{"id":"4919036e-5f8f-4482-8f25-d79850f45fef","arxiv_id":"2507.10108","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A lambda-algebra linear-algebra algorithm gives explicit Singer-transfer preimages for c0, d0, p0, a claimed refutation of a published d0 proof, and an automated G4-invariant basis tool.","lead":"This paper reduces the search for preimages under the Singer algebraic transfer to exact linear algebra over F2 and exhibits explicit preimages for three known Ext classes, including a first explicit preimage for p0. It also argues that a published proof about d0 is invalid and ships a large SageMath framework for G4-invariant computations that re-verifies the author's earlier manual rank-4 results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"p0 preimage is not shown to be primitive: the paper verifies only Sq^1,Sq^2,Sq^4,Sq^8 on a degree-33 element, omitting Sq^16 and Sq^32 required by its own Remark 3.2.","rationale":"The reader's weakest_assumption focused on whether Corollary 2.3 and recursion (4) are exactly the chain-level Singer transfer models, with misquotation or misindexing voiding all witnesses. That is indeed a foundational concern, but the paper gives explicit formulas and enough printed computation to make the transfer model checkable. The more immediate, internal gap is the incomplete primitivity verification, which the reader noted in the rationale but did not make the weakest_assumption. This gap is concrete: the paper's own criterion in Remark 3.2 requires checking all Sq^{2^t}_* up to floor(log2(deg x)), and the p0 and d0 computations stop early. If the missing operations are nonzero, the claimed witnesses are not primitive and do not establish image membership. This is fixable by running the omitted checks, so it does not force rejection, but it does block acceptance as written. The G4-invariant framework is independent, reproduces dim[(QP4)33]^{G4}=1, and is less affected by this issue. The negative claim against Sum's proof depends on the unshipped negative-search code, but the polynomial-level inconsistency (z not in Lambda^3, delta(R)!=0) is displayed and hand-checkable. Overall the verdict remains CONDITIONAL: pending the complete annihilation checks and ideally a runnable code artifact, the central claims are not fully verified.","tokens_in":80675,"tokens_out":3543,"duration_ms":43410,"concrete_test":"Re-run the A-annihilation check for the displayed p0 witness (degree 33) and d0 witness (degree 14) using the identical dual Steenrod action from equation (3), but with the full range t=0..floor(log2(deg(x))): for p0 also compute (x)Sq^16_* and (x)Sq^32_*; for d0 also compute (x)Sq^8_*. If all of these vanish, the primitivity claim is confirmed and the preimage conclusions stand; if any is nonzero, the witness is not in PAH and the transfer-image conclusion is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central positive claim that p0 admits an explicit preimage rests on the assertion that the displayed degree-33 polynomial x lies in PAH33(B(Z/2)^4). The paper verifies (x)Sq^1_*=(x)Sq^2_*=(x)Sq^4_*=(x)Sq^8_*=0 and justifies stopping at s=3 by \"the instability condition.\" But Remark 3.2 states the correct criterion: one must verify (x)Sq^{2^t}_*=0 for t=0,...,floor(log2(deg(x))), which for deg(x)=33 is t=0,...,5, i.e., also Sq^16_* and Sq^32_*. Instability of the cohomology action does not imply vanishing of the dual homology action when 2^t<=deg(x); for example Sq^deg(u)(u)=u^2 is nonzero in cohomology. The same gap affects the d0 witness: for deg(x)=14, floor(log2(14))=3, so Sq^8_* must be checked, but the text checks only j<=2. If any omitted operation is nonzero, the corresponding x is not A-annihilated, the equation phi_k(x)+delta(z)=y no longer implies membership in the Singer transfer image, and both the p0 preimage and the d0 preimage would fail as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a computational framework for the preimage problem of the Singer algebraic transfer inside the lambda algebra. It formulates the search for x and z satisfying φ_k(x)+δ(z)=y as a linear system, applies it to the classes c0, d0, and p0, and claims to exhibit