Repository path: workshop/experiments/E-20260815f-night-formula/design.md · rendered 2026-09-09

Does a repeated formula reach English as a formula?

Frozen 2026-08-15, before either published comparator was opened for this question. The translation limb it hangs on — span D, the first night, log D43–D55 — was frozen first, at commit bb912110.

1. The question, and where it came from

Translating span D forced two decisions the previous three spans did not. D43 had to invent an English form for وأدرك شهرزادَ الصباحُ فسكتَتْ عن الكلامِ المباح — the sentence that closes every night, rhymed, and repeated about a thousand times in the received text. D45 had to decide whether a phrase the copy-text repeats across the seam between the frame and an embedded tale (تحت إبطه, the shroud under an arm, once for the merchant and once for the vizier) must repeat in English. Both decisions rest on one premise, written into the register as V12 and V14:

a formula's whole content is that it is the same words again.

The question this asks of the published record: when a translator carries a repeated formula over, does he carry it over as a formula — the same words each time — or as a sentence to be written afresh each time it comes round?

What this unit teaches about translating literature (subject rule): the night formula is the device that makes a thousand tales one work. Whether that device survives translation is a fact about translating, not about this project's instruments.

2. The baseline the source supplies, and the control inside it

Measured on the copy-text before any English was opened (materials/arabic_boundaries.txt):

• The closing formula is verbatim invariant. Eight instances in the transcribed extent (Hindawi 2022 vol. 1, pp. 11–59), and all eight read وأدرك شهرزاد الصباح فسكتت عن الكلام المباح, letter for letter after vocalisation is stripped. n = 8, distinct types = 1, mean pairwise similarity = 1.000.
• The sentence that follows it is not. Where the copy-text has the sister's reply, it varies at every instance — ما أطيب حديثك، وألطفه وألذه وأعذبه / يا أختي، ما أحلى حديثك وأطيبه، وألذه وأعذبه / ما أحلى حديثك with Dunyazad named / absent altogether at four of the eight.

This is the design's internal control and it is the source's own doing. At one seam the Arabic holds one sentence fixed and lets the next drift. A hand that has understood which is the formula will reproduce that asymmetry; a hand that varies both equally has not.

3. Hands

Both public domain, both already this arm's comparators, neither opened for this question before this design was frozen:

• Lane 1839, PG #34206 (vol. I) — Būlāq.
• Burton 1885, PG #3435 (vol. 1) — Macnaghten (Calcutta II).

The lead's own V12 is n = 1 and is declared, not measured; it cannot be scored for invariance until many nights are rendered.

Declared contamination on the predictions. The lead has seen Burton's rendering of this sentence (translation contamination block, priming event i) and believes from training, not from the text, that Lane suppressed the night divisions. P1, P2 and P4 below are therefore primed and are registered only so the counts are checkable. The finding rests on P3, P5 and P6, on which the lead has no prior.

4. Procedure — positional, so that it does not depend on knowing the words

The enumeration rule must not be built out of a formula the lead has already seen. It is therefore positional:

1. Enumerate night markers in each hand: passages announcing a numbered night. Count them.
2. Window = the three sentences preceding each marker.
3. Normalise: lowercase, strip punctuation and diacritics, collapse whitespace.
4. The hand's formula is defined as the modal sentence of the pooled windows — the most frequent normalised sentence across all windows, found by clustering at token-similarity ≥ 0.60. Nothing about its wording is assumed.
5. Each night's realisation = the sentence in that window closest to the modal form.
6. The reply = the sentence immediately following the realisation inside the window.

If a hand has no night markers, steps 2–6 are undefined for it and that is the retention result.

5. Measures

6. Registered predictions

P3 is directional and both directions are reportable. If similarity is 1.000, the finding is that a formula can cross intact and here is the hand that did it — which would run against ARM-recurrence's Gogol result, where three English hands of four lost a repetition the source insists on. If it is below 1.000, the finding is that even a hand which keeps the formula does not keep it as one.

7. Rhyme rule, stated before any string is looked at

A realisation rhymes if the last stressed vowel-plus-coda of its final word is identical to that of the final word of some earlier colon in the same sentence, and the two words are not the same word. Identical repetition (say … say) is not a rhyme and is coded separately as repetition.

8. Failure criteria

• Fewer than 5 realisations in a hand → its M2 is reported and no comparison is claimed.
• No night markers in a hand → M2–M4 not computed; M1 stands alone.
• If the modal-sentence clustering at 0.60 puts more than one obviously distinct sentence in the same cluster, the threshold is not tuned: the clusters are published and the result is reported at the threshold as frozen, with the anomaly named.
• The copy-text's transcribed extent is incomplete: p. 30 is not transcribed, so at most one Arabic boundary may be missing, and the ninth heading is misnumbered ٨ a second time. Both are limits on M1's denominator and neither touches M2, M3 or M4, which are within-hand.

9. Cost and gates

$0 for the census — the two hands are Project Gutenberg files and every measure is a string count. One pre-run adversarial critic call on this frozen design, one non-Anthropic seat, before any text is opened. Post-run verification recomputes every reported number by an independent implementation with mutation tests. No panel judgment is used and none is needed: nothing here is a quality claim.

