{"id":"ccd58021-725b-47ab-a7e0-7a8f87ee7e51","arxiv_id":"2602.04833","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Continuous eigenvalues of minimal subshifts are characterized by letter-coboundaries along recognizable, decisive S-adic structures.","lead":"This paper gives new conditions that decide when a minimal symbolic dynamical system has a continuous eigenvalue, using hierarchical S-adic descriptions and letter-coboundaries. It unifies and extends earlier eigenvalue characterizations for substitution and S-adic subshifts, including systems with infinite alphabets.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.12's proof relies on a false eventual-constancy of first-letter maps, leaving a genuine gap in Theorem 5.2; a semigroup/diagonal repair seems likely but is not supplied.","rationale":"After reading the manuscript in good faith, the central claim is the bounded-alphabet/decisive coboundary characterization of continuous eigenvalues (Theorems 5.1–5.3 and Proposition 5.7). The decisive case is handled by Lemma 5.9 without the problematic stabilization. The bounded-alphabet case depends on Lemma 5.12. The proof of that lemma contains the false eventual-constancy assertion about the first-letter maps f_{ℓ,k}. I agree with the reader that this is the weakest load-bearing assumption: it is not a mere typo, because the construction of f_ℓ, f∞, and property (iii) is exactly what yields the stabilized coboundary c used to prove Theorem 5.2(1)⇒(2) and Proposition 5.7. I do not believe this falsifies the theorem; the boundedness of c_n and finiteness of the relevant semigroup make a diagonal/idempotent repair plausible, so the appropriate disposition is CONDITIONAL, matching the reader's verdict. The remaining typos and numbering inconsistencies do not change this assessment.","tokens_in":65112,"tokens_out":16116,"duration_ms":160771,"concrete_test":"Verify the repair of Lemma 5.12 on the alternating example: define τ_{2i}: 0→01, 1→10 and τ_{2i+1}: 0→10, 1→01 with c_n≡0. Check that the displayed \"eventually constant\" claim fails by computing f_{0,k}(0) for k≤8. Then replace that claim with the semigroup argument: choose a subsequence n_l so that all block first-letter maps a↦f(τ_{n_l,n_{l+1}}(a)) are equal to a fixed map h; pass to a further subsequence with h idempotent, and verify that the two conclusions of Lemma 5.12 hold. If this succeeds, Theorem 5.2 stands with a corrected proof; if a bounded-alphabet τ and bounded coboundaries are found where no such idempotent subsequence exists, Lemma 5.12 is false and the proof of Theorem 5.2 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 5.12 is the step that lets Theorems 5.2 and 5.7 pass from level-dependent coboundaries c_n to a single stabilized coboundary c. In its proof, the authors assert: \"For each ℓ≥0 and a∈A, the sequence (f_{ℓ,k}(a):k≥ℓ) is eventually constant.\" This is false for bounded alphabets. The maps f_{ℓ,k}=f(τ_{ℓ,k}()) are right-tail products in the finite semigroup generated by a↦f(τ_n(a)); such products can be periodic. Example: A={0,1}, τ_even: 0→01, 1→10 (first-letter map id), τ_odd: 0→10, 1→01 (first-letter map swap). Then f_{0,k}(0) follows 0,1,1,0,0,1,1,0,... and is not eventually constant. Consequently the construction of f_ℓ, f∞, property (iii), and the subsequent proof of c∘τ_{nℓ,nk}=c and estimate (5.28) is unsupported. This is not a counterexample to Theorem 5.2 itself: since |c_n(a)| is bounded and the coboundary spaces are finite-dimensional, a diagonal/semigroup-idempotent subsequence (e.g., indices whose block first-letter maps are a fixed idempotent) should repair the lemma. But as written, the proof of the bounded-alphabet case of Theorem 5.2 and of Proposition 5.7 has a real gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops S-adic characterizations of continuous additive eigenvalues for minimal subshifts. It introduces letter-coboundaries and a new 'decisiveness' condition, and proves in Section 4 general criteria for primitive recognizable directive sequences (Theorems 4.1--4.3). In Section 5, under decisiveness or bounded alphabets, these criteria are refined to local letter-coboundary conditions (Theorems 5.1--5.3), including a stabilized single-coboundary form for bounded alphabets. The paper also relates coboundaries to extension graphs (Theorem 3.14), derives a tower-measure duality for eigenvalues (Proposition 5.5), characterizes balancedness (Proposition 5.7), and applies the