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Semantic Transferability of Generative Rate-Distortion-Perception Systems

Jinwan Jeon, Seoyeon Hwang, Sung Hoon Lim

arXiv:2610.34697v1cs.ITmath.IT

TL;DR

The paper asks which parts of an RDP system remain reusable for prescribed conditional generation. It compares increasingly restrictive transfer strategies and characterizes exact transfer through recurrent-class structure, including tasks supported by compatible reused codes.

  • Problem

    It is unclear which parts of a matched-marginal RDP system can be reused for prescribed semantic generation, since renderer capability, inherited latent statistics, and deployed-code behavior are distinct questions.

    Source: §I · pp. 2–3

  • Method

    The paper compares full redesign with renderer-only, representation-induced, and full-model transfer, analyzing single-letter laws separately from reuse of an actual compatible block code.

    Source: §I · p. 3

  • Results

    Representation-induced transfer is exact precisely when S depends on X only through recurrent-class label C; every compatible reused code supports X-C-S tasks, with additional dependence bounded by I(S;X|C) ≤ R0−H(C).

    Source: §I · pp. 4–5

  • Takeaways & Limitations

    The retained component matters: systems with the same source–reconstruction law can differ in renderer-only transfer feasibility, so the end-to-end test channel alone does not determine that capability.

    Source: §I · pp. 4–5

    The distortion-based transport guarantee for approximate transfer does not establish the vanishing block total variation required for strong synthesis.

    Source: §VI · p. 30

Abstract

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Generative lossy compression combines a rate-limited representation with a stochastic decoder that preserves source-dependent information while generating unresolved detail. We ask what remains reusable when a rate-distortion-perception (RDP) system selected for one operating point is transferred to a prescribed conditional-generation task. We distinguish the single-letter RDP factorization from its block-code implementation and compare Full Redesign with three increasingly restrictive transfer strategies. Renderer-Only Transfer retains only the renderer and redesigns semantic steering. Representation-Induced Transfer additionally retains the semantic latent law induced by the original representation. Full-Model Transfer retains a compatible RDP code, marginalizing over the task conditional law to form its semantic encoder. For finite alphabets, we characterize Renderer-Only Transfer's exact compression-common-randomness rate region, renderer-hull controllability condition, and compression-rate penalty relative to Full Redesign. Under unlimited common randomness, this penalty is the minimum residual semantic information $I(S;U|X)$ forced by the inherited renderer. Representation-Induced Transfer is exact if and only if the semantic condition depends on the source only through the recurrent-class label $C$ of the end-to-end RDP test channel. Every compatible RDP code reused under Full-Model Transfer preserves $C$ and supports every task satisfying $X-C-S$. Any additionally transferable within-class semantic dependence must satisfy $I(S;X|C)\le R_0-H(C)$, where $R_0$ is the inherited compression rate. The bound is exact when $R_0=H(C)$. A scalar Gaussian specialization yields a renderer-noise controllability threshold, a closed-form transfer penalty, and examples of identical Gaussian RDP test channels with different Renderer-Only Transfer capabilities.

I. INTRODUCTION · II. PRETRAINED SYSTEM AND TRANSFER TASK · A. Strong-Perception RDP Coding

The paper asks which conditional-generation tasks and rates remain achievable when a generative RDP system is reused, distinguishing reuse of its renderer, representation, or operational code. It characterizes the limits of these strategies and separates the single-letter RDP factorization from a compatible block-code implementation.

I. INTRODUCTION

  • Generative RDP coding separates source-dependent information preserved in a representation from unresolved detail generated by a stochastic renderer.The paper distinguishes distortion, which ties reconstruction to the source sample, from perception, which constrains reconstructed-sample distributions; correct unconditional generation alone does not ensure a renderer realizes a specific conditional task.Source: §I · pp. 1–2
  • The transfer task fixes q(s,x)=q(s)q(x|s) with q(x)=q0(x): the output marginal stays unchanged while generated outputs acquire the prescribed dependence on S.The three strategies retain progressively more of the pretrained system: the renderer, then its induced semantic latent statistics, then an existing RDP block code; Full Redesign is the benchmark.Source: §I · pp. 2–3
  • Renderer-Only Transfer characterizes task feasibility and the exact compression–common-randomness tradeoff, with a potentially positive rate penalty even under unlimited common randomness.That penalty is the minimum semantic information the renderer’s latent input must carry beyond the generated output.Source: §I · p. 4
  • Representation-Induced Transfer is exact if and only if the semantic condition depends on the source only through the recurrent-class label C.C determines the family of exactly transferable tasks; when transfer is exact, inherited latent statistics attain Full Redesign’s minimum compression rate under unlimited common randomness.Source: §I · pp. 4–5
  • Every compatible RDP code reused under Full-Model Transfer preserves C and supports matched-marginal tasks satisfying X−C−S.The inherited compression rate limits how much additional semantic dependence beyond C a particular implementation can preserve.Source: §I · pp. 4–5
  • In the scalar Gaussian model, finite-rate transfer with unlimited common randomness is possible exactly when target conditional variance exceeds renderer noise variance.The analysis also derives the minimum compression rate and its penalty relative to Full Redesign; discrete and Gaussian examples show identical source–reconstruction laws can have different renderer transfer capabilities.Source: §I · p. 5