explicit preimages for c0 and p0, and a preimage for d0 under the Chon-Ha recursion. It further asserts that the proof in Nguyen Sum's paper [17] that d0 lies in the image of the fourth transfer is false, both because the z used there lies in the wrong lambda-algebra length and because Sum's recursion (4) is allegedly not a model of the transfer. The final part of the paper presents a SageMath algorithm for computing dimensions and bases of the G4-invariant spaces [(QP4)d]^{G4}, with a detailed sample run for d=33 and a claim that this verifies the author's earlier manual rank-4 results.","tokens_in":80758,"tokens_out":8583,"duration_ms":108413,"significance":"If the missing verifications are supplied, the explicit polynomial x for p0 would be a concrete new contribution: a previously unpublished preimage for an indecomposable element in Ext^{4,37}_A, with the duality pairing against the G4-invariant f displayed in detail. The d0 computation also provides an explicit witness that the class is in the image of φ4. The SageMath framework and the extensive output for k=4, d=33 are valuable because they make the hit-problem and invariant computations reproducible in principle, and the paper explicitly ships machine-checkable witness polynomials and a sample execution. However, the central positive claims are not yet fully established: the primitive checks for p0 and d0 omit required Steenrod operations, the refutation of [17] rests on an unproved assertion about formula (4), and the extrapolation from finitely many degrees to a general family of degrees is not justified.","major_comments":[{"comment":"The manuscript verifies (x)Sq^1_* = (x)Sq^2_* = (x)Sq^4_* = (x)Sq^8_* = 0 for the degree-33 element x and says the instability condition makes this sufficient. According to the paper's own criterion in Remark 3.2, one must check (x)Sq^{2^t}_* for t = 0, ..., floor(log2(deg x)), which for deg x = 33 means also Sq^16_* and Sq^32_*. Instability of the cohomology action does not imply vanishing of the dual homology action when 2^t ≤ deg x; the standard example Sq^{deg u}(u) = u^2 shows the distinction. The same gap occurs for the degree-14 d0 witness: floor(log2 14) = 3, so Sq^8_* must be checked, but the text says only 0 ≤ j ≤ 2. Without these checks the displayed polynomials are not proved to lie in PAH_*, and Lemma 3.1 cannot be applied to conclude membership in the image of the transfer.","section":"§3.2 (p0 and d0 cases), Remark 3.2, Eq. (3)"},{"comment":"The refutation of Nguyen Sum's proof depends on two load-bearing assertions: that recursion (4) is not a valid chain-level model of the Singer transfer while Corollary 2.3 is, and that under (4) there is no pair (x,z) with x ∈ PAH14, z ∈ Λ^3, and φ4(x)+δ(z)=y for Sum's representative. Neither assertion is proved in the manuscript. No derivation or citation is given showing that (4) is incompatible with the Chon-Ha formula, and the 'no solution' verdict is reported as a computer result without a rank or consistency certificate. The observation that Sum's z = λ1λ3λ9λ3 + λ1λ3^2λ9 has length four and hence is not in Λ^3 already invalidates the displayed proof; the stronger claim about formula (4) is therefore not needed and should either be proved rigorously or removed.","section":"Remark 3.3"},{"comment":"The claim that it suffices to compute s = 1,2,3,4 and then 'match the monomial patterns' to obtain the general G4-invariant formula is an unproved induction. The uniform structure of the admissible basis for all s is asserted from four observed cases, and no theorem or algorithmic certificate establishes that the pattern persists. Since this inference is used to claim verification of the rank-4 Singer conjecture results in [11,12,13], the paper should either prove the pattern by induction or explicitly restrict the claim to the finitely many computed degrees.","section":"§4.1, paragraph on d = 2^{s+3}+2^{s+1}-3"},{"comment":"The central computational results are not fully auditable as presented. The preimage search is described by pseudocode (Algorithm 0), but the actual SageMath code in the appendix implements the G4-invariant computation; the code used to produce the