{"id":"d232a183-e4f7-4ea2-b208-2cfb0e9d3427","arxiv_id":"2608.04068","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A tail-calibrated soft-output GRAND decoder for finite-memory noise posteriors is introduced, with proven ML, unbiased missing-list estimation, and correlation-aware gains in simulations.","lead":"This paper builds a noise-guessing decoder that works on channels with memory, such as bursty noise or fading, and gives both reliability scores and an automatic stopping rule. It could help low-latency receivers decode short messages without the delay of interleaving.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The missing-list estimator's unbiasedness is proven only for uniform random codebooks, while the main experiments use random linear codes; the unmeasured bias from linear-subspace coupling is the load-bearing gap.","rationale":"The reader's weakest-assumption diagnosis is correct and matches the most load-bearing point: the central estimator's guarantee is proved for a uniform random codebook while the demonstrated setting is a random linear code. I read the paper in good faith: Theorems 5.1, 5.2, 5.4, and 5.5 are internally consistent within their stated assumptions, the finite-state recursion for the partition function and tail masses is standard and correct, and the paper is unusually explicit in Remark 5.3 about the gap. The issue is not internal inconsistency but extrapolation: the numerical section transfers a proven conditional-mean identity to an ensemble where membership indicators are coupled by linearity. The specific coupling via span(L_q) is a concrete mechanism that could produce bias, though its magnitude in the tested regimes is unknown. The proposed test would settle whether the bias is material without requiring a new theoretical result. Since the reader's CONDITIONAL verdict already requests additional analysis of exactly this issue, my stress-test does not move the verdict; it sharpens the condition. No code or data are released, so the numerical results remain unverifiable independently, but that is a reproducibility concern rather than the single most load-bearing technical gap.","tokens_in":26668,"tokens_out":13418,"duration_ms":140276,"concrete_test":"In the Section 6.4 RLC calibration setup with [64,52], instrument the decoder at each stopping index q to compute d_q = dim span(L_q) by Gaussian elimination over F_2, and compute the ambient posterior mass W_q = sum_{z not in A_q, y xor z in span(L_q)\\L_q} pi(z) by enumerating the span (feasible at qmax = 10000 because d_q is small). Compare W_q with p_q T_q over the frames used in Fig. 5. If W_q exceeds 10% of p_q T_q on a non-negligible fraction of frames, the forced-in span codewords alone make the uniform-sampling approximation materially biased and the RLC calibration claims lack support; if W_q is negligible, the RLC coupling is not the bottleneck and the approximation is reasonable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The stopping rule and soft outputs in Sections 4.4-4.5 rest on the estimator bU_q = p_q T_q, whose unbiasedness is Theorem 5.2. That theorem is proved for a uniformly random M-subset codebook: conditional on the queried transcript, the remaining codewords form a uniform K-subset of the N unqueried candidate words. The main numerical results in Sections 6.2-6.4, however, use binary random linear codes, and Remark 5.3 explicitly concedes that the conditional uniformity behind the proof does not hold for an RLC, so p_q T_q is only an approximation there. The concrete failure mechanism is linear-subspace coupling: with L_q the set of detected codewords, every word in span(L_q) is forced to be a codeword with probability 1, whereas p_q = (M-|L_q|)/(2^n-q) assigns it a much smaller probability. Consequently U_q contains a deterministic contribution from span(L_q)\\L_q that the plug-in estimator does not model, and the variance and concentration guarantees of Theorem 5.2 do not transfer. Because the advertised missing-list calibration, tail-stopping behavior, and bitwise APP estimates in the demonstrated RLC setting all depend on this estimator, the unquantified bias of the approximation is the most load-bearing concern. The paper is honest about the limitation, but that does not by itself bound the bias in the experiments; without such a bound, the central numerical claims are point estimates supported by an unproven approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces Tail-Calibrated SOGRAND, a GRAND variant for binary additive channels with finite-memory noise-effect posteriors. The decoder orders candidate noise effects by a finite-memory posterior energy, queries codebook membership, computes the partition function and ambient tail