{"id":"7413cf9f-fb93-404e-86c7-c6153bdb8f47","arxiv_id":"2411.17486","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A cut-elimination based realisability model over proof nets is adequate and complete for MLL and MLL*.","lead":"Proof theorists introduce a new 'daimon' opponent that lets abstract proof graphs interact through cut elimination, defining realisability by which graphs win. They prove that a graph that wins in every interpretation basis is exactly a valid proof for the multiplicative fragment of linear logic, with and without generalised axioms.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 188's proof is a sketch whose load-bearing splitting claim is dismissed with 'one can easily check'; the tensor case of Theorem 88 (MLL completeness) is unsupported as written, so the conditional verdict should hinge on filling this gap.","rationale":"The reader's conditional verdict rests on Proposition 30 and, in the rationale, also names 'the splitting proposition' (Proposition 188). I partially agree. Proposition 30 is foundational, but it is supported by long, concrete proofs in Appendix B (Prop 92 strong normalisation via the (connective, cut) lexicographic measure; Prop 97 confluence of unrelated cuts; Prop 103 commutation of multiplicative cuts; Props 109–114 the delay/anticipate lemmas). The risk there is an undetected error in an intricate case analysis, but the argument is present. Proposition 188 is different: the argument is absent exactly where the work is, in two ways. First, the base-case sentence about empty intersections is incoherent as written. Second, the decisive non-orthogonality claim is dismissed as 'one can easily check'. This matters because Theorem 88 (MLL completeness) is a headline claim and its tensor case needs Prop 188 through Lemma 189. The step is genuinely delicate: orthogonality demands reduction to the single empty daimon, and component-wise convergence to ✠0 is not enough. I traced one small interaction (a‖a) against the merge of two par-nets and it normalises to a sum of several daimons rather than to a single ✠0, so any proof of Prop 188 must track daimon counts and connectivity rather than argue componentwise; the 'easy check' is not evidently routine. The proposed test is an exhaustive bounded computation of exactly these interactions, which either produces a counterexample to Prop 188 (refuting Theorem 88) or verifies the omitted claim on the minimal non-trivial nets. I am not claiming the theorem is false; I am claiming the written argument does not establish a headline result, which is what a conditional verdict should gate on. Hence verdict_should_be is UNCHANGED: the reader's CONDITIONAL is right, and this note identifies concretely which proposition must be expanded.","tokens_in":54,"tokens_out":51537,"duration_ms":957590,"concrete_test":"Implement Definitions 19/22 (cut elimination) and Definition 42 (orthogonality) for nets of at most ten links and run two bounded checks. Let a = b = ⟨⊲✠p1⟩+⟨⊲✠p2⟩+⟨p1,p2⊲⊗p⟩ (one conclusion, two binary daimons), take atoms X ≠ Y⊥, and the legal basis B with ⟦X⟧_B = ⟦Y⟧_B = {a}⊥ (a∉{a}⊥ because a⊥a fails by the (⊗/⊗) clash cut, while ✠1∈{a}⊥). First, seek a witness of non-membership: for R ranging over merges s⊲⊳s′ of one-conclusion nets of size at most three (elements of {a}⊥⊥ /Yright {a}⊥⊥ by Prop 178), check whether (a‖a)::R has a reduction to ✠0; any R with no such reduction shows a‖a∉⟦X‖Y⟧_B, consistent with Prop 188, and one such R is enough.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing gap is Proposition 188 (Appendix G.6), used in Lemma 189 and in the tensor case of Theorem 88, the paper's stated MLL completeness result. Prop 188 asserts a strong splitting property: if a‖b ∈ ⟦A⟧_B‖⟦B⟧_B for every basis B (with A, B variable-disjoint), then a ∈ ⟦A⟧_B and b ∈ ⟦B⟧_B for every basis B. In proof-net and Ludics completeness arguments, such a splitting is normally the hardest step; here the proof is a sketch with three concrete defects. (1) The base case ends with 'Since furthermore ⋂_B ⟦X⟧_B is empty it follows that ⟦X⟧_B‖⟦Y⟧_B is also empty thus the proposition hold', which is incoherent: each ⟦X⟧_B is a type (non-empty for the daimon basis 1), and emptiness of an intersection neither follows nor implies the claimed splitting. (2) The decisive step — 'we assume that a✠∉⟦g(A)⟧_B for some basis ... one can easily