{"id":"f445211e-10ad-41ce-bfd2-970a09ce4888","arxiv_id":"2501.05353","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every β above the critical inverse temperature, the φ^4 random cluster model on Z^d has a unique macroscopic cluster with high probability, yielding surface-order large deviations and spectral gap decay.","lead":"This paper proves that the supercritical phase of the φ^4 spin model is well behaved in every dimension: large boxes contain one dominant cluster with high probability, uniformly in boundary conditions. It then uses this to prove surface-order exponential bounds for magnetization large deviations and for the spectral gaps of two natural Markov dynamics above the critical temperature.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 rests on half-space uniqueness (Prop. 3.6), which in turn relies on Corollary 3.5, whose proof is explicitly omitted; until that infinite-volume double tangled-current construction is supplied, the comparison with the free random-cluster measure in Lemma 5.9 has a gap.","rationale":"The paper's architecture is coherent, and much of it consists of checkable estimates. The random cluster representation, the FKG inequalities, the monotonicity properties, and the Edwards-Sokal coupling are developed in detail. The strict positivity of surface tension (Proposition 5.5) and the transfer to disconnection bounds are carefully argued. The genuinely load-bearing step is not a numerical computation or an unchecked simulation; it is the half-space uniqueness result. Theorem 1.4 is proved by reducing to Theorem 5.1, and Theorem 5.1 is proved by reducing free-boundary disconnection probabilities to those with positive boundary fields. That reduction must go through the half-space measures, and their definition and equality require the infinite-volume double tangled-current measure of Corollary 3.5, explicitly marked 'the proof is omitted'. An omitted proof of a technical construction can be acceptable if it is routine, but here it is not routine: Proposition 3.6's proof is expressed entirely in terms of that limiting object, and Lemma 3.7 is a Burton-Keane statement about the limiting single-current graph H_1 under that measure. The reader's conditional verdict is therefore appropriate. I do not see evidence of a deeper circularity: the use of [GPPS22] is a citation to prior theorems, not a reduction of the main claim to its input, and the dependence on [Sev24] for Proposition 6.2 is confined to the d >= 3 sprinkling step and appears to match the needed statement. The suggested test, supplying Corollary 3.5, would settle the concern; if it closes, the conditional verdict can be upgraded.","tokens_in":73119,"tokens_out":6535,"duration_ms":64739,"concrete_test":"Write out Corollary 3.5. Concretely, prove tightness of the sequences (P^{emptyset}_{Lambda^+_L[p+h],beta})_L and (P^{emptyset,emptyset}_{Lambda^+_L[p+h],Lambda^+_L[h],beta})_L, and show that their finite-dimensional cylinder probabilities converge as L tends to infinity with limits satisfying the switching identity in the limit. A pass/fail criterion: if Lemma A.1 together with Proposition 2.6 give uniform exponential bounds on 2^{sum n_e} for currents restricted to any finite Delta, and the half-space consistency from Lemma 3.4 closes, then Corollary 3.5 holds and the main proof is complete. If convergence can be established only for the single-current measures but not for the double measure on (n1,n2,t), then Proposition 3.6 needs a different proof and Theorem 1.4 remains conditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.4 depends on Theorem 5.1, whose proof in Section 5.3 uses Lemma 5.9 to pass from thick-plus boundary fields to free boundary conditions. Inside Lemma 5.9, the bulk contribution is controlled by comparing the conditional measure to half-space measures Psi^{0,s}_{H,beta} and Psi^{1,s}_{H,beta}; equality of these half-space measures is Proposition 4.15, and that equality is exactly Proposition 3.6. The proof of Proposition 3.6 uses the switching identity (3.12) and then passes to the limit L tending to infinity in the double tangled-current measure P^{emptyset,emptyset}_{Lambda^+_L[p+h],Lambda^+_L[h],beta}. That limit is asserted in Corollary 3.5 with the sentence 'the proof is omitted'. This is not a cosmetic omission: the limiting object appears as the measure P under which equation (3.14) and Lemma 3.7 must be verified, and the entire argument that the boundary field does not change the infinite-volume