Does the triple-correlation bias in the Liouville sums survive to 10⁹, and does per-scale information stay flat?

What it settles. The triple-correlation bias Σλ(n)λ(n+2)λ(n+4) at 10⁹. At 10⁷ the small-p excess is not a residue-class effect — the chi-square across residue classes is flat — but a global √y-scale bias, which the twin-prime harvest identifies as the Haar measure of the scaling flow (see notes/log-weight-haar-measure.md). The run also carries per-scale information to p ≤ 10⁵ and Fourier uniformity to H = (log x)⁴.

The 10⁷ baseline is the falsifier's other half. λ matched a coin to three decimals from H = 8 through H = 65536, spanning the MRT threshold (log x)^{1+ε}. A departure at 10⁹ would be the first structure found; agreement extends direct evidence for the MRT conjecture at its conjectured threshold.

The gate, now cleared. This run carried no prediction and no falsifier, and per OPS those are written before any code runs — the only bug that survived every other check on 2026-09-13 was a stability test that could not fail, and nothing catches that except committing in advance to what a failure would look like. Both are now filed, from the twin-prime harvest (notes/DIGEST-twin-prime-chat.md §6), so the run is open.

Note (2026-09-14): the ~5 GB figure is the whole-array approach, not a wall. The ~5 GB at 10⁹ quoted in liouville_1e9.py's docstring is the whole-array sieve (a uint32 cofactor array during the sieve, int8 λ after). A segmented sieve removes the constraint entirely — it keeps only the base primes to √N = 10⁵ resident and streams cache-sized blocks — so N = 10¹⁰ runs on the M5 with no more memory, and faster for it. Memory is therefore not blocking this run at 10⁹; it blocks only the scale beyond, and the fix is to segment the sieve (second TODO item under Runs), not to buy RAM.

Note (2026-09-16): first-differencing precondition, from the same one chat. The twin-prime harvest (notes/DIGEST-twin-prime-chat.md §4) records that a headline "signal" died to exactly one check — block series must be first-differenced before correlating, or a density trend masquerades as structure. This run correlates block/per-scale series, and liouville_1e9.py does not currently first-difference (checked 2026-09-16), so the precondition is a live gate, not a satisfied one: the per-scale and triple-correlation measurements must first-difference before correlating, or the run reproduces the same spurious-trend error the harvest already paid for. Evidence for the gate is one chat (the twin-prime harvest), not two — the earlier cross-attribution to a second (continued-fractions) chat was wrong and is retracted; no second confirmation is claimed.