check that a✠‖b✠ is not orthogonal to s⊲⊳s′ for all s′∈⟦g(B)⟧_B' — is exactly the polarisation analysis needed to generalise Figure 15 and Remark 90 to nets whose daimon outputs instantiate non-dual atoms; it is not a routine check, because orthogonality requires reduction to a single ✠0 and component-wise convergence is insufficient (parallel pairs can deadlock at ✠0‖✠0). (3) The equivalence 'a✠∈⟦g(A)⟧_B iff a∈⟦A⟧_B' is asserted without proof. Consequence: the tensor step of Theorem 88's induction cannot be verified as written. This is an internal proof gap, not a consensus disagreement. The surrounding paper is unusually careful — Prop 30's rewriting lemmas get full Appendix B proofs, and Figure 14 documents the dependency graph — which makes the one sketchy step that carries a headline theorem stand out.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a realisability model for multiplicative linear logic (MLL and MLL✠) based on orthogonality between untyped proof nets with daimon (generalised axiom) links. The central construction is a cut-elimination system for nets that includes homogeneous and non-homogeneous cuts, an orthogonality relation defined by reduction to the empty daimon, and a type interpretation of formulas. The authors prove adequacy (Theorem 64) and two completeness theorems: Theorem 85 for MLL✠ and Theorem 88 for MLL, the latter applying to nets whose daimons are binary and atomically labelled. The proof is supported by a long appendix ending with a dependency graph. The main claimed contribution is that interactive realisability over nets characterises provability exactly.","tokens_in":99,"tokens_out":11009,"duration_ms":150400,"significance":"The claimed result is significant: if the proof is completed, this is the first complete realisability model for multiplicative linear logic over general proof nets, and the use of non-homogeneous cut elimination to separate geometrical from provability correctness is a genuine novelty. The paper is careful in several respects: it proves the rewriting properties of the new cut-elimination system in detail (Proposition 30 and Appendix B), it provides a proof dependency graph (Figure 14), and the completeness direction is not obtained by fitting parameters but by a bootstrap through the daimon basis 1 and the independent Danos-Regnier criterion. My main concern is concentrated on one load-bearing lemma, Proposition 188, whose proof is a sketch; the surrounding material appears sound.","major_comments":[{"comment":"The base case of the proof of Proposition 188 ends with the sentence 'Since furthermore ⋂_B ⟦X⟧_B is empty it follows that ⟦X⟧_B‖⟦Y⟧_B is also empty thus the proposition hold.' This is logically incoherent. For each basis B, ⟦X⟧_B is a type, and for the approximable basis 1 it contains ✠1, so each of these types is non-empty at a fixed basis. Emptiness of the intersection over all bases does not imply emptiness of the parallel composition at a fixed basis; indeed the parallel composition of two non-empty types contains the parallel sum of any two of their elements, and is generally non-empty. The argument therefore does not establish the claimed splitting a∈⟦X⟧_B and b∈⟦Y⟧_B.","section":"Appendix G.6, Proposition 188, base case"},{"comment":"The decisive step of the proof of Proposition 188 is the assertion 'one can easily check that a✠‖b✠ is not orthogonal to s⊲⊳s′ for all s′∈⟦g(B)⟧_B', together with the unproved equivalence 'a✠∈⟦g(A)⟧_B iff a∈⟦A⟧_B'. This is precisely the polarisation/deadlock analysis needed to generalise Figure 15 and Remark 90 from atomic dualities to arbitrary variable-disjoint formulas; it is not a routine check, because orthogonality requires reduction to a single ✠0 and component-wise convergence is insufficient (parallel pairs may get stuck at ✠0‖✠0). Since Proposition 188 is used directly in Lemma 189 and then in the tensor case of Theorem 88, the proof of the MLL completeness theorem (Theorem 88) is incomplete as written.","section":"Appendix G.6, Proposition 188, main step"}],"minor_comments":[{"comment":"The displayed multiplicative cut-elimination rule contains a typo: the result is written as '⟨p1,q1⊲cut⟩+⟨q2,q2⊲cut⟩', but the second cut should be '⟨p2,q2⊲cut⟩'. The same typo appears in the description of the rule in the text.","section":"Figure 3 and Section 1.2"},{"comment":"The caption of Figure 4 first states