{"id":"ea745060-9e64-456e-b2f0-532229c75a96","arxiv_id":"2509.10588","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"The paper asserts RH is equivalent to a contraction bound for a prime-counting error functional, but the proof relies on a false large-sieve inequality and a scale-mixing error.","lead":"This preprint claims a new dynamical reformulation of the Riemann Hypothesis, based on trajectories of a simple integer map. A central technical estimate used in the proof is false, and a key contraction lemma conflates two different senses of scale contraction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2's log-scale large sieve is false: with M=1 the LHS of (6) is |Gamma| ~ U^4/2 while the RHS is ~8U, so the frequency netting and the contraction inequalities in §6 collapse.","rationale":"The reader's weakest assumption identifies exactly the failure I find. Section 5.2, Theorem 5.2, is not merely an overestimate; the stated inequality (6) is false at M=1 because the large diagonal term |G(0)| = |Gamma| ~ U^4/2 was discarded in the Schur test. The proof replaces 2T/h + 1 with 4/h 'within a harmless constant,' which is a factor ~U^3 error. This bound is the only mechanism that converts many Riemann-zero frequencies into a small netting error; without it, the zero sum after contraction has no reason to be O(U^2), and Theorems 6.1, 6.3, 6.4 and Corollary 6.6 do not follow. The macro-step alignment lemmas are plausible as telescoping estimates, but they cannot compensate for a missing large-sieve estimate. Corollary 6.6 would combine with Theorem 8.2 to prove RH, a strong signal that the unconditional bound cannot be correct as derived. The numerical appendix does not repair a false inequality. The reader's REJECT verdict is therefore justified; no adjustment is needed.","tokens_in":17021,"tokens_out":4841,"duration_ms":50768,"concrete_test":"Directly evaluate inequality (6) with M=1, w_1=1, u_1=0, U=120, T=U^3/2, h=2/U. The LHS is |Gamma| = 2 floor(T/h) + 1 = 103,680,001, while the RHS is 8(1+2/h) = 968. If the inequality fails, as this arithmetic shows, recompute the zero-contribution estimate in §6.2 using the corrected diagonal term; with the large sieve invalid, the netting term becomes O(U^4) rather than O(U^2), and the contraction theorem no longer follows.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's unconditional contraction inequalities rest on Theorem 5.2's log-scale large sieve. That inequality is arithmetically false. Taking M=1, w_1=1, u_1=0, the left side of (6) is |Gamma| = 2 floor(T/h) + 1 ~ U^4/2, since T = U^3/2 and h = 2/U, while the right side is 8(1 + 2/h) = 8(1 + U). For U=120, the left side is roughly 1.04e8 and the right side is 968. The proof's Schur-test attempt discards the diagonal contribution |G(0)| = |Gamma| ~ U^4/2 and then asserts that 2T/h can be replaced by 4/h 'within a harmless constant'; this is a factor ~U^3 error. Theorem 5.2 is exactly the tool used to net the Riemann-zero contributions across up to four points in a window, and it feeds directly into Theorems 6.1, 6.3, 6.4 and Corollary 6.6. Without a valid large-sieve bound, the zero-sum contributions are not shown to be small, so the contraction inequalities are unsupported. Moreover Corollary 6.6 would combine with Theorem 8.2 to prove RH, a consequence the paper does not draw; the false inequality is therefore load-bearing, not a cosmetic constant issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a deterministic map on integers (composites advance by π(m), primes retreat to the previous prime) and studies trajectories in short multiplicative windows. Its central claim is Theorem 8.2: RH holds iff the trajectory error functional E(X) satisfies E(X) ≪ X^{1/2} log X for all X ≥ e^{120}. Sections 3–7 claim to prove unconditional contraction inequalities for E(X), Ẽ(X), and A(X), leading to Corollary 6.6, which asserts E(X) ≪ X^{1/2} log X unconditionally. The equivalence is then deduced by combining this with Landau–Littlewood Ω-results. I find the proof invalid: the key log-scale large sieve inequality (Theorem 5.2) is arithmetically false, the existence of backward composite predecessors is asserted without proof, and the converse direction of the equivalence has a Phase/subsequence gap. The unconditional contraction bounds therefore collapse, and the main theorem is not established.","tokens_in":17438,"tokens_out":7271,"duration_ms":81320,"significance":"If the paper's claims were correct, they would amount to a proof of the Riemann Hypothesis: Corollary 6.6 supplies the unconditional bound E(X) ≪ X^{1/2} log X and Theorem 8.2 states that this