c0, d0, and p0 witnesses and to establish the 'no solution' verdict for Sum's recursion is not provided. Moreover, the asserted identity φ4(x) = y for p0 is not displayed, unlike the d0 case, and the c0 primitive check is asserted rather than shown. For a computational paper, the generating scripts and machine-readable certificates for these specific claims should be included or linked.","section":"§3.1 and Appendix"}],"minor_comments":[{"comment":"The sample workflow states 'Polynomial ring: Z/2[x1,x2,x3,x4,x5]' for k=4; the ring should be Z/2[x1,x2,x3,x4].","section":"§4.2, Example Computational Workflow"},{"comment":"The sentence 'For z ≠ 0, − z ∈ {λ2λ7, λ7λ2}' is malformed; the minus sign is spurious and the intended meaning appears to be 'z ∈ {λ2λ7, λ7λ2}'.","section":"§3.2, c0 case"},{"comment":"The bound in the IsAnnihilated helper function is written as max_t = floor(log2(degree)) + 1 followed by range(max_t - 1); this is consistent with Remark 3.2 but should be stated uniformly to avoid confusion.","section":"Remark 3.2 and Algorithm 0"},{"comment":"The representative y = λ7^2λ5λ14 + λ7^2λ9λ10 + λ7λ11λ9λ6 is imported from Lin [10] with no check that it is a cocycle; a short verification would remove reliance on external transcription.","section":"§3.2, p0 case"},{"comment":"The differential in (2) increases the length of a monomial by one, but this is not stated explicitly; the confusion is visible in Remark 3.3, where the length of z is the key point.","section":"§2, differential δ"}],"recommendation":"major_revision","confidential_remarks":"The claim against [17] is strong and central to the paper's narrative. If formula (4) turns out to be a valid equivalent model of the transfer after suitable admissible reduction, the negative conclusion about Sum's proof would not stand as stated. I recommend that the editor require the authors to provide the actual preimage-search code and certificates, or to restrict the claims to the finitely many degrees actually verified. The paper is a plausible candidate after revision, but the current gaps in primitivity checks and in the comparison of the two recursive formulas are load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a serious computational paper that deserves referee time, but the central primitivity checks are incomplete as written. If that gap is closed, the p0 preimage is a genuine new result.\n\nWhat is actually new: the explicit closed-form φ4 and φ5 expansions, the linear-algebra formulation of the preimage equation, and the explicit 60-term preimage for p0, which Hung–Quynh only proved exists. The paper also gives a concrete mechanical critique of Nguyen Sum's proof: the alleged boundary term has length 4, not 3, so it cannot lie in Λ^3, and the remainder R is shown not to be a cocycle. That critique is credible and does not depend on the author's own computations. The SageMath framework reproduces the author's earlier manual results for the G4-invariant spaces, which is useful verification even if not new.\n\nSoft spots. The biggest is the A-annihilation check. For the degree-33 p0 preimage, the paper checks Sq^1, Sq^2, Sq^4, Sq^8 and stops, citing instability. But Remark 3.2 in the paper itself requires checking up to floor(log2(deg)) = 5, i.e., also Sq^16 and Sq^32. Instability of the cohomology action does not imply vanishing of the dual homology action when the operation degree is ≤ deg(x); the example Sq^deg(u)(u) = u^2 shows the distinction. The same gap appears for d0: degree 14 requires Sq^8, but the paper checks only up to Sq^4. The witnesses are explicit and the missing checks are mechanical, so this is a proof gap rather than a demonstrated false result, but the claim \"x ∈ PAH\" is not established by the displayed computation.\n\nSecond, the code. The appendix claims a complete SageMath implementation, but the listing is truncated and corrupted by missing whitespace; it is not runnable as printed. The streamlined version is \"available on request,\" which is not reproducible. The negative claim about Sum's recursion is said to be computer-verified, but the code is not shipped.