that p1, p2, q1, q2 are fresh positions and then explains that q1 and q2 may be elements of a or b and that p1 and p2 may be elements of q1,...,qn. This is confusing: the 'fresh' naming and the reuse of the same letters for existing positions should be reconciled.","section":"Figure 4 and Remark 24"},{"comment":"Items (2) and (3) of Proposition 30 use the notation 'S c− →·→∗ S′' and 'S→∗· c− →S′' without recalling that '·' denotes relation composition in this paper. Since these commutation statements are used pervasively, a parenthetical reminder at the first use would improve readability.","section":"Proposition 30"},{"comment":"The proof of Theorem 77 invokes the 'counter-proof criterion' of [4] without a proof or a precise statement in the present paper. Since the framework here extends the setting with non-homogeneous cut elimination, a more self-contained treatment of this external criterion would strengthen the paper.","section":"Theorem 77"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is carefully written and the proof infrastructure is unusually detailed, with a proof dependency graph and extensive appendices. The blocking issue is localised: Proposition 188 (Appendix G.6) is the only place where I found a load-bearing gap, but it is used in the proof of Theorem 88, so the MLL completeness result is not established as written. I would ask the authors to supply a full proof of Proposition 188, including the base case and the polarisation analysis, and to state explicitly how the 'one can easily check' step avoids the deadlock issue at ✠0‖✠0. If that lemma is replaced by a rigorous argument, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing you should know: this is not a routine variant of Curien or Beffara. The non-homogeneous cut elimination rules for daimons against tensor and par are the real novelty, and they let the authors state the first completeness theorems for an orthogonality-based realisability model over untyped nets. The architecture is convincing: adequacy holds by induction, testability collapses with typeability under an approximable basis, and the basis 1 is chosen so that realisability reduces to testability, which is then identified with correctness via Danos-Regnier. The rewriting theory in Appendix B is given full proofs, and the proof dependency graph is a nice touch that makes the paper unusually checkable.\n\nNow the soft spot, in proportion to how much it matters. Theorem 88, the MLL completeness result, depends on Proposition 188 for the tensor case. As written, that proposition's proof does not hold up. The base case ends with the claim that an empty intersection of the interpretations of a variable makes the parallel composition empty; that is incoherent, since each interpretation is non-empty for the daimon basis and the parallel composition contains the very net you're analysing. The decisive step, where a supposed failure of a to realise A is used to show a daimon-labelled opponent defeats every test of B, is dismissed with 'one can easily check' — but this is exactly the polarisation analysis that needs spelling out, especially because orthogonality requires reduction to a single daimon and parallel pairs can get stuck. And the equivalence between membership of the daimon skeleton in the ground interpretation and membership of the original net in the type is asserted without proof. This is not a gap in an auxiliary lemma; it is the load-bearing step of the main MLL completeness theorem. As far as I can see, Theorem 85 (MLL* completeness) does not rely on Proposition 188, so that part may well be fine. The rewriting lemmas in Proposition 30 are also heavily used, but those are actually proved in the appendix, so they are less worrying.\n\nI think the paper is worth engaging with seriously. The idea is fresh, the presentation is careful, and the gap in Proposition 188 looks like the kind of thing that could be repaired with an honest and technical proof — or, alternatively, it might reveal a genuine obstruction to the splitting property. The referee should be asked to focus attention there and not to desk-reject the paper. It belongs in the review process, with a clear request to expand or replace that proof.