{"id":"c436efb0-a90f-4bc1-9748-ca2979f4e981","arxiv_id":"2505.05846","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Rigid non-pivotal Temperley-Lieb, Motzkin, and planar rook monoids have smaller representation gaps than their pivotal counterparts, making them worse for cryptographic use.","lead":"This paper defines new non-pivotal versions of three planar diagram monoids and shows their smallest nontrivial representations are exponentially smaller than those of the standard pivotal monoids. The result indicates these new monoids are less suitable for cryptography based on monoid representations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Pivotal Motzkin gap bounds in Theorem 3A.3 are mutually inconsistent and rely on an unavailable thesis; the Motzkin comparison is unsupported as printed.","rationale":"The reader's verdict is CONDITIONAL and identifies the unpublished thesis [Ar25] as the weakest assumption, with an internal inconsistency in Theorem 3A.3 mentioned in the rationale. My independent read finds the same load-bearing point, sharpened: Theorem 3A.3 itself contains an impossible pair of bounds for Gap_K(MoT_n), so the pivotal Motzkin input to the central claim cannot be checked from the manuscript. This is not a manufactured objection; the comparison in Corollary 5C.8 relies on the lower bound, and the paper's table and abstract present the Motzkin comparison as part of the main conclusion. The non-pivotal rMo computations in Section 5 appear to be self-contained, and the rTL and pRo comparisons do not depend on [Ar25], so the failure is localized to one leg of a three-part claim. Because the issue is addressable by supplying the missing thesis or deriving the pivotal Motzkin gap independently, CONDITIONAL remains the appropriate verdict; no change to the reader's recommendation is needed.","tokens_in":32548,"tokens_out":3680,"duration_ms":39222,"concrete_test":"Make [Ar25] available, or independently recompute Gap_K(MoT_n) for n = 1 through 10 using the cell formulas in [KST24] and the GAP/Semigroups code in [St25]. Plot log Gap against n and compare with the two bounds in Theorem 3A.3. If the exponential base is not 9, the 9^n lower bound fails and Corollary 5C.8 collapses. If the computation at any tested n violates the stated 4^n upper bound, the theorem as printed is internally inconsistent; then re-derive the correct upper bound (for example, O(n^c 9^n)) to see whether the discrepancy is only a typo.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central comparison for the Motzkin family is unsupported as written. Theorem 3A.3 asserts for the pivotal truncated Motzkin monoid both f(n)·9^n ≤ Gap_K(MoT_n) and Gap_K(MoT_n) ≤ 2^{-3/2} n^{-3/2} 4^n. These two inequalities cannot hold simultaneously for large n, since 9^n/4^n grows exponentially. Consequently at least one bound in the theorem is wrong, mis-stated, or uses an f(n) that does not satisfy the claimed inequality. Corollary 5C.8 and the table entry 'Mo 2n Gap^{1/n}→9' depend on the 9^n lower bound, which is cited to [Ar25], an unpublished honours thesis with no public text. The rTL and pRo legs of the main claim are derived in this paper from explicit cell-size formulas and are not affected, but the Motzkin leg of Theorem 3B.6 versus Theorem 3A.3 cannot be verified from the available text. This is load-bearing because the abstract and Section 1B claim the non-pivotal monoids are 'largely worse' across all three families; if the pivotal Motzkin lower bound fails, the non-pivotal rMo semisimple gap may still be smaller, but the stated comparison would lose its main support.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines rigid, non-pivotal analogues of the Temperley–Lieb, Motzkin, and planar rook monoids, computes bounds on the sizes of their simple (or cell) representations, and compares the resulting representation gaps and gap ratios with the pivotal versions. It concludes that the non-pivotal monoids are generally worse for cryptographic purposes.","tokens_in":32769,"tokens_out":8006,"duration_ms":71989,"significance":"If the technical claims are correct, the paper provides the first systematic comparison of pivotal versus non-pivotal planar diagram monoids from the point of view of representation gaps, with explicit cell-size formulas and concrete asymptotic bounds. The rTL and rpRo legs rest on self-contained derivations, and the paper ships reproducible code on GitHub for the computer-assisted parts. However, the Motzkin comparison depends on an unpublished thesis, and the headline claims about the rigid Motzkin monoid are stated for the RepGap while only