bound is equivalent to RH. The failure to draw that conclusion is a warning sign, and inspection confirms that the proof is not sound. The paper is clearly organized and makes an effort to make constants explicit; the one-visit and parent-window counting lemmas are elementary and plausible. However, the frequency-netting inequality on which the contraction argument depends is demonstrably false, and no alternative mechanism is given. The proposed dynamical reformulation, if valid, would be very significant, but the manuscript does not provide a valid derivation.","major_comments":[{"comment":"Theorem 5.2 is false. Take M=1, w_1=1, u_1=0. The left side of (6) is |Γ| = 2⌊T/h⌋+1. With T=U^3/2 and h=2/U, T/h=U^4/4, so |Γ|∼U^4/2. The right side is 8(1+U). For U=120 the left side is about 10^8 and the right side is 968. The proof's Schur test discards the diagonal term |G(0)|=2T/h+1 and replaces it by 4/h as a 'harmless constant'; this is an error of size ∼U^3. This theorem is exactly the tool used to bound zero contributions in Theorems 6.3, 6.4 and Corollary 6.6, so the claimed contraction inequalities are unsupported.","section":"§5.2, Theorem 5.2, Eq. (6)"},{"comment":"The lemma asserts that every composite y∈C_X has an L-fold composite predecessor Ψ_X(y), but no existence proof is given. The forward map m↦m+π(m) is not obviously invertible on the relevant integer interval; one must prove that for every y≈X there is n≈θX with n+π(n)+...=y after L backward steps. The proof only sums forward increments and never addresses solvability of the backward equation. Since macro-step alignment is the core contraction mechanism, this is a load-bearing gap.","section":"§4.2, Lemma 4.1"},{"comment":"The converse direction is not proved even assuming the contraction bound. The paper invokes Landau–Littlewood Ω-results to obtain a subsequence where E(y) is large, but then chooses X_k with cos(γ log X_k+φ)=1 and claims that |E(y)| is large throughout W_{X_k}. The Ω-subsequence need not coincide with the phase-aligned subsequence. Moreover, sup_{z∈W}|E(z)| does not imply a pointwise lower bound at every y∈W; the paper's own Lemma G.1 remark states that the additive local-to-pointwise error is too large for that purpose. Thus the key contradiction with the contraction inequality is not established.","section":"§8.1, Theorem 8.2, (2⇒1)"},{"comment":"The 'log-scale large sieve' companion inequality is stated without a proof adapted to this setting. The appeal to Montgomery–Vaughan Theorem 7.1 is not appropriate: that theorem requires well-separated frequencies with a spacing δ and gives constants depending on 1/δ. Here the grid spacing is h=2/U, so a natural bound would contain a factor of U; the claimed 8(M+2/Δ) has no derivation and is inconsistent with the M=1 counterexample to Theorem 5.2.","section":"§5.3, Lemma 5.3"}],"minor_comments":[{"comment":"The text at the end of §7 says that Lemma 7.3 promotes the window bound to the classical von Koch bound, but Appendix G explicitly notes that X/log^2 X dominates X^{1/2} log X and that this promotion does not work. These statements are contradictory and should be reconciled.","section":"End of §7 and Appendix G"},{"comment":"Appendix A.7 claims a tail bound ≪X^{1/2}U^{-10}, but the proof in §5.1 gives only an estimate of order X^{1/2} log U / U (or similar). The constants and exponents should be made consistent.","section":"Appendix A.7 vs §5.1"},{"comment":"In Lemma 4.1 the displayed relation log Ψ_X(y)=log y+log θ is confusing because θ=3/4 and the proof sums negative increments −log(4/3); the notation should clarify the sign, e.g. log y−log(4/3)+O(1/U).","section":"§4.2, notation"}],"recommendation":"reject","confidential_remarks":"The paper's main theorem is invalidated by a simple counterexample in a central lemma (Theorem 5.2). The manuscript also contains an unrecognized corollary: if the unconditional contraction bounds were correct, they would prove RH; the paper does not state this. The remaining gaps (backward predecessor existence, phase-alignment issue in the Ω-result argument) further confirm that the central claim is not salvageable within the manuscript's scope. I see no path to revision that would preserve the claimed result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this paper does not prove RH, and it does not even contain a sound new criterion. The claimed equivalence E(X) << X^{1/2} log X iff RH is a thinly disguised von Koch: E(X) is defined as a supremum over trajectories of E(m), but since every composite can start its own trajectory, E(X) is just max_{m in W_X} E(m). The dynamics drop out.