{"id":"a4e91f2b-4b98-4d1d-a5d1-d5ca5e7a548b","arxiv_id":"2511.01537","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For 3D cubic lattices, the paper claims first-order finite-size signatures of explosive percolation for k≥2 and monotone rigidification efficiency with k, but the proof of the central theorem is arithmetically impossible and the abstract's simulations (L=192) never appear in the body.","lead":"This preprint simulates, and claims to prove, how an Achlioptas edge-selection rule controls explosive connectivity and mechanical rigidity in 3D cubic lattices. The genuinely new observation is that on richly-connected hosts, more choices (k) narrow the rigidity–connectivity gap; the claimed rigorous proofs of a first-order transition are internally inconsistent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem IV.25's merger cascade is arithmetically impossible: a sublinear window of w steps can contain at most w merges, yet the theorem asserts ≥ c*N merges; the proof's switch to w=c0N makes the window linear, not o(N).","rationale":"The reader's verdict (REJECT) is well supported. The single most load-bearing concern is the arithmetic impossibility at the heart of the merger-cascade construction: a sublinear window cannot contain a linear number of merge events. The proof's attempt to repair this by setting w=c0N destroys the sublinearity needed for an o(1) density interval, so Theorem IV.37 loses its premise. This is not a matter of disagreement with external consensus; it is an internal inconsistency in the proof of the paper's own headline claim. The reader's weakest_assumption pointed to the susceptibility bound (Lemma IV.4/Prop IV.7/IV.19), which is also a serious flaw but is downstream of the same cascade machinery or at least independent of it. I choose the merger-cascade counting because it alone suffices to invalidate the connectivity theorem without relying on external results, and because the text itself switches w=o(N) to w=c0N in a single proof. The rigidity theorem is additionally conditional on an unproved assumption, so it cannot rescue the central claims. My recommendation is to keep the REJECT verdict: the numerical data may be honest (and the k=1 benchmark is credible), but the theoretical core does not stand as written.","tokens_in":54623,"tokens_out":3298,"duration_ms":38851,"concrete_test":"Directly test the merger-cascade window: run the k=2 Achlioptas process on the NN host for L=32 and L=64 (or any N), let tc be the first time Smax≥N/2, and count X of inter-component merges in windows [tc−w, tc] for w=N^{2/3} and w=c0N. Since X≤w, the claimed X≥c*N with w=o(N) will fail in every realization; if any window achieving X≥c*N has width Θ(N), then Theorem IV.25 and hence Theorem IV.37's o(1) density window are contradicted. Also verify whether the proof's 'w=c0N while still o(N)' is used explicitly; it cannot be.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central first-order claim (Theorem IV.37) rests on the merger-cascade window (Theorem IV.25), which asserts that for a window [t−, t+] of width w = o(N), the number of inter-component merges satisfies X_{t−,t+} ≥ c*N with c*>0. This is impossible by definition: X is a sum of w merge indicators, so X ≤ w deterministically. If w = o(N), then X = o(N). The proof tries to escape this by writing w = c0N for an arbitrarily small c0 and calling it o(N), but c0N is linear, not sublinear. Proposition IV.33 similarly concatenates N^{1/3} subwindows of width N^{2/3}, giving total length Θ(N), not o(N). Consequently, Theorem IV.28's 'deterministic jump criterion' — which requires a sublinear window to produce a vanishing density interval — has no valid input. The density interval would have width Θ(1), not o(1), so Theorem IV.37's δ>0 jump across an o(1) interval does not follow. A separate sign of the same arithmetic trouble is that Theorem IV.25's proof first chooses w = N^γ for γ∈(0,1), then 'for clarity' sets w = c0N; these are incompatible. Even within Theorem IV.28, the batching argument requires fixed fractions (e.g., εN, αεN) but the pre-cascade cap from Proposition IV.23 is only o(N), so the displayed 'Increase' formula garbles the scaling. Thus the connectivity half of the paper's headline claim is not supported by the proof as written. The rigidity half (Theorem IV.51) is explicitly conditional on unproved Assumption IV.48 and on Lemma IV.49, which is justified only via an ER proxy model; its transfer to the Achlioptas process is hypothesized, not proven. The abstract's own claim of a continuous thermodynamic limit (γ→1.000) is also in direct tension with the Theorem IV.37 jump, and Corollary IV.44 (Riordan–Warnke) would set Δ(k)=0 for fixed k.