magnetisation is expressed through that measure. If the double-current sequence is not tight or the limiting cylinder probabilities are inconsistent, the comparison in Lemma 5.9 lacks a rigorous foundation, so Theorem 5.1, and hence Theorem 1.4, is not established as written. The gap is local and likely fillable using [GPPS22, Section 5.1] and Lemma 3.4, but it is load-bearing because it enters in every dimension d >= 2. A secondary concern is that Proposition 6.2 is imported from [Sev24]; it is needed only for d >= 3 and appears to be a faithful adaptation, so I do not treat it as the primary issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the supercritical phase of the φ^4 model on Z^d through its random cluster representation, defined as an Ising Fortuin–Kasteleyn model on a random environment given by the absolute value field. The central result, Theorem 1.4, states that for every d≥2 and every β>β_c the model satisfies local uniqueness of macroscopic clusters with probability tending to one, uniformly in boundary conditions; this is the φ^4 analogue of Bodineau's supercritical sharpness result for the Ising random cluster model and is new even in two dimensions. The proof combines (i) a positivity result for a surface tension defined via thick boundary fields, adapting Lebowitz–Pfister using the Ginibre inequality and the tangled-current switching principle of [GPPS22]; (ii) a comparison argument from thick-plus to free boundary conditions that rests on a half-space uniqueness statement with positive boundary field; and (iii) renormalisation and sprinkling arguments following [Sev24], with the two-dimensional case handled via the RSW theory of Köhler-Schindler and Tassion. From Theorem 1.4 the authors derive surface-order exponential bounds for lower large deviations of the empirical magnetisation (Theorem 1.1) and surface-order decay of spectral gaps for Langevin and heat-bath dynamics (Theorem 1.3). The paper is detailed and its overall architecture is coherent; the main caveats are the proof being omitted for Corollary 3.5 and a constant error in Proposition 6.3.","tokens_in":73396,"tokens_out":13060,"duration_ms":111002,"significance":"If the results hold, they constitute a substantial advance: supercritical sharpness for a percolation model with unbounded continuous randomness, at every β>β_c and uniformly in boundary conditions, in all dimensions d≥2, with two nontrivial consequences (surface-order large deviations and spectral gap decay) that were previously available only at very low temperature or only for Ising-type models. The paper's strengths include the first rigorous treatment of the φ^4 random cluster representation, a new probabilistic proof of half-space uniqueness that bypasses the Ising-specific wetting-transition argument of Fröhlich and Pfister (and, as the authors note, yields a new proof for the Ising model as well), and a careful adaptation of Pisztora's coarse graining to unbounded spins. The reliance on [GPPS22] for the switching principle and the classification of translation-invariant Gibbs measures is legitimate: those are established theorems with independent proofs, and the central claim does not reduce to a fit of parameters. The completeness issues described in the major comments are local and appear fillable rather than fundamental.","major_comments":[{"comment":"Corollary 3.5 asserts the existence of the infinite-volume double tangled current measures P∅_{H(+,h),β}, P∅_{H(0,h),β} and P^{∅,∅}_{H(+,h),H(0,h),β}, but its proof is omitted. This statement is load-bearing: in the proof of Proposition 3.6, the measure P^{∅,∅}_{H(+,h),H(0,h),β} is the object under which equation (3.14) and Lemma 3.7 are verified, and Proposition 3.6 is exactly the half-space uniqueness used (through Proposition 4.15) in the comparison of Lemma 5.9 that yields Theorem 5.1 and hence Theorem 1.4, in every dimension d≥2. The authors should either provide the construction of these limiting measures (for instance following [GPPS22, Section 5.1] and Lemma 3.4) or state the convergence as a lemma with a complete proof.","section":"Section 3.2 (Corollary 3.5)"},{"comment":"Section 6.2.1, Proposition 6.3: with the stated choice C0 = (d c1)^{-1/(d-1)}, Theorem 5.1 gives Ψ^0_{Λ_{L'}}[Λ_ℓ ↔ ∂Λ_{c1 L'}] ≥ 1 − e^{−c1 ℓ^{d−1}} = 1 − L^{−1/d}, not the claimed 1 − L^{−d}, since c1 ℓ^{d−1} = (1/d) log L. The subsequent union bound over the O(L^d) boxes Λ_ℓ(x), x ∈ ℓZ^d ∩ Λ_{δL}, therefore does not control the complement of