{"id":"1174259e-5668-43ed-8352-fb65f2ce5329","arxiv_id":"1908.02854","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"On variable-exponent sequence spaces, modular-preserving linear maps are shown to be a set map followed by coordinate-wise multiplication by a bounded sequence, while shift-like maps preserve the norm only when the exponent sequence is invariant under the shift.","lead":"A pure-math paper characterizes certain norm-preserving maps on sequence spaces where the exponent used for measuring size changes coordinate by coordinate. It also studies shift-like operators and claims they preserve norms only under restrictive conditions, but one of the key proofs contains a false algebraic step.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7 proof uses false uniqueness of 2^x+2^y=1; non-equal exponent triples satisfy the displayed equation, so the necessity claim is unproved as written.","rationale":"The reader's weakest assumption is exactly the load-bearing flaw. The theorem's necessity proof depends on deriving equality of exponents from 2^x+2^y=1, but that equation's solution set is a curve, not the single point (-1,-1). A local triple of exponent values satisfying the displayed equation with p_j≠p_{θ(j)} shows the inference is invalid. The theorem itself appears likely salvageable through an orbit-decay argument, so this is a proof gap rather than a demonstrated false theorem; nonetheless, as submitted the advertised characterization of S_θ-isometries is unsupported. This confirms the reader's central criticism and does not change the recommendation. Theorem 6 and its proof seem sound and are not the point of failure.","tokens_in":6450,"tokens_out":16155,"duration_ms":186829,"concrete_test":"Run the following analytical check on Theorem 7. Fix an infinite θ-orbit and define q_k=p_{θ^{k+1}(j)}/p_{θ^k(j)}. From the isometry of b_k=2^{-1/p_{θ^k(j)}}e_{θ^k(j)}+2^{-1/p_{θ^{k+1}(j)}}e_{θ^{k+1}(j)}, derive 2^{-q_k}+2^{-q_{k+1}}=1 for all k. Show that if q_0≠1 then q_k alternates between a>1 and h(a)=-log_2(1-2^{-a})<1, with two-step product q_k q_{k+1}<1, so p_{θ^{2m}(j)} tends to 0, contradicting p_n≥1. If this proof succeeds, Theorem 7 is true but the submitted argument must still replace the false uniqueness step. Alternatively, search for a full admissible pair (p,θ) satisfying the recurrence on an infinite orbit with all p_n≥1; any such example would disprove the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The necessity proof of Theorem 7 reduces isometry of the two-point vector b to 2^{-p_{θ(j)}/p_j}+2^{-p_{θ(θ(j))}/p_{θ(j)}}=1 and then invokes the assertion that 2^x+2^y=1 has unique real solution x=y=-1. This assertion is false: x=-2, y=log_2(3/4) gives 1/4+3/4=1. The equation has a one-parameter family of real solutions, all with negative exponents. In fact, the displayed identity alone does not force equality of the p's: taking p_j=3, p_{θ(j)}=6, p_{θ(θ(j))}=6·log_2(4/3)≈2.49 gives 2^{-2}+2^{-0.415}=1 while all p values are ≥1 and p_j≠p_{θ(j)}. Thus the conclusion p_j=p_{θ(j)}=p_{θ(θ(j))} does not follow. A correct proof would need a global orbit argument: iterating the relation forces a nonconstant ratio to alternate and the exponents to decay below 1 along an infinite θ-orbit, contradicting p_n≥1. No such argument is supplied, so the necessity direction of Theorem 7, and hence Corollary 1, is not established by the submitted text. Theorem 6 appears sound; the gap is in the paper's second advertised main result.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies linear norm-preserving operators on variable-exponent discrete Lebesgue spaces ℓ(p_n), under the standing hypotheses that the exponent sequence lies entirely in [1,2) or entirely in (2,∞). Its first main result, Theorem 6, asserts that every isomodular operator has the form (Sx)_n = h(n)(Tx)_n for a regular set isomorphism T and h∈ℓ∞ with |h|≤1. Its second main result, Theorem 7, asserts that for an injective map θ:N→N the induced operator S_θ is an isometry of ℓ(p_n) if and only if p_n=p_{θ(n)} for every n; Corollary 1 then concludes that the unilateral shift is isometric only when the exponent sequence is constant. The proofs use Clarkson-type modular inequalities, density of finitely supported sequences, and a reduction of the shift isometry condition to an exponential equation.","tokens_in":6749,"tokens_out":10971,"duration_ms":118386,"significance":"If both main theorems stand, the paper gives a clean Lamperti-type structure theorem for isomodular operators on variable-exponent sequence spaces and a sharp contrast with the fixed-exponent case. Theorem 6's proof is direct, self-contained, and