semisimple bounds are proved.","major_comments":[{"comment":"The two displayed bounds for Gap_K(MoT_n), namely f(n)·9^n ≤ Gap_K(MoT_n) ≤ 2^{-3/2} n^{-3/2} 4^n, are mutually incompatible for large n because (9/4)^n grows exponentially. As written, at least one of the bounds is wrong or mis-stated; this directly affects Corollary 5C.8 and the Motzkin row of the table in the introduction.","section":"§3A, Theorem 3A.3"},{"comment":"The pivotal Motzkin lower bound is cited to [Ar25], an unpublished honours thesis with no public text. This bound is load-bearing for the paper's central comparison, so the authors should either give a self-contained proof of the 9^n exponential growth or make the thesis available and ensure the quoted statement is correct, especially in light of the inconsistency noted above.","section":"§3A, proof of Theorem 3A.3"},{"comment":"The abstract and Table 1 report Gap^{1/n}→4 and Ratio^{1/n}→1 for the rigid Motzkin monoid, but Theorem 3B.6 and Theorem 5C.6 only establish bounds on the semisimple gap ssGap. Since Gap ≤ ssGap by Lemma 2A.10, these bounds do not determine the exponential growth of the actual RepGap; the claims must be rephrased as semisimple-gap statements or supplemented with genuine RepGap bounds.","section":"§1B and §5C"},{"comment":"The proof that the right cell representations of rTL_n are simple is sketched and contains an internal inconsistency: the text first states that wx ∉ W for a suitable x, but the displayed computation proves xw ∉ W. The proof also assumes without justification that a nontrivial invariant subspace W contains an element with at least one cap. Because this proposition underpins the rTL gap bounds, a complete and correct proof is needed.","section":"§4C, Proposition 4C.1"}],"minor_comments":[{"comment":"The notation '∋ Ratio_K MoT_n' is non-standard and should be replaced by an explicit statement such as 'Ratio_K(MoT_n) ∈ [Ω((1−ε)^n), O((1+ε)^n)]'.","section":"§5C, Corollary 5C.8"},{"comment":"In the displayed formula for |rpRo_n|, the factor 'binom{n+j}{2}^2' should read 'binom{n+j}{k}^2'.","section":"§6B, Proposition 6B.1"},{"comment":"The upper bound for the gap ratio is typeset as '23n/233n/45−5n/4'; this should be 2^{3n/2}3^{3n/4}5^{-5n/4}.","section":"§6C, Theorem 6C.4"},{"comment":"There is a typo: 'the difference wll not play a role' should be 'will not play a role'.","section":"Remark 3B.5"},{"comment":"The bibliography entry [CG23] lists the authors as 'S. Doty and A. Giaquinto', but the citation key begins with 'C'; this mismatch should be fixed.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The dependence on the unpublished thesis [Ar25] for a load-bearing bound is a serious concern for a journal publication; the authors should be asked to include the proof or obtain a public version. The internal inconsistency of Theorem 3A.3 and the ssGap-versus-Gap overclaim in the abstract are likely fixable by careful rewriting, but until they are addressed the central Motzkin comparison is not verifiable from the manuscript alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real paper with new objects and a mostly solid cell-theoretic analysis, but the headline Motzkin comparison is not supported as printed. The stress-test note is right: Theorem 3A.3 asserts both f(n)·9^n ≤ Gap_K(MoT_n) and Gap_K(MoT_n) ≤ 2^{-3/2} n^{-3/2} 4^n, which cannot both hold. And the 9^n lower bound is carried by an unpublished honours thesis [Ar25] with no public text. That is load-bearing for Corollary 5C.8 and the table.\n\nWhat is genuinely new: the rigid Motzkin monoid rMo_n and the rigid planar rook monoid rpRo_n are new (as far as I can tell), and the RepGap computations for rTL_n, rMo_n, rpRo_n are new. The rTL_n cell theory is worked out concretely: left/right cell sizes are explicit binomial sums, Theorem 4C.3 gives matching asymptotics up to constants, and the non-semisimplicity is pinned down. The rpRo_n analysis is clean and gives a semisimple monoid with a clear RepGap estimate. The authors also ship code on GitHub, which is good practice.\n\nSoft spots, in proportion. The TL leg is fine. The planar rook leg is fine. The Motzkin leg is the problem. The abstract says Gap^{1/n} -> 4 for rMo, but the proof only bounds the semisimple RepGap; the actual RepGap is smaller, so this overstates the result. Proposition 4C.1 (simplicity of rTL cell representations) is sketched; it might be fixable but as written it relies on an argument that is not fully detailed. The reliance on [Ar25] is the real issue; if that lower bound is wrong, the Motzkin comparison loses its support.