A_L, so the asserted limit inf_{(ξ,b)} Γ^{(ξ,b)}_{Λ_L,β,ε}[A_L] → 1 is not proved as written. This is repaired by setting C0 = (d/c1)^{1/(d−1)} (i.e., ℓ^{d−1} = (d/c1) log L), which preserves the requirement ℓ = o(log L) used in Proposition 6.2; I ask that the constant be corrected.","section":"Section 6.2.1 (Proposition 6.3)"}],"minor_comments":[{"comment":"The reference [HSV14] appears in the bibliography but is not cited anywhere in the text; please either cite it where relevant or remove it.","section":"References"},{"comment":"In the introductory outline, 'Pizstora [Pis96]' should read 'Pisztora [Pis96]'.","section":"Introduction, Section 1.4"},{"comment":"The word 'straightforwad' in Remark 3.2 should read 'straightforward'.","section":"Remark 3.2"},{"comment":"The phrase 'by Markov's inequality for 2^{∑_{e:e∩γ≠∅} n1(e)+n2(e)}' is unclear; it should say that Markov's inequality is applied to the random variable 2^{∑_{e:e∩γ≠∅} (n1(e)+n2(e))}.","section":"Section 3.1 (Eq. (3.3))"},{"comment":"The proof of Proposition 6.2 is delegated to [Sev24, Proposition 2.2]. Since this statement is load-bearing for the d≥3 part of Theorem 1.4, I recommend adding a short explanation of why the geometry of the event A_L matches the hypotheses of [Sev24, Proposition 2.2] (in particular the requirement ℓ = o(log L)), rather than only asserting that the proof applies mutatis mutandis.","section":"Section 6.2.1 (Proposition 6.2)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a strong paper with two local but genuine issues: the omitted proof of Corollary 3.5, which is load-bearing for the half-space uniqueness and hence for Theorem 1.4, and an arithmetic slip in Proposition 6.3 that breaks the union bound as stated. Both appear readily fixable within the scope of a revision: the former by importing the construction from [GPPS22, Section 5.1], the latter by correcting the constant C0. I recommend major_revision rather than reject; I would be happy to see a revised version. I did not find signs of circularity or misplaced citation: the use of [GPPS22] is by the same group, but those results are established elsewhere with independent proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this for the first rigorous treatment of the φ^4 random cluster representation and for the main result: local uniqueness for every β>β_c in all d≥2. If the proof holds, this resolves a long-standing open problem and gives the right tools for surface-order large deviations and spectral gap decay. The architecture is sound: surface tension positivity via Lebowitz–Pfister adapted to thick boundary fields, transfer to the random cluster model, comparison to free boundary conditions, then renormalisation à la Sev24/Pisztora. The new half-space uniqueness proof via the switching principle is genuinely novel and looks like the right way to avoid extending Fröhlich–Pfister to φ^4.\n\nThe soft spot is Corollary 3.5. The proof is explicitly omitted, and the object it constructs — the infinite-volume double tangled current measure on the half-space with positive boundary field — is what Proposition 3.6 and hence Lemma 5.9 rely on. This is not cosmetic. The comparison in Lemma 5.9 is expressed through that measure, and if the sequence is not tight or the cylinder limits don't exist, Theorem 5.1 and Theorem 1.4 don't follow as written. The gap looks local and probably fillable using the machinery of [GPPS22, Section 5.1] and Lemma 3.4, and I agree with the stress-test note that this is the primary issue rather than the imported Proposition 6.2, which is a faithful adaptation from [Sev24] and only used for d≥3. Still, 'the proof is omitted' on a load-bearing statement is not acceptable in a final version.\n\nI also want to give credit where it's due: the random cluster representation is defined carefully, the Edwards–Sokal coupling is spelled out, and the applications — Theorem 1.1 and Theorem 1.3 — are genuine consequences, not afterthoughts. The reliance on [GPPS22] is citation of established theorems, not circularity. I checked the logical dependencies from Theorem 1.4 back through Section 5 and the chain is coherent except for that one missing proof.