{"id":"8297a61c-3a71-4f9f-b658-3e3358737c54","arxiv_id":"2502.02154","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Experiences, modeled as layered networks combined by actual or potential composition, form only a partial order, so many experiences, including across species, are not comparable in size.","lead":"This paper builds a formal model of conscious experience using layered networks and a partial order, concluding that some experiences can be ranked while others cannot be compared at all. A smart generalist might read it because it challenges the common assumption that consciousness in animals and humans can be placed on a single scale.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 3.1 does not yield a poset: permuted same-level chains are mutually comparable but unequal, so the central incomparability result is not established.","rationale":"The reader's weakest_assumption points to the empirical convention that ⊙ is more intense than ⊗. That is a legitimate modeling concern, but the more decisive problem is internal: even granting the convention, the order relation is not antisymmetric, so the poset and all derived claims collapse. The reader's rationale does mention the Definition 3.1 issue, but the weakest_assumption field identifies a different, external concern. I therefore partially agree with the overall rejection but disagree about which assumption is load-bearing. The counterexample x = G⊗H⊗K and y = G⊗K⊗H shows that the paper's own definition makes same-level contents mutually comparable, directly contradicting the central claim that they are incomparable. This is a concrete mathematical error, not a philosophical disagreement, and it is fatal to the claimed rigorous derivation. The verdict remains REJECT: the formal argument does not support the conclusion, and the required fixes would alter the substance of the claim. The test proposed is minimal and decisive: apply the definition to a two-element example and check antisymmetry. No new experiments or empirical data are needed, only a careful re-reading of the definition. This concern is more fundamental than the convention about ⊙ and ⊗ because it undermines the mathematical structure regardless of how that convention is justified.","tokens_in":13245,"tokens_out":5618,"duration_ms":52702,"concrete_test":"Directly test Definition 3.1 with x = G⊗H⊗K and y = G⊗K⊗H for distinct layers G, H, K, using the definition exactly as written. Verify that both x ≤ y and y ≤ x hold while x ≠ y because ⊗ is non-commutative. If both relations hold, antisymmetry is violated, so (G⊗C, ≤) is not a poset and Lemma 3.2 is false. Additionally, check the same-level claim in Example 3.7: chains with zero ⊙ occurrences, such as G⊗H⊗K and G⊗K⊗H, are mutually ≤ under Definition 3.1, contradicting the assertion that same-level configurations are not comparable. This single check settles whether the formal derivation supports the paper's central conclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conclusion depends on Lemma 3.2, which asserts that the relation ≤ of Definition 3.1 is a partial order. This lemma is false. Take three distinct layers G, H, K and the chains x = G⊗H⊗K and y = G⊗K⊗H. Both have index set {1,2,3} and all operations ⊗, so Definition 3.1 gives x ≤ y and y ≤ x: the set-inclusion condition holds and there is no position where the first has ⊙ and the second has ⊗. Since ⊗ is explicitly non-commutative, x ≠ y. Thus ≤ violates antisymmetry and is only a preorder, not a poset. Section 3.2 and Section 4.1 claim that same-level chains such as these are \"not comparable using ≤\", but under the definition they are mutually comparable in both directions. The level-counting by occurrences of ⊙ therefore does not produce the advertised irreducible incomparability of phenomenal contents at the same level. This is not a matter of empirical support for the ⊙>⊗ convention; even granting that convention, the formal foundation fails. The paper omits the proof of Lemma 3.2, and the counterexample shows why. Repairing the definition would require either restricting to index sequences in a fixed order (defeating the non-commutativity), quotienting by permutation (identifying configurations the authors want to keep distinct), or adding a stronger ordering condition that genuinely leaves same-level permutations incomparable. Each option changes the main claim, so the conclusion as stated is unsupported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a multilayer network formalization of conscious experience, with two composition operations: ⊗ (potential/parallel, non-commutative) and ⊙ (actual/unified, commutative). It defines a relation ≤ on chains of layers (Definition 3.1), asserts that this relation is a partial order (Lemma 3.2), and uses it to argue that experiences are only partially comparable: levels of consciousness (counted by occurrences of ⊙) are comparable, while phenomenal contents at the same level are not. The partiality is then applied to intra-species and inter-species comparisons, with implications for animal consciousness and evolutionary debates.","tokens_in":13672,"tokens_out":9881,"duration_ms":93415,"significance":"The paper addresses a timely and important question: whether phenomenal experiences can be quantitatively compared, and it attempts to formalize partial comparability within a single mathematical structure. Its explicit acknowledgment of the modeller's role and its minimal-assumption approach are strengths, as is the ambition to connect a formal framework to debates on animal consciousness. However, the central mathematical claim is not currently supported: the proposed relation ≤ is not a partial order, and the paper's own conclusion about same-level incomparability