{"id":"5feaa00c-9840-47ef-aa24-4ecd8cebe860","arxiv_id":"2607.23662","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Self-prefix hits of c^m admit exact discrepancy identities, Lambert two-gap candidates, resonance rigidity, and certified O(N^{1-1/ν} polylog N) search; infinitude of 2^m starting with m remains open.","lead":"The paper maps when powers like 2^m start with the digits of m, giving exact identities, rigid arithmetic structure, and a certified sublinear search—while leaving infinitude open. It turns a classical open digit problem into precise counting, gap, and localization theorems a computer can check.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No significant objection identified; the principal remaining risk is verification of the long unformalized development and the shipped (2,10) certificate.","rationale":"The reader’s weakest-assumption discussion correctly locates the external Diophantine inputs, but the manuscript checks the hypotheses needed to invoke them and does not claim consequences at the unresolved critical scale. This is therefore a dependency risk, not a hidden circularity or an apparent inconsistency. The exact candidate/counting statements are elementary and independently checkable, while the certified numerical corollary is supported by a reproducible artifact. I would not move the verdict on the present record.","tokens_in":52661,"tokens_out":2915,"duration_ms":191987,"concrete_test":"Independently clone repository release v1.0.0, verify the recorded SHA-256 digests, install the pinned python-flint==0.6.0 in a clean environment, and rerun the certificate verifier. Require it to recompute the strict safety inequality at M_safe=1,914,818,931,502,442 and its failure at M_safe−1 using fresh 1024-bit Arb enclosures. Any disagreement would weaken Corollary 6.8, though not the exact identities.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I do not find a load-bearing mathematical objection to the central claims as scoped. Theorem 4.1 is an exact telescoping identity, and its equivalences do not smuggle in infinitude or the logarithmic law. The Lambert two-gap and counting results concern candidates rather than hits, and the paper repeatedly preserves that distinction. The quantitative parts do rely on external finite-type Diophantine control—Matveev lower bounds and the Tichy–Turnwald discrepancy theorem—but for the fixed integer/algebraic parameters considered here those are standard applicable inputs, not an unsupported heuristic. The complexity theorem likewise assumes only finite irrationality type, which Matveev supplies effectively in the fixed multiplicatively independent integer case. The paper’s stopping point at the critical scale is explicit and internally consistent. Thus the main residual risk is ordinary verification risk in a lengthy unformalized manuscript, plus possible implementation or interval-arithmetic error in the certified threshold, rather than an identifiable flaw in the argument.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The useful core is not a resolution of van de Lune’s problem, but a clean reorganization of it. Theorem 4.1 turns the count into an exact signed discrepancy for one nonlinear sequence, so infinitude and the log law become precise statements about D rather than folklore. Alongside that, the Lambert W_{-1} layer geometry gives a genuine candidate sequence with an exact count and a global 3–4 gap law for (2,10), and the paper never pretends candidates are hits.\n\nWhat is new is the problem-specific package: coupled inverse-root widths and complete monotonicity, fixed-difference resonance shells, floor-center nesting with endpoint filling and a coherent-skeleton size bound, and an interpolated continued-fraction locator with bit complexity O(N^{1-1/ν} polylog N) under finite type, plus a shipped interval-arithmetic certificate for a safe singleton threshold past about 1.9×10^{15}. Table 1 and §7 keep the logical boundaries honest; that is real discipline, not padding.\n\nSoft spots are proportional and mostly expected. Quantitative sparsity, subcritical moving targets, and the complexity exponent lean on Matveev and Tichy–Turnwald; for fixed algebraic/integer parameters that is standard, not a hidden fit, but if you only care about unconditional critical-scale recurrence you will leave empty-handed. The manuscript is long and unformalized, so ordinary verification risk remains, including the Arb certificate. Significance is moderate: organizational and algorithmic progress inside a classical open OEIS-type question, not a breakthrough on the open problems themselves.