masses by finite-state forward/backward recursions, and estimates the unqueried codebook-restricted denominator by p_q T_q, where p_q=(M-|L_q|)/(2^n-q) is the remaining random-codebook occupancy probability and T_q is the ambient tail mass. This estimate feeds a missing-list stopping rule, blockwise APPs, and bitwise APP LLRs. The paper proves that exact enumeration yields an ML codeword (Theorem 5.1), that p_qT_q is conditionally unbiased with variance and concentration bounds under the fixed-size random-codebook model (Theorem 5.2), a perturbation bound for the plug-in missing-list estimate (Theorem 5.4), and an ambient-tail abandonment error bound (Theorem 5.5). Numerical evaluations compare the exact finite-memory ordering against ORBGRAND, ORBGRAND-AI, and block-product ablations in Gauss-Markov and hard-decision Markov/Gilbert-Elliott noise, and validate missing-list calibration in a fixed-size random-codebook experiment and in an RLC experiment.","tokens_in":26846,"tokens_out":13466,"duration_ms":127792,"significance":"Taken at face value, the random-codebook analysis is the paper's cleanest contribution: Theorem 5.2 gives an exact conditional-mean identity and computable variance and concentration bounds for the missing-list denominator estimator, and the finite-state computation of posterior weights and tail masses is useful for GRAND with correlated noise. The ML property and ambient-tail abandonment bound are elementary but correctly stated. The numerical section reports a consistent ordering in BLER and membership-query counts favoring the matched finite-memory enumerator, and the fixed-size random-codebook calibration (Table I) supports the occupancy model. The main limitation is that the supporting theory for tail calibration is proved for uniformly random M-subset codebooks, while the headline experiments use random linear codes; the paper discloses this in Remark 5.3 but does not quantify the resulting bias. This does not affect the first-hit BLER and membership-query comparisons, but it does leave the tail-stopping and soft-output claims in the demonstrated setting without a theoretical guarantee.","major_comments":[{"comment":"The advertised missing-list calibration, tail-stopping rule, and bitwise APP estimates are used in the main random-linear-code experiments, but Theorem 5.2's conditional-mean, variance, and concentration guarantees are proved only for the fixed-size uniformly random M-subset codebook. Remark 5.3 concedes that conditional uniformity does not hold for a random linear code and that p_q T_q is used only as an approximation. Because the linear-subspace structure forces every word in span(L_q) to be a codeword with probability one while p_q assigns a much smaller occupancy probability, the bias of bU_q in the demonstrated setting is unquantified. The first-hit BLER and membership-query comparisons do not depend on bU_q, but the tail-stopping and soft-output claims (Sections 4.4-4.5, Fig. 5, Fig. 6, Table VI) do. Please either extend the analysis to RLCs, even under a restricted ensemble or with a worst-case bias bound, or explicitly rescope the theoretical claims so that all RLC tail-calibration statements are labeled as heuristic and are accompanied by a quantified empirical bias characterization over the reported operating points.","section":"Remark 5.3, Sections 4.3-4.5 and 6.2"},{"comment":"Theorems 5.2 and 5.4 are stated for a fixed query index q and a fixed transcript event E_q(ell_q), whereas Algorithm 1 and the calibration experiments evaluate bPmiss at the random stopping time q_eta (Algorithm 1, line 18; Section 6.4.2). The proof of Theorem 5.2 in fact goes through conditional on any event that fixes the intersection C^n cap B_q, so the stopping-time selection is not fatal; however, the manuscript should state this explicitly. As written, a reader cannot tell whether the advertised tail-stopping guarantee is meant to cover the random stopping time or only fixed query indices. Please add a remark clarifying that the unbiasedness and relative-error bound hold conditionally on the transcript, including the event {q_eta = q}, and discuss the nonlinearity of f(u) = u/(S_q + u) when converting the unbiasedness of bU_q into a calibration statement for bPmiss at the stopping time.","section":"Theorem 5.2, Theorem 5.4, Algorithm 1, Section 6.4.2"}],"minor_comments":[{"comment":"The expression 'z^n = L^m_{j=1} 1_{B_j}' should use a sum or disjoint-union symbol rather than 'L', because the intervals B_j are disjoint and the sum of indicators is the standard notation for the union of runs.","section":"Section 4.2.1, Eq. (23)"},{"comment":"The zero BLER entries are plotted at 1/(2N) but printed as 0 in Table IV; please add the Wilson upper endpoint in