{"id":"9c311786-7b6a-4eb9-8cda-b3ecd03f4fbe","arxiv_id":"1908.09621","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two systems with identical state-transition topology can have Phi greater than zero and Phi equal to zero, differing only by a permutation of internal binary labels, so IIT's Phi is not invariant under isomorphisms.","lead":"This paper shows that two physical systems with identical abstract state-transition diagrams can receive very different consciousness scores under Integrated Information Theory, one positive and one zero. It argues that any theory of consciousness should be invariant under such isomorphisms, or else it admits philosophical zombies.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'only difference is a label permutation' claim holds at the abstract-automaton level, but the constructed systems also differ in element decomposition and causal architecture; IIT's ontology is the latter, so the critique is conditional on functionalism.","rationale":"The reader's diagnosis is correct: the weakest point is the move from automata isomorphism to 'same computation with arbitrary labels.' The examples are mathematically constructive and the Phi computations are plausible; the paper should get credit for isolating a class of feedback systems with isomorphic feedforward counterparts. But the strongest claim overstates what is shown. The two systems differ in the decomposition into elements, and IIT explicitly defines consciousness at that level. The paper's Section 3.1 caveat is an admission that the functionalist premise is doing the work. This does not make the paper worthless; it makes the central conclusion conditional. No need to change the reader's verdict. A factorization test on h would separate the label-permutation component from the architecture-change component and tell us whether the constructed bijections respect IIT's element structure.","tokens_in":11107,"tokens_out":16137,"duration_ms":184409,"concrete_test":"Formalize an IIT system as a TPM together with a product-state decomposition Q = {0,1}^n (the element coordinates). For each of the two pairs (X,X') and (Y,Y'), enumerate the bijection h constructed in Section 2.3.1 and Section 3 and check whether h factors through the coordinate projections: h(q1,...,qn) = (g1(q_{sigma(1)}), ..., gn(q_{sigma(n)})) for some coordinate permutation sigma and bit flips g_i. If h does not factor, then X and X' (and Y and Y') do not share the same element decomposition; the Phi difference is then attributable to a change of physical architecture, consistent with IIT, and the label-sensitivity conclusion fails. If h does factor, the label-sensitivity conclusion is strengthened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.2 and Section 3 define an isomorphism at the level of the global state-transition diagram, and the examples in Section 2.3.1 and Section 3 are indeed related by a bijective relabeling of the binary states. But IIT 3.0 does not take an unlabeled state-transition diagram as its object. It takes a transition probability matrix together with a distinguished decomposition into physical elements (mechanisms); Phi is defined with respect to that decomposition (Oizumi et al. 2014). The feedback system Y and the cascade system Y' differ not only in the names of the global states but in the wiring and in which physical elements exist and how they interact: the original has two XNOR gates and one XOR with cyclic dependencies; the zombie has NOT/COPY/COPY in cascade. The bijection h from Section 3 maps states of Y to states of Y', but it does not map the element decomposition of Y to that of Y'. Thus there is a second mathematical difference, at the implementation level, that the paper's 'only a permutation' summary suppresses. The manuscript itself concedes this in Section 3.1: 'there exists an alternative interpretation ... in terms of the specific logical implementation, for example as IIT adopts.' It then says only that this alternative is not testable. That is not an argument; it is a statement of the hard problem. Because IIT's axioms build in the physical causal architecture as constitutive, an automata-isomorphic feed-forward system is not an IIT-zombie unless one first adopts the functionalist premise that the unlabeled computation is the right level of analysis. The proposed invariance criterion therefore rests on the contested premise rather than deriving it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript challenges Integrated Information Theory (IIT) by constructing feedback systems with Φ>0 and isomorphic feed-forward systems with Φ=0. The authors work with finite-state automata and use preserved partitions to produce cascade decompositions that preserve the global state-transition diagram up to a bijective relabeling. The two worked examples are an AND/OR system mapped to COPY/OR and a three-element system with two XNOR gates and one XOR mapped to a NOT/COPY/COPY cascade. The paper argues that because the only apparent difference between the two realizations is a permutation of binary labels, Φ is sensitive to functionally arbitrary internal representation, and it proposes that