquisition cap (Lemma 4), ∆V≤kT∆ +JD; by the (f)-form of Lemma 5, ∆ +JD = σM−Σ total≤σ M. Compose.■ (iv).T-average Lemma 6: ∆ +JD isτ-independent by draw exogeneity (Definition 3 and Lemma 1:τ⊥ (M,D,ancillas) ma...
[13]

(2)M n is a measurable function of (M0,D,all ancillas) withM 0 deterministic and the ancillas jointly indepen- dent of (W,M,D) (the per-update freshness composes: each ancilla is jointly independent of the entire history at its introduction, so the pooled randomness is indepen...
[14]

Proof of the flat-task bound and the task optimum Lemma 7.The informed converse of Appendix B 3 uses no (F5′) and gives, for everyM, informed≤kT[ln|X|− H(X|Y,M)]. Blind achievability is carried by the y-clause: the blind conditional distributionsp τ(x|y) are task properties in...
[15]

By (β),DandX τ have identical content, so I(M′ f;Xτ |Y τ) =I(M ′ f;D|Y τ)

Proof of the two-layer correspondence (Proposition 2) (A).By (F5 ′)-stability, the extraction identity (Lemma 3, via Corollary 1) applies to each candidate state:V(M ′ f;δτ,b) =kTI(M ′ f;Xτ|Y τ), independently ofb. By (β),DandX τ have identical content, so I(M′ f;Xτ |Y τ) =I(M...
[16]

Proof of the subject-difference identity (Corollary 2) and Lemma 8 Corollary 2.Under hypothesis (i), Lemma 6 holds at everyτ∈T: ∆Gap τ =kT[∆ +JD−ga(τ)−g b(τ)−g c(τ)]. Average againstT train andT shift and subtract: by hy- pothesis (iii) the joint law of (M,M ′,D) is common to ...
[17]

The uninformative-port lemma Lemma 12(Uninformative port).IfMis jointly inde- pendent ofallvariables of a taskτ(world registers, gate keys, cartridge states), thenGap(M;τ,b) = 0for every b. Proof.UnderM⊥(all task variables), conditioning on M=mleaves the task-side joint law un...
[18]

Witness S: device SH and the proof of Main Theorem III(ii) Device SH.Shared worldW= (X A,XB), one uni- form bit each, independent,pcommon to both tasks. T={τ A,τB}:τ A = (W,uniform,E≡0,G=∅,Man = {XA}∪anc,K= 3,static all-read), flat ∗ withY τA = {XB};τ B symmetric.D= (D A,DB) p...
[19]

Flat part:X,nuniform bits as a single manipulable regis- ter (K= 3, static all-read)

Witness B: device BG and the proof of Main Theorem III(iii) Device BG (Type I).Two independent parts. Flat part:X,nuniform bits as a single manipulable regis- ter (K= 3, static all-read). Gate part: an imported gate of the cartridge type (Appendix D), Type I real- ization, wit...
[20]

All states (F5 ′) (machine-checked; no noise is needed)

Witness A: device AC ′, proof, and the two variants Device AC′ (shadow design).A single task:X= (X1[2 bits],X 2[1 bit]) one manipulable register (joint permutations available,K= 3);Y= a readable, non- manipulable duplicate of thecontentofX 1—soXandY are correlated,I(X;Y) = 2 l...
[21]

D= (copy ofX, J)—clause (β) broken: the record ex- ceeds the faithful observation of the medium.T=δ τ, barbitrary; capacitynbits;F={f X,fJ}

Witness R: device RJ and proof Device RJ.A single flat∗ task:Xone uniform bit (ma- nipulable),Y=∅,K= 3;J:n≥2 uniform bits indepen- dent ofX(junk—n≥2 makes the fit preference strict). D= (copy ofX, J)—clause (β) broken: the record ex- ceeds the faithful observation of the mediu...
[22]

Upper end: supply boundedness—prize plus budget plus bookkeeping

Proof of the gate threshold transition (Theorem 1) (1) Training side.Lower end: the informed branch readsM=s(Θ) and drives the known value to the tar- get by a zero-work conditional permutation (it draws no external work, so it runs atb= 0; statement B1 of Ap- pendix D, whose ...
[23]

Sagawa and M

T. Sagawa and M. Ueda, Physical Review Letters104, 090602 (2010), arXiv:0907.4914

2010 arXiv
[24]

Goldt and U

S. Goldt and U. Seifert, Physical Review Letters118, 010601 (2017), arXiv:1611.09428

2017 arXiv
[25]

Goldt and U

S. Goldt and U. Seifert, New Journal of Physics19, 113001 (2017), arXiv:1706.09713

2017 arXiv
[26]

J. M. Horowitz and M. Esposito, Physical Review X4, 031015 (2014), arXiv:1402.3276

2014 arXiv
[27]