the table or in a footnote so that the finite-sample uncertainty of those zero-error observations is visible.","section":"Table IV and Fig. 3"},{"comment":"The theorem statement uses P_full^e(n) and P_ab^e(n) without defining them; please define them as the block error probabilities of full and abandoned GRAND, respectively, in the statement itself.","section":"Theorem 5.5"},{"comment":"The software-work proxy C_sw assigns unit weights to membership tests, metric evaluations, priority-queue removals, and preprocessing operations; please state whether the qualitative conclusions from Fig. 2 (bottom right) are robust to reasonable alternative weightings.","section":"Section 6.2, software-work proxy"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest and well organized, and the random-codebook analysis is a solid contribution. The main issue is the gap between the random-codebook theory and the RLC experiments used for the headline tail-stopping and soft-output claims. If the authors can either bound the RLC approximation bias or clearly rescope the claims, I would be willing to accept a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a real extension: it adds finite-memory posterior tail computations and bitwise marginals to SOGRAND, and proves a clean conditional-mean/variance/concentration result for the p_q T_q missing-list estimator under a uniform random-codebook model. Second, the main numerical claims live in exactly the setting where that estimator is not proven: random linear codes. The paper says so in Remark 5.3, but it doesn't bound the bias.\n\nWhat is genuinely good: the finite-state recursions for the partition function and ambient tail mass are standard but applied cleanly to GRAND lists; Theorem 5.2 is a correct occupancy argument; Theorem 5.4's perturbation bound on the plug-in missing-list estimate is a nice practical touch. The experiments are careful about what they measure: membership-query counts versus a software-work proxy, Wilson intervals in plots, mismatch sweeps, and a separate validation in the fixed-size random-codebook ensemble where the theory actually applies. The author is strikingly honest, not just in Remark 5.3 but also in the zero-BLER caveats and the model-mismatch table.\n\nThe soft spot, as your stress-test note says, is the linear-subspace coupling. Conditioning on the observed transcript of an RLC does not make the remaining membership events exchangeable; span(L_q) contributes deterministically to the unqueried denominator. So the calibration results in Sections 6.2–6.4 rest on the approximation, and the theory in Section 5 does not cover them. The Table I validation helps, but it is in the uniform-ensemble, not the RLC setting. Secondary issues: no code or data release, and the tables give point estimates without error bars, so the fine-grained BLER ordering in Table II is only as good as the Monte Carlo.\n\nThis is for GRAND-family researchers, especially those working on soft-output and iterative decoding of product/GLDPC codes. It deserves a serious referee; the right outcome would be a conditional accept asking for a code release and some treatment of the RLC gap—either a bias bound under linear-subspace structure or a convincing demonstration that the approximation holds there. The core theory is sound, the limitations are stated, and the problems are addressable.","headline":"Solid, honest extension of SOGRAND to finite-memory posteriors, with clean bounds for a random-codebook estimator; the main gap is that the headline RLC experiments use that estimator as an approximation with no bias bound.","tokens_in":27551,"tokens_out":2738,"would_cite":true,"duration_ms":26897,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B35","94A15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that the missing-list probability in GRAND decoding can be estimated as the ambient posterior tail mass times a random-codebook occupancy factor, and that under exact enumeration the first listed codeword is…","keywords":["guessing random additive noise decoding","soft-output decoding","channels with memory","posterior tail probability","random coding","APP log-likelihood ratio","finite-state recursion","missing-list probability"],"falsifier":"Take a small random linear code and a fixed finite-memory posterior, enumerate all unqueried noise effects to compute the exact U_q for many transcripts, and compare the average with p_qT_q; a systematic deviation beyond the variance bound of Theorem 5.2 would show the random-codebook identity does not transfer to the linear-code setting.","tokens_in":26309,"feed_emoji":"📡","tokens_out":7477,"duration_ms":66231,"temperature":0.7,"pith_summary":"This paper introduces a decoding rule that gives soft