# RUN-029 — Hard-Zeta A-U.2d.3: both second-order coefficients survive a test that would reject a neighbour, the continued fraction is certified without a single logarithm, and the round's one load-bearing citation names the wrong paper

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** Neo.K + Aletheia, `Hard_Zeta_Phase_II_Round_AU2d3_bundle.zip` (source item 47) — Second-Order Survival Closure via Explicit Diophantine Duration, Continued-Fraction Height Capacity, and Sparse-or-Congested B-Renewal. Ships a verification script, a constants JSON, and literature notes.
**Tools:** `src47_survival_closure.py` · `src47_drill.py` · `src47_emit_report_block.py`
**Logs:** `src47-au2d3.json` · `src47-drill.json` · `au2d3-literature-check.json`

**Result: the mathematics holds, and this round is the most checkable one the sweep has met. Its whole exponent chain is exactly rational; its two second-order coefficients are verified by a probe that goes to zero for the round's values and to a nonzero constant for neighbouring ones; and its continued fraction was recomputed by integer comparisons with no logarithm evaluated anywhere. Three findings, none in the mathematics — and the recurring artifact defect of items 35, 45 and 46 does not recur.**

---

## Why this round can be held to a higher standard than the last three

A-U.2d.3 takes exactly **one** external input — a Diophantine exponent for
`α = log2/log3` — and derives everything else from it:

> `ρ★ = 4.1164`,  `μ★ = ρ★ + 1`,  `θ★ = 1/μ★`,  `σ★ = 1/(1 + θ★)`

`4.1164` is a terminating decimal. So `μ★ = 12791/2500`, `θ★ = 2500/12791` and
`σ★ = 12791/15291` are **exact rationals**, and the shipped JSON publishes 100
digits of each. That is the RUN-027 situation again — a quantity the artifact
computes in floating point which is in fact rational — except that here it is not
an incidental observation but the spine of the round.

All three are right to every published digit.

## The two coefficients, and a probe that can reject a neighbour

Sections 9 and 16 do not merely quote constants. They quote **second-order
coefficients of asymptotic expansions**:

> `q⁺ ≳ c₀·y/q + c₁·y/q^{3/2} − q`   and   `q ≥ A/η_β + (κ_rot/η_β^{3/2})·√A`

A wrong *first* coefficient is loud — the expansion stops tracking. A wrong
*second* coefficient is quiet: the leading term still dominates and everything
looks fine. Checking that `c₁` equals `ln2·κ_rot/η_β²` to a hundred digits tests
the symbols, not the expansion.

The discriminating probe is the **scaled residual**, which tends to zero exactly
when the coefficient is right:

| | round's value | value 1% off |
| --- | --- | --- |
| §9 residual at `q = 10¹²` | `1.73e-7` | `−0.00706` |
| §16 residual at `A = 10¹²` | `3.00e-8` | `−0.00500` |

The round's residuals fall by a factor of `10⁴` from `q = 10⁴` to `10¹²`; the
perturbed ones **converge to a nonzero constant** instead. Both coefficients are
right, and the check that says so would have said otherwise about a value 1%
away. That is the difference between confirming a number and testing a claim.

## The continued fraction, with no logarithm evaluated

The artifact's convergents come from mpmath's floor-and-reciprocal loop at 100
digits. Recomputing them at higher precision would agree — and would prove
nothing about method, only about arithmetic.

So they were recomputed with **no floating point at all.** Every step of the
continued fraction of `log₂3` is `⌊log_u v⌋` for exact rationals `u, v` lying in
the multiplicative group generated by 2 and 3, so each partial quotient reduces
to comparing `2^A` against `3^B` — integers. **16 partial quotients certified
this way, up to denominator 10,781,274**, in agreement with the artifact on every
shipped row.

**Where that stops is measured, not asserted.** The integers grow with the
denominators, and the seventeenth term was still running after 200 further
seconds when the budget cut it off. So the run reports 16 certified and says
plainly that the three rows beyond it (`q` up to 397,573,379) are **cross-checked
at four times the artifact's precision, not certified**. A bound is not a
measurement, and a cutoff crossed silently is worse than one named.