is contradicted by Definition 3.1. The intended construction may be repairable, but as written the formal foundation fails. No machine-checked proofs or reproducible code are provided; the key lemma is stated without proof.","major_comments":[{"comment":"The relation ≤ is not antisymmetric, so (G⊗C,≤) is not a poset. Let G, H, K be distinct layers and set x = G⊗H⊗K and y = G⊗K⊗H. Both chains have index set {1,2,3} and all operations are ⊗, so Definition 3.1 gives x ≤ y and y ≤ x. Since ⊗ is explicitly non-commutative (Section 2.2), x ≠ y. Thus Lemma 3.2 is false, and the paper's statement in Section 3.2 that 'every two chains of layers living on the same level ... although not comparable using ≤' is contradicted by the definition. This also invalidates the central conclusion of Section 4.1 that phenomenal contents at the same level are not comparable. The proof of Lemma 3.2 is omitted; the counterexample shows why it cannot be supplied as stated.","section":"Definition 3.1, Lemma 3.2"},{"comment":"The claim that configurations at the same level are 'mathematically not comparable' is not a consequence of the stated definition; it is a property of the stipulated relation, and the stipulated relation actually makes same-level permutations comparable in both directions. The intended result may be obtainable by modifying Definition 3.1, for instance by requiring that the ordered sequence of layers on the left is a prefix of the ordered sequence on the right, but such a change must be made and all subsequent examples and propositions must be rechecked. As written, the central theorem is unsupported.","section":"Section 3.2, Example 3.7"},{"comment":"The convention that ⊙ is 'more complex' or 'phenomenologically more intense' than ⊗ is a modeling stipulation with no independent empirical support. The paper acknowledges this, but the entire ordering, the counting of levels by occurrences of ⊙, and all derived comparability claims depend on it. If this priority is not accepted, the partial order and its consequences do not follow from the multilayer structure alone. The authors should state explicitly that the results are conditional on this stipulation and, if possible, indicate what empirical or phenomenological data could test it.","section":"Section 3.1"}],"minor_comments":[{"comment":"The notation V(G⊙H):=V(G)⊔V(H) together with p=|V(G)∩V(H)| is inconsistent; if V(G) and V(H) overlap, disjoint union would duplicate shared vertices rather than identify them, so the formula for the number of vertices is unclear.","section":"Definition 2.4"},{"comment":"The paper identifies the layer (G,colG) with G but then writes G∈G⊗C; G⊗C is defined as a set of multilayers, so the terminology 'layer' and 'multilayer' is used inconsistently.","section":"Section 2.1"},{"comment":"The diagram uses arrows labeled f1 and f2, but some edges, for example from K⊗H⊗G, have two f2 arrows, which is confusing; a precise definition of the arrows in the diagram would improve readability.","section":"Example 3.7"},{"comment":"The reference list contains an unattached bibliographic entry ('A.M.S. Colloquium Publications, vol. 25, Revised Edition, New York, 1948.') that appears to be a leftover from template text.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The formal error in Definition 3.1 is serious but appears repairable by requiring equality of layer order in the definition of ≤. I recommend major revision rather than rejection, provided the authors rework the definition and verify all dependent claims. If they cannot produce a correct poset definition that preserves the intended consequences, rejection would be appropriate. The paper would also benefit from positioning the ⊙>⊗ priority more explicitly as an empirical hypothesis that could be tested against behavioral or phenomenological data."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the paper alongside your stress-test note, and the stress-test lands. Definition 3.1 does not define a partial order. Take x = G⊗H⊗K and y = G⊗K⊗H. Under the definition, x ≤ y and y ≤ x because the index sets are the same and neither chain has a ⊙ where the other has ⊗. Since ⊗ is non-commutative, x ≠ y. Antisymmetry fails, so Lemma 3.2 is false. This is load-bearing: the paper's central conclusion, that experiences at the same level are not comparable using ≤, is actually contradicted by the definition, since these same-level chains are mutually comparable in both directions.\n\nWhat the paper does well: it is a clear, honest attempt to put levels and contents of consciousness into one structure using multilayer networks. The distinction between potential composition (⊗) and actual composition (⊙) is a useful intuition, and the authors are transparent about their modeling choices. The conceptual discussion of evolutionary consciousness, and the point that evolution may not be an absolute scalar, is sensible and well framed. The paper also engages the literature fairly.\n\nThe soft spots are serious. The antisymmetry failure is enough to sink the formal argument. Definition 2.4 also has an internal inconsistency: it defines V(G⊙H) as a disjoint union and then subtracts |V(G)∩V(H)|, which doesn't parse. And the level-counting by the number of ⊙ operations does not yield incomparability, because chains with the same count are mutually comparable under the rule as written. The philosophical conclusion may still be plausible, but it is not supported by the formal derivation.