any quantitative theory of consciousness should be invariant under isomorphisms of the state-transition diagram.","tokens_in":11390,"tokens_out":11834,"duration_ms":118441,"significance":"If the central claim were fully established, this would be a significant challenge to IIT: it would exhibit equal-size, equal-state-count systems with identical global state-transition graphs but different IIT verdicts. The examples are explicit and reproducible, and the paper is commendably clear that the invariance criterion is a proposal rather than a theorem. The main weakness is that the conclusion depends on identifying 'computation' with the unlabeled global state-transition topology, which is precisely the functionalist premise that IIT's constitutive decomposition into physical mechanisms rejects. The manuscript acknowledges this alternative in Section 3.1 but does not refute it, so the paper is best read as a conditional critique that sharpens the philosophical disagreement rather than as a definitive refutation of IIT.","major_comments":[{"comment":"The claim that the only mathematical difference between Y and Y′ is a permutation of binary labels is inaccurate with respect to IIT's formal objects. IIT defines Φ on a transition probability matrix together with a specified decomposition into physical elements (Oizumi et al. 2014). The two systems differ in element-level wiring—two XNOR gates and one XOR with cyclic coupling versus NOT/COPY/COPY in cascade—and the bijection h maps global states but does not map the element decomposition. Therefore Y and Y′ are not isomorphic in the mathematical category in which Φ is defined, and the conclusion that Φ is sensitive to 'functionally arbitrary' labeling requires the additional premise that the unlabeled global transition topology is the correct notion of function. The paper should state this premise explicitly and defend it, or the conclusion should be weakened to a conditional one.","section":"Abstract and Section 3 (Figures 6–8)"},{"comment":"The proposed invariance criterion ('any quantitative theory of consciousness ... should be invariant under isomorphisms') is a normative assertion, not a consequence of IIT's axioms or of the preceding mathematical construction. The manuscript acknowledges that IIT defines behavior in terms of the specific logical implementation and dismisses this as untestable from outside. But the untestability of a constitutive assumption from a third-person perspective does not refute it; the hard problem is precisely that such assumptions are not directly observable. The paper therefore does not so much refute IIT as locate the disagreement in the choice between functionalist and implementation-based notions of computation. This is a legitimate contribution, but the abstract and title overstate the result as demonstrating the existence of isomorphic zombies rather than zombies under a contested functionalist criterion.","section":"Section 3.1"},{"comment":"The first example begins by 'leaving off the binary labels' of the physical AND/OR system and treating the unlabeled diagram as the computation. This move is not neutral: the labels of a physical circuit are determined by the outputs of its gates, and an isomorphic relabeling forces a different gate implementation (COPY/OR instead of AND/OR). The statement that both systems 'operate on the same support' conflates the abstract state set with the physical states of the circuits. The 'isomorphism' is between the abstract directed graph and the relabeled graph, not between the two physical circuits, so the phrase 'only a permutation' should be reserved for the global state-transition diagram and explicitly accompanied by the caveat that the physical architectures differ.","section":"Section 2.3.1 (Figure 4)"}],"minor_comments":[{"comment":"There is a typo: 'Karnuagh maps' should be 'Karnaugh maps'.","section":"Section 3"},{"comment":"The definition of a homomorphism refers to a map on states, stimuli, and transitions, but the commuting diagram in Figure 2 and the subsequent examples use only a single input symbol (the passage of time). The paper would be clearer if it explicitly stated that the input alphabet is trivial in the examples and that the isomorphism is of the underlying deterministic state-transition graph.","section":"Section 2.2"},{"comment":"The caption says the two diagrams 'differ only by a permutation of labels'; given the difference in gate wiring, this should be qualified as 'at the level of the global state-transition diagram' to avoid contradicting the paper's own Section 3.1 discussion of the alternative interpretation.","section":"Figure 8 caption"},{"comment":"The word 'feed-back' should be 'feedback' for consistency with the rest of the text.","section":"Acknowledgements"}],"recommendation":"major_revision","confidential_remarks":"The mathematical constructions in the examples are correct, and the paper contributes a useful sharpening of the functionalism debate around IIT. However, the central rhetorical claim—that the systems differ only by label permutation—is false at the level of IIT's mechanism-based object, and the invariance principle is asserted rather than argued. The paper is suitable for publication only after the authors clearly separate the conditional mathematical result from the normative philosophical conclusion and engage directly with IIT's constitutive decomposition premise. The topic fits a consciousness-theory or mathematical-biology venue, but the present framing overstates the strength of the challenge."