outputs—reliability estimates and log-likelihood ratios—for GRAND-style noise-guessing decoders when the noise process has memory. The key move is to separate the ambient posterior over all possible noise effects, modeled by a finite-memory energy, from the codebook-restricted denominator that enters the a posteriori probability (APP). After q queries, the unqueried part of that denominator is estimated as the ambient posterior tail mass times the occupancy probability of the remaining random codebook, p_q=(M-|L_q|)/(2^n-q). Under exact enumeration the first listed codeword is ML, and the same tail decomposition yields blockwise and bitwise APP estimates. This matters because it extends soft-output GRAND from memoryless channels to correlated channels without interleaving and replaces a fixed query limit with a principled, tail-calibrated stopping rule.","feed_headline":"GRAND's missing-list probability equals a tail times p_q","feed_subtitle":"A finite-memory decoder makes the first hit ML and calibrates soft-output LLRs on channels with memory.","key_machinery":"The load-bearing object is the finite-memory posterior energy E_{r^n}(z^n) = sum_i alpha_i(z_i;r^n) + sum_i beta_i(z_{i-1},z_i;r^n), normalized by a partition function Z(r^n) computed by a finite-state forward recursion; a forward-backward recursion gives full-space bit masses, and subtracting queried-prefix contributions yields ambient tail bit masses. The random-codebook correction p_q=(M-|L_q|)/(2^n-q) turns the ambient tail mass T_q into the estimator bU_q=p_qT_q for the unqueried denominator contribution, and the same tail decomposition yields bitwise APP LLRs. The burst-atom decomposition of binary patterns into separated runs provides an additive ordering cost, so candidate noise effects can be enumerated in nondecreasing energy by a finite-state search.","core_discovery":"With exact nondecreasing-energy enumeration and no abandonment, the first codeword returned by the membership-query sequence is maximum-likelihood for the likelihood model that defines the posterior energy. Separately, under a fixed-size random-codebook ensemble, after q queries the conditional expectation of the unqueried codebook-restricted denominator contribution U_q equals p_q T_q, where T_q is the ambient posterior mass of the unqueried noise-effect tail and p_q is the fraction of remaining codewords among remaining candidate words; variance and concentration bounds follow from the effective support size of that tail. The decoder therefore stops when the plug-in missing-list estimate bU_q/(S_q+bU_q) falls below a tolerance, and it uses the same posterior-tail decomposition to form blockwise APP estimates and bitwise APP log-likelihood ratios under a fixed 0-over-1 convention. In the tested Gauss-Markov and binary Markov random-linear-code settings, the matched finite-memory ordering gives lower BLER point estimates and fewer average membership queries than the memoryless and approximate-independence orderings tested, at the price of more enumeration work in the software implementation.","pith_inferences":["Because the conditional-mean identity is proved for uniformly random codebooks, applying p_qT_q to random linear codes is an unproven transfer; a code-aware correction that exploits subspace structure could improve calibration further.","The enumeration bottleneck at low SNR suggests the practical value of the method will hinge on faster exact or approximate enumerators for finite-memory energies; if such enumerators appear, the same posterior-tail machinery should scale to longer blocklengths and iterative loops.","The bitwise tail-marginal decomposition could also support list-quality metrics beyond missing-list probability, such as per-position reliability for erasure or retransmission decisions.","A direct small-n experiment comparing p_qT_q with the true conditional mean under uniform-sampling and random-linear ensembles would quantify when linear-code membership coupling matters."],"forward_implications":["With exact energy-ordered enumeration and no abandonment, decoding is ML for the model that defines the posterior energy, so correlated-channel ordering need not sacrifice finite-block optimality.","A decoder that stops when bU_q/(S_q+bU_q) is at most eta keeps the true missing-list probability within the perturbation bound (1+epsilon)eta/(1+epsilon*eta) whenever the estimator's relative error is at most epsilon.","The same posterior-tail decomposition supplies blockwise APP estimates and bitwise APP LLRs, enabling soft-output exchange in iterative decoding of product and GLDPC codes.","If the abandoned ambient posterior tail mass is at most epsilon_n, the abandoned decoder's block-error probability exceeds full GRAND's by at most