Hartich, A

D. Hartich, A. C. Barato, and U. Seifert, Journal of Statistical Mechanics: Theory and Experiment , P02016 (2014), arXiv:1402.0419

2014 arXiv
[28]

Still, D

S. Still, D. A. Sivak, A. J. Bell, and G. E. Crooks, Physi- cal Review Letters109, 120604 (2012), arXiv:1203.3271

2012 arXiv
[29]

Updates that do not touch data are in general not learning

A. Kolchinsky and D. H. Wolpert, Interface Focus8, 35 TABLE V. Notation contract. Primary terms are fixed here; the body uses these terms and no synonyms. The word “fitting” from an early working document is rendered throughout asrecord-correlation increase(the predicate ∆I(M;...

2018 arXiv
[30]

A. B. Boyd, J. P. Crutchfield, and M. Gu, New Journal of Physics24, 083040 (2022), arXiv:2006.15416

2022 arXiv
[31]

Sudo, The thermodynamic value of data: Meaning as enablement under budget constraints (2026), (unpub- lished)

A. Sudo, The thermodynamic value of data: Meaning as enablement under budget constraints (2026), (unpub- lished)

2026
[32]

Sudo, Replacement costs of state and structure infor- mation in opaque thermodynamic gates (2026), (unpub- lished)

A. Sudo, Replacement costs of state and structure infor- mation in opaque thermodynamic gates (2026), (unpub- lished)

2026
[33]

L. J. Fiderer, P. C. Barth, I. D. Smith, and H. J. Briegel, The work capacity of channels with memory: Maxi- mum extractable work in percept-action loops (2025), arXiv:2504.06209

2025 arXiv
[34]

Sudo, Mechanism-resolved second law for multipartite systems: An entropy-production ledger for correlation loss (2026), arXiv:2108.00738

A. Sudo, Mechanism-resolved second law for multipartite systems: An entropy-production ledger for correlation loss (2026), arXiv:2108.00738

2026
[35]

F. G. S. L. Brand˜ ao, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Physical Review Letters 111, 250404 (2013), arXiv:1111.3882

2013 arXiv
[36]

A. B. Boyd, J. P. Crutchfield, M. Gu, and F. C. Binder, New Journal of Physics27, 063901 (2025), arXiv:2402.16995

2025 arXiv
[37]

Caraffa, Dissipative learning: A framework for viable adaptive systems (2026), arXiv:2601.17933

L. Caraffa, Dissipative learning: A framework for viable adaptive systems (2026), arXiv:2601.17933

2026
[38]

Sudo, A robustness spectrum for hidden target spec- ifications: Degeneracy, transfer, and effective leakage in opaque thermodynamic gates (2026), (unpublished)

A. Sudo, A robustness spectrum for hidden target spec- ifications: Degeneracy, transfer, and effective leakage in opaque thermodynamic gates (2026), (unpublished)

2026
[39]

Shettell and A

N. Shettell and A. Auff` eves, Entropic efficiency of bayesian inference protocols (2026), arXiv:2601.17282

2026
[40]

Kolchinsky and D

A. Kolchinsky and D. H. Wolpert, Journal of Statisti- cal Mechanics: Theory and Experiment2017, 083202 (2017), arXiv:1607.00956

2017 arXiv
[41]

Still, Physical Review Letters124, 050601 (2020), arXiv:1705.00612

S. Still, Physical Review Letters124, 050601 (2020), arXiv:1705.00612

2020 arXiv
[42]

Still and D

S. Still and D. Daimer, New Journal of Physics24, 073031 (2022), arXiv:2103.15803

2022 arXiv
[43]

Daimer and S

D. Daimer and S. Still, The physical observer in a szilard engine with uncertainty (2023), arXiv:2309.10580

2023 arXiv
[44]

A. B. Boyd, D. Mandal, and J. P. Crutchfield, Physical Review X8, 031036 (2018), arXiv:1708.03030

2018 arXiv
[45]

Perrier, Watts-per-intelligence: Part i (energy effi- ciency) (2025), arXiv:2504.05328

E. Perrier, Watts-per-intelligence: Part i (energy effi- ciency) (2025), arXiv:2504.05328

2025 arXiv
[46]

Perrier, Watts-per-intelligence part ii: Algorithmic catalysis (2026), arXiv:2604.20897

E. Perrier, Watts-per-intelligence part ii: Algorithmic catalysis (2026), arXiv:2604.20897

2026 arXiv
[47]

Finzi, S

M. Finzi, S. Qiu, Y. Jiang, P. Izmailov, J. Z. Kolter, and A. G. Wilson, From entropy to epiplexity: Rethink- 36 ing information for computationally bounded intelligence (2026), arXiv:2601.03220

2026
[48]