II. PRETRAINED SYSTEM AND TRANSFER TASK

  • Section II frames the pretrained system through strong-perception RDP coding and the new conditional-generation task through channel synthesis.These provide the operational setting for asking which pretrained components can be reused.Source: §II · p. 6

A. Strong-Perception RDP Coding

  • The paper distinguishes a single-letter RDP factorization from its operational block code, defining compatibility by convergence of the code’s actual joint law to the factorization’s ideal law.Compatibility does not require the operational encoder to match q0(u|x); the cited likelihood-encoder construction provides compatible codes under the stated rate inequalities.Source: A. Strong-Perception RDP Coding · pp. 6–7

B. Channel Synthesis and the Full Redesign Benchmark

Full Redesign freely designs the encoder and decoder to synthesize a prescribed finite-alphabet source-output distribution, without retaining any component of the pretrained RDP system.

B. Channel Synthesis and the Full Redesign Benchmark

  • The encoder observes S^n, and the decoder generates X̂^n so that their joint distribution approaches the prescribed transfer target.The target is a product distribution q(s^n, x^n) formed from the finite-alphabet single-letter law q(s, x).Source: B. Channel Synthesis and the Full… · p. 8
  • A synthesis code uses a message and independent shared common randomness, with rates R and R_c, and achievability requires strong synthesis.The encoder and decoder are stochastic; the shared randomness is uniform and independent of S^n.Source: B. Channel Synthesis and the Full… · p. 8
  • Full Redesign is the closure of achievable rate pairs and retains no component of the pretrained RDP system.Its region is characterized in Theorem 1; the supplied passage does not include the region's formula.Source: B. Channel Synthesis and the Full… · p. 8

III. PROBLEM STATEMENT · A. Renderer-Only Transfer · B. Representation-Induced Transfer

The paper studies how a pretrained RDP system can be reused for a prescribed semantic-to-output task while retaining different components. Renderer-Only Transfer retains the renderer but redesigns semantic steering, whereas Representation-Induced Transfer also fixes the latent statistics induced by the pretrained representation.

III. PROBLEM STATEMENT

  • The transfer encoder observes semantic conditions and seeks to synthesize the prescribed joint law of conditions and outputs while retaining selected pretrained components.The paper distinguishes reuse of only the renderer, the renderer plus induced semantic latent law, or an entire compatible RDP code sequence.Source: §III · p. 9

A. Renderer-Only Transfer

  • Renderer-Only Transfer retains the generative renderer and redesigns the condition-to-latent steering, unlike approaches that retain an encoder representation and adapt the decoder.The transfer encoder and latent-generation decoder may be designed to meet the output synthesis criterion through the fixed renderer.Source: A. Renderer-Only Transfer · pp. 9–10
  • Renderer-Only Transfer uses shared common randomness and a message to generate latents, then passes them through the inherited renderer.Its model does not prescribe a latent distribution, and it permits an arbitrary transfer target marginal q(x).Source: §III · p. 9, A. Renderer-Only Transfer · p. 10

B. Representation-Induced Transfer

  • Representation-Induced Transfer fixes the semantic latent law B0 and renderer W0 while redesigning the semantic encoder and latent-generation decoder.B0 represents inherited semantic or class-conditional aggregate latent statistics, not a symbolwise operating law of the original RDP encoder.Source: B. Representation-Induced Transfer · pp. 10–11
  • Representation-Induced Transfer realizes the prescribed target distribution exactly only when the task satisfies its representation-induced compatibility condition.The condition is equation (42) in the paper; when it fails, the Representation-Induced Transfer region is defined to be empty.Source: B. Representation-Induced Transfer · p. 11
  • Whenever the compatibility condition holds, every Representation-Induced Transfer code is also a Renderer-Only Transfer code.Thus the Representation-Induced Transfer region is contained in the Renderer-Only Transfer region for compatible tasks.Source: B. Representation-Induced Transfer · p. 11