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies percolation under the k-choice Achlioptas product rule on two three-dimensional cubic-lattice hosts, the nearest-neighbor (NN) model and the Intra model with face and body diagonals. The authors report numerical simulations (Section III) showing a sharpening connectivity transition for k>=2, susceptibility peaks scaling with exponent near 1, and a rigidity–connectivity gap in the Intra host that shrinks as k increases. The theoretical part claims: (i) a rigorous proof of sublinear merger-cascade windows for k>=2 leading to a first-order connectivity transition (Theorems IV.25, IV.28, IV.37); (ii) monotone delay of the threshold in k (Theorem IV.42); and (iii) a conditional progress-function model for monotonic rigidification efficiency (Theorem IV.51), said to rest on two physically motivated assumptions. The SI provides supporting material including a proxy-model rigidity proof for Erdős–Rényi Intra subgraphs.","tokens_in":55020,"tokens_out":22361,"duration_ms":211474,"significance":"Were the connectivity theorems valid, the paper would be significant: a rigorous first-order jump for fixed k on a fixed-dimensional lattice would sit in tension with established continuity results for Achlioptas processes and would reframe the interpretation of explosive-percolation simulations. The numerical observation of a host-dependent, k-dependent rigidity–connectivity gap is potentially interesting, and the manuscript is transparent in identifying which rigidity results are conditional. Credit is due for attempting a self-contained framework with SI proofs. However, the central proofs contain a factor-N error in the susceptibility bound, an arithmetic impossibility in the merger-cascade window, and a direct contradiction between Theorem IV.37 and Corollary IV.44; the rigidity theorem is explicitly conditional on an unproved assumption. As it stands, the theoretical framework does not support the headline claims, and the numerical evidence alone (L<=10 in the body) is too limited to carry them.","major_comments":[{"comment":"The uniform mesoscopic-sparsity bound chi_L(p) <= eta for all p <= pc,alpha (Prop IV.7, Lemma IV.19) is false. Lemma IV.4's inequality chi_L <= |Cmax|/N is a factor-N error: from sum a_i^2 <= a_max sum a_i = a_max N one obtains chi_L <= |Cmax|, not |Cmax|/N; SI Lemma S3.1 repeats the same algebra error (concluding a_max N/N = |Cmax|/N). Moreover, at the pseudo-threshold E[|Cmax|] = alpha N and chi_L >= S_max^2/N, so E[chi(pc,alpha)] >= alpha^2 N, which is unbounded; Lemma IV.19's own display E[chi] <= N*P_N <= alpha N gives a divergent bound, not a constant K. Theorems IV.21, IV.24, and IV.31 all inherit this false input.","section":"§II.D/§IV.B, Lemma IV.4, Prop IV.7, Lemma IV.19; SI Lemma S3.1"},{"comment":"Theorem IV.25 asserts X_{t-,t+} >= c*N merges in a window of width w=o(N). Since the merge count X is a sum of w indicator variables (Def IV.8), X <= w deterministically; a sublinear window cannot contain a linear number of merges. The proof first fixes w=N^gamma then switches to 'w=c0N ... while still w=o(N)', which is a contradiction in terms: c0N is linear. Proposition IV.33 concatenates N^{1/3} subwindows of width N^{2/3}, total length Theta(N), not N^{2/3} as its statement claims. Proposition IV.35 states 'total width o(N)' but proves W=Bw=Theta(N). Thus Theorem IV.28's jump criterion has no valid sublinear-window input, and Theorem IV.37's o(1)-density jump is unsupported.","section":"§IV.C, Theorem IV.25; Proposition IV.33; Proposition IV.35"},{"comment":"Theorem IV.37 concludes a genuine thermodynamic discontinuity: liminf_L [P_N(p+(L))-P_N(p-(L))] >= delta>0 across density intervals of width ->0, and the text calls this 'a hallmark of a first-order (explosive) transition.' This directly contradicts Corollary IV.44 (Delta(k)=0 for every fixed k) and the abstract's statement that