# RUN-055 — The programme's Diophantine input, read from the review of the paper it cites: the general result covers β, the withdrawal's reason does not hold, and every exponent derived from it re-evaluated at the review's 4.117

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the one external input of the Hard-Zeta rounds A-U.2d.3 to A-U.2d.25, `ρ★ = 4.1164` (an exponent with `‖qβ‖ ≳ q^{−ρ}` for `β = log 3/log 2`), withdrawn at A-U.2d.26 — and the conflict between that withdrawal and this line's RUN-029, which RUN-054 left open
**Tools:** `src74_diophantine_input.py`, `src74_drill.py`
**Logs:** `src74-diophantine-input.json`, `src74-drill.json`
**External records:** `wuwang-2014-zbmath-review.mardi.json`, `diophantine-input-literature-check.json`

**Result: RUN-054 said the conflict "is answered by reading Wu–Wang, not by running anything". The paper is behind a paywall; its zbMATH review is not. Archived byte for byte from MaRDI's mirror of zbMATH Open document 1303.11081, the review says the paper's bound `μ(log 3) < 5.117` *follows from a more general result*: `|p + q₁ log 2 + q₂ log 3| > H^{−4.117}` for every `p, q₁, q₂` with sufficiently large `H = max(|q₁|, |q₂|)`. Both readings were describing one paper. Its headline theorem is about `log 3`, as the withdrawal says; the result it derives that from is a linear form in 1, log 2 and log 3, and with `p = 0` that form bounds `β` directly — `|bβ − a| > (2b)^{−4.117}/log 2`, every step an integer comparison — as RUN-029's survey says. **The withdrawal's reason does not hold for the general result.** What the review prints, `4.117`, is a bound rounded up and does not reach the programme's `4.1164`; the survey's `4.11633052` does. So the conservative input, certified by the review alone, is `ρ_c = 4117/1000`, and this round re-evaluates at it every exponent the rounds derived from `ρ★`: **29 rows across RUN-029, 032, 041 and 042, and 10 levels of RUN-040's dense source floor, each first reproduced exactly from its formula at `ρ★` — 0 disagreements — then evaluated at `ρ_c`.** Every exponent whose role is stated moves in the weaker direction, as a weaker hypothesis must; the five side conditions the arguments need all hold at `ρ_c` (`ρ > 3`; `ρ < 1/α − 1` for every `α` of the dense floor, binding at `1/6`; `4.11633052 ≤ ρ ≤ 7.616`). Not read: the paper itself — its exact exponent, and whether its "sufficiently large H" is effective.**

---

## The conflict, as it stood

| | says | source |
| --- | --- | --- |
| A-U.2d.26 | the Wu–Wang theorem is about `log 3`, not `β` | the programme's own source audit |
| RUN-029 | Wang and Wu improved Rhin's `ρ ≤ 7.616` to `ρ ≤ 4.11633052` for `α = log 2/log 3`, hence for `β` | Fan–Queffélec–Queffélec, arXiv 2303.06850 |
| RUN-054 | the two cannot both be right; reading the paper decides | — |

## What the review says

The record is MaRDI's entity `Q403297`, mirroring zbMATH Open `1303.11081`, DOI `10.1016/J.JNT.2014.03.007`; the review (reviewer Jan Šustek) is its property `P1448`. Its SHA-256 is recorded in the literature file and checked on every run. The gate reads three things out of the review text itself, not out of this report:

| | read from the review |
| --- | --- |
| the linear-form exponent | `4.117` |
| the height | `H = max{|q₁|, |q₂|}` |
| the relation to `log 3` | the `log 3` bound "follows from a more general result" |

## The transfer to β

Take `p = 0`, `q₁ = −a`, `q₂ = b`. Then `|q₁ log 2 + q₂ log 3| = log 2 · |bβ − a|` and `H = max(a, b)`. For `a` the nearest integer to `bβ`: `3⁵ < 2⁸` gives `β < 8/5`, so `a ≤ 8b/5 + 1/2 ≤ 2b` once `b ≥ 2`, and at `b = 1` the nearest integer to `β ∈ (3/2, 2)` is 2. So `H = a ≤ 2b` and

`|bβ − a| > (2b)^{−4.117} / log 2` for `H` sufficiently large.

The form's exponent is an exponent for `β`, with an explicit constant. Each step is an integer comparison, and the gate checks each.

## The three figures

`4.11633052 < 4.1164 < 4.117`, as exact rationals. The survey's figure supports the programme's value; the review's printed figure — an upper bound rounded up — does not reach it. The gate takes the review's, `ρ_c = 4117/1000`, because it is the one this line has read in a record it can archive.