Takahashi and Y

K. Takahashi and Y. Hayashi, Thermodynamic limits of physical intelligence (2026), arXiv:2602.05463

2026 arXiv
[49]

Hasegawa, Thermodynamics of classifiers (2026), arXiv:2605.24365

Y. Hasegawa, Thermodynamics of classifiers (2026), arXiv:2605.24365

2026 arXiv
[50]

Watanabe and R

K. Watanabe and R. Takagi, Nature Communications17, 1857 (2026), arXiv:2504.12373

2026
[51]

Lumbreras, R

J. Lumbreras, R. C. Huang, Y. Hu, M. Fanizza, and M. Gu, Reinforcement learning for quantum processes with memory (2026), arXiv:2603.25138

2026
[52]

Touzo, M

L. Touzo, M. Marsili, N. Merhav, and ´E. Rold´ an, Jour- nal of Statistical Mechanics: Theory and Experiment , 093403 (2020), arXiv:2006.04544

2020 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [9]

Proof of update data processing (Lemma 4) By Lemma 3 applied to both states ((F5 ′)-stability), ∆Gapτ =kT[I(M ′;X|Y)−I(M;X|Y) ]. Supply cap.Chain rule:I(M ′;X|Y)≤ I((M′,M);X|Y) =I(M;X|Y) +I(M ′;X|M,Y), so ∆Gapτ/kT≤I(M ′;X|M,Y); the Markov chain X−D−M ′ given (M,Y) (Lemma 11) and data process- ing giveI(M ′;X|M,Y)≤I(D;X|M,Y). Acquisition cap.The same Marko...

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[2] [1]

Device LB (ledger-blocked correlation) Registers.X 1,X 2: one uniform bit each, independent (the working media of the two tasks).Y= (Y 1,...,Y n): i.i.d. fair bits independent of (X 1,X 2)—world registers outside every manipulable set.M core: a perfect copy ofX 1 (one bit of genuine capital, so that ∆V= 0 is exhibited in a nondegenerate situation withV >0...

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[3] [2]

The ledger device suite (equality and its non-achievements) Device E (reversible copy; equality achievement).W= {X1}(one uniform bit); a single flat∗ task (E≡0,G=∅, Man ={X 1}∪anc,K= 3, static all-read);T=δ τ;D= perfect copy ofX 1;M 0 blank. Update: controlled-copy ofDinto fresh memory (growth form) or into a pre- 27 erased cell (overwrite form); both det...

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[4] [3]

The remaining reten- tion devices: Device KR (key redraw; Type II).As defined in Sec

The retention device suite The alignment witnesses SH, BG, AC′, RJ are defined with their proofs in Appendix C. The remaining reten- tion devices: Device KR (key redraw; Type II).As defined in Sec. VIII B: wired key Θ (frozen, energy-decoupled), pass iffB=s(Θ), uniformd-fold degenerate pass map, prize Fcart, horizonK hor, budgetb;M= copy ofs(Θ); shift re-...

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[5] [4]

Numerical verification Three pure-Python scripts (standard library only), re- executed directly in the main verification pass, cover the devices above; all exit with status 0. Coverage discipline (stated exactly).None of the scripts performs a numerical search over multi-stage adaptive protocols: such suprema are not finitely enu- merable, and the convers...

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[6] [5]

HenceE[W ext] of ev- eryP∈Prot b(Aτ−M,p) is a functional of the non-M marginals alone, and the blind branch does not depend on the coupling structureπ M

The blind branch does not depend onM: full argument A blind protocol has no read port onM(deletion), Mis energetically decoupled (Definition 2(i)), and by clauses (ii)–(iv) it enters no dynamics: every stage map of a blind protocol factorizes as idM⊗ΛW , and the non- Mvariables evolve autonomously. HenceE[W ext] of ev- eryP∈Prot b(Aτ−M,p) is a functional ...

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[7] [6]

We state it once, so that the weight-bearing path depends on a statement rather than on the interior of an imported proof

The conditional chain bound, stated The converse below rests on a conditional form of the chain bound of the companion framework, with the total- correlation term retained. We state it once, so that the weight-bearing path depends on a statement rather than on the interior of an imported proof. Lemma 9(Conditional chain bound, retained form).Let τbe a tas...

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[8] [7]

Step 1.Mis frozen (derived from Defini- tion 2(ii)+(iv)); under convention (α), apply Lemma 9 to theM=mconditional law

Proof of the extraction identity (Lemma 3) Converse (branch upper bounds; valid for every state, no use of (F5′)).WriteX:=X τ,Y:=Y τ. Step 1.Mis frozen (derived from Defini- tion 2(ii)+(iv)); under convention (α), apply Lemma 9 to theM=mconditional law. Step 2 (degenerate local free energies).By (F1), Floc,i =−kTS i, so−β P i ∆F (m) loc,i = P i ∆S(m) i . ...

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