the transition 'is continuous in the infinite thermodynamic limit.' The manuscript never reconciles these claims; the finite-size 'signature' language of Section IV.D and the thermodynamic jump statement of Theorem IV.37 are different assertions, and the proof of the theorem does not bridge them.","section":"§IV.C–D, Theorem IV.37 vs. Corollary IV.44"},{"comment":"Lemma IV.31's proof counts Theta(N^2) inter-component vertex pairs and concludes 'a Theta(N) pool of inter-component host edges,' conflating vertex pairs with bounded-degree host edges; the inter-component edge count is O(Delta N) regardless of pair counts, and a linear lower bound additionally requires average component size O(1), which Proposition IV.23 does not provide (it guarantees only o(N) component sizes; with size N^{2/3}, inter-component edges can be sublinear). Lemma IV.22 similarly asserts linearity under hypotheses that yield only N/S. Proposition IV.23 and Lemma IV.29 define tc,alpha through the expectation P_N but treat it as an almost-sure hitting time; the stated contradiction with 'the definition of tc,alpha' is therefore invalid.","section":"§IV.C, Lemma IV.22, Lemma IV.31, Proposition IV.23"},{"comment":"Theorem IV.51 is stated unconditionally, but its proof begins 'Conditional on the validity of the Monotonic Density assumption (Theorem IV.48)' — Assumption IV.48 is an unproved assumption, not a theorem. Lemma IV.47, the non-increasing property of P(s), is deferred to Appendix S5 and proved there only under Assumption S5.3, the same monotone-density assumption. Lemma IV.49 / Hypothesis IV.50 is established only for an Erdős–Rényi proxy (Theorem S4.3) and then explicitly hypothesized to carry over to the Achlioptas process. Moreover, the proof invokes the coupling of Theorem IV.42, whose candidate sets are sampled from edges unused in either process and hence do not reproduce either process's marginal distribution. The monotone-efficiency claim is therefore unsupported at every load-bearing step; the theorem statement should be made explicitly conditional.","section":"§IV.G, Theorem IV.51; Appendix S5; Theorem IV.42"},{"comment":"Several steps in the cascade are asserted rather than proved. Theorem IV.21's proof contains 'Details are routine', 'A double counting argument ... shows', and 'This handles the small-s case', and its union bound over small sets is admitted to be 'superpolynomially large if taken literally'; no valid bound is supplied. Lemma IV.24 and Lemma IV.32 infer that the product rule selects an inter-component edge whenever any sampled candidate is inter-component, but the rule selects the minimum product; an intra-component candidate inside a small component can have a smaller product, so the claimed per-step merge probability p* is not established. These gaps are inherited by Theorem IV.25 and Proposition IV.33.","section":"§IV.C–B, Theorem IV.21; Lemmas IV.24, IV.32"}],"minor_comments":[{"comment":"The abstract's 'massive-scale simulations up to L=192 (N approx 7e6) with 20,000 independent realizations' do not appear in the main text or SI; Section III reports L=1,...,10 with 1,000 realizations and Tables S1/S2 cover L<=10. The headline finite-size-scaling claims should be tied to data actually presented.","section":"Abstract vs. Section III"},{"comment":"The abstract states that for k>=8 the peak-susceptibility exponent is gamma=1.000; Tables S1 and S2 report, e.g., k=8: gamma=1.118 for both hosts, and Intra k=17-32: gamma between 0.978 and 0.998. The stated exact value 1.000 is not what the tables show.","section":"Abstract vs. Tables S1/S2"},{"comment":"The final step of SI Lemma S3.1, 'a_max N/N = |Cmax(t)|/N', is a factor-N algebra error; the correct consequence is |Cmax(t)| (see Major Comment 1). The SI proof should be corrected in any revision.","section":"SI Lemma S3.1"},{"comment":"The proof of Theorem IV.51 refers to 'Theorem IV.48' for the Monotonic Density assumption, and Lemma IV.49's justification is labelled 'Justification' based on a proxy model. Assumptions and hypotheses should be labelled consistently so the reader can track what is proved vs. assumed.","section":"Theorem IV.51 proof; Assumption IV.48; Hypothesis IV.50"},{"comment":"Lemma IV.14 claims |M_s-M_{s-1}|<=1 for the Doob martingale, but its proof abandons M and proves