The independence of that route is itself measured: the gate **counts the integer
comparisons it performed** and refuses if the count is too low. That field exists
because the drill needed to be able to break the claim — see below.

---

## Looked for and not found

**The artifact defect of items 35, 45 and 46 does not recur.** The shipped JSON's
top-level keys and row keys match what the shipped script writes, exactly, in
both directions. Three occurrences in a row does not make a fourth, and reporting
a streak that has broken would be worse than never having counted it.

**And a specific, quantitative hypothesis about the Diophantine input was raised
and refuted.** The value `4.1164` is close to `μ(ln 3) ≤ 5.1163051 − 1`, a
published bound on a *different* number, and the round would collapse
quantitatively if that were the source: `θ★` would move from `0.19545` to about
`0.116`, taking every exponent with it. Fan–Queffélec–Queffélec's own text
settles it — they give Rhin's bound for `α = log2/log3` as `ρ ≤ 7.616` and state
that Wang and Wu improved it to `ρ ≤ 4.11633052`, for that same `α`. The round's
value is correctly sourced, and `4.1164 > 4.11633052` by `1737/25000000` — so
"conservatively" is right too, checked as exact rationals.

A web search summary said otherwise. **A search summary is not a source.**

## Finding 1 — one digit, in one constant, and a mechanism for the rest

`η_β = 1/(6 ln 2)` is published to 100 decimals; the hundredth is `8` where the
true value has `7` followed by `4`. Confirmed at 2000 digits. Same class as item
45 (RUN-027), second occurrence, and far milder — one digit in one of four
transcendental constants, the other three being correctly rounded.

The per-row `primitive_error = p − βq` is the interesting half. It is published
to **100 significant digits at every convergent** while the computation behind it
supports between **83 and 100**, and the shortfall is not arbitrary: it is the
cancellation of subtracting two quantities of size `βq` to leave one of size
`1/q⁺`. That costs about `log₁₀(β·q·q⁺)` digits — which predicts the measured
agreement **at every row, to within 2.0 digits, always conservatively**.

That prediction is a check in the gate, not a remark in this file. "Over-published"
is an adjective; `100 − log₁₀(β·q·q⁺)` is something a reader can re-derive.

The two capacity fields are unaffected — no cancellation, and correct at every
published digit.

## Finding 2 — the load-bearing citation names a different paper

The notes cite:

> Lihong Wang, Qiang Wu, *On the irrationality measure of `log₂ 3`*, Journal of
> Number Theory **142** (2014), 264–273, DOI: 10.1016/j.jnt.2014.03.007.

Crossref, for that DOI: title **"On the irrationality measure of log 3"**,
authors **Qiang Wu, Lihong Wang**. Volume, pages, year and DOI match exactly. The
title does not, and the authors are reversed.

`log 3` → `log₂ 3` is a one-character slip, and in most papers it would be worth
a footnote. Here it is worth more, for one reason: **it is precisely the
substitution that makes the citation self-justifying.** This round's entire
quantitative content rests on the exponent applying to `log2/log3` rather than to
`ln 3`, which are different numbers with different literatures. A reader who goes
to check that — the one check that matters — finds the title already asserting
it, and stops.

**The exponent is fine.** It applies to `log2/log3`, established above from
Fan–Queffélec–Queffélec directly rather than from the misstated title. What fails
is the citation, at exactly the point where a reader would rely on it.

Not established: whether that paper itself contains the `log2/log3` exponent, or
whether Fan–Queffélec–Queffélec's reference is a different Wang–Wu work. The
publisher returns 403 and the survey's bibliography was not retrievable. Matching
volume, pages and DOI make it the same paper; its contents were not read, and
this run does not pretend otherwise.

## Finding 3 — the withdrawn citation recurs a fourth time, beside the paper it defers to

`arXiv:2605.13886` (Niu), **withdrawn 2026-05-20**, is cited for the **fourth
bundle running** (A-U.2d, A-U.2d.1, A-U.2d.2, A-U.2d.3).