\n\nWhat is good: the one-visit and parent-window lemmas in Section 3 are elementary and correct, and the author is honest about constants, including numerical audits. The map itself is a nice curiosity.\n\nThe soft spots are load-bearing. Theorem 5.2, the log-scale large sieve, is plainly false. Take M=1, u_1=0, w_1=1. Then the left side of (6) is |Gamma| ~ U^4/2, while the right side is 8(1+U). For U >= 120 this is off by a factor of U^3. The proof's step replacing 2T/h = U^4/2 by 4/h = 2U \"within a harmless constant\" is not harmless; it is the whole term. This theorem feeds directly into the contraction inequalities in Section 6.\n\nSecond problem: Lemma 4.1 conflates additive and multiplicative contraction. After L = floor(log(4/3) U) backward steps, each decreasing log by about 1/U, the log decreases by about log(4/3) ~ 0.2877. So a point near X lands near 0.75X, not near X^{0.75}. The claimed error O(1/U) cannot bridge a gap of size about (U - log(4/3)) - 0.75U ~ 0.0377U. So the core-overlap lemma fails, and the iteration closure never gets going.\n\nAlso note: if Corollary 6.6 were true, it would combine with Theorem 8.2 to prove RH. The author does not draw this conclusion, which suggests the contraction inequalities were never actually checked at the level of consequences.\n\nConclusion: the reformulation is not new, and the unconditional machinery is broken in a way that is easy to verify. The paper is not coherent on its own terms, so I would not spend referee time on it. Desk reject. If the author isolated the Section 3 window lemmas as a small note, that might be citable, but not this paper as a whole.","headline":"The dynamical reformulation is just von Koch's criterion in disguise, and the claimed unconditional contraction inequalities collapse: Theorem 5.2's large sieve is false and the macro-step lemma moves X to 0.75X, not X^{3/4}.","tokens_in":17923,"tokens_out":6425,"would_cite":false,"duration_ms":71796,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N05","37A25","37E05","11N56"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a dynamical reformulation of the Riemann Hypothesis: the trajectory error functional E(X) is bounded by X^(1/2) log X if and only if RH holds.","keywords":["Riemann Hypothesis","dynamical system","trajectory","prime-counting error","contraction inequality","explicit formula","large sieve","integer map"],"falsifier":"Evaluate equation (6) directly for M=1 with U=120: the left side is |Γ| ≈ U^4/2 ≈ 9.3×10^7, while the right side is 8(1+U) = 968; if the inequality does not hold numerically, the paper's contraction proof fails at this step. A reader could also compute E(X) for moderate X (e.g., 10^6 to 10^8) and check whether the claimed contraction inequality E(X) ≤ (5/6)E(X^(3/4)) + 100 X^(1/2) log X is satisfied on random trajectories.","tokens_in":16879,"feed_emoji":"🔢","tokens_out":6448,"duration_ms":63149,"temperature":0.7,"pith_summary":"The paper introduces a discrete dynamical system on the integers: composite numbers jump forward by π(m), and primes jump backward by the gap to the previous prime. Error functionals E(X) aggregate the prime-counting error π(m) − Li(m) over composite visits inside short multiplicative windows. The central claim is that the bound E(X) ≪ X^(1/2) log X holds for all large X exactly when the Riemann Hypothesis is true. The forward direction follows from the classical von Koch bound under RH; the converse invokes Landau–Littlewood Ω-results to show that any off-critical zero forces oscillations large enough to break the contraction inequality. A sympathetic reader would care because this converts an analytic statement about zeros into a stability property of a simple, explicitly computable integer map.","feed_headline":"Equivalence claimed: integer trajectories contract exactly when RH holds","feed_subtitle":"A simple map's prime-counting error growing as X^(1/2) log X would prove and be proved by the Riemann Hypothesis.","key_machinery":"The central object is the map a(m) = m + π(m) for composites and a(m) = m − prevprime(m) for primes, together with the error functional E(X) defined as the supremum over trajectories of the sum of E(m) = π(m) − Li(m) over composite hits inside the one-visit window W_X = [X, (1 + 0.1/log X)X]. The key carrying mechanism is the macro-step alignment lemma (Lemma 4.1), which shows that tracing a trajectory backward through L = ⌊log(4/3) log X⌋ composite steps maps the scale X to X^(3/4) with controlled distortion, plus the frequency-netting lemma (Theorem 5.2), a log-scale large-sieve inequality over a grid of spacing h = 2/log X that bounds the zero sum from the smoothed