\n\nThird, the entire pipeline depends on Corollary 2.3 (from Chon–Ha) and the quoted recursion (4) from Sum. The paper does not independently verify these transcriptions. This is a moderate concern; a misquote would void the critique, though the length argument against Sum's boundary term stands on its own.\n\nFor whom: anyone working on the Singer transfer, the hit problem, or explicit Ext_A computations. It is a computational contribution rather than a deep conceptual one. I would send it to a serious referee, with the expectation that the primitivity checks be completed, the code hosted in a repository, and the key formulas double-checked against the sources. The author is honest about the method, but the verification gap and reproducibility issue are real and fixable.","headline":"A serious computational paper with a real new preimage for p0, but the primitivity checks are incomplete as written and the code is not reproducible; deserves peer review with major revision.","tokens_in":81506,"tokens_out":2606,"would_cite":true,"duration_ms":31339,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55T15","55S10","55S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper turns preimage-finding for the Singer transfer into solvable linear systems over the two-element field, gives an explicit preimage for the indecomposable p0 that earlier work left implicit, and claims that a cited proof that d0…","keywords":["Singer algebraic transfer","lambda algebra","Adams spectral sequence","Peterson hit problem","Steenrod algebra","G_k-invariants","preimage problem","SageMath"],"falsifier":"Run the paper's own SageMath routine (Listing 1) on the 36-term polynomial $q_{4,3}$ using recursion (4): if the Adem-reduced image matches $d_0^* + R$ and $\\delta(R) = 0$, or if a legitimate $z \\in \\Lambda^3$ with $\\delta(z) = R$ can be produced, the refutation of [17] fails. Independently, run the paper's linear-system search for the [17] representative of d0 under the Corollary 2.3 recursion: if any pair $(x, z)$ with $x \\in PAH_{14}(B(\\mathbb{Z}/2)^4)$ and $z \\in \\Lambda^3$ satisfies $\\varphi_4(x) + \\delta(z) = y$, the claim that the proof is irreparable is wrong. The same search for p0 is also checkable by substitution: the displayed polynomial x must satisfy $(x)\\mathrm{Sq}^1_* = (x)\\mathrm{Sq}^2_* = (x)\\mathrm{Sq}^4_* = (x)\\mathrm{Sq}^8_* = 0$ and $\\varphi_4(x) = \\lambda_7^2\\lambda_5\\lambda_{14} + \\lambda_7^2\\lambda_9\\lambda_{10} + \\lambda_7\\lambda_{11}\\lambda_9\\lambda_6$.","tokens_in":80212,"feed_emoji":"🧮","tokens_out":18136,"duration_ms":173616,"temperature":0.7,"pith_summary":"The paper is trying to turn the preimage problem for the Singer algebraic transfer into routine computation: given a class in the $\\lambda$-algebra cohomology of the Steenrod algebra, equivalently a cell of the Adams spectral sequence, decide whether the transfer detects it and, if so, exhibit an explicit witness. The transfer is realized by a recursion transcribed from [5], and the search for a preimage x and a coboundary z is written as the linear equation $\\varphi_k(x) + \\delta(z) = y$ over $\\mathbb{F}_2$, which a computer can solve and filter by the condition that x is annihilated by the positive-degree Steenrod operations. On this basis the paper exhibits an explicit preimage for the indecomposable p0 that earlier work detected only non-constructively, verifies c0 and d0 against the [5] model, and claims that the proof in [17] that d0 is detected is false because it runs inside a different recursive rule, where the claimed correction term does not even belong to the correct $\\lambda$ space and the residual is not a cocycle. The same project produces a SageMath pipeline that recomputes the $G_4$-invariant spaces of the rank-4 Singer conjecture from scratch, confirming $\\dim[(QP_4)_{33}]^{G_4} = 1$ with an explicit basis. If correct, the paper converts two previously manual procedures, transfer preimage construction and $G_k$-invariant computation, into reproducible tools and flags a concrete proof in the literature as invalid.","feed_headline":"Explicit preimages found for Singer transfer; a cited proof is false","feed_subtitle":"Linear-algebra search yields concrete witnesses