New this time: the withdrawal notice says the work was withdrawn because *Rozier
and Terracol* already contain the enumeration — and **these notes cite Rozier and
Terracol two entries below**, accurately, with the right journal reference. The
bundle carries both the withdrawn paper and the paper it was withdrawn in favour
of, and says nothing about the relation. Still not load-bearing anywhere.

---

<!-- BEGIN GENERATED measured block: python code/src47_emit_report_block.py -->

**The exponent chain, exactly.** `rho_star` is a terminating decimal, so every exponent the round derives from it is a rational number and the published digits can be compared with an exact expansion rather than with a better approximation:

| | exact value | 100 published digits |
| --- | --- | --- |
| `θ★ = 1/(ρ★+1)` | `2500/12791` | exact to every published digit |
| `σ★ = 1/(1+θ★)` | `12791/15291` | correctly rounded at the last digit |
| `1 − σ★` | `2500/15291` | exact to every published digit |

`ρ★ = 10291/2500` exceeds the `4.11633052` the round cites by `1737/25000000`, so "conservatively" holds — a larger exponent is the weaker assumption. Checked as exact rationals, not as floats.

**The two second-order coefficients.** The scaled residual must vanish for the round's value and must *not* vanish for a neighbouring one — otherwise the probe confirms rather than tests.

| | `1e4` | `1e12` | falls ≥1000× | a 1%-wrong value still vanishes |
| --- | --- | --- | --- | --- |
| §9, `c₁ = ln2·κ_rot/η_β²` | `0.00173552` | `1.73126e-7` | `True` | `False` |
| §16, `κ_rot/η_β^{3/2}` | `0.000300373` | `3.00283e-8` | `True` | `False` |

The 1%-perturbed residuals go to `-0.00706435` and `-0.00499765` instead of to zero.

| what | measured against | value |
| --- | --- | --- |
| partial quotients of `log₂3` certified by integer comparison alone | `2^A` vs `3^B`; no logarithm evaluated | `16` |
| …largest denominator so certified | where the integers stop being cheap | `10781274` |
| …integer comparisons actually performed | so the independence is measured, not claimed | `68` |
| shipped rows disagreeing with the recomputed convergents | 7 integer-certified, 3 cross-checked at 400 dps | `0` |
| continued-fraction tax violations `1/(q+q⁺) < p−βq < 1/q⁺` | 6 rows decided by exact rationals, the rest at 400 dps | `0` |
| upper convergents missing from the JSON / wrongly present | the filter checked in **both** directions, by parity | `0 / 0` |
| …parity confirmed independently by the exact bracket on | indices where the bracket is tight enough to judge | `14` |
| transcendental constants over-published | of 4, against their closed forms at 400 dps | `1` |
| constants disagreeing beyond rounding | a different number, not a rounding complaint | `0` |
| shipped row values that are **wrong** | as opposed to over-published | `0` |
| fewest significant digits correct in any row field | `p − βq` at the largest convergent | `83` |
| rows where the cancellation model mispredicts that | `log10(β·q·q⁺)`, required conservative and within 4 digits | `0` |
| elementary-inequality violations | `2^D−1 ≥ (ln2)D`, `2^{gd}−1 ≥ g(2^d−1)`, `Qα−L = αD`, `D < 1` | `0` |
| renewal sum-bound failures | `Σ(2j+1)^θ` against its integral, R up to 3000 | `0` |
| interval-colouring trials using more colours than the max overlap | 300 trials, 14 distinct overlap counts, 292 with a genuine overlap | `0` |
| defects planted / caught by the check named for each | 1 of the entries is a robustness property; 0 malformed | `20 / 20` |

**Provenance — and this time it holds.** Top-level keys in the script but not the JSON: `none`. Row keys in the JSON but not the script: `none`. The shipped JSON **is** what the shipped script produces (`True`), so the item-35 class does not recur at its fourth look.