epsilon_n, so rates below the noise entropy rate remain achievable.","In the reported Gauss-Markov and binary Markov experiments, the matched finite-memory ordering lowers BLER point estimates and average membership-query counts relative to the tested ORBGRAND-AI and block-product orderings."],"supporting_citations":[{"why":"Supplies the GRAND membership-query decoding principle and the ML property of the first hit under exact noise-effect ordering, which Theorem 5.1 extends to finite-memory posteriors.","marker":"[10]"},{"why":"Supplies the SOGRAND random-codebook APP and missing-list estimation scheme whose denominator calculation this paper extends to finite-memory posterior tails.","marker":"[14]"},{"why":"Provides the ORBGRAND reliability-ordered enumeration and the membership-query complexity model used as a baseline and evaluation metric.","marker":"[7]"},{"why":"Provides the ORBGRAND-AI local-block approximate-independence ordering that the numerical experiments compare against.","marker":"[9]"},{"why":"Gives the finite-state forward-backward recursion used to compute full-space bit masses and tail bit masses.","marker":"[35]"},{"why":"Supplies the K-shortest-path/list-Viterbi enumeration procedure used to produce the exact energy-ordered sequence on the finite-state trellis.","marker":"[34]"},{"why":"Motivates the finite-memory noise-effect model by showing correlated noise can be exploited without interleaving.","marker":"[8]"}],"fun_headline_variants":["Finite-memory GRAND: first hit is ML, tails calibrate LLRs","Tail-calibrated GRAND beats correlation-blind decoders","Memory-aware GRAND makes soft-output exact via tail masses","GRAND with tails: finite-memory posterior, ML first hit","Soft-output GRAND improved: tail mass controls missing-list and LLRs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that unqueried candidate words behave like a uniformly random subset; the paper's main experiments use random linear codes, where that premise is known to fail and p_qT_q is used only as an approximation.","fun_headline_variants_meta":{"raw":{"variants":["Finite-memory GRAND: first hit is ML, tails calibrate LLRs","Tail-calibrated GRAND beats correlation-blind decoders","Memory-aware GRAND makes soft-output exact via tail masses","GRAND with tails: finite-memory posterior, ML first hit","Soft-output GRAND improved: tail mass controls missing-list and LLRs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1473,"prompt_tokens":1107,"completion_tokens":366,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":723,"completion_tokens_details":{"reasoning_tokens":272}},"tokens_in":723,"tokens_out":366,"duration_ms":3694,"temperature":1.0,"reasoning_tokens":272,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:46:45.636759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small random linear code and a fixed finite-memory posterior, enumerate all unqueried noise effects to compute the exact U_q for many transcripts, and compare the average with p_qT_q; a systematic deviation beyond the variance bound of Theorem 5.2 would show the random-codebook identity does not transfer to the linear-code setting.","supporting_citations":[{"cited_title":"Capacity-achieving guessing random additive noise decoding,","cited_arxiv_id":null,"evidence_quote":"Supplies the GRAND membership-query decoding principle and the ML property of the first hit under exact noise-effect ordering, which Theorem 5.1 extends to finite-memory posteriors."},{"cited_title":"Soft-output (so) grand and iterative decoding to outperform LDPC codes,","cited_arxiv_id":null,"evidence_quote":"Supplies the SOGRAND random-codebook APP and missing-list estimation scheme whose denominator calculation this paper extends to finite-memory posterior tails."},{"cited_title":"Ordered reliability bits guessing random additive noise decoding,","cited_arxiv_id":null,"evidence_quote":"Provides the ORBGRAND reliability-ordered enumeration and the membership-query complexity model used as a baseline and evaluation metric."},{"cited_title":"Using channel correlation to improve decoding-ORBGRAND-AI,","cited_arxiv_id":null,"evidence_quote":"Provides the ORBGRAND-AI local-block approximate-independence ordering that the numerical experiments compare against."},{"cited_title":"Optimal decoding of linear codes for minimizing symbol error rate (corresp.),","cited_arxiv_id":null,"evidence_quote":"Gives the finite-state forward-backward recursion used to compute full-space bit masses and tail bit masses."},{"cited_title":"Keep the bursts and ditch the interleavers,","cited_arxiv_id":null,"evidence_quote":"Motivates the finite-memory noise-effect model by showing correlated noise can be exploited without interleaving."}],"review_version":2}