for Adams classes c0, d0, p0 and automates rank-4 invariant checks.","key_machinery":"The machinery has two stages. The first is the chain-level equation $\\varphi_k(x) + \\delta(z) = y$ in the $\\lambda$ algebra $\\Lambda$, the bigraded differential algebra with generators $\\lambda_t$, $t \\geq 0$, Adem relations (1) and differential (2), whose admissible monomials of length k form the chain complex computing $\\mathrm{Ext}_{\\mathcal A}(\\mathbb{Z}/2,\\mathbb{Z}/2)$. The Singer transfer is represented on $H_*(B(\\mathbb{Z}/2)^k)$ as a map to $\\Lambda^k$ through the generating-function correspondence of [5], and its recursive form (Corollary 2.3) expresses $\\varphi_k$ of a divided-power monomial $a^{(t_k)}_k \\cdots a^{(t_1)}_1$ as a sum of $\\lambda_i$ terms feeding $\\varphi_{k-1}$ on elements transformed by the right action of the dual Steenrod operations, with the single-generator rule $a^{(t)}(\\mathrm{Sq}^j_*) = \\binom{t-j}{j}a^{(t-j)}$. Writing the target cocycle y, the unknown x in the divided-power basis, and the unknown z in the length-$(k-1)$ admissible $\\lambda$ basis makes $\\varphi_k(x) + \\delta(z) = y$ a finite $\\mathbb{F}_2$-linear system; the algorithm then filters the solution space by the $à$-primitive condition $(x)\\mathrm{Sq}^j_* = 0$ for $j = 2^t$, $t = 0, \\dots, \\lfloor \\log_2 \\deg x \\rfloor$. The second stage is the $G_k$-invariant computation: a weight-vector filtration of $P_k = \\mathbb{Z}/2[x_1, \\dots, x_k]$ stratifies the cohit space $QP_k$ by weight, an admissible basis is read off from an echelon form of the hit matrix, $\\Sigma_k$-components are found by adjacency under the adjacent transpositions $\\rho_1, \\dots, \\rho_{k-1}$, and $G_k$-invariants are obtained by imposing the transvection $\\rho_k(x_k) = x_k + x_{k-1}$ through kernel computations, with duality to the transfer domain supplied by the coefficient pairing between divided-power homology and polynomial cohomology.","core_discovery":"Stated as the author would state it to a fair reader, the central claim is that the transfer's preimage problem is a solvable linear-algebra problem in the $\\lambda$-algebra model: a class $[y] \\in \\mathrm{Ext}^{k,*}_{\\mathcal A}(\\mathbb{Z}/2,\\mathbb{Z}/2)$ lies in the image of $\\varphi_k$ exactly when $\\varphi_k(x) + \\delta(z) = y$ admits a solution with $x$ an $\\mathcal A$-annihilated element of $H_*(B(\\mathbb{Z}/2)^k)$ and $z$ a length-$(k-1)$ $\\lambda$ element. The recursive description of the transfer from [5] (Corollary 2.3) turns the equation into a finite matrix system over $\\mathbb{F}_2$, and the paper solves it for three indecomposables. The outputs are explicit: a preimage $x \\in PAH_8(B(\\mathbb{Z}/2)^3)$ for c0 with $z = 0$; a preimage $x \\in PAH_{14}(B(\\mathbb{Z}/2)^4)$ with $z = \\lambda_3^2\\lambda_9 + \\lambda_3\\lambda_9\\lambda_3$ for d0; and a preimage $x \\in PAH_{33}(B(\\mathbb{Z}/2)^4)$ with $z = 0$ for p0, the p0 witness being a preimage that the earlier detection arguments of [8] did not make explicit. The paper further claims that the proof in [17] that d0 lies in the image of the fourth transfer is false: that paper's recursion (4) places the $\\lambda$ factor differently from the model of [5], its correction term $z = \\lambda_1\\lambda_3\\lambda_9\\lambda_3 + \\lambda_1\\lambda_3^2\\lambda_9$ does not lie in $\\Lambda^3$, and with its representative $d_0 = d_0^*$ the image of its own $q_{4,3}$ is $d_0^* + R$ with $\\delta(R) \\neq 0$, so no $z \\in \\Lambda^3$ can close the equation. The paper is explicit that the statement that d0 is detected is not what fails, since it gives a working preimage and notes an earlier alternative proof in [7], and that what collapses is the published argument. Alongside this, the paper claims its SageMath algorithm reproduces the $G_4$-invariant spaces previously computed by hand for the rank-4 Singer conjecture, including $\\dim[(QP_4)_{33}]^{G_4} = 1$, and prints an explicit basis polynomial for each invariant space.","pith_inferences":["Model-dependence is the paper's sharpest edge but also its narrowest: the no-solution verdict is computed with one cocycle representative under one recursion, so