core/docs/research/math-phase-4-2-case-first-traces-2026-07-25.md

Math Phase 4.2 — case-first traces for 0000 / 0001 / 0148 / 0082

Baseline reproduced: holdout_dev: correct=5 wrong=0 refused=495 (n=500) — unchanged from the 2026-07-24 measurement, five days and one arc later.

Deliverable: the per-case trace math-reader-phase-4-1-status-2026-07-24.md asked for, plus a corpus-frequency measurement that changes the recommendation. All four named cases are blocked by surface gaps that are among the rarest in the 500. Building for them is the overfit move ADR-0251 prohibits — and this document is the first time that has been quantified rather than argued.

Nothing was changed in the reader. See §5.

1. Where all four refuse — one seam

Every one of the four fails at the same place, with the same shape of message:

candidate_graph: recognizer matched but produced no injection for statement:
  '<first statement>'  (category=...)

Traced through generate.recognizer_match.match(statement, registry):

case	category assigned	parsed_anchors	injection
0000	DISCRETE_COUNT_STATEMENT	()	()
0001	DISCRETE_COUNT_STATEMENT	()	()
0148	DESCRIPTIVE_SETUP_NO_QUANTITY	()	()
0082	DISCRETE_COUNT_STATEMENT	()	()

So the recognizer classifies each statement and then extracts no grounded anchor. _is_comparative_multiplicative_v1_surface is False for all four, so none of them ever reaches inject_comparative_multiplicative — the injector that would handle "twice as many X as Y". The gate is _COMPARE_MULT_ANCHOR_RE (generate/math_candidate_parser.py:1229), anchored ^…$, so it must match the whole statement:

^(?P<actor>ENTITY)\s+VERB\s+(?:a\s+)?(?P<anchor>twice|thrice|half|quarter|third)
 \s+as\s+many\s+(?P<unit>\w+(?:\s+\w+)?)\s+as\s+(?P<reference>REF)\s*\.?$

2. Gap isolation — one variable at a time

Each gap isolated by mutating a known-good baseline in exactly one dimension. This is the evidence, not the regex reading:

probe	matches	isolates
Aria has twice as many credits as Emily.	YES	baseline
… as many school credits as … (2-word unit)	YES	unit slot admits 2
… as many high school credits as … (3-word)	no	unit slot caps at 2 words
… as many pounds of lobster as …	no	of-headed unit phrase
… as many credits than Emily.	no	than connector
… as Emily, who has twice … as Spencer.	no	trailing relative clause
At a station, Aria has twice …	no	leading adjunct
The number of people was twice the total number counted.	no	copula + bare twice the N
Mary has twice the amount of carrots as green beans.	no	twice the amount of X as Y
… as many credits as the two other students combined.	no	aggregate reference

3. Per-case traces

Case 0000 — expected 140.0

Aria has twice as many high school credits as Emily, who has twice the number of high school credits as Spencer. If Emily has 20 credits, what's twice the total number of high school credits the three have?

Blocked by two gaps in one sentence: a 3-word unit (high school credits) and a trailing relative clause carrying a second comparative. Solving needs a chained resolution (Emily=20 → Spencer=10, Aria=40), a summation (70), and a final ×2 on the aggregate. Four capabilities.

Case 0001 — expected 480.0

Hooper Bay has twice as many pounds of lobster than the two other harbors combined. If the other two harbors have 80 pounds of lobster each, how many pounds of lobster are the three harbors holding?

Blocked by four gaps: than where the template requires as; an of-headed unit (pounds of lobster); an aggregate reference (the two other harbors combined); and a multi-token proper name — extract_proper_noun_subject returns 'Hooper', not 'Hooper Bay', which alone would fail the injector's narrow actor binding (actor != actor_token).

Case 0148 — expected 1500.0

At a people counting station, the number of people counted on the first day was twice the total number counted on the second day. If 500 people were counted on the second day, how many people were counted on the two days?

The only one classified DESCRIPTIVE_SETUP_NO_QUANTITY — the recognizer does not see a count statement at all. Three gaps: a leading adjunct; a noun-phrase quantity as subject (the number of people counted on the first day) rather than an entity; and a copula + bare twice the total number form with no as many … as at all.

Case 0082 — expected 2.0

Mary uses plastic grocery bags that can hold a maximum of twenty pounds. She buys 4 pounds of green beans, 6 pounds milk, and twice the amount of carrots as green beans. How many more pounds of groceries can Mary fit in that bag?

The refusing statement is not the comparative at all — it is the capacity frame (bags that can hold a maximum of twenty pounds), a relative-clause capacity assertion with a spelled-out cardinal. The comparative arrives in the next sentence in the twice the amount of X as Y form. Then the question wants capacity − Σ(4, 6, 8). Four capabilities, none of which is the v1 template.

4. The measurement that changes the recommendation

Frequency of each blocking surface feature across all 500 holdout cases:

surface feature	cases	% of 500
aggregate reference (combined / together / altogether)	53	10.6%
leading adjunct (At/In/On/During …,)	28	5.6%
twice/N times as many … as (any unit width)	29	5.8%
twice the number/amount/total of	13	2.6%
copula + was/is twice	13	2.6%
capacity frame (hold / maximum of / capacity)	14	2.8%
— of the 29, unit is >2 words	1	0.2%
— of the 29, connector is than	2	0.4%

The gap blocking case 0000 affects one case in five hundred. The than gap blocking case 0001 affects two.

That is the whole finding. "Widen the unit slot from two words to three" and "admit than alongside as" are precisely the per-case pattern growth ADR-0251 and reader-arc-overfit-inventory-2026-07-19.md exist to prohibit — and until now the prohibition rested on a prior about overfitting. It now rests on a count.

The forms that actually recur are different ones: aggregate reference (53), leading adjunct (28), twice the number/amount of (13), copula-twice (13).

Honest limit on these numbers. They are surface-feature counts, not conversion estimates. A case containing combined is not thereby convertible — §3 shows every one of these problems needs three or four capabilities at once, so a case can carry a frequent feature and still strand on the other three. The counts bound the opportunity; they do not predict yield. Treating them as yield is the exact error the arc already made once, when tune-set parse-rate gain was read as capability and shipped answers that were wrong on the exam.

5. What was deliberately not done

No reader change was made. The plan's Lane 4 exit criterion is the traces plus wrong=0 holding, not a shipped extension, and three things argue against shipping one from this session:

1. §4 shows the named cases' gaps are the rarest in the corpus. Building them is the prohibited move, now with a number attached.
2. The sealed 1,319-case test split is the final arbiter and must not be read. A change justified only by movement on the open dev set has exactly the evidentiary shape ADR-0251 was written against.
3. The four cases are each 3–4 capabilities deep. There is no "minimum reader extension" that converts one of them — which is itself the strongest form of the arc's own conclusion that real GSM8K statements are individually multi-capability.

wrong=0 still holds: correct=5 wrong=0 refused=495.

6. Recommendation

Phase 4.2 as scoped — case-first on 0000/0001/0148/0082 — should be recorded as traced and declined, not attempted. The tracing was the right instruction; its result is that these four are the wrong four.

If the reader arc resumes, the measured entry point is aggregate reference (53 cases, 10.6%) — the single most common blocking surface in the corpus, and the one whose semantics (the two other harbors combined → Σ over a named complement set) is a genuine compositional capability rather than a template widening. Scope it the same way this document scopes: isolate the gap, count its reach, and require that the conversion evidence be a capability argument, not a parse-rate delta.

Relates to project-generalization-arc.