bounded differences for the partial-sum martingale S_u instead. Theorem IV.15's inequality is correct via S_u; the lemma statement should be revised or removed.","section":"Lemma IV.14/Theorem IV.15"},{"comment":"The claim that gamma 'rapidly approaches 1' for k>=2 is weakened by the values in Tables S1/S2 for k=2 (gamma ~ 1.18) and k=17-32 (gamma slightly below 1); the text should discuss deviations from gamma=1 rather than assert approach. Also, the figure captions label susceptibility as chi_max for the Intra model while the text defines chi'_max; please make notation consistent.","section":"Section III.A and Fig. 4 caption"}],"recommendation":"reject","confidential_remarks":"The gulf between the abstract (L=192, 20,000 realizations, gamma=1.000) and the manuscript body (L<=10, 1,000 realizations; gamma between 0.978 and 1.118) is concerning; if the L=192 data exist, they should be in the paper. The fundamental contradiction between Theorem IV.37 and Corollary IV.44 — a thermodynamic jump vs. Delta(k)=0 — appears to reflect a confusion between finite-size signatures and thermodynamic statements, and it is not a presentational slip. I also note that the manuscript's own SI contains the factor-N algebra error that produces the false susceptibility bound, suggesting that the proof chain has not been checked symbolically. If the authors can supply the missing L=192 data and repair the arithmetic, a resubmission focused on the numerical phenomenology might be viable; the theoretical framework as it stands is not."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: this paper is not ready as a theory paper, but there is a real empirical kernel worth keeping.\n\nWhat's actually new and worth credit: the numerical finding that the rigidity–connectivity gap in the Intra model shrinks monotonically as k increases, while the NN model shows no gap and admits layered shear flexes. The explicit construction of the layered shear flex (Thm S1.28) is a clean, deterministic obstruction, and the k=1 benchmark against the known cubic lattice threshold checks out. If the paper were just those observations, it would be a decent numerical contribution.\n\nThe soft spots are not minor. The headline merger-cascade theorem (Thm IV.25) asserts c*N merges in a window of width w=o(N), but there is at most one merge per step, so X ≤ w. The proof tries to escape by setting w = c0N, which is linear, not sublinear. The susceptibility bound in Lemma IV.4 is off by a factor of N relative to the paper's own SI Lemma S3.1 — the claimed mesoscopic sparsity bound is false because the inclusive susceptibility includes the giant component. Theorem IV.37's jump also directly contradicts Corollary IV.44, which cites Riordan–Warnke and sets the jump to zero. The abstract claims L=192, N≈7×10^6, 20,000 realizations, and γ=1.000 for k≥8, but the body and tables use L≤10 with 1,000 realizations and report γ(8)=1.118. That mismatch is hard to explain. The rigidity branch is explicitly conditional on Assumption IV.48, which is unproved and essentially encodes the conclusion.\n\nThe paper is not incoherent in every part — the coupling argument for monotonic delay is reasonable — but the load-bearing theoretical claims are not supported as written. The arithmetic error alone sinks the connectivity proof; the contradiction with known results sinks the first-order claim; the factor-N slip sinks the mesoscopic framework.\n\nWho is this for? Someone working on explosive percolation or rigidity transitions might want to know about the numerical trends, and the shear obstruction is a useful result. But I wouldn't cite the theory. I would send it to a serious referee because the empirical content deserves checking and the authors might salvage a weaker version, but as submitted I'd expect a reject or major revision.","headline":"Interesting numerical observations and a clean shear obstruction, but the theoretical core is broken — the headline theorems don't survive contact with the paper's own definitions.","tokens_in":55701,"tokens_out":3037,"would_cite":false,"duration_ms":34664,"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":"The paper claims that for k≥2 the product-rule process on cubic lattices jumps discontinuously in connectivity across a vanishing density window, while richer hosts