{"id":"ecfef603-b510-4953-af6d-8988fbf0d08b","arxiv_id":"2502.08502","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The capacity-distortion function for ISAC under logarithmic loss is exactly characterized for binary and Gaussian models, including non-degraded regimes via a state-splitting trick.","lead":"This paper derives the fundamental tradeoff between communication rate and sensing distortion for integrated sensing and communications (ISAC) systems when the sensing receiver outputs a soft estimate scored by logarithmic loss. It gives exact formulas for binary-symmetric and Gaussian channel models, revealing when communication and sensing rates decouple and when they do not.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5 rests on an unproven single-letter characterization for squared error; if (86)-(88) fail, the Gaussian closed-form collapses.","rationale":"The reader's weakest assumption is the binary extremal inequality (112) in Appendix B. I checked the proof of (112): the Lagrangian bound in (113) is valid, the reduction to J(alpha) is correct, and the second-derivative analysis of J(alpha) (including the sign of mu and nu) supports the claimed convexity/concavity of R_B(D). The state-splitting algebra in Theorem 5 is also internally consistent: (99) follows from the Gaussian decomposition, and substituting N2->N1, NS->N2+NS-N1, D->D' into (93) reproduces the original formula exactly. The entropy power inequality step in Appendix C is sound. The actual soft spot is the single-letter characterization (83)-(88) for squared error, which is the foundation of Theorem 5 and is asserted without a proof. Because the headline claim of the paper is this exact Gaussian formula, the missing proof is a load-bearing gap. I would not reject the paper; the gap is likely fillable by adapting standard broadcast-channel converse techniques. But the current manuscript does not contain that proof, so the verdict should be conditional on providing it.","tokens_in":22729,"tokens_out":35615,"duration_ms":324179,"concrete_test":"Derive the converse for (86)-(88) directly: for an arbitrary length-n code with average squared error <= D, define U(t)=(Y1^{t-1},Y2_{t+1}^n), V(t)=Y2^{t-1} as in Appendix A, prove (i) R <= I(X;Y1|U)+I(U;Y2)+delta, (ii) (1/n) sum_t E[(S(t)-hat{S}(t))^2] >= E[var(S|U,V,Y2)], (iii) the distribution p_{UVXY1Y2S} factorizes as p_{UVX} p_{Y1Y2S|X}, then consolidate U,V and take n->infty in the Gaussian limit. If the resulting single-letter program matches (86)-(88), Theorem 5 is confirmed; if an extra auxiliary variable or an additional distortion term survives, formula (93) must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The strongest claim, Theorem 5's closed-form C'_G(D,P), is derived entirely from the single-letter expressions (83)-(88), which are asserted without proof. The text says the characterization 'closely resembles' the log-loss counterpart and cites a 'simple extension' of Corollaries 1 and 2, but those corollaries are proved only for logarithmic loss over finite alphabets. For squared error, the converse in Appendix A cannot be reused verbatim: it relies on H(S|U,V,Y2) as the distortion lower bound, while squared error requires a variance lower bound, and the Gaussian case requires discretization/weak-convergence steps that are only footnoted. The state-splitting trick and the log-loss upper bound both presuppose that any achievable rate satisfies R <= I(X;Y1|U)+I(U;Y2) with var(S|U,Y2) <= D for some U satisfying U--X--(Y1,Y2,S). If the true single-letter form required an additional auxiliary variable (as in C_sym(D) for log loss) or a different distortion constraint, formula (93) would not be the exact fundamental limit. This is a gap, not a demonstrated error, but it is the most load-bearing unproven step in the paper.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the capacity-distortion tradeoff for an integrated sensing and communication (ISAC) system with a communication receiver and a sensing receiver, where the sensing distortion is measured by logarithmic loss (both sequence-wise and symbol-wise). The main results are: (i) lower and upper bounds on the capacity-distortion function C(D), which coincide when the two receiver channels are degraded in either order; (ii) an explicit characterization of C(D) for a binary symmetric channel pair with additive Bernoulli state (Theorem 3); and (iii) Gaussian counterparts, including a closed-form capacity-distortion-power function under squared error distortion obtained via a state-splitting argument (Theorem 5). The paper also develops original extremal inequalities in the appendices.","tokens_in":22939,"tokens_out":10589,"duration_ms":97180,"significance":"If correct, the results constitute substantial progress on the information-theoretic limits of ISAC. The paper provides the first exact characterization of the log-loss capacity-distortion function in degraded broadcast scenarios, resolves a previously open non-degraded binary case, and offers a closed-form expression for the Gaussian squared-error case that was not previously known. The extremal inequalities in Appendices B and C are technically nontrivial and appear to be proven carefully. The connection between log-loss soft estimation and conventional hard estimation is also