# Independent proof arbitration

Dispositions apply only to listed targets in the original frozen edition. A broken proof is not a refuted theorem. Later-edition decisions are separate.

Usable arbitration: 1/1. Author response: not_supplied.

Prior accounting $5.579903; arbitration accounted/reserved $0.340696; campaign budget $25.00.

Uncovered queued targets: eq:cI3bnd, eq:cI41bnd, eq:cI42bnd, eq:cIQ11bnd, eq:cIQ12bnd, eq:cIQ1def, eq:cIQ2bnd, eq:cIQ3def, eq:cKzBnd, eq:cRfuIs, eq:cSfbnd1, eq:decayMtrx, eq:eqeq, eq:fincK, eq:gls, eq:hatRfuBnd, eq:modBndaEqAp, eq:modBndaNeqAp, eq:nmUpToM, eq:vgIs, lem:GGq, lem:kEllD, lem:spec0, lem:spec1, lem:spec2, lem:spec3, lm_generation, lm_local, lm_transference, narrative:103198, narrative:104579, narrative:105898, narrative:109314, narrative:110408, narrative:112352, narrative:114852, narrative:115416, narrative:118709, narrative:119956, narrative:122162, narrative:123431, narrative:123918, narrative:13050, narrative:144695, narrative:47002, narrative:55546, narrative:68030, narrative:71322, narrative:86590, narrative:92045, narrative:93454, narrative:98161, narrative:99181, th_sg, thm:CircMeth, thm:IQ0, thm:IQM, thm:IQX, thm:RNU, thm:Vin, thm:cMNis, thm:cMNtoMNU, thm:specGap

## thm:Main — gemini-4-argon

Delivery: completed. Scoped proof disposition: broken. Statement: unresolved.

In the displayed proof of Theorem thm:Main assuming Theorem thm:RNU, there are two substantive gaps against the stated target. First, Theorem thm:RNU establishes the singular-series lower bound and exceptional-set estimate O(N^{1-\eta}) only on the dyadic interval \frac{1}{2}N < n < N, whereas the proof of Theorem thm:Main defines Z and W over all admissible n < N. Because \cR_N(n) is localized to scale TX^2 = N, \cR_N(n) and \cR_N^U(n) vanish for small n \ll N, so |W^c| \ll N^{1-\eta} is literally false for W defined on all admissible n < N; a restriction to (N/2, N) followed by dyadic summation over scales is required but omitted. Second, Theorem thm:Main claims that \eta > 0 is effectively computable, whereas Theorem thm:RNU only asserts existence of \eta > 0 and the manuscript never proves effective computability. The Möbius-tail estimate and power-saving algebra on a single dyadic interval are valid in the full manuscript context. Regarding chronology, the full manuscript bibliography cites Vinogradov's preprint arXiv:1204.5498v1 (April 2012, prior to the 2012-05-24 cutoff), though the external preprint text is not supplied in the packet.

New repair (conditional on Theorem thm:RNU, eq:fFTbnd, eq:Uis, eq:TXNis, and Lemma 2.12 / eq:binQuad for the existence of a power-saving exponent): For a scale parameter M, define Z(M) to be the set of admissible n in (M/2, M) for which \cR_M(n) = 0, and W(M) to be the set of admissible n in (M/2, M) satisfying eq:RNUbnd at scale M. By the Möbius-tail estimate and Theorem thm:RNU applied on (M/2, M), |Z(M) \cap W(M)| \ll_\vep M^{1+\vep-\fu} and |Z(M) \cap W(M)^c| \le |W(M)^c| \ll M^{1-\eta_R}, where W(M)^c denotes the complement inside admissible n in (M/2, M). Taking \vep = \fu/2 and \eta' = \min(\eta_R, \fu/2) > 0 gives |Z(M)| \ll M^{1-\eta'}. Since \cR_M(n) \neq 0 implies n \in \sB by eq:cRNis and eq:binQuad, every admissible exception n \in (M/2, M) \setminus \sB lies in Z(M). Summing over dyadic scales M = N/2^k for 0 \le k \le \lfloor \log_2 N \rfloor + 1 yields #(\{n < N : n \in \sA \setminus \sB\}) \ll \sum_{k \ge 0} (N/2^k)^{1-\eta'} \ll N^{1-\eta'}. However, the full statement of Theorem thm:Main remains unresolved in this packet because (1) Theorem thm:Main additionally claims that \eta > 0 is effectively computable, which is neither assumed in Theorem thm:RNU nor proved in the manuscript (the Appendix explicitly notes that explicit bounds are not pursued), and (2) upstream external inputs (such as Vinogradov 2013 / arXiv:1204.5498v1) are not supplied in full text for direct verification.