tools to balancedness, factor complexity, Tijdeman's theorem, and several worked examples, including a sharpness example for the necessity of the coboundary criteria.","tokens_in":65499,"tokens_out":12376,"duration_ms":122671,"significance":"If the results hold, this is a substantial extension of Host's coboundary approach to S-adic shifts, covering infinite alphabet rank under decisiveness and providing concrete, testable eigenvalue criteria. The paper is largely self-contained, with detailed proofs, a new structural notion (decisiveness), and useful connections between extension graphs and coboundaries. However, a load-bearing intermediate claim in the proof of Lemma 5.12 is currently false, and this blocks Theorems 5.2 and Proposition 5.7 as written. The likely fix is local, but it must be supplied before the central claims of Section 5 are established.","major_comments":[{"comment":"The proof of Lemma 5.12 relies on the assertion that, for each fixed ell and a, the sequence (f_{ell,k}(a): k>=ell) of first-letter maps is eventually constant. This is false in general: the maps f_{ell,k}=f(tau_{ell,k}(.)) are products in the finite semigroup generated by a |-> f(tau_n(a)), and such products can be periodic. For example, take A={0,1}, tau_0(0)=10, tau_0(1)=01 (first-letter map is the swap), tau_1(0)=01, tau_1(1)=10 (first-letter map is the identity), and alternate. Then f_{0,k}(0) is 1 for odd k and 0 for even k, so it is not eventually constant. The subsequent construction of f_ell, f_infty, property (iii), and the invariance c = c o tau_{n_ell,n_k} used in (5.28) is therefore unsupported. This gap affects Theorem 5.2 and Proposition 5.7, both central claims. A diagonal/idempotent-selection argument over the finite semigroup may repair the lemma, but a complete proof m","section":"Section 5.4, Lemma 5.12"}],"minor_comments":[{"comment":"The internal numbering is inconsistent: statements labelled Theorem 5.1, Theorem 5.2, Theorem 5.3, Theorem 6.2 and Theorem 4.3 are subsequently referred to as 'Lemma 5.1', 'Lemma 5.2', 'Lemma 5.3', 'Lemma 6.2', and 'Lemma 4.3' in the proofs. This makes the paper difficult to read and should be harmonized.","section":"Throughout"},{"comment":"Equation (7.4), L_n(X_sigma x Y) = L_n(X_sigma) x L_{2^n}(Y), is dimensionally incorrect as written; a product subshift over A x A has length-n words with n letters in each coordinate. What is needed and what the construction actually gives is a relation involving L_{2^n}(X_sigma) x L_{2^n}(Y). This appears to be a typo rather than a structural gap, but it should be corrected.","section":"Section 7.1, Eq. (7.4)"},{"comment":"The abstract advertises an application to the Thue--Morse system in the rational base 3/2, but I could not locate this example in the body. Please either include the promised application or amend the abstract.","section":"Abstract"},{"comment":"The notation 'finite alphabet rank' is defined via a subsequence, while Theorem 5.2 and Proposition 5.7 assume bounded alphabets along all n. A sentence explaining that contraction allows passage between these assumptions would help the reader.","section":"Section 2.3, Definition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The only mathematical blocking point I see is the false intermediate claim in the proof of Lemma 5.12. The rest of the paper is coherent and the framework is valuable. I recommend major revision rather than rejection because the repair is plausible; in the revised version I would check the proof of Lemma 5.12 carefully and also verify the abstract/body consistency regarding the rational-base 3/2 example."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is worth your time. It delivers a genuinely useful framework for continuous eigenvalues of minimal subshifts via S-adic representations, introducing decisiveness for directive sequences and a local letter-coboundary calculus. The dimension formula for the coboundary space in terms of extension graphs (Theorem 3.14) is clean, and the applications—linear involutions, the 3/2 rational-base Thue–Morse, bounded discrepancy, and a new proof of Tijdeman’s theorem—are solid illustrations. I agree with the reader that the main theorems are plausible and that the re-derivations of Host and Durand–Frank–Maass are not just restatements.