\n\nWho is this for: probabilists and mathematical physicists working on percolation, random cluster models, and dynamical spin systems. It deserves a serious referee. Send it to review, but the referee should require the authors to either prove Corollary 3.5 or restructure the argument so that Proposition 3.6 doesn't depend on an unproved existence statement.","headline":"Real progress on supercritical φ^4, but the half-space uniqueness step has a load-bearing omitted proof that needs to be supplied before the main theorem is fully established.","tokens_in":74014,"tokens_out":1714,"would_cite":true,"duration_ms":17144,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","82B20","82B26"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every β above criticality and every d≥2, the φ⁴ random cluster model has a unique macroscopic cluster with high probability, uniformly in boundary conditions; surface-order large deviations and exponential spectral-gap decay follow.","keywords":["φ⁴ model","supercritical phase","random cluster model","local uniqueness","surface order large deviations","spectral gap","tangled currents","surface tension"],"falsifier":"Direct simulation can test Theorem 1.4 itself: for $d=2$ (and $d=3$) at $\\beta$ modestly above $\\beta_c$, estimate $\\inf_\\# \\Psi^\\#_{\\Lambda_{10L},\\beta}[U(L)]$ from the free-boundary $\\varphi^4$ random cluster measure; if the local-uniqueness probability does not tend to one as $L$ grows, the main claim is false. The structurally weakest point to attack is the omitted proof of Corollary 3.5: attempting to construct the half-space limiting tangled-current measures directly, and checking whether the two half-space magnetisations of Proposition 3.6 are equal, would either complete or break the boundary-condition comparison on which the argument rests.","tokens_in":72847,"feed_emoji":"🧲","tokens_out":20572,"duration_ms":168835,"temperature":0.7,"pith_summary":"The paper proves that the supercritical phase of the $\\varphi^4$ model is well behaved in a precise percolation sense: for every inverse temperature $\\beta$ above the critical value $\\beta_c$ and in every dimension $d\\ge 2$, a unique macroscopic cluster crosses large boxes with probability tending to one, uniformly in the boundary conditions. This local-uniqueness statement is the $\\varphi^4$ analogue of classical supercritical sharpness results for Bernoulli percolation and the Ising random cluster model, and it is the missing input needed to run non-perturbative renormalisation arguments throughout the supercritical regime. From it the authors derive two concrete consequences: the empirical magnetisation obeys surface-order large deviations (the probability of falling far below the spontaneous magnetisation decays like $e^{-cn^{d-1}}$), and both the Langevin and heat-bath dynamics of the model have spectral gaps decaying like $e^{-cn^{d-1}}$. The proof gives the first mathematical treatment of a random cluster (Fortuin–Kasteleyn) representation of $\\varphi^4$ — an Ising random cluster model in the random environment created by the absolute value field — and uses random tangled currents to compare boundary conditions, a step that is new even for the Ising model.","feed_headline":"Above critical β, one giant cluster rules the φ⁴ model","feed_subtitle":"For every d≥2, this single fact drives surface-order large deviations and exponential spectral-gap decay for φ⁴ dynamics.","key_machinery":"The load-bearing object is the $\\varphi^4$ random cluster measure $\\Psi_{\\Lambda,\\beta}$ (Definition 4.1): sample $a=|\\varphi|$ from the absolute value field, then conditionally draw an Ising random cluster configuration with edge weights $p_{xy}=1-e^{-2\\beta a_x a_y}$; under the Edwards–Sokal coupling, spin correlations become connectivity events, so 'the supercritical spin phase is well behaved' becomes a percolation statement. Three mechanisms carry the proof. (1) The random tangled current representation of [GPPS22] and its switching principle (Theorem 2.13): this is the tool that compares measures with different boundary conditions, yielding the thick-boundary approximation of the plus state (Proposition 3.1) and the uniqueness of half-space measures with positive boundary field (Proposition 3.6, resting on the construction in Corollary 3.5). (2) A surface tension $\\tau_\\beta$ for $\\varphi^4$, defined through Dobrushin-type boundary fields on a logarithmically thick boundary; its strict positivity for every $\\beta>\\beta_c$ (Proposition 5.5) adapts the [LP81] argument via the Ginibre inequality, and transfers — through the coupling — into an exponential disconnection bound for the free-boundary random cluster measure (Theorem 5.1). (3) Two routes from disconnection to local uniqueness: in $d=2$, the general Russo–Seymour–Welsh theory of [KST23] for FKG measures; in $d\\ge 3$, the [Sev24] route of Bernoulli sprinkling, stochastic domination of the sprinkled measure by the model at slightly higher $\\beta$ (Proposition 6.5), slab percolation, and an onion-peeling argument. Finally the [Pis96] coarse-graining scheme converts local uniqueness into the surface-order large-deviation bound, with [LSS97] supplying the product-measure domination at each renormalisation step.","core_discovery":"The central claim of the paper is Theorem 1.4: for every $d\\ge 2$ and every $\\beta>\\beta_c$, the probability of the local-uniqueness event $U(L)$ — there is a cluster crossing the annulus $\\Lambda_{8L}\\setminus\\Lambda_L$, and any two crossing paths in $\\Lambda_{4L}\\setminus\\Lambda_{2L}$ are connected inside $\\Lambda_{8L}\\setminus\\Lambda_L$ — tends to one under the $\\varphi^4$ random cluster measure $\\Psi^\\#_{\\Lambda_{10L},\\beta}$, uniformly over all boundary conditions $\\#$. In words, exactly one macroscopic cluster governs every large region of the supercritical phase. The theorem is the engine for two applications. Theorem 1.1 establishes surface-order exponential bounds for the lower large deviations of the empirical magnetisation: for every $\\beta>\\beta_c$ and $0<\\delta<m^*(\\beta)$, the probability that $|m_{\\Lambda_n}|\\le m^*(\\beta)-\\delta$ lies between $e^{-Cn^{d-1}}$ and $e^{-cn^{d-1}}$, while upward deviations remain of volume order. Theorem 1.3 states that the spectral gaps of Langevin and heat-bath dynamics on $\\Lambda_n$ decay as $e^{-cn^{d-1}}$ in the entire supercritical regime. A further consequence (Proposition B.2) identifies the boundary fields $h_L\\le p_{\\Lambda_L}$ with $L^{d-1}h_L\\to\\infty$ whose finite-volume measures converge to the infinite-volume plus state. The result is new even in dimension two, where the random cluster representation has no tractable dual.","pith_inferences":["The random cluster definition of Section 4 works for any even single-site measure with super-Gaussian tails, so the same proof strategy should transfer to other unbounded-spin models and to dilute random cluster models — the Blume–Capel case flagged in Remark 4.3 is the most immediate test case the paper leaves open.","The surface-order lower bound in Theorem 1.1 has the right order but no identified rate constant; a full Wulff theorem for $\\varphi^4$ that pins the constant to the surface tension $\\tau_\\beta$ is the natural next step, and Theorem 1.4 supplies the coarse-graining backbone such a theorem would need.","Because Theorem 1.4 is uniform in boundary conditions and Theorem 1.3 gives spectral gaps of order $e^{-cn^{d-1}}$, the supercritical regime is hard for any sampler uniformly in boundary data: an exponential relaxation barrier is intrinsic to the model, not an artifact of boundary conditions.","Combining Theorem 1.3 with the known positivity of the Langevin spectral gap below $\\beta_c$ yields a sharp dynamical phase transition at $\\beta_c$; a further question the paper does not touch is whether the dynamics also exhibit a cutoff phenomenon in the supercritical phase, as the Ising model does."],"forward_implications":["For every $\\beta>\\beta_c$ and every $\\delta\\in(0,m^*(\\beta))$, the lower large deviations of the empirical magnetisation on $\\Lambda_n$ have surface order: $e^{-Cn^{d-1}} \\le \\nu_{\\Lambda_n,\\beta}[|m_{\\Lambda_n}|\\le m^*(\\beta)-\\delta] \\le e^{-cn^{d-1}}$, whereas upward deviations are of volume order $e^{-cn^d}$.","Both the Langevin and the heat-bath dynamics for $\\varphi^4$ on $\\Lambda_n$ have spectral gaps bounded by $e^{-cn^{d-1}}$ for all $\\beta>\\beta_c$, extending the very-low-temperature result of [CGW22] to the whole supercritical regime and providing the slow side of a sharp dynamical phase transition at $\\beta_c$.","Under renormalisation the $\\varphi^4$ random cluster model is comparable to highly supercritical Bernoulli percolation: the unique giant component is ubiquitous and every other component is logarithmically small, which is precisely the structure the coarse-graining argument uses.","The infinite-volume plus state $\\nu^+_\\beta$ arises as the limit of finite-volume measures with boundary fields that may decay to zero — $L^{d-1}h_L\\to\\infty$ with $h_L\\le p_{\\Lambda_L}$ — so the plus phase does not require maximal boundary conditions (Proposition B.2).","The infinite-volume free and wired random cluster measures coincide for every $\\beta\\ge 0$ (Proposition 4.13), so the random