{"id":"964fa597-f926-4d97-bf97-a1cebcaee70b","arxiv_id":"2607.26096","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For every odd q>=3, reduced singular instanton homology of the pretzel knot P(-2,3,q) has rank q+2, and naive pillowcase Floer rank differs by one differential controlled by the determinant.","lead":"This paper proves a rank formula for instanton knot homology of an infinite pretzel family and computes geometric correction terms in a symplectic model. It is a testbed for how two major homology theories for knots are related.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The pillowcase half depends on the unproved Hypothesis 4.1; all computed ranks, the deficiency law, and the cochains inherit this condition.","rationale":"I examined Theorem 1.1 and found it robust: the Alexander-polynomial lower bound and Khovanov upper bound are both external and correctly assembled, so the unconditional claim does not depend on the pillowcase model. The weakest point of the paper is exactly what the reader identified: the pillowcase computations are all conditional on Hypothesis 4.1. The paper is unusually transparent about this, separating proved, computed, and conjectural content. My read adds only a note that even within the model, the Maurer–Cartan verification for the computed cochains is reported tersely (µ1-closedness only), and the combinatorial section does not explicitly settle whether repeated-vertex polygons are counted; this is a secondary point that a code audit could quickly clarify. It does not change the verdict: CONDITIONAL remains the right assessment, because the unconditional theorem can be accepted while the pillowcase claims await a proof or independent audit of the model.","tokens_in":12042,"tokens_out":23793,"duration_ms":236076,"concrete_test":"For the q=7 configuration, take the triangle counted in §5.2 (vertices: generator pair x,y and self-crossing s) and attempt to construct the corresponding holomorphic quilt in the pillowcase smooth stratum using Zhang's methods (arXiv:2409.19744, arXiv:2504.09284), adapted to the orbifold points. If the quilt count differs from the winding-number count, Hypothesis 4.1 is false for k=2; if it matches, the model is confirmed for this configuration and the deficit-side law gains support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The unconditional Theorem 1.1 is sound: the L1 bound from Proposition 2.1 and Manion's upper bound close, and both inputs are external. The load-bearing risk is entirely on the pillowcase side. Every naive rank, the deficiency law (Computation 1.2), and the bounding-cochain computations (Computation 1.3) are counts of immersed polygons computed under Hypothesis 4.1 (§4.1), which asserts that these winding-number counts equal the A∞ operations µ_k of the immersed Fukaya algebra of the pillowcase. The paper itself states (register (c), §1.5) that no proof at this generality is known and that dSRS covers only embedded loops in surfaces. If Hypothesis 4.1 fails — e.g., because orbifold points or self-intersections introduce additional holomorphic polygons not captured by the winding conditions — the naive ranks and the MC-validity of the cochains would change, and the deficiency law would be unsupported. This concern does not touch Theorem 1.1, but it is exactly what makes the pillowcase claims conditional. A secondary internal point: the MC verification for the q=7 and q=11 cochains is reported only as µ1-closedness (§5.2, §5.3); under Hypothesis 4.1 one should also confirm that all higher µ_k with repeated vertices from the support vanish, as §4.1 does not explicitly state that only distinct-vertex polygons are counted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper has two distinct parts. The first proves Theorem 1.1: for every odd q ≥ 3, the reduced singular instanton knot homology of the pretzel knot P(-2,3,q) has rank q+2. The proof is a squeeze: Proposition 2.1 computes the Alexander polynomials of the family via a Conway skein recursion, showing their L1 norm is q+2, and Manion's closed-form reduced Khovanov homology gives the matching upper bound q+2. The second part works in the immersed-curve pillowcase model of Herald--Kirk--Smith. It reconstructs the relevant Lagrangians, computes naive Lagrangian--Floer ranks for q = 3,5,7,11,13, formulates a deficiency law (Table 1, Computation 1.2), and computes bounding cochains for q = 5,7,11 that repair the rank to q+2 (Computation 1.3). The paper explicitly separates what is proved unconditionally, what is computed within the combinatorial model, and what remains conjectural.","tokens_in":12407,"tokens_out":10434,"duration_ms":109608,"significance":"Theorem 1.1 is a clean, family-wide rank formula and a good illustration of the Alexander-polynomial/Khovanov squeeze; it is unconditional, elementary modulo [Man18] and [Hir01], and appears to be new in this uniform form. The pillowcase part is potentially valuable: these would be the first explicitly computed