C. Full-Model Transfer … 1) Exact Transfer Region:

The section contrasts retaining an operational RDP code with retaining only its renderer, distinguishing guarantees for the actual block implementation from the renderer-constrained channel-synthesis region. Full-Model Transfer fixes the inherited code and its rates, while Renderer-Only Transfer restricts admissible decoder channels to those in the renderer’s output hull.

C. Full-Model Transfer

  • Full-Model Transfer reuses a compatible RDP code sequence, including its encoder, latent codebook, common-randomness mechanism, and renderer, with compression and common-randomness rates fixed rather than optimized for the task.The semantic encoder is determined by the task distribution and original code.Source: C. Full-Model Transfer · p. 11
  • The single-letter RDP test channel does not describe every operational block code; failure of its transfer condition alone does not establish that the actual code fails.The strong RDP criterion constrains output distribution and average distortion, not independent coordinate-wise implementation of the test channel.Source: C. Full-Model Transfer · p. 12
  • Theorem 8 guarantees transfer for every compatible pretrained block code when the single-letter condition holds, and bounds the semantic dependence any transferable task can retain.The frozen implementation is guaranteed to preserve the nominal recurrent-class invariant, while additional capability can depend on compression rate and operational structure.Source: C. Full-Model Transfer · p. 12

IV. MAIN RESULTS

  • The main results progress from redesigned steering with a retained renderer, to retained representation-induced latent statistics, and finally to reuse of an operational RDP code.The final analysis identifies a task-independent transferable family and bounds additional semantic dependence preserved by a particular implementation.Source: §IV · p. 13

A. Renderer-Only Transfer

  • Renderer-Only Transfer retains only W0 while redesigning the semantic encoder, synthesis auxiliary, and condition-to-latent mapping, isolating the cost of retaining the generator.Source: A. Renderer-Only Transfer · p. 13

1) Exact Transfer Region:

  • For finite alphabets, Theorem 2 characterizes the Renderer-Only Transfer region using the usual channel-synthesis architecture but restricting admissible decoder channels.The theorem permits an auxiliary V with |V| ≤ |S||X| + 1.Source: 1) Exact Transfer Region · p. 14
  • Retaining W0 adds no coding resource: each conditional decoder row must lie in its output hull C(W0), separating unrealizable targets from realizable but less rate-efficient factorizations.Source: 1) Exact Transfer Region · p. 14

2) Semantic Controllability: · 3) Price of Transfer Relative to Full Redesign: · 4) Unlimited Common Randomness:

Renderer-Only Transfer is possible exactly when the fixed renderer can realize the target’s conditional output laws, but feasibility need not mean zero compression-rate penalty. With unlimited common randomness, the penalty is the least residual semantic information forced into the renderer input beyond what the output reveals.

2) Semantic Controllability:

  • A target is controllable through the fixed renderer exactly when each requested conditional output law can be generated through its renderer hull.The single-letter criterion is the existence of a semantic-to-latent channel p(u|s) inducing the target through W0.Source: 2) Semantic Controllability · p. 14
  • The renderer can reproduce its unconditional base output law, but this does not guarantee that it can realize every conditional law with that mixture.Unconditional realism and conditional controllability are distinct requirements.Source: 2) Semantic Controllability · p. 14

3) Price of Transfer Relative to Full Redesign:

  • At fixed common-randomness rate, transfer has zero price exactly when some Full Redesign boundary optimizer uses only decoder output laws in the renderer hull.A finite positive price means a compression-rate penalty despite exact feasibility; +∞ denotes lack of controllability.Source: 3) Price of Transfer Relative to Full… · p. 15
  • Zero price at one operating point does not require the inherited renderer to implement every decoder channel; one Full Redesign optimizer within its hull suffices.Source: 3) Price of Transfer Relative to Full… · p. 15

4) Unlimited Common Randomness:

  • For a controllable target with unlimited common randomness, the minimum Renderer-Only Transfer compression rate is the least I(S;U) over feasible renderer-input channels.The unlimited randomness constraint makes the sum-rate constraint inactive.Source: 4) Unlimited Common Randomness · pp. 15–16
  • The unlimited-randomness compression-rate penalty over Full Redesign equals the minimum residual semantic information I(S;U|X) required by a feasible renderer input.Full Redesign requires only I(S;X), while the inherited renderer may force the code to distinguish latent states that become indistinguishable after rendering.Source: 4) Unlimited Common Randomness · pp. 15–16
  • A controllable target has zero unlimited-randomness transfer price exactly when some feasible U carries no semantic information beyond X, so S and U are conditionally independent given X.Source: 4) Unlimited Common Randomness · p. 16

5) Universal No-Loss Renderers: … 1) Operational Region:

Universal no-loss transfer requires a renderer capable of realizing every conditional decomposition of the base output marginal, not merely the marginal itself. Renderer ordering depends on output-hull inclusion, while Representation-Induced Transfer synthesizes a fixed latent target whose compatibility with the prescribed output law is a separate requirement.

5) Universal No-Loss Renderers:

  • Universal controllability requires C(W0) to contain the entire probability simplex on X0, including every point mass, beyond the base perception requirement q0(x) ∈ C(W0).Thus, unconditional perceptual adequacy does not guarantee support for arbitrary conditional-generation tasks with the same output marginal.Source: 5) Universal No-Loss Renderers · p. 17

6) Renderer Ordering:

  • If W1 ⪯in W2, then ΓW2(q) ≤ ΓW1(q); for finite alphabets, this ordering holds exactly when C(W1) ⊆ C(W2).Input simulation means W2 can reproduce W1 by simulating its input, preserving the transfer law and rates.Source: 6) Renderer Ordering · p. 17
  • Finite-alphabet renderers with the same output hull have identical Renderer-Only Transfer regions for every target, even if their latent parameterizations differ.A renderer with a larger output hull cannot be worse for Renderer-Only Transfer.Source: 6) Renderer Ordering · pp. 17–18

B. Representation-Induced Transfer

  • Representation-Induced Transfer retains W0 and the semantic latent law induced by q0(u|x), then designs a new synthesis code for p(s,u) = q(s)B0(u|s).The scheme does not execute q0(u|x) symbol by symbol.Source: B. Representation-Induced Transfer · p. 18

1) Operational Region:

  • With representation-induced compatibility, the minimum compression rate is exactly the rate required to synthesize (S,U).The latent target is q(s)B0(u|s), and applying W0 after synthesis requires no additional rate.Source: 1) Operational Region · p. 18
  • The latent target yields the prescribed output law only when (B0W0)(x|s) = q(x|s); no synthesis code for that fixed target can repair incompatibility.Compatibility ensures that decoder-local postprocessing by W0 produces the prescribed q(s^n,x^n) law.Source: 1) Operational Region · p. 18

2) Basic Invariances Under Matched Marginals: · 3) Exact Transfer and the Recurrent-Class Invariant: · 4) Semantic Information and Universal Interfaces:

Representation-Induced Transfer preserves distortion and the unconditional output law under matched marginals, but exact semantic transfer is limited by the recurrent-class label C of the inherited channel. C is also a rate-optimal interface for exactly transferable tasks and the minimum information every universal exact-semantic interface must retain.

2) Basic Invariances Under Matched Marginals:

3) Exact Transfer and the Recurrent-Class Invariant:

  • Exact transfer can randomize X within a recurrent class but cannot change C; if T0 is irreducible, C is constant, whereas if T0 is the identity, C = X.A task that distinguishes source symbols within one recurrent class is outside the exact-transfer family.Source: 3) Exact Transfer and the… · p. 19, 4) Semantic Information and Universal… · p. 20
  • Representation-Induced Transfer is exact if and only if X and S are conditionally independent given the recurrent-class label C; thus C is the greatest exactly transferable semantic variable.Equivalently, q(s|x) = q(s|c(x)); every exactly transferable semantic variable is a function or stochastic degradation of C.Source: 3) Exact Transfer and the… · p. 20

4) Semantic Information and Universal Interfaces:

  • C itself and its coarsenings are exactly transferable, so a task may use less semantic information than the full recurrent-class label.The three-class example defines S = 1{C = 3}, which is exactly transferable although S differs from C.Source: 4) Semantic Information and Universal… · p. 20
  • Under unlimited common randomness, choosing V = C achieves the optimal compression rate I(S;X) for every fixed exactly transferable task.A particular task may require only a coarser function of C.Source: 4) Semantic Information and Universal… · p. 20
  • H(C) is the maximum exactly transferable semantic information, attained by S = C.Every exactly transferable S satisfies X − C − S, so data processing bounds its information about X by H(C).Source: 4) Semantic Information and Universal… · p. 21
  • C is a minimal universal exact-semantic interface: every interface supporting all exactly transferable tasks must preserve at least the information in C.In the three-class example, two transferable coarsenings together determine C, so an interface for arbitrary future exact-transfer tasks cannot merge recurrent classes.Source: 4) Semantic Information and Universal… · p. 21

5) Optimality When Representation-Induced Transfer Is Exact: … 1) Class Consistency of Compatible Implementations:

Representation-Induced Transfer is rate-optimal when exact and admits distortion-based approximation guarantees. Full-Model Transfer preserves recurrent-class semantics across compatible codes and bounds how much additional within-class semantic dependence a fixed-rate code can transfer.

5) Optimality When Representation-Induced Transfer Is Exact:

  • When Representation-Induced Transfer is exact, its inherited latent law is optimal even relative to Full Redesign under unlimited common randomness.Redesigning semantic steering cannot lower the minimum compression rate, though that representation-level rate may be below the base rate R0 of a particular inherited code.Source: 5) Optimality When… · pp. 21–22

6) Approximate Transfer Guarantees from RDP Distortion:

  • For every matched-marginal target, the unchanged transfer law’s average conditional transport cost is bounded by the base distortion budget.The coupling uses the inherited kernel T0, leaving the renderer unchanged.Source: 6) Approximate Transfer Guarantees… · p. 22
  • Under Hamming distortion, the average conditional total-variation error is small when base distortion is low, even if exact-transfer conditions fail.For general distortion, the guarantee is transport cost; total variation requires an additional relation to Hamming distortion, and the bound is average rather than pointwise or block-level.Source: 6) Approximate Transfer Guarantees… · p. 22

C. Full-Model Transfer

  • Full-Model Transfer keeps the compatible code and its rates fixed, and compatible strong-RDP implementations of the guaranteed task family exist at strict interior rate inequalities.Matched marginals alone do not preserve source-semantic dependence; the theorem supplies additional task conditions for success.Source: C. Full-Model Transfer · p. 22

1) Class Consistency of Compatible Implementations:

  • The latent U, source X, and reconstruction X̃ share the same recurrent-class label almost surely, and this consistency carries coordinatewise to compatible block-code implementations.Each supported latent value has a unique class label.Source: 1) Class Consistency of Compatible… · p. 23
  • Every compatible strong-RDP code preserves the recurrent-class sequence and transfers matched-marginal tasks satisfying X-C-S; additional within-class dependence is bounded by I(S;X|C)≤R0−H(C), with exactness iff X-C-S when R0=H(C).The guaranteed family applies without task-specific modification of the deployed code at strict interior RDP rates.Source: 1) Class Consistency of Compatible… · pp. 23–24
  • A compatible rate-one code can transfer S=X exactly even when C is constant, showing a particular code may preserve within-class information beyond the guaranteed invariant.In the example, I(S;X|C)=R0−H(C)=1, while Representation-Induced Transfer is not exact.Source: 1) Class Consistency of Compatible… · pp. 24–25

V. EXAMPLES AND GAUSSIAN SPECIALIZATION · A. A Controllable Task With a Positive Price of Transfer

This example distinguishes three transfer strategies: Representation-Induced Transfer fails, Renderer-Only Transfer succeeds at a positive price, and Full Redesign ranks above both. The renderer-hull endpoints and the task’s output symbols characterize the example’s transfer constraints.

A. A Controllable Task With a Positive Price of Transfer

B. Same RDP Test Channel, Different Renderer-Only Transfer Capability · C. Scalar Gaussian Specialization

The examples show that an end-to-end RDP test channel can conceal differences in Renderer-Only Transfer capability, while Representation-Induced Transfer follows the semantic dependence induced by the representation. In the scalar Gaussian case, renderer noise sets the controllability threshold and affects transfer price.