cluster representation has a unique thermodynamic limit."],"supporting_citations":[{"why":"Supplies the random tangled current representation, switching principle, Ginibre and FKG inequalities, and regularity estimates on which Sections 2–3 (boundary-condition comparisons) rest.","marker":"[GPPS22]"},{"why":"The Ising-model strategy this proof follows globally (Dobrushin boundary on a thick boundary, comparison from thick-plus to free random cluster measure) and the analogue supercritical sharpness result for the Ising random cluster model.","marker":"[Bod05]"},{"why":"The surface tension positivity argument, adapted via the Ginibre inequality, that yields $\\tau_\\beta>0$ for every $\\beta>\\beta_c$ (Proposition 5.5).","marker":"[LP81]"},{"why":"Provides the $d\\ge 3$ route: local uniqueness under Bernoulli sprinkling, domination of the sprinkled measure by a slightly hotter measure, slab percolation, and the onion-peeling uniqueness argument.","marker":"[Sev24]"},{"why":"General Russo–Seymour–Welsh crossing bounds for FKG measures, used in dimension two to convert the disconnection bound of Theorem 5.1 into circuit crossings and local uniqueness.","marker":"[KST23]"},{"why":"Domination by product measures, applied at each renormalisation step (slab percolation, coarse graining, weak plus measure) to compare the model with Bernoulli percolation.","marker":"[LSS97]"},{"why":"The coarse-graining scheme that converts local uniqueness (Theorem 1.4) into the surface-order large-deviation bound (Theorem 1.1).","marker":"[Pis96]"},{"why":"Large-deviation estimates for highly supercritical Bernoulli percolation (Lemma 7.5) quantifying that non-giant clusters contribute a negligible mass, used in Proposition 7.4.","marker":"[DP96]"},{"why":"The standard random cluster toolbox — Edwards–Sokal coupling, FKG, monotonicity, domain Markov property — on which the $\\varphi^4$ random cluster definitions and comparison arguments are built.","marker":"[Gri06]"},{"why":"Established spectral-gap decay for $\\varphi^4$ at very low temperatures; Theorem 1.3 extends this to every $\\beta>\\beta_c$, so it is the baseline the dynamical application must improve upon.","marker":"[CGW22]"}],"fun_headline_variants":["φ⁴ supercritical: one giant cluster rules every region","Local uniqueness of macroscopic clusters in φ⁴ for β>βc","φ⁴ model above βc: exact one macroscopic cluster governs","Single-cluster dominance in φ⁴ supercritical phase proved","φ⁴: supercritical phase is well behaved, one cluster wins"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on an assertion that is stated without proof: Corollary 3.5 in Section 3.2, which postulates the existence and convergence of infinite-volume (double) tangled current measures on the half-space with positive boundary field, is followed by the sentence 'the proof is omitted,' and this construction underpins the half-space uniqueness of the spin measure (Proposition 3.6) used to compare plus and free boundary conditions on the way to Theorem 5.1 and hence Theorem 1.4.","fun_headline_variants_meta":{"raw":{"variants":["φ⁴ supercritical: one giant cluster rules every region","Local uniqueness of macroscopic clusters in φ⁴ for β>βc","φ⁴ model above βc: exact one macroscopic cluster governs","Single-cluster dominance in φ⁴ supercritical phase proved","φ⁴: supercritical phase is well behaved, one cluster wins"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000671,"raw_usage":{"total_tokens":3112,"prompt_tokens":1053,"completion_tokens":2059,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":1968}},"tokens_in":669,"tokens_out":2059,"duration_ms":15917,"temperature":1.0,"reasoning_tokens":1968,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:12:32.267203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Direct simulation can test Theorem 1.4 itself: for $d=2$ (and $d=3$) at $\\beta$ modestly above $\\beta_c$, estimate $\\inf_\\# \\Psi^\\#_{\\Lambda_{10L},\\beta}[U(L)]$ from the free-boundary $\\varphi^4$ random cluster measure; if the local-uniqueness probability does not tend to one as $L$ grows, the main claim is false. The structurally weakest point to attack is the omitted proof of Corollary 3.5: attempting to construct the half-space limiting tangled-current measures directly, and checking whether the two half-space magnetisations of Proposition 3.6 are equal, would either complete or break the boundary-condition comparison on which the argument rests.","supporting_citations":[],"review_version":1}