\n\nThe soft spot is real. Lemma 5.12 asserts that after relabeling, the first-letter maps f_{ℓ,k}(a) are eventually constant in k. That is false in general, as the stress-test example with alternating two-letter substitutions shows. The lemma is load-bearing for Theorem 5.2 and Proposition 5.7: it is how the level-dependent coboundaries c_n get stabilized into a single c. As written, the proof of the bounded-alphabet case has a gap. I do not think this sinks the paper. The convergence hypotheses give you boundedness of c_n, finite-dimensional coboundary spaces, and a compactness argument; a diagonal/semigroup-idempotent subsequence should repair the lemma. But the repair has to be supplied. The rest of Section 5 depends on this step, so the authors need to address it head-on, not as a minor typo.\n\nMinor issues: a few numbering inconsistencies (Theorem vs Lemma labels) and typos, but nothing that affects the mathematics.\n\nVerdict: definitely send to peer review. The new machinery and the examples are worth refereeing, and the gap is a fixable local flaw rather than a fundamental error. I would not desk-reject. For my own work, I’d cite the decisiveness notion and the extension-graph dimension result even before the gap is patched. Reading group: I’d bring it up, especially to discuss whether the diagonal repair actually goes through.\n\nBest.","headline":"Substantial S-adic eigenvalue paper with a genuine but repairable proof gap in Lemma 5.12; the central theorems are plausible and the new machinery is worth engaging.","tokens_in":65981,"tokens_out":1861,"would_cite":true,"duration_ms":21078,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B10","37A30","68R15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Continuous eigenvalues of minimal S-adic subshifts are characterized by letter-coboundaries under decisiveness or bounded alphabet rank.","keywords":["S-adic subshifts","continuous eigenvalues","letter-coboundaries","extension graphs","decisiveness","symbolic discrepancy","balancedness","Kakutani-Rohlin towers"],"falsifier":"Take A={0,1} and alternate the substitutions τ0(0)=01, τ0(1)=00 and τ1(0)=10, τ1(1)=00 at every level; compute the first-letter maps f_{0,k}(a) = first letter of τ0∘τ1∘... applied to a. If these maps do not become eventually constant in k for both letters, Lemma 5.12's stabilization claim fails for a bounded-alphabet primitive recognizable sequence, and the proof of Theorems 5.2/5.7 must be repaired or replaced, even if the eigenvalue characterization itself survives.","tokens_in":65020,"feed_emoji":"🔁","tokens_out":6638,"duration_ms":65478,"temperature":0.7,"pith_summary":"This paper seeks to characterize exactly which real numbers are continuous eigenvalues of a minimal symbolic dynamical system when the system is presented as an S-adic subshift—an infinite composition of substitutions. The central answer is that, for a primitive recognizable S-adic structure that is either decisive or has bounded alphabets, a number α is an eigenvalue precisely when the tower heights of the Kakutani–Rohlin partitions approach α modulo 1 after being corrected by a letter-coboundary at every level. This matters because it turns an analytic spectral question into a finite combinatorial one: checking how height functions behave on the language of the subshift, or equivalently on return words. The same machinery yields eigenvalue groups expressed as integer combinations of tower-base measures, a criterion for letter-balanced substitutions, and rational-only eigenvalues for constant-length finite-rank directive sequences.","feed_headline":"Coboundaries decide which frequencies are continuous eigenvalues","feed_subtitle":"For decisive or finite-rank S-adic systems, checking return words tells you the full continuous spectrum.","key_machinery":"The letter-coboundary is the central object: a morphism c:A*→R vanishing on every return word to a letter, equivalently satisfying c(a)=ρ(b)−ρ(a) whenever ab lies in the language. It converts the global eigenvalue equation into local linear conditions on adjacent letters. The extension graph Γ_X(ε), whose vertices are left and right copies of the alphabet and whose edges are length-2 words, encodes all letter-coboundaries: their real vector space has dimension r−1 for r connected components. Decisiveness is the third load-bearing tool: it says the first/last letter of τ_n(b) is constant on connected components of the level-(n+1) extension graph, which makes the Kakutani-Rohlin partitions gen","core_discovery":"The paper's discovery is that the obstruction to a height sequence h_n converging to α modulo 1 is always a