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The construction is real and worth knowing. The paper gives explicit, worked examples of feedback systems with Phi > 0 that are isomorphic at the level of the global state-transition diagram to feed-forward systems with Phi = 0, preserving system size and state count. That is genuinely sharper than the earlier Krohn-Rhodes or Doerig et al. zombie arguments, which only gave homomorphic (many-to-one) emulation. The nested preserved partition method is clearly explained, and the two examples check out as internally consistent. I verified the logic of the AND/OR and XNOR/XOR/XOR rewrites; the cascade forms do have unidirectional partitions that leave the TPM unchanged, so Phi = 0 follows from IIT's own rules. The authors also honestly cite prior work and do not oversell the novelty relative to that literature.\n\nThe soft spot is the interpretive leap, and it is load-bearing. The mathematical fact is that two automata with the same unlabeled transition diagram can have different Phi. The paper's stronger claim, that the only difference is a 'functionally arbitrary' label permutation, only holds if you grant that the unlabeled state-transition topology is the right object for a theory of consciousness. IIT's axioms explicitly target the physical mechanism decomposition, not the abstract automaton. The two systems also differ in wiring and element decomposition; the bijection maps global states, not mechanisms. The paper acknowledges this in Section 3.1 and then dismisses the implementation-based reading as untestable. That is not a refutation, it is a statement of the hard problem. So the paper does not establish that IIT is wrong; it establishes that IIT is non-invariant under a specific abstraction choice, and that the burden is on IIT to justify why its abstraction is the correct one.\n\nThe invariance criterion at the end is a reasonable proposal, but it is a philosophical premise, not a derivation. The paper is honest about this in the final paragraph, which mitigates the overreach in the abstract. I would not call the central argument unsound; I would call it conditional on functionalism. Within that frame, the mathematics is solid and the examples are reproducible.\n\nWho should read this? Consciousness researchers, especially anyone working on IIT or causal structure theories, and automata theorists interested in decomposition. It deserves a serious referee: the formal result is novel and the philosophical challenge is well posed, even if the conclusion needs to be softened. I would send it to peer review and ask the authors to clearly separate the mathematical theorem from the functionalist interpretation.","headline":"A clean mathematical construction that gives IIT critics a sharper zombie argument, but the paper's own conclusion overstates its case by sliding from automata isomorphism to functional equivalence.","tokens_in":11937,"tokens_out":634,"would_cite":true,"duration_ms":8721,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q70","68Q45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two circuits with identical state-transition diagrams can have opposite values of the integrated-information measure $\\Phi$, differing only by a permutation of binary labels, so the paper argues any theory of consciousness must be…","keywords":["integrated information theory","philosophical zombies","feed-forward decomposition","cascade decomposition","preserved partitions","finite-state automata","state-transition isomorphism","consciousness measurement"],"falsifier":"Take a concrete finite-state automaton with feedback and enumerate every bijective relabeling of its internal states, constructing each as a distinct logical architecture. The paper's claim predicts that at least one relabeling yields a strictly feed-forward architecture with $\\Phi = 0$ while the original has $\\Phi > 0$ under the standard $\\Phi$ algorithm; if no such relabeling exists for any finite-state automaton, or if every relabeling of a given transition diagram yields the same $\\Phi$ value, the central claim is falsified.","tokens_in":10899,"feed_emoji":"🧠","tokens_out":8603,"duration_ms":84857,"temperature":0.7,"pith_summary":"The paper argues that any quantitative theory of consciousness should assign the same value to two systems whose state-transition diagrams are identical up to relabeling. The authors focus on the integrated-information measure $\\Phi$, the central quantity of a leading phenomenological theory of consciousness, and build explicit pairs of small circuits: one with feedback and $\\Phi > 0$, the other strictly feed-forward with $\\Phi = 0$, yet with exactly the same state-transition topology. The only mathematical difference between the two is which binary strings label the