### finding-94be3d9bf03ed68b4408 — confirmed_error (substantive)

In the full source preamble, \foh is defined as \frac{1}{2}. Theorem thm:RNU states that for \frac{1}{2}N < n < N, \fS(n) \gg_\vep N^{-\vep} for n \in \sA and eq:RNUbnd holds for admissible n in (\frac{1}{2}N, N) except on a set of cardinality \ll N^{1-\eta}. However, the proof of Theorem thm:Main defines Z and W over all admissible n < N ('let Z be the "exceptional" set of admissible n<N for which \cR_N(n)=0. Futhermore, let W be the set of admissible n<N for which \eqref{eq:RNUbnd} is satisfied'). Because \cR_N(n) and \cR_N^U(n) are localized via \gU(2x/X)\gU(y/X) and \g \in \fF_T to values \ff(2x,y) of order TX^2 = N, \cR_N(n) and \cR_N^U(n) vanish for small n \ll N. Consequently, for W defined over all admissible n < N, Theorem thm:RNU does not imply |W^c| \ll N^{1-\eta} (and in fact |W^c| \asymp N for a fixed N). The proof omits restricting Z and W to the dyadic interval \frac{1}{2}N < n < N and summing over dyadic scales.

Hypotheses: Checked against the full statement of Theorem thm:RNU and the definitions of \cR_N (eq:cRNis) and \cR_N^U (eq:cRNhatUis) in Section 3.1. Theorem thm:RNU explicitly restricts its hypotheses and conclusion to \frac{1}{2}N < n < N.

Original source: "Note also from Theorem \\ref{thm:RNU} that $|Z\\cap W^{c}|\n\\le |W^{c}|\\ll N^{1-\\eta}$."

Later edition: not_checked. No later edition was supplied for comparison.

Later source: ""

Affected targets (arbiter assessment): thm:Main

### finding-a0a39e9cc39cb243796d — packet_context (none)

This finding is a context note confirming the elementary Möbius-tail estimate \sum_{n<N} |\cR_N(n) - \cR_N^U(n)| \ll T^\gd X^2 / U. In the full manuscript (e.g. Section 3.1 and Lemma lem:spec3 in Section 5.1, eq:1), sums over \fF_T are taken with the pair multiplicity (\g_1, \g_2) from eq:fFdef, and applying the Lax-Phillips ball bound to \|\g_1\| < 2T_1 and \|\g_2\| < 2T_2 with T = T_1 T_2 (eq:TT1T2) bounds the multiplicity-counted family by \ll T_1^\gd T_2^\gd = T^\gd. Since \supp(\gU) \subset [1,2], 2x and y lie in [X, 2X], so for each u \ge U there are O(X/u) choices for 2x and O(X/u) choices for y (and zero choices for u > 2X), yielding \ll T^\gd \sum_{U \le u \le 2X} (X/u)^2 \ll T^\gd X^2 / U.

Hypotheses: Consistent with the definitions of \fF_T (eq:fFdef), T = T_1 T_2 (eq:TT1T2), eq:fFTbnd, and \gU in Section 3.1.