nonzero bounding cochains on Conway-sum tangles, and the q=5/q=7 examples give the first instances acting by cancellation. The manuscript is unusually candid about the status of its claims and ships reproducible code. However, the pillowcase ranks, the deficiency law, and the bounding-cochain computations all depend on the unproved Hypothesis 4.1 and on perturbation-dependent enumerations checked at only two perturbations per member. Their status is experimental and model-dependent, not theorem-level, and the presentation should not let the unconditional theorem lend borrowed certainty to the pillowcase claims.","major_comments":[{"comment":"All pillowcase results — the naive ranks of Table 1, Computation 1.2, and Computation 1.3 — are computed under Hypothesis 4.1, which asserts that the winding-number polygon counts of §4.1 compute the higher operations µ_k of the immersed Fukaya algebra of the pillowcase. This extends de Silva–Robbin–Salamon from embedded loops in surfaces to immersed curves in an orbifold and to k ≥ 2, and the paper states that no proof at this generality is known. Since the pillowcase computations are headline contributions, they should not be presented as facts about the actual pillowcase Floer homology. I recommend either adding a theorem that reduces the needed cases to a checkable condition, or explicitly relabeling the pillowcase results as model-dependent experimental evidence throughout the abstract, introduction, and Section 5. Theorem 1.1 is not affected.","section":"§4.1, Hypothesis 4.1"},{"comment":"For q=7 and q=11, the Maurer–Cartan verification is reported only as µ1-closedness of the support crossing. Equation (2) requires the vanishing of all higher terms µ_k(b,...,b). Since b is a single self-intersection point for q=7, the only possible higher terms involve repeated occurrences of the same vertex. Section 4.1 does not explicitly state that polygon counts require distinct vertices, and the code description does not resolve this. Such terms could be nonzero. Please give explicit counts for µ_k(s,...,s), k ≥ 2, for the chosen cochain support s, or prove that no such polygons exist under the definition used.","section":"§5.2, §5.3; Eq. (2)"},{"comment":"The sign law 'naive rank − rank I♮ = 2 sgn(det K − 3)' is presented as a 'sharp experimental law' but is fitted to exactly five members q=3,5,7,11,13. The row q=13 is labeled 'predicted' even though its naive rank is already computed. There is no out-of-sample check: the sign and threshold were read off after the computations. This is honest but weaker than the wording suggests. I recommend either computing additional members (e.g. q=17,19) after fixing the prediction, or rephrasing the statement as a conjecture supported by five examples rather than a law.","section":"Computation 1.2, Table 1"}],"minor_comments":[{"comment":"The elimination leading to the three-term recursion (5) is not shown. The displayed relations a_q = a_{q−2} + sz b_{q−1} and b_{q−1} = b_{q−3} + sz a_{q−2} require an additional use of the skein relation for b_{q−3}; please include the two-line algebra so the reader can verify the recursion without reconstructing it.","section":"§2.1"},{"comment":"For reproducibility, state the perturbation amplitudes and the locality-window size used in the polygon enumerations. Figure 1 gives ε ≈ 0.05–0.07 for one case, but the window of consecutive blue-arc positions in §4.2 is not quantified. The code is provided, but these parameters should be in the text.","section":"§3.2, §6.4"},{"comment":"The q=13 row should distinguish what is computed (naive rank 17, rank I♮ = 15) from what is predicted (the sign of the correction). The word 'predicted' in the correction column is confusing because the naive rank is computed in the same table.","section":"Table 1"},{"comment":"The sentence 'This proves Computation 1.3(i)' would be less likely to confuse if it read 'establishes within the model' or 'verifies', in keeping with the paper's own register distinction.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"Theorem 1.1 is solid and, in my view, would justify publication even if the pillowcase sections were drastically softened. My recommendation of major revision is driven by the conditional status of the headline pillowcase claims under Hypothesis 4.1 and by the gap in the Maurer–Cartan verification for q=7 and q=11. I would be satisfied by a revision that adds the missing higher-product checks, makes the model-dependence of the pillowcase results explicit in the abstract and introduction, and either adds more members or softens the 'law' language."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one solid takeaway is Theorem 1.1: rank I^#(P(-2,3,q)) = q+2 for all odd q>=3, proven by a squeeze between the L1 norm of the Alexander polynomial and Manion's reduced Khovanov rank. The skein recursion for the Alexander polynomials looks right—the Lehmer match at q=7 is a nice check—and the proof is genuinely unconditional. That part deserves to be accepted as a new, clean result.