## Every exponent, re-evaluated

The rounds' exponents are rational functions of one number: `θ = 1/(ρ+1)`, `σ = 1/(1+θ)`, `κ₁₃ = (ρ+1)/(ρ+1+θ)`, `λ₁₃ = 1/(ρ+1+θ)`, `χ(σ) = (5σ−4)/3`, `ζ(σ) = (σ+1)/3`, `ψ(σ) = (σ−(1−θ))/θ`, and A-U.2d.12's dense floor `μ(α) = (θ−α)/(1−α)`. The earlier gates archived each at `ρ★` as an exact rational. This gate reads every archived row, refuses any whose name it has no formula for, reproduces the archived value exactly from the formula at `ρ★`, and only then evaluates at `ρ_c`.

| family | at `ρ★` | at `ρ_c` |
| --- | --- | --- |
| `θ` | `2500/12791` | `1000/5117` |
| `σ` | `12791/15291` | `5117/6117` |
| `1 − σ` | `2500/15291` | `1000/6117` |
| `ρ/(ρ+1)` | `10291/12791` | `4117/5117` |
| `κ₁₃` | `163609681/169859681` | `26183689/27183689` |
| `λ₁₃` | `31977500/169859681` | `5117000/27183689` |
| `χ(σ)` | `2791/45873` | `1117/18351` |
| `ζ(σ)` | `28082/45873` | `11234/18351` |

| | count |
| --- | ---: |
| archived rows read (RUN-029, 032, 041, 042) | 29 |
| …of them `ρ`-free (the 4/5, 2/5, 3/5 of the controlled continued-fraction route) | 5 |
| dense-floor levels read (RUN-040: six hierarchy levels, four inherited `α`) | 10 |
| archived values not reproduced by their formula | **0** |
| rows refused for want of a formula | **0** |
| exponents moved the wrong way under the weaker hypothesis | **0** |

**The direction is the point.** `ρ_c > ρ★` is a weaker hypothesis, so every exponent of a lower bound must fall and every exponent of an upper bound must rise: the survival duration `L(y) ≳ y^θ` falls from `θ★` to `θ_c`, the sparsity bound `R ≤ N^σ` and the support bound `M ≲ N^{κ₁₃}` rise, the dense floor falls at all ten levels. Eight rows sit in places whose role neither the report nor the gate states — log exponents, the partial-quotient pressure — and are re-evaluated without a direction claimed.

**The side conditions.** All hold at `ρ_c`: `ρ > 3`, which is `σ > 4/5` and `χ(σ) > 0` (RUN-041, RUN-042); `θ > α` for every `α` of the dense floor, binding at `α = 1/6`, i.e. `ρ < 5` (RUN-040); `ρ ≥ 4.11633052` and `ρ < 7.616` (RUN-029).

**What does not move.** RUN-043's core (`Δ_N + δ_N ≥ (1−o(1)) log₂ N`, the record bounds) and RUN-044's (the forced `N^{1/5−o(1)}` gap, the bank identity) consume no `ρ`; the controlled exponents 4/5, 2/5 and 3/5 are the continued-fraction route's and are unchanged. The `o(1)` terms consume A-U.2d.12's constant `C_ε`, which depends on `ρ` only through the existence of some finite exponent.

## What this changes for this line's earlier reports

RUN-054 listed eight reports carrying `ρ★`-derived exponents (RUN-029, 032, 036, 040, 041, 042, 043, 044), each verifying the chain and none certifying the source. Their arithmetic stands. Their source now has a reading: the general Wu–Wang result covers `β`, at `4.117` by the review and at `4.11633052` by the survey. The exponents at `ρ★` hold if the paper's exact exponent is at most `4.1164`, which the survey says it is and this line has not read; at `ρ_c` they hold on the review alone. RUN-036 to RUN-039's exponents `(6θ−1)/5`, `(9θ−1)/8`, `(45θ−4)/41` and `(θ−α)/(1−α)` are the dense floor at `α = 1/6, 1/9, 4/45, 1373/25856`, and are among the ten levels above.

## The drill

**12 defects planted, 12 caught by the check named for each, 2 controls undisturbed, the gate restored byte for byte.** One of them was a finding before it was a catch: with the review's exponent unreadable, the first version fell back to `ρ = 0`, divided by it and died without writing a log — red, but a crash, and a crash is not a catch. The gate now refuses with a log that says why.

| planted defect | went red |
| --- | --- |
| the review is read for a different exponent | review-exponent |
| the record is hashed against the wrong file | review-hash |
| the height is read as a sum of the coefficients | review-height |
| the transfer bounds `β` by `3/2` | transfer |
| the survey's figure is placed above the review's | order |
| `θ` is taken as `1/ρ` | reproduced |
| `κ₁₃` is written with a minus | reproduced |
| the dense floor adds `α` | reproduced |
| a row loses its formula | refused |
| the survival exponent is called an upper bound | direction |
| the backbone condition is demanded at `ρ > 5` | side |
| `ψ` is measured from `θ` instead of `1 − θ` | family |

Suite totals after this round, from `code/suite_totals.py`: 57 drills, 1472 planted defects, 1472 caught by their named check, 128 controls undisturbed.

## What this round does not claim

* **The paper is not read.** Every statement about Wu–Wang is a statement about its zbMATH review and, for `4.11633052`, about Fan–Queffélec–Queffélec's survey. Its exact exponent, and whether its "sufficiently large H" is effective, are `UNMEASURED`.
* **The rounds' asymptotic theorems are not proved here.** What is re-evaluated is their exponent algebra, which RUN-041 and RUN-042 had already verified as identities in the symbols over random rational parameters.
* **The programme's own finite-continued-fraction route (RUN-054) is not superseded.** It needs no global exponent and stands on its own; this round restores the global route alongside it, at a weaker value.
* **No exponent is claimed optimal.**