{"id":"76b0e334-e725-49eb-b595-380250dc3ca4","arxiv_id":"2606.09031","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence, uniqueness, and canonical description via random limit currents of the measure of maximal fiber entropy for random skew products on compact complex surfaces, with entropy equal to the Furstenberg exponent.","lead":"The paper proves that random skew products on a compact complex surface, driven by a measure on its automorphism group, have a unique measure of maximal fiber entropy under logarithmic integrability and non-elementary generation conditions. This measure is built from random limit currents and has fiber entropy equal to the Furstenberg exponent from the random cohomology action.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the non-elementary assumption as weakest is accurate, and the abstract-only limitation is already reflected in the UNVERDICTED verdict. No further load-bearing concern emerges from the claim as stated.","tokens_in":1666,"tokens_out":247,"duration_ms":11818,"concrete_test":"Extract the precise statement of the main theorem (including all hypotheses) and compare it line-by-line with the construction of the measure in the section that invokes the random limit currents; confirm that the entropy equality is derived directly from the definition of the Furstenberg exponent without additional hidden assumptions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states a theorem whose hypotheses (logarithmic integrability of μ and non-elementary generation by supp(μ)) are the standard ones used to guarantee positive Furstenberg exponent and non-degenerate random limit currents in this setting. The claimed equality between fiber entropy and Furstenberg exponent, as well as the canonical description via Cantat–Dujardin currents, follows the expected outline for such results; no internal gap or unsecured step is visible from the given statement.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that for a compact complex surface X and Borel probability measure μ on Aut(X) satisfying a logarithmic integrability condition with supp(μ) generating a non-elementary subgroup, the associated skew product admits a unique invariant measure of maximal fiber entropy. This measure is described canonically via the random limit currents of Cantat and Dujardin, its fiber entropy equals the Furstenberg exponent of the random action on cohomology, and the measure is mixing under an additional exponential moment assumption.","tokens_in":1766,"tokens_out":375,"duration_ms":14856,"significance":"If the result holds, it canonically identifies the measure of maximal fiber entropy for these random skew products and equates its entropy to the Furstenberg exponent, providing a concrete link between random dynamics, limit currents, and cohomology actions on complex surfaces. The use of standard hypotheses to guarantee non-degenerate currents and the equality itself are strengths that connect entropy directly to the linear action without fitted parameters.","major_comments":[],"minor_comments":[{"comment":"The introduction should include an explicit numbered statement of the main theorem (existence, uniqueness, canonical description, and entropy equality) with a forward reference to the section containing the proof.","section":"Introduction"},{"comment":"Clarify the precise definition of 'fiber entropy' for the skew product (e.g., via the disintegration over the base or as an integral of conditional entropies) and confirm it matches the quantity whose maximality is asserted.","section":null},{"comment":"Add the full bibliographic reference for the Cantat–Dujardin random limit currents at first mention, rather than leaving it implicit.","section":"Abstract"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments appear in the report.","responses":[],"tokens_in":1130,"tokens_out":45,"duration_ms":7841,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result is that for a measure μ on Aut(X) with the usual log-integrability and non-elementary support conditions, the skew product has a unique invariant measure of maximal fiber entropy. It is described explicitly via the Cantat-Dujardin random limit currents, and its fiber entropy equals the Furstenberg exponent of the induced action on cohomology. Under an exponential moment assumption the measure is mixing.\n\nThis is a direct extension of the deterministic picture to the random setting. The canonical description via the currents is the part that feels most useful; it turns an existence statement into something one can work with. The equality to the Furstenberg exponent is the cleanest link if the derivation goes through.\n\nThe hypotheses are the standard ones already used to get positive exponents and non-degenerate currents, so they do not look like an artificial restriction. The mixing statement under exponential moments is a natural strengthening.\n\nThe main limitation visible from the abstract is that the actual proofs are not here to check. The equality and uniqueness steps are the load-bearing parts, and without seeing how the currents are used to construct the measure or how the entropy computation avoids circularity, it is hard to gauge how much new work is required. No obvious gaps in the outline, but that is all one can say.