# Source-role adjudication for the Baier–Bansal follow-up

Date: 13 October 2026. This is a bounded AI source adjudication, not human gold or an assessment of the quasi-Riemann theorem. Inputs are `selected-cases.json`, `retrieval-cases.json`, `retrieval-run/paired-summary.json` and the frozen native manuscript (SHA256 `42a5ee0febca59fd1def55cfd6c6808c322ef4237d7726303ea52f711deac6a3`). No correctness conclusion about the manuscript is drawn here.

## Case and operative distinction

The relevant follow-up is `quasi-riemann-retrieval-1`, whose consumer is the entire manuscript and whose named candidate prerequisite is the **whole Baier–Bansal reference**, not the manuscript's internal planar additive large-sieve lemma. Its short excerpt, native lines 99–113, states the main claim but contains no occurrence of that reference. An excerpt-only `insufficient_context` answer is supported: neither use nor non-use is recoverable from that passage alone.

For the full-source question, I retain the source annotation **`background`** when `proof_input` means an external proposition actually applied as a premise of the written argument. The response `proof_input` is an unsupported promotion under that definition. This is a role-classification disagreement, not a mathematical objection to the reference or to the manuscript.

## Exact source evidence

- Native lines **144–150** call the planar additive large sieve classical, say **“compare”** Huxley and Baier–Bansal's Poisson proof, and immediately announce a direct Eisenstein-lattice proof for the required normalization.
- Native lines **3485–3495** repeat that methodological comparison while introducing the balanced additive estimate. They again state that the lattice proof is included locally.
- Native lines **3497–3514** state the internal `lem:planar-additive-sieve` with the Eisenstein lattice's separation, row-norm and measure conventions. Native lines **3516–3580** supply its proof: bandlimited majorization, local Poisson summation, separated-lift counting and a coefficient estimate. The citation is not used as a theorem invocation in that proof.
- A literal scan finds only two in-text occurrences of `BaierBansal`, at **148** and **3494**, and its bibliography entry at **16466**. Both in-text occurrences have the same comparison role. The bibliography identifies the reference's Gaussian-integer setting `Z[i]`; this is distinct from the locally proved Eisenstein-lattice statement.

These facts distinguish a **real internal proof input** (the manuscript's planar additive large-sieve lemma) from a **reference to a comparable proof method** (Baier–Bansal). Classifying the latter as background does not imply that it had no intellectual influence, nor that its proof or general method is irrelevant. It says the written argument supplies the proposition locally rather than importing the cited theorem as its operative premise.

## Scope ambiguity and interpretation of the observed answer

The state has a broad manuscript consumer and a whole-paper prerequisite. It does not ask about a single numbered external theorem and a single local consumer. A looser classifier could use `proof_input` to include a referenced proof pattern or methodological antecedent. Under that meaning the disagreement is partly taxonomic. For this experiment, the source-grounded role distinction should be stated explicitly: **an applied theorem is a proof input; a comparison to an independently reproduced argument is background**. The citation title or the fact that a large sieve is needed does not by itself establish an external dependency.

The earlier full-context question with lines 144–152 supplied classified this same reference as background, while the omission follow-up with the full manuscript classified it as proof input. That demonstrates sensitivity to the supplied excerpt/prompt state despite the identical long source. It does not isolate whether the failure arose from retrieval, context prioritization, choice semantics or ordinary sampling variability. No rationale or evidence of a precise retrieval process is supplied by a choice-only answer.

## Reporting recommendation

Report the follow-up as **7/8 full-context agreement with AI source annotations**, with this one disagreement reviewed at the source level. The eight cases are a post-hoc context-omission follow-up, not held-out tasks, independent human gold, or a mathematical correctness test. Changes toward an AI annotation measure label agreement. They do not measure proof verification or establish general long-context superiority. The excerpt's information deficit, the broad consumer and the differing role semantics should remain visible beside aggregate counts.

## Source

[Pinned native manuscript](https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/build/paper.tex), named line spans and literal citation scan inspected locally on 13 October 2026. The cited Baier–Bansal theorem's own proof was not re-audited for this role check; no such audit is necessary to establish how the manuscript cites it.