a useful conceptual contribution. However, the Gaussian squared-error result depends on a single-letter characterization that is asserted without proof, and this is a load-bearing gap that must be addressed before the claims can be fully accepted.","major_comments":[{"comment":"The single-letter characterization of C'_G(D,P) under squared error distortion is asserted without proof. The text states that it 'closely resembles' the log-loss counterpart and footnote 2 refers to a 'simple extension' of Corollaries 1 and 2, but those corollaries are proved only for logarithmic loss over finite alphabets. For squared error, the converse requires a lower bound on the conditional variance rather than the conditional entropy, and the Gaussian case requires discretization and weak-convergence steps that are not given. Since Theorem 5's closed-form formula (93) and the state-splitting argument rest entirely on (86)-(88), this is a load-bearing gap. The authors should either provide a full proof of (83)-(88) (including the continuous-alphabet converse) or give a precise citation of a prior result that establishes it.","section":"Section IV-B, Eqs. (83)-(88)"},{"comment":"The state-splitting equivalence used for the subcase N1 ∈ (N2, N2+NS) is stated in a single sentence: 'In light of (99), the original ISAC system with distortion constraint D is equivalent to the new ISAC system with distortion constraint D''.' Although (99) gives the linear relationship between the average conditional variances of S and S', the proof does not explicitly show that the two capacity-distortion-power functions are the same under the mapping (100), i.e., that achievability in one system implies achievability in the other at the corresponding distortion level. This equivalence is essential for the proof of Theorem 5 and should be elaborated in detail.","section":"Section IV-B, Proof of Theorem 5"}],"minor_comments":[{"comment":"There are multiple typos in the notation paragraph: 'sufficienty' should be 'sufficiently', 'discret' should be 'discrete', and 'contiunous' should be 'continuous'.","section":"Section II, notation paragraph"},{"comment":"The phrase 'adaption of [12, Theorem 1]' should be 'adaptation of [12, Theorem 1]', and the word 'achievability' is misspelled as 'achievability' in several places.","section":"Section III-A, Theorem 1 proof"},{"comment":"The sentence 'the timesharing variable can be obsorbed into the auxiliary random vaiable U' contains two typos: 'obsorbed' should be 'absorbed' and 'vaiable' should be 'variable'.","section":"Section III-B, after Theorem 2"},{"comment":"The caption misspells 'symbol-wise' as 'sybmol-wise'.","section":"Section IV-A, Fig. 2 caption"},{"comment":"The phrase 'average power contraint' should be 'average power constraint', and 'incorprating' should be 'incorporating'.","section":"Section IV-B, Eq. (62) and following text"},{"comment":"Reference [5] has an erroneous leading '5.' in the author list, and reference [3] is missing a closing period after 'detection'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the scope of the journal and the main ideas are original and significant. The principal concern is the unproven single-letter characterization for the squared-error Gaussian case, which is the foundation of Theorem 5. If the authors can supply a proof or a precise citation of a prior proof, I would be willing to reconsider. The state-splitting equivalence should also be described in more detail. The remaining issues are presentation-level and can be fixed during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid information-theory paper with two genuinely new exact results—the binary-symmetric capacity-distortion function under log loss (Theorem 3) and the Gaussian squared-error capacity-distortion-power function (Theorem 5) via a state-splitting trick. The bounds in Theorems 1–2 are extensions of [12] to log loss, but the matching conditions in Corollaries 1–2 are clean, and the binary evaluation is new.\n\nThe state-splitting in Theorem 5 is the cleverest part. Splitting S into S' + S'' so that the effective sensing noise has variance N1 turns the previously open non-degraded case N1 in (N2, N2+N_S) into the already-solved subcase. I checked the algebra around equations (98)–(100) and it works. If the single-letter characterization in (83)–(88) is granted, the proof of Theorem 5 is complete.\n\nThe soft spots are real but not fatal. First, Theorem 1's achievability is only sketched, as an \"adaptation\" of [12, Theorem 1]. For an ISAC audience that is probably fine, but a referee will want the differences spelled out. Second, and more important: the single-letter characterization for the squared-error case, equations (83)–(88), is asserted without proof. The paper calls it a 'simple extension' of Corollaries 1 and 2, but those corollaries are proved for logarithmic loss over finite alphabets. For squared error you need a variance lower bound instead of the conditional-entropy lower bound, plus the Gaussian discretization/weak-convergence argument. The footnote cites standard techniques, but the converse for the squared-error constraint is not actually written down. This is the main load-bearing step for Theorem 5, and it is a gap in the written proof—not a demonstrated error. The achievability direction follows from the same superposition coding, and a standard broadcast-channel converse with U as the time-sharing variable should work, so I expect it is true. The extremal inequality (112) in Appendix B is dense, but I followed the second-derivative analysis and it appears correct; it is the engine for the binary result.