\n\nThis is for people already working in random complex dynamics or ergodic theory on surfaces. A reader who knows the Cantat-Dujardin currents and Furstenberg exponents will get the most out of it. It is worth sending to referees; the statement is precise enough and the objects are well-motivated that a careful check could confirm a solid incremental result.","headline":"Cohen gives a canonical maximal fiber entropy measure for random skew products on complex surfaces and equates its entropy to the Furstenberg exponent.","tokens_in":2255,"tokens_out":407,"would_cite":false,"duration_ms":13511,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A unique measure of maximal fiber entropy exists for skew products arising from random automorphisms on compact complex surfaces.","keywords":["random skew products","maximal entropy measures","complex surfaces","fiber entropy","Furstenberg exponent","limit currents","random dynamics","automorphisms"],"falsifier":"An explicit measure μ whose support generates a non-elementary subgroup but for which either multiple distinct measures achieve the same fiber entropy or the entropy value differs from the Furstenberg exponent.","tokens_in":2561,"feed_emoji":"","tokens_out":680,"duration_ms":24945,"temperature":0.7,"pith_summary":"The paper proves that when a Borel probability measure on the automorphism group of a compact complex surface satisfies logarithmic integrability and its support generates a non-elementary subgroup, the associated skew product has a unique invariant measure maximizing fiber entropy. This measure is constructed canonically from random limit currents, and its entropy value equals the Furstenberg exponent of the induced random action on cohomology. Under an added exponential moment condition the measure is also mixing. A sympathetic reader would care because the result supplies a concrete, computable object for the long-term statistical behavior of iterated random holomorphic maps on surfaces.","feed_headline":"Unique maximal fiber entropy measure for random surface automorphisms","feed_subtitle":"Skew products from non-elementary random actions on compact complex surfaces have a canonical invariant measure whose entropy equals the Fur","key_machinery":"Random limit currents, which canonically construct the measure of maximal fiber entropy and equate its entropy to the Furstenberg exponent.","core_discovery":"Let X be a compact complex surface. The skew product associated to a Borel probability measure μ on Aut(X) admits a unique invariant measure of maximal fiber entropy, assuming that μ satisfies a logarithmic integrability condition and that supp(μ) generates a non-elementary subgroup of Aut(X). We describe this measure canonically in terms of the random limit currents constructed by Cantat and Dujardin, and show that its fiber entropy is equal to the Furstenberg exponent of the associated random action on cohomology. Under an exponential moment assumption, we prove that it is mixing.","pith_inferences":["The same construction could be tested on other surfaces where analogous random limit currents have been shown to exist.","Explicit computation of the Furstenberg exponent for concrete finitely generated subgroups might now yield numerical values for maximal fiber entropies.","The mixing property under exponential moments suggests that correlation decay estimates could be derived from the same current-based description."],"forward_implications":["The fiber entropy of the unique maximal measure equals the Furstenberg exponent of the random cohomology action.","The measure is mixing whenever μ also satisfies an exponential moment condition.","The construction applies to every Borel probability measure on Aut(X) meeting the stated integrability and generation hypotheses.","Uniqueness of the maximal fiber entropy measure follows directly from the non-elementary generation assumption."],"fun_headline_variants":["Maximal fiber entropy measure unique for random skew products on surfaces","Random limit currents canonically define max fiber entropy measure on surfaces","Fiber entropy equals Furstenberg exponent in non-elementary random actions","Max fiber entropy measure is mixing under exponential moment condition"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That the support of μ generates a non-elementary subgroup of Aut(X), which is needed to produce well-defined non-degenerate limit currents and guarantee uniqueness.","fun_headline_variants_meta":{"raw":{"variants":["Maximal fiber entropy measure unique for random skew products on surfaces","Random limit currents canonically define max fiber entropy measure on surfaces","Fiber entropy equals Furstenberg exponent in non-elementary random actions","Max fiber entropy measure is mixing under exponential moment condition"]},"model":"grok-4.3","cost_usd":0.007487,"raw_usage":{"total_tokens":3320,"prompt_tokens":597,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":74865500,"prompt_tokens_details":{"text_tokens":597,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2655,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":597,"tokens_out":68,"duration_ms":20641,"temperature":1.0,"reasoning_tokens":2655,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T15:00:56.332340+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit measure μ whose support generates a non-elementary subgroup but for which either multiple distinct measures achieve the same fiber entropy or the entropy value differs from the Furstenberg exponent.","supporting_citations":[],"review_version":1}