appears correct; I found no gap in it, and it is a genuine contribution. The proof of Theorem 7, however, rests on a numerically false uniqueness claim, so the second advertised main result and Corollary 1 are not established as submitted. The paper is clearly written and the derivation is not circular; the flaw is a localized algebraic error that a correct global orbit argument may repair.","major_comments":[{"comment":"The necessity argument contains a numerically false uniqueness assertion. After deriving 2^{-p_{θ(j)}/p_j} + 2^{-p_{θ(θ(j))}/p_{θ(j)}} = 1, the text states that 2^x+2^y=1 has the unique real solution x=y=-1. This is false: for example, x=-2 and y=log_2(3/4) give 2^{-2}+2^{log_2(3/4)} = 1/4+3/4 = 1. In fact the equation has infinitely many real solutions. The conclusion p_j=p_{θ(j)}=p_{θ(θ(j))} therefore does not follow from the displayed identity. Concretely, if p_j=3, p_{θ(j)}=6, and p_{θ(θ(j))}=6 log_2(4/3)≈2.49, all exponents are at least 1, the identity holds, and p_j ≠ p_{θ(j)}. Because this step is the entire necessity proof, Theorem 7 and its Corollary 1 are not established as submitted. A repair would require controlling the full θ-orbit of j, for example by iterating the relation and deriving a contradiction with p_n≥1, and no such argument is supplied.","section":"Section 4, proof of Theorem 7"},{"comment":"The two directions of the theorem are mislabeled and the sufficiency direction is not finished. The paragraph beginning 'For necessity' assumes p_{θ(n)}=p_n and computes ||S_θ a||, but it stops without the reindexing step needed to conclude ||S_θ a||=||a||; the paragraph beginning 'For sufficiency' is actually the beginning of the necessity argument. As a result, as written the proof does not prove either direction cleanly, although the sufficiency part is easily completed.","section":"Section 4, proof of Theorem 7"}],"minor_comments":[{"comment":"In the display for ||S_θ a||, the variable in the infimum is written 'θ>0' but should be 'λ>0'.","section":"Section 4"},{"comment":"The proof of the sufficiency direction should explicitly reindex the sum over n∈θ(N) by m=θ^{-1}(n) and use p_{θ(m)}=p_m to identify ||S_θ a|| with ||a||.","section":"Section 4"},{"comment":"The citation marker '[?][Theorem 9.2.12]' is left dangling and must be completed.","section":"Section 3, last paragraph"},{"comment":"The assertion that ||b||=1 'is only possible' because p_j,p_{θ(j)}∈[1,∞) deserves a one-line justification via the strict monotonicity of λ^{-p_j}/2+λ^{-p_{θ(j)}}/2 in λ.","section":"Section 4, Theorem 7 necessity paragraph"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized to the necessity proof of Theorem 7. I am not recommending reject because the theorem may well be true and a global orbit argument appears to be a plausible repair; however, the false uniqueness claim is a serious mathematical error and the second main result must be reproved before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here is my read. The paper has one solid result and one broken proof: Theorem 6 on isomodular operators works, but Theorem 7's necessity argument rests on the false claim that 2^x+2^y=1 has unique real solution x=y=-1.\n\nThe genuinely good part is Theorem 6. On ell(p_n) with all p_n<2 or all p_n>2, any isomodular operator is a weighted composition map: (Sx)_n = h_n (Tx)_n, with T a regular set isomorphism and |h_n|<=1. The proof goes through Clarkson-type inequalities to show orthogonality preservation, then builds T from supports of basis images and h from the images themselves. It is a clean generalization of Lamperti's fixed-exponent theorem, and the isomodular class is a reasonable new notion. The exposition is clear and the literature is handled honestly, aside from one dangling citation marker after Theorem 6.\n\nNow the soft spot, and it is not minor. In the necessity part of Theorem 7, the author derives that isometry of a two-point vector forces 2^{-p_theta(j)/p_j}+2^{-p_theta(theta(j))/p_theta(j)}=1 and then invokes the supposed uniqueness of solutions to 2^x+2^y=1. That uniqueness is false: for example, x=-2 and y=log2(3/4) give 1/4+3/4=1. More importantly, you can choose p_j=3, p_theta(j)=6, p_theta(theta(j))=6*log2(4/3) approx 2.49, all >=1, satisfying the equation while p_j != p_theta(j). So the displayed condition does not force the exponents to be equal. The necessity direction, and with it Corollary 1 on the unilateral shift, is unproved as written. Repairing this would require a global orbit argument; none is supplied. The sufficiency direction is actually the easier half, and the proof labels for 'necessity' and 'sufficiency' are swapped.