\n\nRecommendation: send it to a serious referee. The paper is for the linear logic semantics community, and it will be read with care by anyone working on realisability or proof nets. I'd cite it if the gap gets fixed; I probably still would cite it for the cut-elimination rules and the MLL* completeness, even if Theorem 88 is not fully supported in this version.","headline":"A genuinely new and mostly careful realisability model for MLL/MLL* over nets, but the stated completeness for MLL rests on Proposition 188, whose proof is too sketchy to check as written.","tokens_in":60656,"tokens_out":2517,"would_cite":true,"duration_ms":27162,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03F52","03B47"],"pacs":[],"model":"deepseek-v4-flash","headline":"The central claim is that a cut-free net proves a sequent exactly when it survives interaction with every opponent in every interpretation basis.","keywords":["linear logic","realisability","orthogonality","proof nets","daimons","cut elimination","completeness","multiplicative fragment"],"falsifier":"Try to construct a cut-free net $S$ and sequent $\\Gamma$ such that $S\\in \\llbracket \\Gamma\\rrbracket_B$ for every basis $B$ while $S$ does not represent an MLL✠ proof of $\\Gamma$; Theorem 85 says none exists. A more local check is to search for a reduction sequence where the factorisation $\\to^* = \\to^*_{\\text{mult}}\\cdot \\to^*_{\\neg\\text{mult}}$ fails, or where a $(\\text{✠}/\\otimes)$-cut cannot be commuted left of another step without changing the normal form.","tokens_in":59535,"feed_emoji":"🔗","tokens_out":8950,"duration_ms":81889,"temperature":0.7,"pith_summary":"This paper aims to show that provability in multiplicative linear logic can be recovered purely from interaction between untyped proof nets. It builds a realisability model in which formulas denote types, sets of nets closed under bi-orthogonality, and two nets are orthogonal when their cut-elimination interaction reaches the empty daimon. The novelty is a cut-elimination procedure for generalised axioms: a daimon link cut against a tensor or par link gets new rewriting rules, making the daimon an adaptive opponent that never stops answering. The paper proves adequacy and completeness for MLL✠, multiplicative linear logic with generalised axioms, and then derives completeness for standard MLL. The upshot is that geometrical correctness and provability correctness both become interactively testable in one uniform model.","feed_headline":"Untyped nets interact to decide provability in MLL","feed_subtitle":"A new cut-elimination step makes interactive realisability complete for both MLL✠ and MLL.","key_machinery":"The machinery is orthogonality defined by cut elimination on untyped multiplicative nets, extended by non-homogeneous rules for cuts between a daimon link $\\langle \\vdash_{\\text{✠}} p_1,\\dots,p_n\\rangle$, a generalised axiom with no premises and any number of conclusions, and a tensor or par link. Two nets interact by pairing their ordered conclusions with cut links; they are orthogonal if the interaction rewrites to the empty daimon $\\text{✠}_0$. The load-bearing rewriting properties are collected in Proposition 30: strong normalisation, a factorisation of every reduction as multiplicative steps followed by non-multiplicative steps, and commutation lemmas saying that irreversible (✠/par)-cuts can be delayed while reversible (✠/⊗)-cuts can be anticipated. These lemmas justify the orthogonality relation and the type constructions that carry the model.","core_discovery":"The central claim is Theorem 85: for a cut-free net $S$ and a sequent $\\Gamma$, if $S$ belongs to $\\llbracket \\Gamma \\rrbracket_B$ for every interpretation basis $B$, then $S$ proves $\\Gamma$ in MLL✠. Theorem 88 is the same statement for MLL, restricted to nets whose daimons are binary and atomically labelled. With the adequacy theorem, this makes the orthogonality model complete as well as adequate: the nets realising a sequent in all bases are exactly the proof nets of the system. The proof runs through a particular basis, $1$, which maps each atom to the bi-orthogonal closure of the single-output daimon, and through a decomposition argument showing that any realiser in such a basis must be testable by the sequent and therefore correct.","pith_inferences":["Beyond the paper, the same orthogonality recipe could be tried on richer fragments of linear logic, with the cut-elimination commutation lemmas as the main obstacle; the paper itself treats only multiplicatives.","The