sequence of letter-coboundaries, provided the S-adic presentation is decisive or of finite alphabet rank. Letter-coboundaries are morphisms that vanish on return words, equivalently functions of the form c(a)=ρ(b)−ρ(a) on adjacent letters; they capture exactly the oscillatory corrections that cannot be removed by changing the approximating eigenfunction. Under decisiveness, Theorem 5.1 gives the equivalence sup_u ||c_n(u)−α h_n(u)||→0; under bounded alphabet rank, a single coboundary at one level suffices (Theorem 5.2), and under a strong local recurrence hypothesis a summable version","pith_inferences":["A natural testable extension is an algorithmic eigenvalue sieve: for finite-rank S-adic systems, enumerate candidate values from integer combinations of tower-base measures and check the coboundary criterion on finitely many levels; the paper's examples suggest this is feasible even for non-proper substitutions.","The letter-coboundary dimension theorem suggests a conjugacy-independent spectral invariant: although the extension graph of the empty word is not conjugacy-invariant, the dimension of the coboundary space it controls may interact with stable subspaces in a way that gives balance and eigenvalue obstructions for morphic systems.","The constant-length result implies that any primitive recognizable constant-length S-adic subshift with an irrational continuous eigenvalue must have unbounded alphabet rank, a statement that can be checked against existing examples."],"forward_implications":["For any minimal subshift with a primitive recognizable S-adic structure that is decisive or has bounded alphabet rank, eigenvalues are exactly integer linear combinations α=Σ w_a μ(B_n(a)) of tower-base measures for large n, refining the known inclusion E(X)⊆I(X).","Constant-length directive sequences of finite alphabet rank r have only rational eigenvalues, of the form p/(q|τ_{0,n}|) with 1≤q≤r.","A primitive aperiodic substitution generates a letter-balanced subshift exactly when its stable subspace plus its coboundary space has codimension 1, giving a new characterization of letter balance.","The extension-graph counting argument recovers the factor-complexity lower bound p_X(n) ≥ (n−1)(t−1)+|A| for transitive subshifts, with equality forcing all extension graphs to be trees.","Under any of the sufficient conditions (C1)-(C5), the eigenvalue criterion reduces to the classical tall-tower condition ||α h_n(u)||→0 and gives an eigenvalue representation as an integer vector on tower bases."],"fun_headline_variants":["Coboundaries unlock continuous spectrum in minimal subshifts","How letter coboundaries determine continuous eigenvalues","Coboundaries are the key to S-adic spectral classification","Continuous eigenvalues: coboundaries decide in S-adic systems","Letter coboundaries govern continuous spectrum of minimal subshifts"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The bounded-alphabet stabilization lemma—that after relabeling, the first-letter maps f_{ℓ,k}(a) eventually become constant in k for each level ℓ and letter a—is load-bearing; if it fails, the reduction of infinitely many level coboundaries to a single letter-coboundary in Theorems 5.2 and 5.7 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Coboundaries unlock continuous spectrum in minimal subshifts","How letter coboundaries determine continuous eigenvalues","Coboundaries are the key to S-adic spectral classification","Continuous eigenvalues: coboundaries decide in S-adic systems","Letter coboundaries govern continuous spectrum of minimal subshifts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":2962,"prompt_tokens":741,"completion_tokens":2221,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":2148}},"tokens_in":485,"tokens_out":2221,"duration_ms":16064,"temperature":1.0,"reasoning_tokens":2148,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T04:26:47.421446+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take A={0,1} and alternate the substitutions τ0(0)=01, τ0(1)=00 and τ1(0)=10, τ1(1)=00 at every level; compute the first-letter maps f_{0,k}(a) = first letter of τ0∘τ1∘... applied to a. If these maps do not become eventually constant in k for both letters, Lemma 5.12's stabilization claim fails for a bounded-alphabet primitive recognizable sequence, and the proof of Theorems 5.2/5.7 must be repaired or replaced, even if the eigenvalue characterization itself survives.","supporting_citations":[],"review_version":1}