\n\nSo my bottom line: Theorem 6 is worth publishing; Theorem 7 is not established. I would still send this to a referee rather than desk-reject, because the structural theorem is real and the flaw is concrete enough that a revision might fix it. I would not cite the paper in its current form, and I would only mention it in a reading group as a cautionary example of a plausible-looking but false algebraic step.","headline":"One solid isomodular structure theorem; the shift characterization is built on a false uniqueness claim about 2^x+2^y=1, so Theorem 7 and Corollary 1 are unproved as submitted.","tokens_in":7230,"tokens_out":6028,"would_cite":false,"duration_ms":56827,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46B04","47B37","47C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"On variable-exponent sequence spaces whose exponents lie on one side of 2, isomodular operators are weighted coordinate permutations; shift-type operators are isometries only when the exponents are invariant under the shift.","keywords":["variable-exponent Lebesgue spaces","isometries","isomodular operators","shift operators","sequence spaces","regular set isomorphism","norm-preserving operators","unilateral shift"],"falsifier":"Test the necessity direction of Theorem 7 on two-coordinate vectors: choose an injection $\\theta$ and exponents with $2^{-p_{\\theta(j)}/p_j}+2^{-p_{\\theta(\\theta(j))}/p_{\\theta(j)}}=1$ but $p_j\\neq p_{\\theta(j)}$, which is possible because $2^x+2^y=1$ has solutions other than $x=y=-1$ (for example $x=-2$, $y=\\log_2(3/4)$). If the vector $b=2^{-1/p_j}e_j+2^{-1/p_{\\theta(j)}}e_{\\theta(j)}$ still satisfies $\\|S_\\theta b\\|=\\|b\\|$, the claimed necessity fails; the paper's proof contains no other step that rules this out.","tokens_in":6256,"feed_emoji":"🔁","tokens_out":11860,"duration_ms":113803,"temperature":0.7,"pith_summary":"This paper tries to describe the linear maps that preserve the norm on variable-exponent sequence spaces, where the exponent changes from coordinate to coordinate. Its main structural result says that for exponent sequences confined to one side of 2, any operator preserving the modular sum $\\varrho(a)=\\sum_n |a_n|^{p_n}$ must act by permuting the coordinate axes through disjoint support sets and multiplying each output coordinate by a number of modulus at most one. That recovers the fixed-exponent classification as a special case, since on classical spaces every isometry is modular-preserving. The paper also asks when operators induced by injective maps of $\\mathbb{N}$, including the unilateral shift, are isometric, and claims they are isometric exactly when the exponent is constant along the map's orbits. The upshot is that two flexible-looking operator classes reduce to rigid, checkable conditions on the exponent sequence.","feed_headline":"On variable-exponent spaces, isometries fold into weighted relabelings","feed_subtitle":"For exponents below 2 or above 2, norm-preserving operators become permutations with bounded weights.","key_machinery":"The machinery is the modular $\\varrho(a)=\\sum_n |a_n|^{p_n}$ together with a two-sided inequality for scalar pairs: with exponents on one side of 2, $\\varrho(a+b)+\\varrho(a-b)=2\\varrho(a)+2\\varrho(b)$ holds exactly when $a$ and $b$ have disjoint supports. An isomodular operator preserves this equality case, so it sends orthogonal basis vectors to vectors with disjoint supports; those supports assemble into a regular set isomorphism $T$, and the images of the basis vectors define the multiplier $h$. For shift-type operators, the argument isolates a two-coordinate vector and reduces the norm equality to the equation $2^x+2^y=1$ in the exponents.","core_discovery":"The central claim is that norm geometry on these variable-exponent spaces is controlled by the same combinatorial data as in the fixed-exponent case, provided the exponents stay on one side of 2. Theorem 6 states that if $p_n\\in[1,2)$ for every $n$ or $p_n\\in(2,\\infty)$ for every $n$, and $S$ is isomodular, preserving $\\varrho(a)=\\sum_n |a_n|^{p_n}$, then there is a regular set isomorphism $T$ of $\\mathbb{N}$ and a bounded function $h$ with $(Sx)_n=h(n)(Tx)_n$ and $|h(n)|\\le 1$. Theorem 7 states that the operators $S_\\theta$ built from an injective map $\\theta$ are isometries exactly when $p_n=p_{\\theta(n)}$ for all $n$; in particular the unilateral shift is isometric precisely when the exponent sequence is constant. The paper explicitly leaves open whether every isometry is isomodular, so the first theorem is a conditional structural result while the second is meant to be unconditional.","pith_inferences":["This reader's