# Bourgain–Kontorovich: interim source assessment, 18 October 2026

**All submitted jobs are terminal; the collector finished normally. The density-one theorem remains unresolved by this campaign. No new paid requests or public verdict have been issued.**

This note separates collected model findings from source checks by the coordinating
assistant. The latter are AI-assisted mathematical checks, not human grading or
formal verification. The reviewed edition remains arXiv 1205.4416v2, including Varjú's
appendix, with external-input cutoff 24 May 2012.

## Delivery and cost

| Model/stage | Usable reports / attempts | Accounted token cost (USD) |
| --- | ---: | ---: |
| Sol local reviews | 64 / 66 | 3.201094 |
| Opus max local reviews | 0 / 3 | 1.151704 |
| Fable max local review | 0 / 1 | 0.927345 |
| Gemini local reviews | 2 / 3 | 0.299760 |
| Gemini independent arbitration | 2 / 4 | 1.090650 |
| Total | 68 / 77 | **6.670553** |

The panel therefore delivered 66/73 usable assessments; independent arbitration
delivered 2/4. Opus and Fable each exhausted every 32,768-token allowance entirely
on thinking, without a report. Two Sol and one Gemini local responses failed source
quotation validation. Two arbitration responses were rejected for simultaneously
claiming a supported original proof and confirming an error. These are failed
assessments, not mathematical votes. The cost includes all these attempts, uses
conservative token accounting and is not invoice reconciliation. The approved
ceiling was $25. No automatic retries, repairs or publication occurred.

## Returned arbitrations require mathematical interpretation

The accepted arbitrations flag the main-theorem reduction (`thm:Main`) and a
minor-arc estimate (`eq:cIQ12bnd`). Raw reports label their supplied proofs broken;
that model label alone does not justify a paper-level red disposition.

### Dyadic reduction: routine completion available

Source: expanded manuscript lines 1483–1497 and 1500–1565. The premise `thm:RNU`
controls admissible n in (M/2,M), whereas the displayed reduction defines its sets
using all n<M. This is a literal scope mismatch. Conditional on the stated upstream
bounds, its correction is straightforward: define the exceptional set separately
on each interval (M/2,M), use the existing tail argument there, and obtain
E(M) = O(M^(1-sigma)) for a fixed 0<sigma<min(eta_R,u/2,1/2). Sum over M=N/2^k
above the fixed asymptotic threshold. The geometric series is O(N^(1-sigma)); the
O(log N) boundary integers and bounded initial range are absorbed into this bound.
Here E(M) counts actual missing admissible curvatures; a nonzero scale-M
representation implies an actual curvature by the manuscript's encoding.

Thus the missing dyadic summation does not by itself defeat the density-one
conclusion. This check supplies only the local reduction conditional on its
upstream estimates, not those estimates themselves.

### Minor-arc exponent: choose the existential saving small enough

Source: `lem:spec3`, expanded lines 2228–2251; parameter choices at 3756–3770;
`eq:cIQ12bnd` and its proof at 4712–4835. The disputed estimate uses
Q^(-1/4) sum_{g<=C Q} g^(-3/4-theta) = O(Q^(-theta)), valid for 0<theta<1/4.
The displayed setup states only 0<eta_0<1, but the counting lemma's existential
saving may be weakened: choose theta=min(eta_0,1/8) before setting U and H.
For q>=1, q^(-eta_0)<=q^(-theta), so the same counting conclusion remains valid.
The integral comparison gives sum_{g<=C Q} g^(-3/4-theta)=O_theta(Q^(1/4-theta));
Q>=Q_0 then gives O(Q_0^(-theta)). With U=Q_0^(theta^2/100) and
H=Q_0^(theta/10), the combined exponent is

6 theta^2/100 + 9 theta/40 - theta = 3 theta^2/50 - 31 theta/40 < 0.

This removes the parameter objection locally, conditional on the counting lemma
and other stated prerequisites. The preceding prose already specifies the
pairwise coprimality used in the modulus count; keeping that restriction until
after the inner q-count is bounded, then dropping it by positivity, clarifies the
display. These routine completions do not establish the counting lemma itself.

### Effectivity: unresolved, not disproved

The main theorem says the exponent is effectively computable. The accepted
arbiter objects that numerical bounds are not supplied and quotes the appendix's
statement that explicit bounds are not pursued (expanded lines 5329–5334).
Not computing an explicit number does not show that no effective procedure exists.
The reduction needs effective upstream gap/counting exponents, which still require
an audit. This finding remains an unresolved quantitative obligation; it is not a
counterexample to density one or a demonstration of ineffectivity.

## Important remaining work

Sol also raises more technical concerns about congruence-orbit counts, stabilizer
indices, uniform spectral smoothing, quadratic-form identities and Fourier tails.
These did not receive usable independent arbitration in the four selected cases.
Most queued targets also request omitted context, not demonstrated manuscript
errors. We have not promoted those objections into accepted defects, nor treated
the number of flagged packets as independent evidence against the paper.

The panel's local packets are conditional and do not constitute independent
full-paper reviews. External theorem texts and transitive dependence on the earlier
Bourgain–Fuchs work remain incompletely audited. No paper-level green/yellow/red
classification is warranted by this interim note.

[Neutral reading packet](../reading.md) · [Local reports](../dependency-review/review-report.md)
· [Main-reduction arbitration](../arbitration-000/arbitration-report.md)
· [Minor-arc arbitration](../arbitration-003/arbitration-report.md)