B. Same RDP Test Channel, Different Renderer-Only Transfer Capability

  • For S = X, realization A is uncontrollable, while realization B is controllable and has zero Renderer-Only Transfer price at every common-randomness rate.The two realizations induce the same end-to-end test channel, source–reconstruction law, Representation-Induced Transfer map, and recurrent-class invariant.Source: B. Same RDP Test Channel, Different… · pp. 26–27
  • The example shows that the end-to-end RDP test channel determines Representation-Induced Transfer but not Renderer-Only Transfer, leaving the internal factorization operationally relevant.Both realizations are valid RDP factorizations at the common base point, where R0 = 1 is minimum under zero class distortion.Source: B. Same RDP Test Channel, Different… · pp. 26–27

C. Scalar Gaussian Specialization

  • In the scalar Gaussian model, the end-to-end channel determines semantic attenuation, whereas renderer noise determines which semantic conditionals remain reachable.The specialization uses real alphabets, memoryless rendering, i.i.d. sources and targets, and squared-error base distortion.Source: C. Scalar Gaussian Specialization · p. 27
  • A Gaussian target is representable through the renderer if and only if |ρ| ≤ β0; the required mutual information diverges as |ρ| approaches β0.The threshold reflects the renderer’s irreducible noise; the unlimited-common-randomness transfer price tends to zero as β0 approaches 1.Source: C. Scalar Gaussian Specialization · p. 28
  • Representation-Induced Transfer depends on ρ0 = α0β0, and exact transfer is possible only when ρ0 = 1 or for the independent task ρ = 0.For other targets, the induced output has correlation ρρ0 with S.Source: C. Scalar Gaussian Specialization · p. 28
  • For a fixed nontrivial Gaussian target, smaller base distortion preserves more mutual information about S, approaching the target value as D0 decreases to zero.This measures retained dependence; exact transfer still requires preserving the full target distribution.Source: C. Scalar Gaussian Specialization · pp. 28–29
  • Two Gaussian factorizations with the same end-to-end test channel can differ in Renderer-Only Transfer: a target can be infeasible through one renderer yet feasible at finite rate through the other.The shared product αiβi fixes the test channel, but differing βi values yield different renderer controllability thresholds.Source: C. Scalar Gaussian Specialization · p. 29

1) Numerical Illustrations: · VI. CONCLUDING REMARKS

Gaussian examples show that an RDP system’s internal factorization can affect later semantic controllability even when its end-to-end behavior is unchanged. The concluding remarks highlight this design implication and identify approximate transfer as an open problem.

1) Numerical Illustrations:

  • At fixed base distortion D0, increasing the original common-randomness rate Rc,0 enlarges the finite-rate transfer range |ρ| < ρmax through the inherited renderer.The horizontal axis is the original system’s common-randomness rate; larger base distortion makes the renderer less controllable at Rc,0 = 0.Source: 1) Numerical Illustrations · p. 29
  • With ρ0 = 0.5, factorizations with β1 = 0.6 and β2 = 0.9 have different feasible transfer ranges: targets with 0.6 < |ρ| < 0.9 are feasible only through the second renderer.The systems share the same end-to-end RDP test channel, while the larger renderer coefficient also gives a smaller transfer rate for 0 < |ρ| < β1.Source: 1) Numerical Illustrations · pp. 29–30, Figure 2 · p. 30
  • The Gaussian illustrations vary base distortion across D0 ∈ {0.4, 0.8, 1.2, 1.6} to show how original-system resources relate to semantic-correlation controllability.Panel (a) plots the semantic-correlation boundary along the minimum-compression-rate base RDP boundary.Source: 1) Numerical Illustrations · pp. 29–30, Figure 2 · p. 30

VI. CONCLUDING REMARKS

  • The examples show that future semantic controllability depends on internal representation and renderer choices, not solely on the original system’s end-to-end RDP performance.When future reuse is anticipated, the initial design could account for both compression and expected transfer tasks; jointly optimizing them within specified latent interfaces or renderer architectures remains open.Source: §VI · p. 30
  • For Representation-Induced Transfer, exact synthesis requires each target conditional distribution to be preserved by the inherited kernel, so targets outside that invariant class can fail exact transfer.The base distortion bound controls average single-letter transport cost between actual and desired conditional output distributions, but does not alone establish the vanishing block total variation required for strong synthesis.Source: §VI · p. 30
  • An operational theory of approximate transfer remains open: given an allowed semantic mismatch, it should determine which additional tasks are feasible and their required compression and common-randomness rates.This requires connecting the base-distortion transport-cost guarantee for a fixed transfer law to strong synthesis at the block level.Source: §VI · p. 30

APPENDIX A PROOF OF THE RENDERER-ONLY TRANSFER REGION … B. Proof of Theorem 8

These proofs establish the Renderer-Only Transfer rate region and characterize when inherited renderers and RDP codes can support transferred semantic tasks. The key boundary is the recurrent-class label: it is the greatest exactly transferable semantic variable, and it constrains additional within-class dependence under Full-Model Transfer.