completeness proof for MLL deliberately exploits geometrically incorrect opponents, suggesting that incorrectness can serve as a computational resource for separating provability from mere geometry rather than as noise to be filtered out.","A concrete extension would be to map out which pairs of non-equivalent sequents are separated by specially chosen non-approximable bases; the paper shows one such pair, $X,X^\\perp$ versus $X,Y$, leaving the general separation question open."],"forward_implications":["A cut-free net that realises a sequent in every basis must be an actual proof, so the model rules out universal realisers that are geometrically correct but not provable.","Completeness transfers from MLL✠ to MLL, so the same interactive criterion recognises ordinary multiplicative proofs once daimons are restricted to binary, atomically labelled links.","The factorisation of cut elimination gives correctness testing a phase structure: multiplicative interactions can be resolved first and daimon interactions afterwards, which is what allows tests to probe provability as well as geometry.","For cut-free nets in an approximable basis, testability by a sequent and correct typeability coincide, aligning the semantic notion of realisability with the syntactic notion of proof."],"supporting_citations":[{"why":"supplies the original proof-net formalism and the multiplicative cut elimination that this paper extends with daimons.","marker":"[7]"},{"why":"provides the paraproof-net setting, the counter-proof criterion, and the atomic completeness phenomenon this paper generalises.","marker":"[4]"},{"why":"supplies the switching correctness criterion whose partitions are turned into interactive tests.","marker":"[5]"},{"why":"gives the untyped proof-structure syntax and cut-elimination basis for the nets considered here.","marker":"[9]"},{"why":"contributes the pole-based realisability idea that the paper adapts to a symmetric, net-based orthogonality.","marker":"[12]"}],"fun_headline_variants":["Net realisability now complete for MLL* and MLL","Cut elimination unlocks complete realisability for nets","Realising a sequent in all bases means it is provable","Interactive nets characterise provability in MLL and MLL*","Untyped nets: realisability equals provability exactly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model collapses if any of the rewriting lemmas for the new non-homogeneous cut elimination fails—specifically strong normalisation, the factorisation into multiplicative then non-multiplicative steps, or the commutation properties that let reversible (✠/⊗)-cuts be anticipated and irreversible (✠/par)-cuts be delayed.","fun_headline_variants_meta":{"raw":{"variants":["Net realisability now complete for MLL* and MLL","Cut elimination unlocks complete realisability for nets","Realising a sequent in all bases means it is provable","Interactive nets characterise provability in MLL and MLL*","Untyped nets: realisability equals provability exactly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000292,"raw_usage":{"total_tokens":1612,"prompt_tokens":762,"completion_tokens":850,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":378,"completion_tokens_details":{"reasoning_tokens":774}},"tokens_in":378,"tokens_out":850,"duration_ms":8050,"temperature":1.0,"reasoning_tokens":774,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:04:48.992069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Try to construct a cut-free net $S$ and sequent $\\Gamma$ such that $S\\in \\llbracket \\Gamma\\rrbracket_B$ for every basis $B$ while $S$ does not represent an MLL✠ proof of $\\Gamma$; Theorem 85 says none exists. A more local check is to search for a reduction sequence where the factorisation $\\to^* = \\to^*_{\\text{mult}}\\cdot \\to^*_{\\neg\\text{mult}}$ fails, or where a $(\\text{✠}/\\otimes)$-cut cannot be commuted left of another step without changing the normal form.","supporting_citations":[{"cited_title":"Proof-nets: The parallel syntax for p roof-theory","cited_arxiv_id":null,"evidence_quote":"gives the untyped proof-structure syntax and cut-elimination basis for the nets considered here."},{"cited_title":"Realizability in classical logic","cited_arxiv_id":null,"evidence_quote":"contributes the pole-based realisability idea that the paper adapts to a symmetric, net-based orthogonality."}],"review_version":1}