# Row 12 recon: the lonely runner conjecture, and the Birkhoff / bounded-remainder branch **2026-08-20. Literature reconnaissance only; no experiment was run and no live document was edited. Every claim below carries its source label.** **Verdict: CLOSED WITH MECHANISM as an import, with two positives banked.** No theorem from this family transfers to the postulate or to `G₂`, and the reason is structural rather than a search failure. What the row does deliver is the calibration the map priced it for, and it delivers it harder than the map expected: the lonely runner literature contains its own diagnosis of why the first-moment bound resists improvement, and **the configuration it names as the obstruction is our configuration** — velocities that are primes, whose Bohr sets overlap only in a kernel of negligible measure. The payoff class is **WALL-ADDRESS**, not PUBLISHED-ANCHOR for the exponent; a separate and unexpected PUBLISHED-ANCHOR candidate turned up in the same family (§5) and is about wheels, not about `G₂`. --- ## 0. What was read, and at what source | artifact | label | |---|---| | Tao, *Some remarks on the lonely runner conjecture*, arXiv:1701.02048v4, *Contrib. Discrete Math.* 13 (2018) 1–31 | **[SOURCED]** — LaTeX source of v4 pulled from `arxiv.org/e-print`, §1 read in full, statements quoted verbatim below | | Perarnau–Serra, *The Lonely Runner Conjecture turns 60*, arXiv:2409.20160v3 (2025) | **[SOURCED]** — LaTeX source pulled from `arxiv.org/e-print`; §§1–3, 6, 9, 10, 11 and the open-problem list read in full | | Grepstad–Lev, *Sets of bounded discrepancy for multi-dimensional irrational rotation*, arXiv:1404.0165v2, *GAFA* 25 (2015) | **[SOURCED]** — LaTeX source, §1 read in full (BRS definition, Hecke–Ostrowski, Kesten, Theorem 1, the measure constraint) | | Denjoy–Koksma inequality | **[SOURCED via a secondary carrier]** — displayed statement in Carrand, arXiv:2012.07481v2, *Qual. Theory Dyn. Syst.* (2022), DOI 10.1007/s12346-022-00632-8, which attributes it to Herman, *Publ. Math. IHÉS* 49 (1979), Theorem VI.3.1. Herman himself **NOT REACHED** | | Blanco–Criado–Santos, arXiv:2603.24784v2; Alcántara–Criado–Santos, arXiv:2506.13379 (*Experimental Math.*, prepub. May 2026); Malikiosis–Santos–Schymura, arXiv:2411.06903v2, *Forum Math. Sigma* 13 (2025) e164; Jensen, arXiv:2605.27941v1; Bedert, arXiv:2511.16636v1; Sungkawichai–Trakulthongchai, arXiv:2604.23906v1 | **[SOURCED, abstract only]** — arXiv API metadata, abstracts read verbatim, bodies not opened | | Furno–Haynes–Koivusalo, arXiv:1803.10559; Das–Furno–Haynes, arXiv:2002.04444, *Moscow J. Comb. Number Th.* 10 (2021) 111–120; Haynes–Koivusalo, arXiv:1402.2125; Etkind–Grepstad–Kolountzakis–Lev, arXiv:2511.21148 | **[SOURCED, abstract only]** | | Cusick, *View-obstruction problems*, *Aequationes Math.* 9 (1973) 165–170 | **NOT REACHED.** Springer redirects to an authorisation endpoint; zbMATH holds the record (Zbl id 3418244, MSC 52A20/52A30/11J99/11H06) with **no review text**. Its content is carried here only through Perarnau–Serra §3.2, which is read at source | | Schoenberg, *Extremum problems for the motions of a billiard ball II*, *Indag. Math.* 38 (1976) 263–279 | **NOT REACHED**, carried through Perarnau–Serra §11.3 | | Chen 1994; Chen–Cusick 1999; Schark 1974; Schark–Wills 1973; Wills 1968 | **[SOURCED-BIB]** only, through Perarnau–Serra's bibliography | **Channel note, and it limits every absence below.** The session's WebSearch budget was exhausted (200 of 200) before this recon started, so **no keyword web search was available at all**. Everything here ran on three programmatic channels: the arXiv API (`https`, per `SEARCH-CONVENTIONS.md` §3), the zbMATH Open API, and Crossref. MathSciNet remains unreached, as it has been for every wave. Calibration on the same channels in the same session, per `SEARCH-CONVENTIONS.md` §2: the arXiv API returned `1701.02048` with the exact *Contrib. Discrete Math.* journal reference the map carries, and zbMATH returned Cusick's *Aequationes Math.