**Literature.** `arXiv:2605.13886` has been withdrawn since 2026-05-20 and is cited for the **4th bundle running**, alongside the paper its withdrawal notice defers to.

The round's one load-bearing citation gives the title as `log_2 3`; Crossref gives `On the irrationality measure of log 3`, by Qiang Wu and Lihong Wang. Volume, pages, year and DOI match exactly.

Every figure above is emitted by `code/src47_emit_report_block.py` from the gate logs and the archived literature record. None is typed into this file, and `report_block_guard` holds that to a snapshot of what the block actually reads.

<!-- END GENERATED measured block -->

---

## The instrument

**Drill 20/20 by the check named for each, both controls clean, no malformed
mutations in the final pass.** Four things it found are worth stating, because
three of them were defects in my own tooling and one is a new category.

**A defect that changes no number at all.** D6 swaps the integer-certified
continued fraction for the high-precision one. Every partial quotient it produces
is *identical*, so every field in the report agrees — including the check that
the two methods agree, which becomes a comparison of a thing with itself. Nothing
in the design could see it. The gate now counts the integer comparisons it
actually performed, and that is the only field that moves. **The independence
claim became measurable precisely because the drill had to be able to break it**;
before that it was a sentence in a docstring.

**The restore control fired, and it was right.** `read_text` followed by
`write_text` translates LF into CRLF on Windows, so the drill was rewriting every
line of the file it had just promised to restore. The byte-exact control caught
it on the first full pass. That control has been quiet for six items; this is
what it was for. The harness now works in bytes. (The same round trip is in
`src46_drill.py`, which stayed green only because its gate file was already CRLF.)

**A list the gate reported and gated on that no input could fill.** The exponent
chain's `wrong_digits` filtered for a verdict of `WRONG` — and the shared
comparison routine only ever returned `OVER-PUBLISHED`. The empty list looked
like a clean result for as long as nobody asked what would put something in it.
Fixed by drawing the distinction properly: an exactly-rational quantity has no
precision excuse, so a mismatch beyond last-digit rounding is simply wrong.

**Two near-misses in the accusing direction, on the same comparison.** Deciding
whether a published decimal is *wrong* or merely *rounded* went wrong twice while
this file was being written: first by comparing against a truncation, which calls
a correctly-rounded last digit an error; then by reading the reference digit *at*
the last published position instead of the one *after* it, which misses the carry
— `…173219729` correctly rounds to `…17322`, and reading the `1` rather than the
`9` calls that wrong. Both would have published a finding against correct
arithmetic. There is now one implementation, doing exact rational rounding once.

**Three defects were re-aimed after the pre-flight named them, not the checks.**
`want=2` selected the same two convergents the correct call does; slicing that
filtered list from the front did too, because it has exactly two entries; and
flipping the sign of a quadratic root changed nothing, because **both** roots
solve the quadratic. Each would previously have been scored as a check that
missed. This is the second item running where the pre-flight paid for itself.

## Route map

`ROUTE_MAP v2.3` continues the line; item 48 is the New-Chat Handoff bundle
rather than `AU2d4`, so the run of file-order-agrees-with-route-map ends at nine
and the next item is a different kind of object.

## What this run does not claim

1. That the B-survival theorem holds. `L(y) ≳ y^{0.19545}` quantifies over
   suffix-minimum B-injections, and RUN-023 measured **0** surviving crossings
   below `2·10⁵`. Nothing in the sweep has yet met one.
2. That the Diophantine exponent is correct. It is an external input, verified as
   *reported by the survey the round cites*, not re-derived. Nothing here checks
   Wang–Wu's proof.
3. That the sparsity and congestion theorems hold. Their arithmetic, their
   exponent identities and the interval-colouring step they use were checked; the
   dynamical hypotheses they quantify over were not, because no instance exists.
4. That the continued fraction is certified beyond `q = 10,781,274`. Three shipped
   rows lie past it and are cross-checked, not certified.
5. That the Wu–Wang paper does or does not contain the `log2/log3` exponent. Its
   text was not readable from here.