validation/paper2/SETTLING_RULE.md

Appendix C — the settling rule, as pre-registered and as applied

Written from the analysis plan, experiments/r1_turnover.py, experiments/curve1_spacing.py and DATES_TABLE.json. Every threshold is named with the date it entered.

The rule as pre-registered

A construct is settled at a checkpoint if the kept fraction of its 5% carrier set, measured against a partner checkpoint, crosses and stays above a threshold.

element	value	where it comes from
threshold θ	0.7	the contract's fallback, inherited from the pilot (A31)
threshold grid tested	0.50 to 0.95 in steps of 0.01	r1_turnover.THETA_GRID, so θ's choice is a sensitivity and not a hinge
partner	one checkpoint at a token factor inside the band
factor band	(1.25, 2.00)	construct_token_loss.FACTOR_BAND
crossing	"crosses θ and stays above it to the end of the series"	A31

What went wrong with it, and when

A31's form was vacuous at the end of the series. Twelve constructs dated at one and the same checkpoint, pretrain_iter_0364000 (1.527 T tokens). The reference itself is healthy — its twin ceiling runs 0.7486 / 0.7436 / 0.7466 / 0.7414 / 0.7400 across its neighbours and its twin agreement 0.9245 / 0.9242 / 0.9249 / 0.9225 / 0.9203 — so the checkpoint is not the cause.

The spacing is. Curve 1's level falls monotonically with the gap it is read at, Spearman −0.837 over 1,584 (point, partner) pairs for those twelve constructs:

step factor	median ratio
1.20–1.35×	0.7382
1.35–1.50×	0.7128
1.50–1.65×	0.6680
1.65–2.10×	0.6113

And the twelve dates sit exactly where the available spacing shortens: at indices 29–31 the only in-band partner is ≈1.56×, and at index 32 it is 1.332×, because the 1.5× partner (2.29 T) is past the end of training. The same construct at index 29 reads 0.611 at 1.56× and 0.717 at 1.25× — a gap of 0.106, larger than the whole "jump" at index 32 (0.119). Index 32 is also the last point of the series, so "stays above θ to the end" has nothing left to stay above.

The fixed-step form has the same defect in a different place: its dates concentrate at index 33 (pretrain_iter_0455000), 13 of 13 in the product space, whose fixed partner is the smallest gap in the tail (29,865 steps) and which is likewise the last point of its series. Only 56% of the pretraining points carry the nominal 1,000-step spacing at all.

The rule as corrected

A date needs all of:

1. an observed crossing of θ = 0.7 — not a series already above it;
2. at least one checkpoint below θ before the crossing;
3. at least two checkpoints after it, so the crossing can be seen to hold;
4. agreement between two partner spacings — the ratio and the fixed form must both date it;
5. and, in curve1_spacing.py's strengthened form, the ratio stays above θ at every spacing available at every index from the crossing to the end, not only at the partner the band happened to pick.

A date in the last two checkpoints is censored, not a date.

The band is split into four for clause 5 — [[1.25, 1.35], [1.35, 1.5], [1.5, 1.65], [1.65, 2.0]] — because one threshold cannot serve a 1.25× and a 2.0× window.

What the corrected rule yields

spacing	outcomes over 169 constructs
ratio	{"censored": 159, "transient": 10}
fixed	{"transient": 150, "censored": 6, "dated": 13}
both agreeing (the rule)	0 of 169

The zero is the conjunction, not an absence of signal: the fixed spacing dates 13 constructs and the ratio spacing dates none, so nothing survives clause 4. DATES_TABLE.json prints both columns per construct with a c1_why_no_date field, so this can be checked row by row.

Across the six cuts of B14 — the top 1%, 2% and 5%, and entries above 1, 2 and 3 σ₀, sets of 744 to 14,424 neurons — the rule dates 0 of 169 in every band.

The position classes

Curve 4's series are keyed construct|class|role|piece. The classes and their counts on the pretraining branch:

class	series
governed	4,587
obliged	43
scope	158

Table C.1

Rendered from DATES_TABLE.json as DATES_TABLE.md — one row per construct, both curve-1 spacings, curves 2, 3 and 6, curve 4's role counts, and the reason no settling date exists. Curve 5 has no per-construct date: it is defined on the 14,196 construct pairs.