\n\nThe pillowcase half is a different animal. The authors do the field a favor by labeling it honestly: the deficiency law (Computation 1.2), the bounding cochains (Computation 1.3), and all naive ranks depend on Hypothesis 4.1, which asserts that winding-number counts of immersed polygons in the orbifold pillowcase compute the A-infinity operations of the immersed Fukaya algebra. This is unproved at the needed generality, and the paper says so. That makes the deficiency law a well-motivated conjecture verified in five examples, not a theorem. The bounding cochains are also fitted: the search enumerates supports until the deformed rank hits the independently known target q+2. That's fine as a computational discovery, and the paper is transparent about it, but readers should not treat the cochains as uniquely predicting the CHKK naturality data—the authors themselves note the excess case has fifty-five solutions.\n\nA secondary technical point a referee should chase: the Maurer-Cartan verification for q=7 and q=11 is reported as mu1-closedness, and for q=5 as vacuity. The paper tests pairs and triples 'within the locality window' but never states explicitly whether polygons with repeated vertices at the same crossing are counted or excluded. In the self-products mu_k(b,...,b) those terms can exist and can affect the MC equation. The code may handle this, but the text should say so.\n\nAlso worth noting: one non-generic perturbation at q=7 was identified and excluded, and the q=3 value is taken from HHK rather than recomputed. This is honest practice, not a flaw.\n\nOverall: the unconditional theorem is solid, the computational model is state-of-the-art for this program, and the paper's separation of proved/computed/conjectural is exemplary. It deserves serious peer review—the referee can verify Theorem 1.1, audit the reproducibility package, and push for clarification on the repeated-vertex polygon issue. This is the kind of paper that should be in the literature, with its conditions made explicit.","headline":"Theorem 1.1 is a clean, new, unconditional rank formula; the pillowcase half is honest, conditional computation that should be read as evidence within a model, not as theorem.","tokens_in":12842,"tokens_out":2882,"would_cite":true,"duration_ms":32344,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R58","57K10","57K18","57K31","53D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every odd q≥3, the pretzel knot P(−2,3,q) has reduced singular instanton homology of rank exactly q+2.","keywords":["singular instanton homology","pretzel knots","pillowcase","Lagrangian Floer homology","bounding cochains","Alexander polynomial","Khovanov homology","Atiyah-Floer conjecture"],"falsifier":"For the unconditional theorem, find an odd q≥3 where the Alexander polynomial L1-norm or the reduced Khovanov homology rank is not q+2, or compute rank I♮ directly and obtain a different value; for the pillowcase side, compute the true immersed Floer homology of the pillowcase Lagrangians for q=11 analytically, or compute the naive rank for q=17 and check whether naive−I♮ equals +2.","tokens_in":11923,"feed_emoji":"🔗","tokens_out":5384,"duration_ms":51650,"temperature":0.7,"pith_summary":"This paper proves that for the entire family of pretzel knots P(−2,3,q) with odd q≥3, the reduced singular instanton knot homology has rank q+2. The proof is a squeeze: the Alexander polynomials of the family, computed in closed form with all coefficients 0 or ±1, give the lower bound q+2, and closed-form reduced Khovanov homology gives the matching upper bound. Using this unconditional answer as a yardstick, the paper measures a symplectic counterpart: naive Lagrangian–Floer homology in the pillowcase misses the true rank by exactly one differential, with the sign of the error governed by the knot determinant. It then computes explicit bounding cochains that repair the deficiency in both directions—cancelling and creating a differential—for the first time on Conway-sum tangles. This matters because it makes the conjectural correction in the pillowcase program concrete and shows how rigid or flexible those corrections are.","feed_headline":"Instanton homology of (−2,3,q) pretzels equals q+2","feed_subtitle":"Closed-form Alexander and Khovanov bounds agree for every odd q; pillowcase Floer theory misses by exactly one differential.","key_machinery":"The proof rests on the squeeze inequality ℓ(K) ≤ rank I♮(K) ≤ dim Kh_r(K), fed by two closed forms: the Alexander polynomials of the family, derived from a Chebyshev-type skein recursion and having L1-norm q+2, and the reduced Khovanov homology of 3-strand pretzels, with total rank q+2. On the pillowcase side, the machinery is immersed-curve polygon counting by winding numbers: the differential and higher products μ_k are computed as counts of immersed bigons, triangles, quadrilaterals, and larger polygons with convex corners and winding conditions, and the