rigidify monotonically more efficiently as k grows.","keywords":["explosive percolation","Achlioptas process","product rule","cubic lattice","rigidity percolation","rigidity-connectivity gap","finite-size scaling","first-order transition"],"falsifier":"On the NN lattice with k=2, evaluate the inclusive susceptibility χ_L at the pseudo-threshold pc,α: because the largest component has fraction ≥ α there, χ_L ≥ α²N grows with L, directly contradicting the O(1) bound asserted by Proposition IV.7 and Lemma IV.19 — the premise on which the merger-cascade window and Theorem IV.37 rest; this one measurement, together with an extrapolation of the order-parameter jump across the claimed window (Theorem IV.37 says ≥ δ>0, Corollary IV.44 says 0), settles the matter.","tokens_in":54259,"feed_emoji":"🧊","tokens_out":17137,"duration_ms":168784,"temperature":0.7,"pith_summary":"The paper asks whether a purely local rule — at each step, sample k candidate bonds and add the one minimizing the product of the sizes of the two components it would join — can sharpen two global properties of a 3D cubic lattice: the suddenness of the connectivity transition and the mechanical efficiency of rigidification. For connectivity it claims that for every k≥2 a 'merger-cascade window' of sublinear width accumulates a linear number of inter-component mergers, so the giant-component fraction jumps by a macroscopic amount across an infinitesimal density interval — a first-order, explosive transition — and that the threshold is monotonically delayed as k grows. For rigidity it claims that on the richly-connected Intra host (axis, face-diagonal, and body-diagonal bonds) increasing k monotonically increases the expected rigidity gain of each added bond, shrinking the rigidity–connectivity gap; a conditional progress function linking local product score to global mechanical utility is offered as the mechanism, while the sparse nearest-neighbor host is shown to be blocked from global rigidity by layered shear flexes. A sympathetic reader would care because these results would convert a cheap local heuristic into a design principle: more choice buys sharper transitions and mechanically cheaper rigidification, with implications for growing or printing stiff lattice structures bond by bond.","feed_headline":"k≥2 choices trigger a first-order jump in cubic-lattice percolation","feed_subtitle":"A merger cascade crosses the transition in a vanishing density window; on rich hosts, extra choice cuts wasted bonds.","key_machinery":"The central object is the product score s(e)=|C(u)|·|C(v)| — the product of the sizes of the two components a candidate bond would join — with the rule adding the sampled bond of minimal score. Three devices carry the argument: a mesoscopic-sparsity lemma asserting the inclusive susceptibility stays uniformly bounded up to the pseudo-threshold pc,α, guaranteeing all components are sublinear just before the transition; a martingale concentration bound on windowed merge counts, plus a counting lemma on inter- versus intra-component missing edges, showing a linear number of merges land in a sublinear step window (the merger cascade); and, for rigidity, the conditional progress function P(s)=E[r","core_discovery":"The central claim (Theorem IV.37): on bounded-degree cubic-lattice hosts, for every k≥2 there is a size-independent δ>0 and a density window of vanishing width across which the expected largest-component fraction rises by at least δ — a genuine first-order jump in the thermodynamic limit, produced by a merger-cascade window (Theorem IV.25) in which Θ(N) inter-component mergers occur in o(N) steps. The companion claim (Theorem IV.51): under two stated assumptions, the expected single-step rigidity gain is non-decreasing in k, explaining the numerically observed shrinkage of the rigidity–connectivity gap on the richly-connected Intra host. The paper also proves the connectivity threshold is mo","pith_inferences":["Editorial inference: the paper's own quoted theorem for fixed k (Theorem IV.43) and its Corollary IV.44 — the transition is continuous and the jump satisfies ∆(k)=0 in the thermodynamic limit — sit in tension with Theorem IV.37's positive-jump claim; read sympathetically, the durable content is the finite-size first-order crossover at realistic lattice