\n\nThis is for people who work on leading digits, shrinking targets, or certified Diophantine search. Citations look appropriate; no circularity burden on the exact identities. I would send it to referees. Worth engaging if that is your neighborhood; skip if you only wanted a proof that S_{2,10} is infinite.","headline":"Solid structural and algorithmic package on self-prefix digits that honestly stops short of infinitude for 2^m.","tokens_in":42009,"tokens_out":477,"would_cite":true,"duration_ms":22266,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63","11J70","11J71","11Y16"],"pacs":[],"model":"grok-4.5","headline":"Self-prefix powers reduce to one signed discrepancy and a two-gap candidate law, with a certified sublinear search whose infinitude for 2^m still open.","keywords":["leading digits","shrinking targets","Lambert W function","discrepancy","resonance","continued fractions","certified search","self-prefix"],"falsifier":"Decide the sign of d_+(x_j)−(z_j−x_j) for the explicit Lambert roots x_j,z_j of the (2,10) layer equations: infinitely many negative signs prove S_{2,10} infinite; a persistently positive liminf at second order (or higher) would prove it finite.","tokens_in":41880,"feed_emoji":"🔢","tokens_out":1109,"duration_ms":22062,"temperature":0.7,"pith_summary":"The paper studies when c^m begins with the digits of m itself (or the analogous inequality for real c>1). It rewrites the event as a one-sided shrinking target of width about 1/m and proves an exact identity that turns the hit count into the signed discrepancy of a single fixed interval for the nonlinear sequence nα−log_b(n+1). For c≥2, Lambert W_{-1} inversion produces a rigid candidate sequence with an eventual two-gap law and an exact count; for base-10 powers of 2 the gaps are only 3 or 4. Actual hits obey fixed-difference shells and arithmetic-chain rigidity, and at floor resonance centers endpoint hits force every intermediate term. For multiplicatively independent integer pairs the paper builds an interpolated continued-fraction locator whose bit cost is O(N^{1−1/ν} polylog N) whenever the irrationality exponent of {log_b c} is finite, with an explicit machine-checkable safe threshold for (2,10). Infinitude of the classic sequence and the conjectured logarithmic growth remain open; the work isolates exactly what must be proved at the critical scale.","feed_headline":"When 2^m starts with m: exact count identity, still open","feed_subtitle":"A signed-discrepancy formula and two-gap candidates isolate what must be proved; search is now sublinear and certified.","key_machinery":"The exact signed-discrepancy identity (Theorem 4.1): A_{c,b}(N)=log_b(N+1)+B_{α,b}(N)−ρN+{y_N}. It converts the original count into discrepancy of one fixed interval and isolates both open questions; Lambert W_{-1} layer endpoints and the interpolated convergent-block locator carry the geometry and the algorithm.","core_discovery":"For c≥2 the self-prefix count equals log_b(N+1) plus the signed discrepancy of one fixed arc for y_n=nα−log_b(n+1), so infinitude and the logarithmic law are equivalent to that discrepancy being unbounded or o(log N). Lambert inversion supplies an exact candidate geometry (eventual gaps in {⌊1/ρ⌋,⌊1/ρ⌋+1}, closed counting formula), while resonance and continued-fraction packing give arithmetic rigidity and a certified sublinear search.","pith_inferences":["The same discrepancy identity suggests that any future one-sided approximation theorem for the Lambert roots at scale 1/j would settle both open problems at once.","The certified locator can be reused as a black-box filter for other self-referential or moving-target digit problems once a finite-type bound is known.","Because candidate gaps are only 3 or 4 for (2,10), exhaustive verification of candidates up to very large M is limited by the critical phase test, not by candidate density."],"forward_implications":["Infinitude of S_{c,b} is exactly equivalent to log_b(N+1)+D_{α,b}(N) being unbounded.","The logarithmic law A∼log_b N is exactly equivalent to D_{α,b}(N)=o(log N).","For (2,10) every large enough block of length ≤44 699 994 contains at most one index that needs exact verification.","Multiplicatively dependent integer pairs are completely classified and already obey A=log_b X+O(1).","Fixed differences and long arithmetic progressions of hits are confined to explicit resonance shells and force continued-fraction convergents."],"fun_headline_variants":["Exact discrepancy ties 2^m self-prefixes to one arc","Signed identity: 2^m starts with m iff discrepancy unbound","Lambert candidates for 2^m self-prefixes have gaps 3 or 4","Sublinear certified search for self-leading digits of c^m","Resonance forces full chains in self-referential prefixes"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The power-saving and sublinear-complexity claims rest on finite Diophantine type for log_b c coming from lower bounds on linear forms in logarithms and a discrepancy theorem for nα+β log n; if that type control fails, those exponents collapse while the exact identities can still stand.","fun_headline_variants_meta":{"raw":{"variants":["Exact discrepancy ties 2^m self-prefixes to one arc","Signed identity: 2^m starts with m iff discrepancy unbound","Lambert candidates for 2^m self-prefixes have gaps 3 or 4","Sublinear certified search for self-leading digits of c^m","Resonance forces full chains in self-referential prefixes"]},"model":"grok-4.5","effort":"low","cost_usd":0.0052,"raw_usage":{"total_tokens":1493,"prompt_tokens":885,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":52004000,"prompt_tokens_details":{"text_tokens":885,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":528,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":885,"tokens_out":80,"duration_ms":8225,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T18:48:48.846457+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Decide the sign of d_+(x_j)−(z_j−x_j) for the explicit Lambert roots x_j,z_j of the (2,10) layer equations: infinitely many negative signs prove S_{2,10} infinite; a persistently positive liminf at second order (or higher) would prove it finite.","supporting_citations":[],"review_version":2}