it refutes that specific argument rather than establishing that no representative could be found in either model; the cohomology class, not the cocycle, is the stable object.","Testable extension: the same matrix formulation applies unchanged to any Ext class, so the method's limit is only the size of the admissible lambda basis and the divided-power basis; a natural next check is a class at a bidegree where the two recursive models would predict different membership.","Editorial reading: the divergence between recursion (4) and Corollary 2.3 is a warning about hand computations of the transfer more broadly, since any published preimage computed without fixing the chain-level model is exposed to the same kind of collapse documented here for [17].","The pairing $\\langle x, f \\rangle = 1$ between the p0 preimage and the 16-term $G_4$-invariant gives a concrete template: each global invariant should pair with a dual primitive, so the invariant-space algorithm doubles as a generator of explicit Ext classes, which could populate further cells of the Adams chart."],"forward_implications":["All three target classes, c0, d0, and p0, are detected by the Singer transfer, with explicit primitive witnesses that a reader can check by substitution into the identity $\\varphi_k(x) + \\delta(z) = y$.","The published proof in [17] that d0 lies in the image of the fourth transfer is asserted to be false: its recursion (4) is not the chain-level model of the transfer, its auxiliary element z falls outside $\\Lambda^3$, and with its representative of d0 the residual R is not a cocycle, so no correcting z exists.","The detection statement for d0 is not itself overturned, since the paper supplies a valid preimage under the [5] recursion and notes an alternative proof in [7]; the correction concerns the proof, not the theorem.","The preimage of p0, previously known only through the non-constructive detection arguments of [8], is now given explicitly in $PAH_{33}(B(\\mathbb{Z}/2)^4)$, making the membership of p0 in the transfer's image independently checkable.","The SageMath pipeline reproduces the manual rank-4 computations, including $\\dim[(QP_4)_{33}]^{G_4} = 1$ with an explicit 16-term basis, so the $G_4$-invariant results of the author's earlier papers are verifiable by any reader with enough memory."],"supporting_citations":[{"why":"Supplies the invariant-theoretic description of the transfer and the recursive formula (Corollary 2.3) inside which all preimage computations and both verdicts are produced.","marker":"[5]"},{"why":"The cited proof that d0 lies in the image of the fourth transfer is the refutation target: its recursion (4), its representative of d0, and its element z are tested and found invalid.","marker":"[17]"},{"why":"Provides the cocycle representatives of c0, d0, and p0 that serve as the right-hand sides y of the linear systems.","marker":"[10]"},{"why":"Earlier, non-constructive detection of p0 by the transfer; the paper's explicit preimage fills the gap left by this work.","marker":"[8]"},{"why":"Earlier verification by a different method that c0 is in the image of the third transfer, cross-checking the algorithm's output for c0.","marker":"[2]"},{"why":"An alternative earlier derivation that d0 is detected, cited to show the statement survives even though the proof in [17] collapses.","marker":"[7]"},{"why":"Introduces the transfer and the injectivity conjecture whose rank-4 case motivates the $G_4$-invariant space computations.","marker":"[15]"},{"why":"Introduces the lambda algebra, the chain model in which the transfer's representation and the equation $\\varphi_k(x) + \\delta(z) = y$ live.","marker":"[4]"},{"why":"Supplies the uniform structural description and the bound $\\dim(QP_4)_d \\leq 315$ used to justify the degree sweep for the $G_4$ computation.","marker":"[16]"}],"fun_headline_variants":["Linear algebra solves Singer transfer preimages, refutes a proof","Explicit preimage for p0 in Singer transfer, refutes a proof","Proof that d0 lies in transfer image is false, preimages found","SageMath automates Gk-invariant spaces for Singer transfer"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hinges