sizes rather than a literal infinite-size dis","Editorial inference: a discriminating test the authors did not report is a per-step growth-rate histogram of the largest component; if the merger cascade is real, the increments should concentrate into a heavy-tailed spike inside the claimed sublinear window, which would remain visible even if the infinite-size jump does not survive.","Editorial inference: the monotonic-efficiency mechanism, if correct, should generalize to any sufficiently dense non-bipartite host whose bond directions span 3D (the property that makes the Intra host special) and should fail on planar or bipartite hosts — a host-by-host prediction checkable in 2D analogs.","Editorial inference: the intermediate-k sharpening implies an optimal choice parameter k_opt(L) that should grow with system size; mapping k_opt(L) across hosts would convert the theoretical picture into a practical fabrication heuristic — add just enough local choice to buy the stiffness gain, since sharpness degrades beyond k_opt."],"forward_implications":["For k≥2, the cubic-lattice connectivity transition is claimed to be genuinely first-order: a macroscopic fraction of vertices join the giant component inside a density window whose width tends to zero (Theorem IV.37).","More choice monotonically delays percolation: pc,α(k) is non-decreasing in k, proved for every finite lattice by coupling processes with different k on shared randomness (Theorem IV.42).","On the Intra host, rigidity becomes monotonically cheaper with k: expected rank gain per added edge rises and expected redundant edges fall, so the rigidity–connectivity gap shrinks — from about 0.42 at k=1 to 0.25 at k=32 at the largest simulated size.","On the NN host, no finite density can rigidify the lattice: explicit layered-shear flexes survive any o(N) bond additions, so the local rule cannot overcome the geometric obstruction.","Finite-size signatures claimed to accompany the first-order behavior are bimodal order-parameter distributions and susceptibility scaling with exponent γ→1 for k≥2; transition sharpness is non-monotonic in k, peaking at intermediate k before large-k stepwise growth sets in."],"fun_headline_variants":["First-order jump in cubic-lattice percolation for any k≥2","Explosive percolation: k≥2 gives a true jump in lattices","Goldilocks rigidity: intermediate k optimizes global stiffness","Merger cascade window makes percolation jump macroscopic","Local choice, global jump: k≥2 percolation in cubic lattices"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the inclusive susceptibility — the average component size of a random vertex, giant component included — stays bounded by a constant independent of lattice size right up to the pseudo-threshold pc,α, which cannot hold at pc,α itself since the giant's α-fraction alone contributes at least α²N, and which rests on a lemma (|Cmax|/N ≥ χ_L) that the paper's own supplementary lemma S3.1 proves in the opposite direction (χ_L ≤ |Cmax|).","fun_headline_variants_meta":{"raw":{"variants":["First-order jump in cubic-lattice percolation for any k≥2","Explosive percolation: k≥2 gives a true jump in lattices","Goldilocks rigidity: intermediate k optimizes global stiffness","Merger cascade window makes percolation jump macroscopic","Local choice, global jump: k≥2 percolation in cubic lattices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000372,"raw_usage":{"total_tokens":1845,"prompt_tokens":785,"completion_tokens":1060,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":965}},"tokens_in":529,"tokens_out":1060,"duration_ms":10261,"temperature":1.0,"reasoning_tokens":965,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:21:19.198764+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the NN lattice with k=2, evaluate the inclusive susceptibility χ_L at the pseudo-threshold pc,α: because the largest component has fraction ≥ α there, χ_L ≥ α²N grows with L, directly contradicting the O(1) bound asserted by Proposition IV.7 and Lemma IV.19 — the premise on which the merger-cascade window and Theorem IV.37 rest; this one measurement, together with an extrapolation of the order-parameter jump across the claimed window (Theorem IV.37 says ≥ δ>0, Corollary IV.44 says 0), settles the matter.","supporting_citations":[],"review_version":1}