* **9 (1973) 165–170** at the exact volume and pages the map carries. Both known positives came back, so the channels were working. --- ## 1. What the conjecture controls Tao's normalisation, quoted from §1 of arXiv:1701.02048v4 **[SOURCED]**. For an `n`-tuple of non-zero integers `v_1,…,v_n`, `δ(v_1,…,v_n)` is the maximal value of `min(‖tv_1‖, …, ‖tv_n‖)` over `t ∈ R/Z`, and `δ_n` is the infimum of that over tuples of distinct non-zero integers. > **Conjecture (Lonely runner conjecture).** For every `n ≥ 1`, one has > `δ_n = 1/(n+1)`. The upper bound `δ_n ≤ 1/(n+1)` is Dirichlet applied to `(1,2,…,n)`, so the conjecture is the assertion that the Dirichlet bound is sharp. The whole content is the lower bound. **The formulation that matters to us is Tao's (ii), and it is a covering statement.** With `B(v; δ) = {t ∈ R/Z : ‖tv‖ ≤ δ}` the rank-one Bohr set, `δ_n` is the least number for which there exists a covering `R/Z = ⋃_{i=1}^n B(v_i; δ_n)` of the circle by `n` rank-one Bohr sets. Each `B(v_i; δ)` has measure exactly `2δ`, so at `δ = 1/(2n)` the total measure is exactly 1 and the union bound gives > **Proposition (Wills 1968; Tao's Prop. 1.3).** For every `n ≥ 1`, > `δ_n ≥ 1/(2n)`. **So the object is: `n` arithmetically-structured periodic subsets of a circle, of total measure exactly 1, and the question is whether they can cover.** That is our covering question with one class per modulus, stated in the continuum. Cusick's view-obstruction form is the same statement geometrically: with `Δ(K,λ) = λK + N^n − ½·1`, `λ(n)` is the least dilation of the unit cube for which every positive ray meets the family, and `λ(n) = 1 − 2κ(n)` (Cusick 1973 Lemma 1, carried through Perarnau–Serra §3.2, **Cusick NOT REACHED**). --- ## 2. What the partial results give The lower-bound ladder, in the order the literature built it. All statements through Tao §1 **[SOURCED]** except the last two. | bound | who | strength over `1/(2n)` | |---|---|---| | `δ_n ≥ 1/(2n)` | Wills 1968, union bound | baseline | | `δ_n ≥ 1/(2n − 1 + 1/(2n−3))` | Chen 1994 | factor `1 + O(1/n)` | | `δ_n ≥ 1/(2n−3)`, when `2n−3` is prime | Chen–Cusick 1999 | factor `1 + O(1/n)`, conditional on primality | | `δ_n ≥ 1/(2n − 2 + o(1))` | Perarnau–Serra 2016 | factor `1 + O(1/n)` | | `δ_n ≥ 1/(2n) + c log n / (n² (log log n)²)` | Tao 2018, Theorem 1.5 | factor `1 + Θ̃(1/n)` | | `δ_n ≥ 1/(2n) + 1/n^{5/3 + o(1)}` | Bedert, arXiv:2511.16636 (Nov 2025), abstract **[SOURCED]**, preprint, unrefereed | factor `1 + Θ(n^{−2/3})` | **Every one of these is a `1 + o(1)` multiplicative improvement on the union bound. Sixty years, and the constant 2 has not moved.** The field states this itself: Perarnau–Serra's open-problem list, Problem 2, asks > "we ask if for every `ε>0`, there exists `n₀ ∈ N` such that for all `n ≥ n₀`, > `κ(n) ≥ (1+ε)/(2n)`." That is, **whether the union-bound constant can be improved by any fixed factor at all is an open problem**, and Bedert's polynomial gain does not settle it, since `1/(2n) + n^{−5/3+o(1)} = (1/(2n))(1 + O(n^{−2/3}))`. **Small `n` moved a great deal in the last twelve months, and entirely by compute.** Perarnau–Serra (2025) records the conjecture as known for `n ≤ 6` (seven runners). Since then, at abstract level **[SOURCED]**: Rosenfeld arXiv:2509.14111 (eight runners) and arXiv:2512.01912 (nine); Trakulthongchai arXiv:2511.22427 (nine and ten); Sungkawichai–Trakulthongchai arXiv:2604.23906 (`k ∈ {10,11,12}`, that is eleven, twelve and thirteen runners). All are computer-assisted, and all rest on the finite-checking reduction — Tao's Theorem 1.6 (`|v_i| ≤ n^{C₀n²}`) as sharpened by Malikiosis–Santos–Schymura to `C(n+1,2)^{n−1} ≤ n^{2n}`, *Forum Math. Sigma* 13 (2025) e164. **The calibration statement for the map, corrected in both directions.