\n\nOverall: the paper deserves a serious referee, not a desk reject. The referee should insist that the Motzkin comparison be either proved in the paper or stated as conditional on [Ar25], and that Theorem 3A.3 be corrected. The TL and pRo results are worth publishing on their own.","headline":"New rigid diagram monoids with a solid TL/planar-rook analysis, but the Motzkin comparison has a load-bearing inconsistency and an unavailable key reference.","tokens_in":33356,"tokens_out":3104,"would_cite":true,"duration_ms":29817,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18D10","20M30","05A16","94A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that rigid, non-pivotal versions of the Temperley–Lieb, Motzkin, and planar rook monoids have representation gaps exponentially smaller than their pivotal counterparts, making them generally worse for cryptographic…","keywords":["representation gap","diagram monoids","Temperley–Lieb monoid","Motzkin monoid","planar rook monoid","rigid monoidal categories","non-pivotal categories","monoid cryptography"],"falsifier":"Compute the smallest nontrivial simple representation of the pivotal Motzkin monoid $\\mathrm{Mo}_n$ for $n = 10$ through $20$ by direct Gram-matrix rank calculation: if the exponential growth factor is below $9^n$, then Corollary 5C.8's comparison collapses.","tokens_in":1897,"feed_emoji":"🔐","tokens_out":3520,"duration_ms":100189,"temperature":0.7,"pith_summary":"The paper asks whether rigid-but-not-pivotal versions of familiar planar diagram monoids are better or worse as cryptographic building blocks, where 'better' means having large representation gaps: large minimal dimensions of nontrivial simple representations, which resist linear-algebra attacks. The authors construct non-pivotal analogs of the Temperley–Lieb, Motzkin, and planar rook monoids and compute asymptotic bounds for their RepGaps and gap ratios. Their central claim is that, contrary to the intuition that a more complicated category should yield a harder algebra, all three non-pivotal monoids have representation gaps that are exponentially worse than their pivotal counterparts. If correct, the conclusion matters for protocol design: among these families, pivotal planar diagram monoids remain the safer candidates, while the non-pivotal ones are comparatively easy to attack by linear methods.","feed_headline":"Relaxing pivotal symmetry weakens diagram monoids for crypto","feed_subtitle":"Temperley–Lieb, Motzkin, and planar rook analogs show smaller representation gaps than their pivotal versions.","key_machinery":"The machinery is cell theory for finite monoids, using Green's relations to decompose diagrams by the number of through strands. For the rigid Temperley–Lieb monoid $\\mathrm{rTL}_n$, every diagram factors as $\\tau \\circ 1_k \\circ \\beta$ with $k$ even through strands, giving right cells of size $\\binom{n-1}{(2n-k)/2}$ and explicit simple-representation dimensions. For the rigid Motzkin monoid, through-strand sequences form a poset and right-cell sizes are computed via Catalan $k$-fold convolutions, simplified with a hypergeometric Chu–Vandermonde identity. For the rigid planar rook monoid, the same sequence combinatorics gives binomial cell sizes and proves the monoid is semisimple over any field. Truncation, by taking Rees quotients that keep only the dominant through-strand interval, is used so that the smallest nontrivial simple representation determines the gap.","core_discovery":"On the paper's own terms, the discovery is that passing from a pivotal to a non-pivotal rigid planar diagram monoid lowers the exponential growth of the RepGap. For the rigid Temperley–Lieb monoid, the gap lies between $\\Omega(n^{-1/2} 2^n)$ and $O(n^{-1/2} 2^n)$, whereas the pivotal TL monoid has gap between $\\Omega(n^{-5/2} 4^n)$ and $O(n^{-3/2} 4^n)$. For the rigid Motzkin monoid, the semisimple gap lies between $\\Omega(n^{-3/2} 4^n)$ and $O(n^{-1} 4^n)$, while the pivotal Motzkin gap has a lower bound of $\\Omega(9^n)$. For the rigid planar rook monoid, the gap is $O(n^{-1/2} 2^n)$ versus a pivotal lower bound of $\\Omega(n^{-1/2} 4^n)$, and its gap ratio decays at most like $0.87^n$. These comparisons support the paper's conclusion that the rigid non-pivotal monoids are generally worse for cryptographic purposes, despite the rigid Temperley–Lieb monoid having a slightly better gap ratio.","pith_inferences":["Editorial inference: the recurring $2^n$-versus-$4^n$ pattern suggests a broader principle—breaking pivotal symmetry may halve the exponential base of the smallest simple representation, so other two-colour or oriented diagram monoids could show similar drops if