on the recursive model of the transfer: every preimage found and every no-solution verdict is computed inside the recursion of Corollary 2.3 (transcribed from [5]) as the true chain-level model and the differing recursion (4) attributed to [17] as a false one, so a misprint or misapplication in either model, or an unchecked cocycle representative imported from [10], would void the witnesses and the refutation alike.","fun_headline_variants_meta":{"raw":{"variants":["Linear algebra solves Singer transfer preimages, refutes a proof","Explicit preimage for p0 in Singer transfer, refutes a proof","Proof that d0 lies in transfer image is false, preimages found","SageMath automates Gk-invariant spaces for Singer transfer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001045,"raw_usage":{"total_tokens":4630,"prompt_tokens":1417,"completion_tokens":3213,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1033,"completion_tokens_details":{"reasoning_tokens":3136}},"tokens_in":1033,"tokens_out":3213,"duration_ms":27804,"temperature":1.0,"reasoning_tokens":3136,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:44:09.035544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's own SageMath routine (Listing 1) on the 36-term polynomial $q_{4,3}$ using recursion (4): if the Adem-reduced image matches $d_0^* + R$ and $\\delta(R) = 0$, or if a legitimate $z \\in \\Lambda^3$ with $\\delta(z) = R$ can be produced, the refutation of [17] fails. Independently, run the paper's linear-system search for the [17] representative of d0 under the Corollary 2.3 recursion: if any pair $(x, z)$ with $x \\in PAH_{14}(B(\\mathbb{Z}/2)^4)$ and $z \\in \\Lambda^3$ satisfies $\\varphi_4(x) + \\delta(z) = y$, the claim that the proof is irreparable is wrong. The same search for p0 is also checkable by substitution: the displayed polynomial x must satisfy $(x)\\mathrm{Sq}^1_* = (x)\\mathrm{Sq}^2_* = (x)\\mathrm{Sq}^4_* = (x)\\mathrm{Sq}^8_* = 0$ and $\\varphi_4(x) = \\lambda_7^2\\lambda_5\\lambda_{14} + \\lambda_7^2\\lambda_9\\lambda_{10} + \\lambda_7\\lambda_{11}\\lambda_9\\lambda_6$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the invariant-theoretic description of the transfer and the recursive formula (Corollary 2.3) inside which all preimage computations and both verdicts are produced."},{"cited_title":"Sum, On the determination of the Singer transfer , Vietnam J","cited_arxiv_id":null,"evidence_quote":"The cited proof that d0 lies in the image of the fourth transfer is the refutation target: its recursion (4), its representative of d0, and its element z are tested and found invalid."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the cocycle representatives of c0, d0, and p0 that serve as the right-hand sides y of the linear systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier, non-constructive detection of p0 by the transfer; the paper's explicit preimage fills the gap left by this work."},{"cited_title":"Boardman, Modular representations on the homology of power of real projective space , in Algebraic Topology: Oaxtepec 1991, ed","cited_arxiv_id":null,"evidence_quote":"Earlier verification by a different method that c0 is in the image of the third transfer, cross-checking the algorithm's output for c0."},{"cited_title":"Ha, Sub-Hopf algebras of the Steenrod algebra and the Singer transfer , Geom","cited_arxiv_id":null,"evidence_quote":"An alternative earlier derivation that d0 is detected, cited to show the statement survives even though the proof in [17] collapses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the transfer and the injectivity conjecture whose rank-4 case motivates the $G_4$-invariant space computations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the lambda algebra, the chain model in which the transfer's representation and the equation $\\varphi_k(x) + \\delta(z) = y$ live."},{"cited_title":"Sum, The hit problem for the polynomial algebra of four variables , arXiv:14121709","cited_arxiv_id":null,"evidence_quote":"Supplies the uniform structural description and the bound $\\dim(QP_4)_d \\leq 315$ used to justify the degree sweep for the $G_4$ computation."}],"review_version":1}