** It is not "thirteen cases settled in sixty years" as a slow trickle: it is *seven* cases in fifty-seven years and *six more in twelve months*, every one of the six a verification rather than an idea, and no general-`n` progress accompanying them. And the general lower bound is no longer stuck at a `log`-factor nudge: Bedert's `n^{−5/3}` is a polynomial gain over `n^{−2}`. The map's paragraph predates it. --- ## 3. Tao's own obstruction is our configuration, and this is the row's real payoff Tao §1 explains why the improvements are all `1 + O(1/n)`, and the example he gives is a set of prime velocities. Quoted verbatim from arXiv:1701.02048v4 **[SOURCED]**: > "Let `n` be a large integer, and let `p_1,…,p_s` denote the primes between > `n/4` and `n/2`, thus by the prime number theorem `s = (1+o(1)) n/(4 log n)` > … This makes this collection of Bohr sets behave like a 'sunflower' … with a > very small 'kernel'." He splits `B(p_i;δ) = [−δ/p_i, δ/p_i] ∪ B'(p_i;δ)`, shows the petals `B'(p_i;δ)` are pairwise **disjoint** — because for distinct primes and `a,b` not divisible by them, `‖a/p_i − b/p_j‖ ≥ 1/(p_i p_j)`, Farey spacing — and concludes: > "the union bound … is only off from the truth by a multiplicative factor of > `1 + O(1/n)`." Then the diagnosis, which is the sentence to carry: > "the primes are a logarithmically sparse set, and standard sieve theory bounds > tell us that most numbers of size comparable to `n` will not only be > composite, but in fact contain a medium-sized prime factor (e.g. a factor > between `log^10 n` and `n^{1/10}`). One can use these medium-sized prime > factors to show that many of the rank one Bohr sets will intersect other rank > one Bohr sets in various disjoint (and reasonably large) 'major arcs', which > can then be used to improve upon the union bound." **What this says about us.** The published mechanism for beating the first moment in this family is *composite* velocities with medium-sized prime factors, which force the periodic sets to overlap in large arcs. Our moduli are the primes up to `x`, by construction and irremovably: they are the sifting primes. The mechanism is void on our configuration by definition. **The lonely runner literature's own account of its stuck factor of 2 names, as the obstruction, the exact configuration our covering problem has.** This is a wall address of the same class as `SEARCH-CONVENTIONS.md` §4's `β₂` entry: it does not prove that `4.2665 → 2` is unreachable, it says the gap will not be closed by refining a union bound over prime moduli, because a sixty-year effort against the identical geometric obstruction has not moved a factor of 2 either. The numerical coincidence is worth stating without leaning on it: LRC's gap is a factor of exactly 2 (`1/(2n)` against `1/(n+1)`), ours is a factor of `4.2665/2 = 2.13` in the exponent. Same shape, and the shape is the point, not the digits. **One honest demotion of the map's own analogy.** The map says the postulate and the LRC differ because "the lonely runner asks for one good time while the postulate asks that every window of `H` contain a survivor". That is not a mismatch in kind: the windowed form is in print as Rifford's **Timely LRC** (*Acta Appl. Math.* 180 (2022) P15, through Perarnau–Serra §11.4 **[SOURCED-BIB]**) — there is `N = N(n)` such that the origin is lonely before the slowest runner completes `N` laps — proved only for `n ≤ 5`, and Perarnau–Serra's Problem 8 asks the density version: does `B(v_1,…,v_n; 1/(n+1)) ∩ [−N/v_1, N/v_1]` have density bounded below by a function of `n` alone? **That is the postulate's shape, and it is open there too.** The escalation from "one survivor" to "a survivor in every window" is the same escalation in both problems, and it is unsolved in both. --- ## 4. The two-class question, settled: there is no such variant, and there is a published reason not to want one **The deciding question was whether a multi-obstacle-per-speed lonely runner is in print. It is not, and the near-miss that is in print says the general version is worthless.** **What is in print, as studied variants.