10. Pre-run adversarial critic pass — NEEDS REDESIGN, 15 findings, 4 BLOCKING, all 15 accepted

One call, seat openai/gpt-5.6-terra (P1 per config/models.md), provider OpenAI, finish_reason: stop, 3,144 in / 4,393 out, $0.030287 against a declared ceiling of $0.13. Raw: critic.json. The critic saw §§1–9 and the stored Arabic; it saw no English, which is what made findings 1, 3 and 4 possible — it checked the design's baseline against the source and the design lost.

Every finding is accepted. Four of them change what will be computed, and one changes what may be claimed. The amendments below are binding and replace the sections they name.

A1 — the enumeration rule is replaced (findings 1, 2, 8, 9)

The critic showed, from the Arabic alone, that §4's three-sentence pre-marker window does not contain the formula at C1, C2, C4, C5 or C7 — the formula is followed there by the sister's reply, Shahrazad's answer and the king's thought before the next heading arrives. The rule was wrong on the source it was derived from. It is replaced by unsupervised recurrence discovery over the whole volume, which is positional in no way at all:

1. Segment each text into sentences on . ? ! followed by whitespace, with a frozen exception list (Mr. Mrs. Dr. St. vol. p. pp. etc. i.e. e.g. and single capital initials); drop segments under 6 tokens; strip headings, chapter rules and PG front/back matter at the declared boundary strings.
2. Normalise: lowercase, strip all non-letter characters, collapse whitespace.
3. Cluster every sentence in the volume by single-link agglomeration at difflib.SequenceMatcher(None, tokens_a, tokens_b).ratio() >= 0.60 — the metric is named, the linkage is named, ties broken by first occurrence in text order.
4. The hand's formula is the largest cluster, and its modal form is the most frequent exact normalised string inside it. The top five clusters are published so the ranking is checkable.
5. Assignment threshold, and an explicit unassigned category (finding 2): a night boundary is coded recovered only if a sentence at that boundary clears 0.60 against the modal form; otherwise not recovered. Coverage is reported separately from invariance.

A validation gate stands in front of all of it (finding 1's remedy): the identical pipeline is run on the Arabic copy-text first, and must return the closing formula as its largest cluster. If it does not, the pipeline is reported as failed, no English number is claimed from it, and the study falls back to locator extraction with the failure published.

A2 — the baseline claim in §2 is wrong and is withdrawn (findings 3, 15)

The critic checked §2 against the stored Arabic: C2 and C4 are the same reply once punctuation is stripped, which is the design's own normalisation. The claim that the reply "varies at every instance" is false and is withdrawn. The reply baseline will be computed through the same pipeline as everything else and reported as it comes out. §2's formula claim is also narrowed to what the material supports: the eight instances extracted from this transcription are identical after the stated normalisation — not a claim about the received text's thousand repetitions.

A3 — the control is restricted to a matched subset (finding 4)

"The sentence that follows" is four different things in the Arabic: an immediate sister reply (C1, C2, C4, C5), a night heading with no reply at all (C3, C6, C8), and a narrative sentence about the king's day with the sister's request deferred (C7). The control is restricted to the immediate-reply subset, n = 4, said out loud to be small and non-uniform, and matched in English to the sentence immediately following a recovered realisation. M3 requires ≥ 3 usable observations in a hand or it is not computed.

A4 — what may be concluded is narrowed (findings 5, 7)

• No mental-state claim. §2's "a hand that varies both equally has not understood which is the formula" is withdrawn. What is measurable is textual behaviour: the translation preserves, or does not preserve, the source's contrast in recurrence.
• The recension confound is named and it governs M1. Lane translated Būlāq, Burton translated Macnaghten; the baseline is Hindawi 2022. A missing English boundary may be the base text rather than the translator, exactly as RS-20260814 §6a already found for this arm at a different locus. M1 is therefore descriptive only and is reported against the Hindawi extent, never as a translator's choice. M2 and M3 are within-hand and are untouched by this — which is why the primary sits there.

A5 — the predictions are restated with quantifiers, and one is downgraded (findings 10, 11, 12, 13)

Finding 13 is the sharpest and is accepted with a downgrade. The lead has seen one instance of Burton's formula, in one grep, before span A. That reveals his wording; it cannot reveal whether he varies it, which is what P3 measures. But "unprimed" was too strong. P3 is therefore confirmatory for Lane and descriptive for Burton, and P4 is descriptive for Burton outright. No blinded analyst is affordable at $0.28 of remaining daily headroom, so the downgrade is taken rather than papered over.

A6 — M5 is operationalised (finding 12)

Source locators fixed: وأخذ كفنه تحت إبطه (the merchant, inside the tale) and وطلع الوزير بالكفن تحت إبطه (the vizier, in the frame), both in span D. In each hand the corresponding passages are retrieved by content, and coded into four pre-registered categories: (a) exact lexical repetition · (b) partial lexical repetition (≥ half the content words shared) · (c) same sense, different wording · (d) no recurrence. V14 is met only by (a).

A7 — M4 stops claiming to be a string count (finding 14)

Rhyme is reported as three separate columns, not one binary: (i) end-word identity (mechanical), (ii) orthographic rhyme — shared final letter-sequence from the last vowel of the final word (mechanical), (iii) the lead's phonological reading, flagged internal-judgment-only and published string by string. No pronunciation authority is asserted.