constructed along the same lines.","One stress test is to compute RepGaps for the full untruncated monoids or for different truncation widths; the paper's conclusions are stated for the chosen truncations, and it is not yet known whether the exponential separation survives under all reasonable truncation choices.","Because the rigid Temperley–Lieb monoid is characteristic-free while the pivotal one changes dramatically in prime characteristic, the cryptographic ranking drawn in characteristic zero may not transfer to protocols over finite fields; positive-characteristic analysis would settle that.","If the unpublished $9^n$ Motzkin bound is independently verified, the same cell-size formulas could be sharpened to give explicit constants, turning the asymptotic $\\Omega$ and $O$ bounds into $\\Theta$ estimates."],"forward_implications":["If the bounds are correct, a cryptosystem built on the rigid Temperley–Lieb monoid can be reduced to linear algebra of dimension at most $O(n^{-1/2} 2^n)$, while the pivotal version offers dimension at least $\\Omega(n^{-5/2} 4^n)$; the rigid version is exponentially easier to attack.","For the Motzkin monoid, even the semisimple dimension, which is an upper bound for the true simple dimension, grows only like $4^n$ in exponential base for the rigid version, compared with a $9^n$ lower bound for the pivotal version, so the rigid version cannot offer comparable security.","For the planar rook monoid, the rigid gap ratio decays like $0.87^n$, meaning the gap becomes exponentially small relative to the square root of the monoid size, the weakest profile among the monoids studied.","The three cases together support the paper's general statement that pivotal planar diagram monoids are better suited than their non-pivotal analogs for the cryptographic applications considered.","The comparison uses the paper's more generous RepGap definition, which agrees with the earlier definition on the reference monoids, so the conclusion is not an artifact of a stricter gap definition."],"supporting_citations":[{"why":"Introduces the RepGap and gap-ratio framework for diagram monoids and supplies the pivotal Temperley–Lieb and planar rook bounds and the truncation method.","marker":"[KST24]"},{"why":"Supplies the $9^n$ lower bound on the pivotal Motzkin RepGap; this bound is load-bearing for Corollary 5C.8, but the thesis is unpublished.","marker":"[Ar25]"},{"why":"Gives the Catalan $k$-fold convolution formula used to compute right-cell sizes in the rigid Motzkin monoid.","marker":"[LS25]"},{"why":"Provides sandwich cellularity and Gram-matrix methods used to prove simplicity of cell representations and to compute simple-representation dimensions.","marker":"[Tu24]"},{"why":"Supplies the definitions of rigid and pivotal monoidal categories and the uniqueness of duals used in Corollary 3B.4.","marker":"[EGNO15]"},{"why":"Records the isomorphism between the planar partition monoid and even-strand Temperley–Lieb monoids, which justifies excluding it from the study.","marker":"[HR05]"}],"fun_headline_variants":["Non-pivotal diagram monoids shrink representation gaps","Pivotal diagram monoids keep larger crypto gaps","Rigid planar monoids lose crypto strength without pivotal symmetry","Smaller gaps make non-pivotal monoids poor for crypto","Crypto gaps diminish in non-pivotal diagram monoids"],"cache_read_input_tokens":35456,"weakest_assumption_plain":"The comparison depends on the unpublished lower bound that the pivotal Motzkin monoid's RepGap grows like $9^n$; if that bound is wrong or overstated, the paper's main conclusion for the Motzkin case loses its support.","fun_headline_variants_meta":{"raw":{"variants":["Non-pivotal diagram monoids shrink representation gaps","Pivotal diagram monoids keep larger crypto gaps","Rigid planar monoids lose crypto strength without pivotal symmetry","Smaller gaps make non-pivotal monoids poor for crypto","Crypto gaps diminish in non-pivotal diagram monoids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1818,"prompt_tokens":854,"completion_tokens":964,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":881}},"tokens_in":470,"tokens_out":964,"duration_ms":9503,"temperature":1.0,"reasoning_tokens":881,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:56:17.332527+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the smallest nontrivial simple representation of the pivotal Motzkin monoid $\\mathrm{Mo}_n$ for $n = 10$ through $20$ by direct Gram-matrix rank calculation: if the exponential growth factor is below $9^n$, then Corollary 5C.8's comparison collapses.","supporting_citations":[],"review_version":1}