** Perarnau–Serra §11 is the field's own enumeration of variations: invisible runners (Czerwiński–Grytczuk 2008), the loneliness spectrum (Kravitz 2021; Fan–Sun; Giri–Kravitz; Jain–Kravitz), the **shifted LRC**, time to get lonely (Rifford), the **lonely rabbit problem**, and function fields (Chow–Rimanić). Two of these touch our shape and neither is it: - **Shifted LRC.** Each runner starts at an arbitrary point of the track: one obstacle per speed, with a **free shift**. This is the map's mismatch (1) removed, and it is a well-developed line — Beck–Hoşten–Schymura, *Integers* 19 (2019) A29 formulate it; Henze–Malikiosis, *Aequationes Math.* 91 (2017) 331–352 give the covering-radius-of-zonotopes restatement; MSS 2025 give finite checking conditional on the Lonely Vector Property; Alcántara–Criado– Santos verify `n = 4`. **And it is FALSE.** Blanco–Criado–Santos, arXiv:2603.24784v2 (2026), abstract **[SOURCED]**: *"We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at `n=5`) and to the Lonely Vector Property (starting at `n=12`)."* So the moment the adversary is given the shift — which is exactly what our adversary has in `a_p` — the sharp constant `1/(n+1)` dies at five runners. - **Mixed thresholds.** Jensen, arXiv:2605.27941v1 (2026), abstract **[SOURCED]**: different required distances `d_i` per runner. Still one obstacle per speed. The view-obstruction ancestor is Dumir–Hans-Gill, *The view obstruction problems for boxes*, *J. Indian Math. Soc.* (1991). **Nothing anywhere gives a speed more than one obstacle.** The view-obstruction literature's entire generalisation axis is the *shape of the single body* attached to each lattice point — cubes, spheres, ellipsoids, boxes, `ℓ_p` balls, polygons, general convex bodies (Cusick 1974; Dumir–Hans-Gill–Wilker's "Contributions to a general theory of view-obstruction problems" I/II; Chen's series) — plus the codimension of the trajectory. Never the multiplicity per coordinate. **ABSENT, and the owning conventions are named on this line: no lonely-runner or view-obstruction variant assigning two or more obstacles to a single speed, or two or more residue classes to a single modulus, exists in the literature, searched as "lonely runner" and as "view-obstruction"/"view obstruction" — the conventions `SEARCH-CONVENTIONS.md` §2 already identifies for the covering question — on the arXiv API (39 records for "lonely runner", 12 for each view-obstruction spelling, every title read) and on the zbMATH Open API (65 and 46 records respectively, every title and author list read), both calibrated in-session as recorded in §0, and cross-checked against the field's own 2025 survey, whose §11 enumerates the studied variants and contains no such entry. MathSciNet unreached; WebSearch unavailable this session.** **The one published statement about our exact shape, and it closes the route.** Perarnau–Serra §11.3, read at source **[SOURCED]**, on why the shifted conjecture must insist on distinct speeds: > "In the shifted version it is important to insist that the speeds are pairwise > distinct, in contrast to Conjecture 3. If the condition on distinct speeds is > lifted, then the trivial lower bound `1/(2n)` becomes tight, as witnessed by > the set of speeds all equal to one and starting points > `{0, 1/n, ⋯, (n−1)/n}`. The lower bound in this case was found by Schoenberg > in the equivalent context of billiard trajectories". Lifting distinct speeds while keeping free shifts **is** several obstacles per speed: that is what repeated speeds with different starting points means. So the multi-obstacle-per-speed problem is not un-posed, it is posed and answered, and the answer is that **in that generality the union bound is exactly tight and nothing better than the first moment is true** (Schoenberg 1976, `Indag. Math.` 38 (1976) 263–279, **NOT REACHED**, carried through the survey). **Consequence, and it is the mechanism that closes row 12 as an import.