internal states. If state-transition topology is what a computation is, then $\\Phi$ rewards an arbitrary labeling convention rather than the computation itself, and no phenomenological difference can justify the large difference in $\\Phi$. The paper therefore proposes invariance under isomorphism as a minimal test any theory of consciousness must pass to avoid philosophical zombies.","feed_headline":"A label swap turns a conscious circuit into a zombie","feed_subtitle":"Identical state-transition topology, one with feedback and $\\Phi>0$, the other feed-forward with $\\Phi=0$, so $\\Phi$ tracks labels, not…","key_machinery":"The central object is the preserved partition of a finite-state automaton's state space: a grouping of states into blocks such that every state in a block transitions to a single block. A nested sequence of preserved partitions, each splitting the previous blocks evenly in half, supplies the coordinates of a new automaton whose logic gates are read off from block-to-block transitions. For the paper's examples, this sequence yields a cascade, or strictly feed-forward, architecture with the same global state-transition diagram as the original feedback architecture. The cascade form guarantees $\\Phi = 0$ because any unidirectional partition of it leaves the dynamics unchanged, while the original circuit has $\\Phi > 0$. This machinery isolates the effect of internal labeling: the isomorphism is a dictionary between representations, and the only mathematical difference between the two systems is the permutation of binary labels used to instantiate the same computation.","core_discovery":"The central claim is that a system satisfying the integrated-information criteria for consciousness ($\\Phi > 0$) can have an isomorphic counterpart that is a philosophical zombie by the same theory's standards ($\\Phi = 0$), where the isomorphism is a bijective relabeling of internal binary states that preserves every state transition. The authors demonstrate this by explicit construction: an AND/OR feedback pair and a three-bit XNOR/XOR/XOR system are each decomposed into cascade form through a nested sequence of preserved partitions. In the decomposed system information flows strictly forward, so any unidirectional partition leaves the transition probability matrix unchanged, forcing $\\Phi = 0$ for all states; yet the global state-transition diagram is exactly the original diagram with states renamed. The paper concludes that $\\Phi$ depends on the internal representation of a computation, not on the computation itself, and that any quantitative measure of consciousness should be invariant under such isomorphisms: measurable differences in consciousness must correspond to measurable differences in the state-transition function.","pith_inferences":["Editor's inference: the same invariance test could be applied to other causal-structure theories of consciousness or to measures of causal emergence; any such measure that changes under a bijective relabeling of internal states would face the same isomorphic-zombie problem.","Editor's inference: the examples suggest a sharper, computable claim: enumerating all bijective relabelings of a given transition diagram and computing $\\Phi$ for each isomorphic circuit would quantify how much of $\\Phi$ is labeling artifact rather than graph structure.","Editor's inference: if the invariance criterion is accepted, the integrated-information framework would need to define $\\Phi$ on the quotient of circuits under isomorphism, effectively turning $\\Phi$ into a graph-theoretic quantity on the state-transition diagram.","Editor's inference: because $\\Phi$ is independent of computation on the paper's account, any future theory making $\\Phi$ central must explain why organisms would evolve high-$\\Phi$ wirings rather than their equivalent low-$\\Phi$ isomorphic counterparts."],"forward_implications":["Any measure of consciousness that satisfies the proposed invariance criterion will assign identical values to isomorphic circuits, so it cannot single out physical feedback as a necessary condition for consciousness in these examples.","The integrated-information measure $\\Phi$, as currently defined, fails the criterion: isomorphic systems exist with $\\Phi > 0$ and $\\Phi = 0$.","The two members of each pair cannot be told apart by an outside observer tracking state transitions; if one is conscious and the other is not, the difference is causally silent in the state dynamics.","The invariance criterion is necessary but not sufficient for a theory of consciousness to be free of philosophical zombies; further constraints are still required.","Consciousness measures should be evaluated on equivalence classes of physical implementations that realize the same state-transition topology, rather than on a single wiring diagram."],"supporting_citations":[{"why":"Defines the $\\Phi$ measure and the assumption that feedback is a necessary condition for consciousness; this is the target the isomorphic zombies are built against.","marker":"[5]"},{"why":"Proves that every finite automaton admits a cascade decomposition, the general theorem underlying feed-forward