\n\nOverall this is a genuine contribution with honest treatment of its open case (Gaussian log-loss for N1 in (N2, N2+N_S)). My recommendation: send to peer review. In revision, require the authors to prove the squared-error single-letter characterization (83)–(88) fully, or to state it as a lemma with a complete converse and explicit Gaussian extension. Without that, Theorem 5 lacks a foundation; with it, the paper is a clean accept. I'd bring it to reading group.","headline":"Exact binary and Gaussian ISAC limits with a clever state-splitting trick; Theorem 5's single-letter starting point is asserted, but the paper deserves a serious referee.","tokens_in":23493,"tokens_out":2661,"would_cite":true,"duration_ms":28724,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A15","94A17","94A24"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives a closed-form capacity-distortion-power function for Gaussian integrated sensing and communications under squared-error distortion, covering the previously open non-degraded regime.","keywords":["integrated sensing and communications","capacity-distortion function","logarithmic loss","broadcast channel","superposition coding","extremal inequality","state estimation","Gaussian channel"],"falsifier":"Fix binary parameters, say $\\beta_1=0.24$, $\\beta_2=0.2$, $\\beta_S=0.1$, and search over all $p_{UX}$ with $|\\mathcal{U}|\\leq|\\mathcal{X}|+1$ and $\\lambda\\geq 0$ to evaluate both sides of (112); a single distribution violating the claimed upper bound would refute Theorem 3. For the Gaussian formula, the same test can be run by numerically maximizing the single-letter expression in (86)–(88) for $N_1\\in(N_2,N_2+N_S)$, e.g. $N_1=2.5$, $N_2=2$, $N_S=1$, $P=1$, and comparing with (93) at several $D$ values.","tokens_in":22511,"feed_emoji":"📡","tokens_out":9120,"duration_ms":83193,"temperature":0.7,"pith_summary":"This paper derives the fundamental tradeoff between how fast a transmitter can send digital messages and how accurately a separate sensing receiver can estimate an environment state, when the sensing receiver is required to output a soft (posterior-like) estimate scored by logarithmic loss. The central object is the capacity-distortion function $C(D)$, and the paper gives matching lower and upper bounds whenever the channel to the communication receiver is degraded relative to the sensing receiver, or vice versa. For two concrete channel models the bounds are evaluated exactly: a binary-symmetric pair with additive Bernoulli state, and a Gaussian pair under squared-error distortion. The Gaussian result is a closed-form capacity-distortion-power function that covers the previously open non-degraded regime in which neither receiver's channel is degraded with respect to the other. If correct, these formulas settle the information-theoretic cost of sensing in those settings and show that partial decoding at the sensing receiver is an emergent feature of the optimum.","feed_headline":"Gaussian ISAC capacity-distortion function derived in closed form","feed_subtitle":"The formula covers the previously open non-degraded regime, giving the rate cost of each sensing accuracy.","key_machinery":"The load-bearing machinery is superposition coding with an auxiliary random variable $U$: the transmitter splits the message into a common part decodable by both receivers and a private part decodable only by the communication receiver, which yields the single-letter rates $I(X;Y_1|U)+I(U;Y_2)$ and the distortion $H(S|U,Y_2)$. Matching upper bounds use additional auxiliary variables and standard converse steps, and when one channel is degraded the bounds collapse to simple maximizations over $p_X$ or $p_{UX}$. Exact evaluation rests on two extremal inequalities: a binary-entropy inequality (112) that controls $I(X;Y_1|U)+I(U;Y_2)-\\lambda H(S|U,Y_2)$, and a Gaussian entropy-power inequality variant (125) used for the Gaussian converse. The Gaussian squared-error result adds a state-splitting trick that splits $S$ into two independent Gaussians and rewrites the non-degraded parameter range $N_1\\in(N_2,N_2+N_S)$ as an equivalent degraded system with effective sensing-noise variance $N_1$, so the previously solved case applies.","core_discovery":"On its own terms, the paper's central claim is that the capacity-distortion function for ISAC under logarithmic loss is characterized by a superposition-coding expression $C(D)=\\max_{p_{UX}}\\min\\{I(X;Y_1),I(X;Y_1|U)+I(U;Y_2)\\}$ subject to $H(S|U,Y_2)\\leq D$, and that this expression is tight in the two degraded orders. The decisive new results are exact evaluations. For the Gaussian channel $Y_1=X+Z_1$, $Y_2=X+Z_2+S$ with independent noises $Z_1\\sim\\mathcal{N}(0,N_1)$, $Z_2\\sim\\mathcal{N}(0,N_2)$, state $S\\sim\\mathcal{N}(0,N_S)$, power constraint $P$, and squared-error distortion, Theorem 5 states that $C'_G(D,P)=\\frac{1}{2}\\log\\left(\\frac{P+N_2+N_S}{N_1 N_S^2}\\left((N_1-N_2)N_S+(N_2+N_S-N_1)D\\right)\\right)$ for $N_1