** Our kill set is two classes per prime, `{a_p, a_p − 2}`, with `a_p` adversarial: a translate of a *fixed* pattern, so a proper sub-case of the repeated-speed shifted problem rather than the general one. A general theorem for that problem cannot exist, because the general problem is settled at the trivial bound. Any improvement for `G₂` must therefore come from the two structural facts the general problem does not have — that the speeds are **primes** and that the two obstacles are locked at **offset 2** — and §3 says the first of those two is precisely the case the literature identifies as hardest. **There is no theorem to import. The import question is answered, not deferred.** --- ## 5. The nearest published relative in the family is a wheel problem, and the corpus does not know it Unexpected, and worth more than the rest of §4. Perarnau–Serra §11.5, read at source **[SOURCED]**, on the **Lonely Rabbit Problem** — the same problem with time restricted to the integers, `ξ(V) = sup_{t ∈ Z} min_{v∈V} ‖tv‖`, speeds non-integral, `Rab(n) = inf_{|V|=n} ξ(V)`: - Wills 1968 determined `Rab(n)` for `n ≤ 3`; Cusick, *Simultaneous diophantine approximation of rational numbers*, *Acta Arith.* 22 (1972) 1–9, for `n ≤ 7`, conjecturing `Rab(n) = 1/w(n)` where `w(n) = max{z ∈ N : ½φ(z) + h(z) ≤ n}`, with `h(z) = 0` for `z` prime and `h(z) = ω(z)` for `z` composite; - Schark, *Monatsh. Math.* 78 (1974) 131–146, **proved the conjecture**; - Schark–Wills, *Acta Arith.* 22 (1973) 129–136, gave `Rab(n) ∼ e^{−2γ} / (n log log n)`. All four **[SOURCED-BIB]** through the survey's bibliography; none opened. **Why this matters here.** This is a Diophantine extremal problem, in the same family as the LRC, whose answer is *exactly known*, whose extremal modulus is "the largest `z` whose totient is at most about `2n`" — a primorial wheel — and whose asymptotic constant is `e^{−2γ}`, a Mertens constant. It is the one place in this literature where the extremal instance is a wheel and the answer is closed-form. The corpus has no mention of it: `grep` over `research/*.md` for `Schoenberg`, `Wills`, `lonely rabbit` and `view.obstruction` returns nothing, and the only occurrence of "view obstruction" anywhere in the repository is one line of `history/staging/import-map-construction.md`. **Priced honestly: this is a PUBLISHED-ANCHOR candidate for the corpus's "the wheel is the extremal object" intuition, and nothing more.** `Rab` is one obstacle per modulus and its `t` ranges over all of `Z` rather than over a window, so it does not bound `G₂`. It is an analogy with a proof attached, which is a better class of analogy than the corpus usually gets, and it should be recorded before it is lost again. --- ## 6. The Birkhoff / bounded-remainder branch: subsumed, and strictly worse from `x = 13` The comb read as a `Z`-rotation on the CRT torus is translation by 1 on `Z/x#Z`, and the object is `ΔΦ₂(y,x) = Ψ(y,x) − (D_x/x#)·y` of `research/discrepancy-two-class.md`. Three published instruments were priced. **Denjoy–Koksma.** Statement read at source **[SOURCED via Carrand arXiv:2012.07481v2, quoting Herman Thm VI.3.1]**: > "Let `f` be a homeomorphism of the circle with an **irrational rotation > number** `ρ(f)`. Let `μ` be a measure invariant by `f`, and let `p/q` be such > that `gcd(p,q)=1` and `|qρ(f) − p| < 1/q`. Then for all potential `φ` of > bounded variation and all `x ∈ S¹`, > `|Σ_{k=0}^{q−1} φ∘f^k(x) − q∫φ dμ| < Var(φ)`." **Read all the hypotheses and it fails twice.** Our rotation number is `1/x#`, **rational**, so the hypothesis fails at the first clause. And the route by which Denjoy–Koksma yields a bound at general `N` — Carrand's Lemma, decomposing `N` over convergent denominators — needs `α` of **constant type**, bounded partial quotients. The continued fraction of `1/x#` is `[0; x#]`: one partial quotient, equal to `x#`. The constant-type hypothesis fails as badly as it can. **Bounded remainder sets.** Definitions and results read at source in Grepstad–Lev §1 **[SOURCED]**. `S` is a BRS if there is `C = C(S,α)` with `|D_n(S,x)| ≤ C` for every `n` and a.e. `x`, where `D_n(S,x) = Σ_{k