deformed differential is corrected by bounding cochains satisfying a Maurer–Cartan equation.","core_discovery":"The central discovery is the unconditional rank formula rank I♮(P(−2,3,q)) = q+2 for all odd q≥3, obtained by combining a closed-form family of Alexander polynomials (Lehmer-like, with coefficients in {0,±1}) with closed-form reduced Khovanov homology; the two bounds coincide. On the symplectic side, within the immersed-curve combinatorial model of the pillowcase, the paper finds an experimental deficiency law: the naive Lagrangian–Floer rank differs from the true instanton rank by 2·sgn(det K−3), so the naive theory is too small by one differential for q=5,7, exact for q=3, and too large by one differential for q≥11. It then computes bounding cochains that correct the naive theory: a unique","pith_inferences":["If the deficiency law holds for all odd q coprime to 3, the discrepancy between naive pillowcase Floer homology and true instanton homology would be a value in {−1,0,1} controlled by the determinant, possibly provable by a curve-level argument once the combinatorial model is upgraded to a theorem.","The rigidity asymmetry suggests that at excess members the deformed Floer homology may depend on the choice of bounding cochain, so the invariant-theoretic cochain (if it exists) must be selected by additional structure; this could be tested by computing a naturality-fixed cochain for q=11 from a different tangle decomposition and comparing ranks.","A direct analytic computation of the pillowcase Lagrangian Floer homology for q=11 or q=13 would either confirm or refute Hypothesis 4.1 and the deficiency law; computing the naive rank for q=17 (det=11) in the model would test the law's extrapolation.","The coincidence of the deficit regime with the integral-homology-sphere members (det=1) hints that the correction mechanism may be tied to the presence of binary-dihedral characters at the pillowcase seams, and could be studied via the cut-and-paste resolution of the seam circles."],"forward_implications":["The rank formula gives the exact size of reduced singular instanton homology for every odd q≥3, extending the previously known q=5 case to the whole family without any conjecture.","If the deficiency law holds beyond the tested members, the naive pillowcase Floer homology is never off by more than one differential for this family, with the sign determined by det K−3 (equivalently, by the number of binary-dihedral traceless characters).","The computed bounding cochains are the first explicit nonzero cochains on Conway-sum tangles, and the first acting by cancellation; they realize both directions of the conjectured repair within one family.","Cancellation cochains are unique, while creation cochains are numerous, so rank-matching alone cannot pin down the tangle's canonical cochain at excess members; additional naturality data must be invoked.","The correction acts through different polygon orders within one family (a quadrilateral for q=5, a triangle for q=7), so the polygon order is not an invariant of the direction of the correction."],"fun_headline_variants":["Pretzel instanton rank proved: q+2 for all odd q","Bounding cochains correct pillowcase Floer deficiency","First computed bounding cochains on Conway-sum tangles","Pillowcase Floer off by one differential, fixed by cochains","Exact rank for (−2,3,q) pretzels via Alexander+Khovanov"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The pillowcase computations and the deficiency law rest on the unproved identification that winding-number counts of immersed polygons compute the higher operations of the immersed Fukaya algebra of the pillowcase orbifold; if that hypothesis fails, the computed ranks and bounding cochains do not describe genuine Floer theory.","fun_headline_variants_meta":{"raw":{"variants":["Pretzel instanton rank proved: q+2 for all odd q","Bounding cochains correct pillowcase Floer deficiency","First computed bounding cochains on Conway-sum tangles","Pillowcase Floer off by one differential, fixed by cochains","Exact rank for (−2,3,q) pretzels via Alexander+Khovanov"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000335,"raw_usage":{"total_tokens":1815,"prompt_tokens":984,"completion_tokens":831,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":728,"completion_tokens_details":{"reasoning_tokens":735}},"tokens_in":728,"tokens_out":831,"duration_ms":7906,"temperature":1.0,"reasoning_tokens":735,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T01:43:02.358541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the unconditional theorem, find an odd q≥3 where the Alexander polynomial L1-norm or the reduced Khovanov homology rank is not q+2, or compute rank I♮ directly and obtain a different value; for the pillowcase side, compute the true immersed Floer homology of the pillowcase Lagrangians for q=11 analytically, or compute the naive rank for q=17 and check whether naive−I♮ equals +2.","supporting_citations":[],"review_version":2}