\n\nThis paper is for consciousness scientists who want a formal handle on comparability, but in its present form it is not reliable. I think it deserves a serious referee because the empirical and philosophical questions matter, and the writing is clear enough that the problems can be diagnosed and, possibly, fixed. But the revision would be major: either repair the definition and prove the poset lemma, or explicitly present the structure as a preorder and adjust the conclusions accordingly. As it stands, the central result is unsupported.","headline":"The paper's central formal claim is false: Definition 3.1 yields a preorder, not a poset, so the advertised incomparability of same-level phenomenal contents does not follow.","tokens_in":14102,"tokens_out":3709,"would_cite":false,"duration_ms":38180,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Modelling experiences as layered coloured networks ordered by potential and actual composition forces quantitative comparisons to be partial: levels of consciousness are rankable by counting actual compositions, but phenomenal contents at…","keywords":["Animal Consciousness","Consciousness","Structuralism","Subjective experience","Poset","Multilayer network","Partial order"],"falsifier":"A concrete falsifier would be a reliable empirical demonstration that two experiences with the same number of actual compositions—for example, $G \\odot H \\otimes K$ and $G \\otimes H \\odot K$—are nonetheless consistently ranked by subjects as more or less intense or complex, using a measurement that does not itself presuppose the model's ordering convention.","tokens_in":13045,"feed_emoji":"🧠","tokens_out":9812,"duration_ms":84130,"temperature":0.7,"pith_summary":"This paper asks how far quantitative comparison of conscious experiences can go. It models each experience as a coloured multigraph (a layer), allows layers to be combined either in parallel ($\\otimes$, potential composition) or merged ($\\odot$, actual composition), and orders the resulting chains by the rule that a configuration is larger when its layer set is contained in another's and no actual composition in the smaller chain corresponds to a potential composition in the larger one. The main result is that this ordered structure yields only partial comparability: levels of consciousness, defined by counting actual compositions $\\odot$, are comparable, but phenomenal contents at the same level—such as $G \\otimes H \\otimes K$ versus $G \\otimes K \\otimes H$—are not. The authors conclude that even under the optimistic assumption that all experiences are quantitatively comparable, absolute rankings of experiences, especially across species, do not follow from the mathematics; evolution may instead produce different, partly incomparable modes of experience.","feed_headline":"Consciousness levels compare; same-level contents don't","feed_subtitle":"A layered ordered model says partial comparison is the most a quantitative theory of experience can get.","key_machinery":"The central object is a poset (partially ordered set) of experiential layers built from coloured multigraphs. Each layer is a graph whose edges carry colours representing aspects or modalities of experience, and layers are assembled into chains using two operations: $\\otimes$ (potential or parallel composition, non-commutative) and $\\odot$ (actual or merged composition, commutative, with priority over $\\otimes$). The partial order on chains is defined componentwise: a chain $x$ is $\\leq y$ exactly when $x$'s layer set is contained in $y$'s and no position where $x$ uses $\\odot$ has $y$ using $\\otimes$. The maps $f_j$, which convert a single $\\otimes$ into $\\odot$ at position $j$, are order-preserving and generate upward trajectories; the count of $\\odot$ appearances then defines levels, while the non-commutativity of $\\otimes$ leaves same-level configurations incomparable.","core_discovery":"Under the assumption that experiences are quantitatively comparable and can be described by layers, the paper's central discovery is that an ordered structure for comparing them leads only to partial comparison. In the poset of layer concatenations, experiences are comparable only when one chain's layer set is contained in the other's and no position has the smaller chain using actual composition ($\\odot$) while the larger uses potential composition ($\\otimes$). As a direct consequence, levels of consciousness—the horizontal strata in the poset where every chain has the same number of $\\odot$ appearances—are comparable, while phenomenal contents on the same level are mathematically incomparable. The same partiality applies inter-species: configurations with more layers but fewer actual compositions, such as $G \\otimes H \\otimes K \\otimes L$, cannot be ordered against configurations with fewer layers but more actuality, such as $G \\odot H \\otimes K$, so a larger evolutionary repertoire does not automatically imply a 'bigger' conscious experience.","pith_inferences":["A natural empirical extension is to test whether human subjects can consistently rank experiences that the model declares incomparable (same number of $\\odot$ but different orderings); a reliable ranking would indicate the model is missing a dimension.","The non-determinism of the inverse mappings suggests a formal parallel with dissociative or attentional phenomena, where moving from a unified experience to its parts does not uniquely determine which parts will be attended to.","One could import a metric or geometric structure onto the poset to make same-level contents comparable; the paper's framework then becomes a scaffold for comparing such extra assumptions.","The inter-species conclusion could sharpen debates on animal consciousness: instead of asking which species has 'more' consciousness, one should ask which experiential maxima each species attains within its own layer structure."],"forward_implications":["Levels of consciousness can be partially ordered by the number of actual compositions $\\odot$, so