REVIEW 4 major objections 3 minor 20 references
A Dynamical Criterion Equivalent to the Riemann Hypothesis
T0 review · 4 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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}. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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
What would settle it
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.
Extended reading notes
Core claim
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
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (4)
- [§5.2, Theorem 5.2, Eq. (6)] 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.
- [§4.2, Lemma 4.1] 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.
- [§8.1, Theorem 8.2, (2⇒1)] 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.
- [§5.3, Lemma 5.3] 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.
minor comments (3)
- [End of §7 and Appendix G] 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.
- [Appendix A.7 vs §5.1] 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.
- [§4.2, notation] 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).
Circularity Check
The main RH equivalence is a windowed restatement of the classical von Koch/Landau–Littlewood criterion: by Lemma 3.1 the trajectory functional E(X) collapses to sup over single π(m)−Li(m) values, so the 'dynamical' reformulation is a repackaging. No load-bearing self-citation; the invalid contraction machinery is a correctness error, not circularity.
-
renaming known result
[Section 2 (definition of E(X)), Lemma 3.1, Theorem 8.2]
"We define error functionals by aggregating the classical error term E(y)=π(y)−Li(y) along composite visits of a trajectory: E(X)=sup_trajectories Σ_{m∈W_X∩trajectory, comp.} E(m). ... Lemma 3.1 (One-visit uniqueness): any trajectory {x_k} can contain at most one composite element inside the one-visit window W_X. ... Theorem 8.2: The following are equivalent: 1. RH holds. 2. For all X≥e^120, the trajectory error functional satisfies E(X)≪X^{1/2} log X."
By Lemma 3.1, the supremum over trajectories of sums of E(m) reduces to the supremum of the single classical error E(m) over composite m in W_X. Thus E(X) is, by construction, just a short-window supremum of π(m)−Li(m). Theorem 8.2 therefore restates the classical von Koch criterion (with Landau–Littlewood supplying the converse) in trajectory language; the dynamical map a(m) contributes no additional structure beyond the trivial one-visit count. Presenting this as a new dynamical reformulation is a renaming of a known equivalent of RH rather than a derivation from the dynamics.
full rationale
No load-bearing self-citation was found: the OEIS self-citations (refs. [2]–[4]) are motivational only, and the analytic inputs (Dusart, Trudgian, Iwaniec–Kowalski, Montgomery–Vaughan, Titchmarsh) are external and legitimate. The central circularity concern is pattern 6: the 'dynamical criterion' E(X) is defined directly from the classical error E(m)=π(m)−Li(m), and Lemma 3.1 collapses the trajectory supremum to a pointwise window supremum, so Theorem 8.2 is essentially the classical von Koch equivalence in a new notation. That is a repackaging, not an independent derivation from the dynamics. Separately, the paper's unconditional contraction claim (Corollary 6.6) would imply the bound E(X)≪X^{1/2}log X; combined with Theorem 8.2 this would prove RH, a consequence the paper does not draw. The contraction chain rests on Theorem 5.2's log-scale large sieve, equation (6), which is arithmetically false: for M=1 the left side is |Γ|~U^4/2 while the right side is ~8U, and the proof replaces |G(0)|~U^4/2 by 4/h~2U as 'harmless'. This invalidates the frequency netting and the contraction inequalities, but it is a mathematical error rather than a circularity. Accordingly the circularity score reflects the repackaging of the known equivalence (4), not a self-citation chain or a definitional tautology at the maximal level.
Assumptions & free parameters
free parameters (6)
- one-visit window width constant =
0.1
- parent window width constant =
2
- contraction ratio theta =
3/4
- contraction factor alpha =
5/6
- error constant B =
100
- overlap constant c0 =
1/6
assumptions (5)
- standard math Dusart explicit bounds for pi(x)
- domain assumption Smoothed explicit formula with kernel W(t) = (1+t^2)^{-3} and truncation T = U^3/2 has remainder at most 10X^{1/2}
- standard math Landau-Littlewood Omega results
- ad hoc to paper Every y in the core C_X has an L-fold composite predecessor Psi_X(y)
- ad hoc to paper Large-sieve bound on a uniform grid with constant independent of point spacing
Cite this review
Pith. "Pith review of A Dynamical Criterion Equivalent to the Riemann Hypothesis." pith.science (2026). https://pith.science/paper/MWCPFVVJ
@misc{pith2026250910588,
author = {Pith},
title = {Pith review of: A Dynamical Criterion Equivalent to the Riemann Hypothesis},
year = {2026},
howpublished = {\url{https://pith.science/paper/MWCPFVVJ}},
note = {Machine review of arXiv:2509.10588}
}
abstract
We introduce a discrete dynamical system on the integers, defined by moving a composite $m$ forward to $m+\pi(m)$ and a prime $p$ backward to $p-\mathrm{prevprime}(p)$. This map produces trajectories whose contraction properties are closely tied to the distribution of primes. We prove unconditional contraction inequalities for error terms derived from these trajectories, using explicit remainder bounds from the smoothed explicit formula. Building on this, we show that the Riemann Hypothesis is equivalent to a sharp contraction condition: the trajectory error functional satisfies $E(X)\ll X^{1/2}\log X$. The forward implication follows directly from von Koch's classical bound under RH. For the converse, we invoke the Landau--Littlewood $\Omega$-results, which guarantee that any off-critical zero forces oscillations large enough to violate the contraction inequality. This establishes a new dynamical reformulation of RH: the critical-line conjecture is equivalent to the assertion that integer trajectories remain uniformly contracted at the scale $X^{1/2}\log X$, equivalently, RH holds. The perspective is distinct from earlier analytic equivalents, as it arises from stability properties of a simple deterministic system rather than from Hilbert space or Dirichlet polynomial approximations.
Reference graph
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