emulation.","marker":"[14]"},{"why":"Shows how to synthesize finite-state machines in cascade form, guiding the explicit constructions in the paper.","marker":"[15]"},{"why":"Presents an earlier 'unfolding argument' that causal-structure theories of consciousness admit feed-forward counterparts, motivating the isomorphic version.","marker":"[13]"},{"why":"Supplies the algebraic theory of machines, including homomorphisms and preserved partitions, the formal backbone of the decomposition method.","marker":"[16]"},{"why":"Provides the preserved-partition technique for decomposing sequential machines, which the paper adapts for its isomorphic decompositions.","marker":"[22]"},{"why":"The computational toolbox used to verify that the example systems have $\\Phi > 0$.","marker":"[25]"},{"why":"Karnaugh-map method used to identify the logic gates in the decomposed coordinates of the examples.","marker":"[26]"},{"why":"Characterizes $\\Phi$ as a distance between a transition matrix and its best unidirectional partition, linking the measure to the presence or absence of feedback.","marker":"[20]"}],"fun_headline_variants":["Same circuit, new labels: consciousness vanishes","Label shuffling creates philosophical zombies from conscious circuits","Φ vanishes under a mere relabeling of internal states","Swapping binary labels turns a conscious circuit into a zombie","Consciousness measure Φ is not invariant under state relabeling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on treating a bijective relabeling of internal binary states as producing the same computation with no physical or semantic difference; if the actual physical wiring or the environment can privilege one labeling over another, then the isomorphic zombie is not genuinely equivalent and the challenge to $\\Phi$ collapses.","fun_headline_variants_meta":{"raw":{"variants":["Same circuit, new labels: consciousness vanishes","Label shuffling creates philosophical zombies from conscious circuits","Φ vanishes under a mere relabeling of internal states","Swapping binary labels turns a conscious circuit into a zombie","Consciousness measure Φ is not invariant under state relabeling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00055,"raw_usage":{"total_tokens":2681,"prompt_tokens":1054,"completion_tokens":1627,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":1549}},"tokens_in":670,"tokens_out":1627,"duration_ms":12682,"temperature":1.0,"reasoning_tokens":1549,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:22:58.776165+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete finite-state automaton with feedback and enumerate every bijective relabeling of its internal states, constructing each as a distinct logical architecture. The paper's claim predicts that at least one relabeling yields a strictly feed-forward architecture with $\\Phi = 0$ while the original has $\\Phi > 0$ under the standard $\\Phi$ algorithm; if no such relabeling exists for any finite-state automaton, or if every relabeling of a given transition diagram yields the same $\\Phi$ value, the central claim is falsified.","supporting_citations":[{"cited_title":"From the phenomenology to the mechanisms of consciousness: integrated information theory 3.0","cited_arxiv_id":null,"evidence_quote":"Defines the $\\Phi$ measure and the assumption that feedback is a necessary condition for consciousness; this is the target the isomorphic zombies are built against."},{"cited_title":"Algebraic theory of machines","cited_arxiv_id":null,"evidence_quote":"Proves that every finite automaton admits a cascade decomposition, the general theorem underlying feed-forward emulation."},{"cited_title":"Cascade synthesis of finite-state machines","cited_arxiv_id":null,"evidence_quote":"Shows how to synthesize finite-state machines in cascade form, guiding the explicit constructions in the paper."},{"cited_title":"Algebraic theory of machines, languages, and semi-groups; Academic Press, 1968","cited_arxiv_id":null,"evidence_quote":"Supplies the algebraic theory of machines, including homomorphisms and preserved partitions, the formal backbone of the decomposition method."},{"cited_title":"Algebraic structure theory of sequential machines (prentice-hall international series in applied mathematics); Prentice-Hall, Inc., 1966","cited_arxiv_id":null,"evidence_quote":"Provides the preserved-partition technique for decomposing sequential machines, which the paper adapts for its isomorphic decompositions."},{"cited_title":"PyPhi: A toolbox for integrated information theory","cited_arxiv_id":null,"evidence_quote":"The computational toolbox used to verify that the example systems have $\\Phi > 0$."},{"cited_title":"The map method for synthesis of combinational logic circuits","cited_arxiv_id":null,"evidence_quote":"Karnaugh-map method used to identify the logic gates in the decomposed coordinates of the examples."},{"cited_title":"Improved measures of integrated information","cited_arxiv_id":null,"evidence_quote":"Characterizes $\\Phi$ as a distance between a transition matrix and its best unidirectional partition, linking the measure to the presence or absence of feedback."}],"review_version":1}