explicit formula. These","core_discovery":"The paper's core discovery is Theorem 8.2: RH holds if and only if the trajectory error functional satisfies E(X) ≪ X^(1/2) log X for all X ≥ e^120. The proof runs through a chain of unconditional estimates—one-visit and parent-window lemmas limiting how often a trajectory hits a window, macro-step alignment showing that L ≍ log X composite steps contract the scale from X to X^(3/4), and a frequency-netting lemma that controls the sum over Riemann-zero contributions using an explicit smoothed formula with cubic-log truncation and a grid-based large sieve. Iterating the contraction inequalities yields the unconditional bound E(X) ≪ X^(1/2) log X (Corollary 6.6), and the paper then argues that","pith_inferences":["The frequency-netting inequality (6) appears numerically suspect: for a single point M=1 with U = log X, the left side is |Γ| ≈ U^4/2 while the right side is 8(1+U), a gap of order U^3; if this bound fails, the unconditional contraction theorem and the equivalence are not established. This is an editorial inference from a direct check, not a claim the paper makes.","The converse direction leans on the classical Landau–Littlewood Ω-results rather than on the new dynamical machinery; the genuinely new content is the claim that the trajectory contraction bound is strong enough to capture zero oscillations, which is what would need independent verification.","A testable extension would be to compute E(X) for X up to a large computational limit using the trajectory definition, measuring whether the growth exponent stays at 1/2 log X; this would provide empirical support or a fast falsifier for the equivalence.","If the contraction bound can be proven unconditionally by a different method—say via a correct large-sieve estimate—then the framework would reduce RH to a finite-checkable constant, not just an abstract equivalence."],"forward_implications":["If correct, RH becomes equivalent to verifying a deterministic bound on integer trajectories, giving a new computational probe: the contraction inequality can be tested numerically at increasingly large scales.","The forward implication means the von Koch bound |π(x) − Li(x)| ≪ x^(1/2) log x follows directly from the dynamical contraction property, so the trajectory system 'hears' RH.","The converse shows that any off-critical zero would appear as a violation of the contraction bound on infinitely many logarithmic windows, making the absence of such violations a sharp zero-location criterion.","The explicit constants (X_0 = e^120, θ = 3/4, α = 5/6, B = 100) make the claimed inequalities checkable at finite scales, at least in principle.","Because the paper's unconditional contraction inequalities, if valid, would themselves imply RH via the converse, the framework suggests a direct route from dynamical stability to zero location."],"fun_headline_variants":["RH iff integer trajectory error shrinks to X^1/2 log X","Prime-tracking map: RH iff contraction bound holds","Dynamical reformulation: RH equals X^1/2 log X contraction","Integer walk contraction is equivalent to RH","Prime dynamics: RH restated as trajectory stability"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the log-scale large-sieve bound in Theorem 5.2 (equation (6)), which the contraction proof uses to control the Riemann-zero sum over up to four window points; if that inequality is false, the unconditional contraction bound and the equivalence with RH collapse.","fun_headline_variants_meta":{"raw":{"variants":["RH iff integer trajectory error shrinks to X^1/2 log X","Prime-tracking map: RH iff contraction bound holds","Dynamical reformulation: RH equals X^1/2 log X contraction","Integer walk contraction is equivalent to RH","Prime dynamics: RH restated as trajectory stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3005,"prompt_tokens":768,"completion_tokens":2237,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":2154}},"tokens_in":512,"tokens_out":2237,"duration_ms":15826,"temperature":1.0,"reasoning_tokens":2154,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T18:15:30.695564+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate equation (6) directly for M=1 with U=120: the left side is |Γ| ≈ U^4/2 ≈ 9.3×10^7, while the right side is 8(1+U) = 968; if the inequality does not hold numerically, the paper's contraction proof fails at this step. A reader could also compute E(X) for moderate X (e.g., 10^6 to 10^8) and check whether the claimed contraction inequality E(X) ≤ (5/6)E(X^(3/4)) + 100 X^(1/2) log X is satisfied on random trajectories.","supporting_citations":[],"review_version":1}