inference: the false uniqueness step in the proof of Theorem 7 does not disprove the theorem, but it means the paper has not established the claimed characterization of shift isometries; a different argument would be needed.","This reader's inference: a natural next test is the mixed-exponent regime where $p_n$ takes values on both sides of 2, since the equality case of the two-sided inequality fails there and the paper's mechanism for building the set isomorphism no longer applies.","This reader's inference: for surjective isometries, the permutation-plus-multiplier form may hold without the isomodular hypothesis, because surjectivity might force modular preservation; that direction is not pursued in the paper."],"forward_implications":["If Theorem 6 is correct, every isomodular operator on such spaces is a weighted coordinate permutation, fixed by a partition of $\\mathbb{N}$ into disjoint support sets and a pointwise multiplier bounded by 1.","Specializing to constant exponents recovers the classical fixed-exponent isometry structure for $p\\neq 2$ as a special case.","If Theorem 7 is correct, isometric shifts are rare: the unilateral shift is an isometry only when the exponent sequence is constant, and a power of the shift is isometric only for periodic exponent sequences whose period divides the shift length.","Together the theorems give a practical test: compare exponents along orbits of the coordinate map to check whether a shift-type operator is isometric, and check support disjointness and the multiplier bound for an isomodular operator."],"supporting_citations":[{"why":"Supplies the original norm inequalities for exponents above and below 2 whose equality case detects disjoint supports.","marker":"[1]"},{"why":"Establishes the variable-exponent space construction, the norm-modular relation, and density of finitely supported sequences.","marker":"[2]"},{"why":"Provides the fixed-exponent isometry structure theorem and the expanded norm-inequality corollary that the proof generalizes to variable exponents.","marker":"[3]"}],"fun_headline_variants":["Variable-exponent isometries shrink to weighted relabelings","When exponents avoid 2, isometries are weighted permutations","Shift isometries force constant exponent sequences","Isometries on variable-exponent spaces: weighted bijections","One-sided exponents turn isometries into bounded weighted maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the shift-isometry characterization rests on the claim that if $2^x+2^y=1$ then the only real solution is $x=y=-1$; that claim is false, so this part of the paper does not establish its conclusion.","fun_headline_variants_meta":{"raw":{"variants":["Variable-exponent isometries shrink to weighted relabelings","When exponents avoid 2, isometries are weighted permutations","Shift isometries force constant exponent sequences","Isometries on variable-exponent spaces: weighted bijections","One-sided exponents turn isometries into bounded weighted maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00045,"raw_usage":{"total_tokens":2259,"prompt_tokens":926,"completion_tokens":1333,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":1251}},"tokens_in":542,"tokens_out":1333,"duration_ms":13939,"temperature":1.0,"reasoning_tokens":1251,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:33:36.903206+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the necessity direction of Theorem 7 on two-coordinate vectors: choose an injection $\\theta$ and exponents with $2^{-p_{\\theta(j)}/p_j}+2^{-p_{\\theta(\\theta(j))}/p_{\\theta(j)}}=1$ but $p_j\\neq p_{\\theta(j)}$, which is possible because $2^x+2^y=1$ has solutions other than $x=y=-1$ (for example $x=-2$, $y=\\log_2(3/4)$). If the vector $b=2^{-1/p_j}e_j+2^{-1/p_{\\theta(j)}}e_{\\theta(j)}$ still satisfies $\\|S_\\theta b\\|=\\|b\\|$, the claimed necessity fails; the paper's proof contains no other step that rules this out.","supporting_citations":[{"cited_title":"Uniformly convex spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the original norm inequalities for exponents above and below 2 whose equality case detects disjoint supports."},{"cited_title":"Lebesgue and Sobolev Spaces with Variable Exponents","cited_arxiv_id":null,"evidence_quote":"Establishes the variable-exponent space construction, the norm-modular relation, and density of finitely supported sequences."},{"cited_title":"On the Isometries of Certain Function-Spaces","cited_arxiv_id":null,"evidence_quote":"Provides the fixed-exponent isometry structure theorem and the expanded norm-inequality corollary that the proof generalizes to variable exponents."}],"review_version":1}