Original source: "T^{\\gd}            \n{X^{2}\\over U}"

Later edition: not_checked. No later edition was supplied for comparison.

Later source: ""

Affected targets (arbiter assessment): thm:Main

### finding-5e7e13d1bcc21dc161e4 — packet_context (none)

This finding is a context note confirming that on the dyadic interval (N/2, N) covered by Theorem thm:RNU, the exponent arithmetic is valid: since U = N^\fu with \fu > 0 (eq:Uis) and TX^2 = N (eq:TXNis), choosing \vep = \fu/2 in N^{1-\eta} + N^{1+\vep}/U yields a power saving N^{1 - \min(\eta, \fu/2)}. Moreover, the implication \cR_N(n) \neq 0 \implies n \in \sB is established in Section 2.2 of the full manuscript (Lemma 2.12, eq:binQuad and eq:ffToInProd).

Hypotheses: Checked against Lemma 2.12 (eq:binQuad), eq:ffToInProd, eq:TXNis, eq:Uis, and Theorem thm:RNU in the full manuscript.

Original source: "which\n is a power savings since $\\vep>0$ is arbitrary."

Later edition: not_checked. No later edition was supplied for comparison.

Later source: ""

Affected targets (arbiter assessment): thm:Main

### finding-34cafcc1e4a10af2e9bc — confirmed_error (substantive)

Theorem thm:Main explicitly states that the exponent \eta > 0 in O(N^{1-\eta}) is 'effectively computable'. However, in the reduction to Theorem thm:RNU, Theorem thm:RNU only states 'There exists an \eta > 0', and the proof of Theorem thm:Main assuming Theorem thm:RNU provides no argument for effective computability. Inspecting the full manuscript confirms that effective computability is never mentioned or proved anywhere after the statement of Theorem thm:Main: Theorems thm:CircMeth, thm:specGap, and thm:Vin are all stated as existence results without effective bounds on the spectral gap \vep(\G) or counting exponent \gT < \gd, and the Appendix explicitly remarks 'we do not pursue explicit bounds'.

Hypotheses: Checked against the full text of the manuscript, including Theorem thm:RNU, Theorem thm:CircMeth, Section 4 (Theorems thm:specGap and thm:Vin), Section 5 (Lemmata lem:spec0-lem:spec2), Section 6 (eq:Q0is, eq:fuIs), and the Appendix.

Original source: "$O(N^{1-\\eta})$, where $\\eta>0$ is effectively computable."

Later edition: not_checked. No later edition was supplied for comparison.

Later source: ""

Affected targets (arbiter assessment): thm:Main

### finding-3f2d492685ad2dff96d1 — unresolved (none)

In the local reduction of Theorem thm:Main to Theorem thm:RNU and eq:fFTbnd, no external result is directly invoked in the displayed proof step (eq:fFTbnd is an internal display cited to Lax-Phillips 1982 or eq:Vin). Upstream, Theorem thm:Vin (eq:Vin) is deduced from \cite[Thm 2.2]{Vinogradov2013}. Inspecting the full manuscript's bibliography (omitted from the reviewer's local section packet) resolves the citation metadata: \bibitem[Vin13]{Vinogradov2013} cites 'Princeton University Thesis 2012, {\tt arxiv:1204.5498v1}', which dates to April 2012, prior to the 2012-05-24 received cutoff. However, because the external text of arXiv:1204.5498v1 is not supplied in the review packet, Theorem 2.2 of Vinogradov2013 and its exact hypotheses cannot be directly inspected in the supplied material, leaving the external input verification unresolved.

Hypotheses: Checked against Section 4.4 (Theorem thm:Vin) and the bibliography (\bibitem[Vin13]{Vinogradov2013}) in the full reviewed source.

Original source: "The following theorem follows easily              \nfrom \\cite[Thm 2.2]{Vinogradov2013}."

Later edition: not_checked. No later edition was supplied for comparison.

Later source: ""

Affected targets (arbiter assessment): thm:Main, thm:Vin