A. Achievability

  • Renderer-Only Transfer achieves rates R > I(S;V) and R + Rc > I(S,X;V) using a synthesized channel that operates through the prescribed renderer W0.The construction first selects an effective channel factoring through W0, then implements it by generating a renderer input from V.Source: A. Achievability · p. 31

B. Converse

  • The converse single-letterizes any Renderer-Only Transfer code while preserving the factorization S − V − U − X through W0, yielding the region’s rate bounds.Strong synthesis controls coordinate dependence and permits passage to a limiting law with the exact target marginal.Source: B. Converse · pp. 31–32

APPENDIX B

  • Renderer controllability is equivalent to the renderer producing every point mass on the source support; under this condition, Renderer-Only Transfer and Full Redesign have the same rate region.The proof links controllability to inclusion of the full probability simplex in the renderer hull, then constructs renderer inputs that reproduce each source symbol deterministically.Source: Appendix B · pp. 32–33

APPENDIX C

  • The recurrent-class label C is the greatest exactly transferable semantic variable: every exactly transferable S must satisfy X − C − S, and C itself is transferable.The proof decomposes the stationary source transition law into closed recurrent classes and shows each class has a unique stationary distribution.Source: Appendix C · p. 33

A. Proof of Lemma 2

  • For every renderer input u, the source and renderer-output supports lie within one recurrent class, which yields class preservation between source and reconstruction.This support property is established using stationarity and closure of the source support under the transition law.Source: A. Proof of Lemma 2 · p. 34

B. Proof of Theorem 8

  • Compatible Full-Model codes preserve the recurrent-class label between source and reconstruction, enabling transfer when the task’s semantic condition depends on the source only through C.The proof applies class preservation to the task kernel and uses the base code’s near-perfect perception property to establish the transfer claim.Source: B. Proof of Theorem 8 · pp. 34–35
  • For the case R0 = H(C), the converse forces the conditional mutual information I(S; X|C) to equal zero, equivalently X − C − S; part (ii) proves achievability.Source: B. Proof of Theorem 8 · p. 35

APPENDIX E OPTIMALITY OF THE COMMON BASE RATE · APPENDIX F

Appendix E proves that the common compression rate R0 = 1 is optimal. Appendix F characterizes Gaussian transfer representability by |ρ| ≤ β0 and establishes the corresponding operational rate bounds.

APPENDIX E OPTIMALITY OF THE COMMON BASE RATE

  • R0 ≥ 1 in the limit, proving that the common compression rate R0 = 1 is optimal.The converse uses independent uniform class bits and the Markov relation between the source classes, message and common randomness, and reconstructed classes.Source: Appendix E · p. 36

APPENDIX F

  • |ρ| ≤ β0 is necessary and sufficient for the fixed Gaussian renderer to represent the target, and the representing steering law is unique.Its conditional variance is β0^2 − ρ^2, with zero variance allowed at equality; uniqueness follows from injectivity of Gaussian convolution.Source: Appendix F · p. 36
  • At |ρ| = β0, the steering law is deterministic and I(S; U) = ∞, excluding any finite compression rate.The law reduces to U = (ρ/β0)S almost surely.Source: Appendix F · pp. 36–37
  • For |ρ| > β0, no sequence of Renderer-Only codes can strongly synthesize the transfer target.The limiting characteristic-function modulus would exceed one, which is impossible.Source: Appendix F · p. 36
  • For |ρ| < β0, the operational compression-rate threshold is I(S; U): rates above it are achievable with sufficiently large finite common randomness, and rates below it are excluded.The converse gives lim inf R_n ≥ I(S; U), while the synthesis construction achieves every R > I(S; U).Source: Appendix F · p. 37
  • As common randomness tends to infinity, the Gaussian synthesis region yields (134), and its difference from the rate bound in (133) gives the transfer penalty in (135).The supplied passage identifies these relations but does not state their explicit formulas.Source: Appendix F · pp. 36–37

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