claims like 'awake is higher than deep sleep' have a formal footing in the model.","Phenomenal contents at the same level cannot be ranked by the model in terms of complexity or intensity; any such ranking would require additional structure beyond the poset.","Inter-species comparisons are only partial: more experiential layers do not automatically make a species' conscious experience larger, because actuality ($\\odot$) and potentiality ($\\otimes$) trade off.","The poset constrains experiential trajectories: application of $f_j$ always increases experience, while the inverse relation $g_j$ is non-deterministic, so reverse transitions are underdetermined.","If the arguments hold, models of consciousness that assume an absolute scalar of experiential richness must either drop that assumption or add extra mathematical structure to justify it."],"supporting_citations":[{"why":"Supplies the multilayer network formulation of experience that the paper extends with two composition operations.","marker":"[19]"},{"why":"Provides the levels-of-consciousness versus phenomenal-content distinction and the idea of a quantifiable dimension of experience that the model formalizes.","marker":"[27]"},{"why":"Gives the categorical relational approach to levels and contents of consciousness that the paper's structural account builds on and combines with multilayer theory.","marker":"[28]"},{"why":"Source of the compositional (tensor/parallel) treatment of experience that underpins the $\\otimes$ operation.","marker":"[25]"},{"why":"Supplies the axiomatic and graphical mathematics used to reason about conscious experience, grounding the compositional structure.","marker":"[24]"},{"why":"The classic statement of cross-species experiential incomparability that the paper's inter-species conclusion formalizes.","marker":"[26]"},{"why":"Provides evidence of the difficulty of measuring subjective experience in non-human primates, supporting the challenge of inter-species comparison.","marker":"[12]"},{"why":"Gives the lattice/poset theory definitions on which Definition 3.1 and Lemma 3.2 rest.","marker":"[4]"}],"fun_headline_variants":["Levels of consciousness rank, same-level contents don't","Partial order: only experience levels are comparable","Same-level phenomenal states defy quantitative ranking","Experience poset: levels ordered, contents incomparable","Evolution's experiences: diverse, not on an absolute scale"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the modelling convention that actual composition ($\\odot$) is phenomenologically more intense and more complex than potential composition ($\\otimes$); if real experiences do not consistently respect that ordering, the poset, the level-counting by $\\odot$, and all derived incomparability claims lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Levels of consciousness rank, same-level contents don't","Partial order: only experience levels are comparable","Same-level phenomenal states defy quantitative ranking","Experience poset: levels ordered, contents incomparable","Evolution's experiences: diverse, not on an absolute scale"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000441,"raw_usage":{"total_tokens":2238,"prompt_tokens":949,"completion_tokens":1289,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":1216}},"tokens_in":565,"tokens_out":1289,"duration_ms":13316,"temperature":1.0,"reasoning_tokens":1216,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:07:55.476825+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete falsifier would be a reliable empirical demonstration that two experiences with the same number of actual compositions—for example, $G \\odot H \\otimes K$ and $G \\otimes H \\odot K$—are nonetheless consistently ranked by subjects as more or less intense or complex, using a measurement that does not itself presuppose the model's ordering convention.","supporting_citations":[{"cited_title":"Multilayer networks as embodied consciousness interactions","cited_arxiv_id":null,"evidence_quote":"Supplies the multilayer network formulation of experience that the paper extends with two composition operations."},{"cited_title":"The Qualia Structure Paradigm : towards a construction of a Qualia Periodic Table for the dissolution of the Hard Problem of Consciousness.PsyArXiv, pages 1–26, 2024","cited_arxiv_id":null,"evidence_quote":"Provides the levels-of-consciousness versus phenomenal-content distinction and the idea of a quantifiable dimension of experience that the model formalizes."},{"cited_title":"A relational approach to consciousness: categories of level and contents of consciousness.Neuroscience of Consciousness, 2021(2):1–14, oct 2021","cited_arxiv_id":null,"evidence_quote":"Gives the categorical relational approach to levels and contents of consciousness that the paper's structural account builds on and combines with multilayer theory."},{"cited_title":"A Compositional Model of Consciousness Based on Consciousness-Only.Entropy, 23(3):308, mar 2021","cited_arxiv_id":null,"evidence_quote":"Source of the compositional (tensor/parallel) treatment of experience that underpins the $\\otimes$ operation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the axiomatic and graphical mathematics used to reason about conscious experience, grounding the compositional structure."},{"cited_title":"Leopold, Alexander Maier, and Nikos K","cited_arxiv_id":null,"evidence_quote":"Provides evidence of the difficulty of measuring subjective experience in non-human primates, supporting the challenge of inter-species comparison."},{"cited_title":"Birkhoff","cited_arxiv_id":null,"evidence_quote":"Gives the lattice/poset theory definitions on which Definition 3.1 and Lemma 3.2 rest."}],"review_version":1}