Birch and Swinnerton-Dyer read the rank of an elliptic curve as a growth exponent — the slope of log ∏_{p≤X} N_p/p against log log X. That is the same instrument runs 051–056 use on carpets: a global exponent fitted from local counts across a few decades of scale, where pair slopes drift. On curves whose ranks are known, over four decades of X, does the slope read the rank, and does the drift behave like a finite-size correction that shrinks with scale?

Sealed 2026-09-29, before any a_p was computed for this run (codifier). Known at sealing: the ranks of the five curves (Cremona's tables; checked by control ii), the history of the conjecture (the slopes were found numerically in the 1960s and converge slowly), and runs 051–056. Not known: any product value or slope at these ranges. No attribution for the convergence theory is relied on; the prediction is made from the instrument's geometry, not a cited theorem.

Why it is a test and not an illustration. The whole span of four decades of X is 0.69 in ln ln X. With a rank-one difference worth only 0.69 in y across that span, the oscillating part of y(X) — driven by the curve's L-function zeros — could reorder neighbouring ranks or pull a slope half a unit off. Clause (2)'s 0.35 is a bet that it does not; the falsifier's 0.7 is where the reading would be unusable. Clause (3) is the transferable question: in runs 053–055 the carpet pair slopes drifted monotonically with L, and the scoring treated that drift as finite-size. Here the true exponent is known, so the drift's direction can be checked against an answer.

What the outcomes mean. Confirmed: the slope-of-a-log-log-fit instrument reads a known exponent at this range, and its pair-slope drift shrinks toward the answer — support for reading the carpets' drift the same way. Falsified on order or size: four decades is too short for this instrument even with an exact answer available, and the carpet readings, which span log L over a similar number of decades, deserve the same doubt. Unanticipated on (3) alone: the drift is not a monotone correction, and "the asymptotic value may be lower" (053's caveat) is not a safe reading of drift.

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Scored 2026-09-29 — confirmed

Results: R28_bsd_euler/RESULTS057.md, slopes057.json (commit 651d50e). PARI 2.17.2 via cypari2 2.2.4; one forprime + ellap sweep to 10^8 per curve, 212–218 s each, five in parallel.

What it settles. Over four decades of X — 0.69 in ln ln X — the five-point fit reads every rank to within 0.3 and orders all five. The instrument works at this range when it is given five sizes.

What it does not, stated because the outcome word would otherwise carry it. Clause 3 passed as written (first pair slope versus last), but its gloss does not hold. The pair slopes are not a correction shrinking toward r; they oscillate by up to ±1 between neighbouring decades — 5077a1 goes 1.83, 3.53, 2.93, 3.84, 2.02 — and the first-versus-last comparison passed partly by where the oscillation happened to land. A single pair slope is not a reliable reading of the exponent here; the multi-point fit is.

What it means for the carpets. Runs 053–055 read 1/ν from three sizes, with two pair slopes, and 053 and 055 read the downward drift of those pair slopes as "the asymptotic value may be lower". This run shows that with a known answer, pair slopes can swing both ways, so two of them cannot distinguish a monotone drift from an oscillation. The carpet pair slopes move by 0.02–0.05, far less than here, so the carpets' fitted values are not in doubt from this — but the drift caveats in 053 and 055 are unsupported readings, not findings. The next: asks for the size that would tell them apart.

Pointers corrected 2026-10-05 (codifier; no sealed field, score or number is changed). The script: line was written with the seal, before the script existed, and read: script: R28_bsd_euler/ (to be written by the compute chat). It now names what was written and committed: R28_bsd_euler/run057.sh, R28_bsd_euler/euler057.py, R28_bsd_euler/analyze057.py. Found by sync_cases.py's record-field check (docs/REPO-AUDIT.md 5.4).