Do Tier 1–2 sources confirm that Ibn al-Banna's Talkhīṣ contains lattice multiplication, and that the 12th-c. Lilavati-commentary 'Kapat-sandhi' is the same method?

history-of-mathematicslattice-multiplicationislamic-golden-agetransmission-of-knowledgeverificationcalculating-instruments

This capture researches the open question at question-verify-lattice-multiplication-maghreb-india-primary, itself raised against claim-lattice-multiplication-india-maghreb-italy-transmission-chain. That note's two load-bearing attributions — that Ibn al-Banna's Talkhīṣ aʿmāl al-ḥisāb (late 13th c., Maghreb) contains the lattice/gelosia method, and that 'Kapat-sandhi' in a 12th-century Lilavati commentary is the same method — rest entirely on Tier 4 Wikipedia. This capture went looking for Tier 1–2 confirmation of both. It found partial, complicating evidence rather than confirmation.

Claim: The most thorough modern scholarly study of the Talkhīṣ's actual contents does not mention lattice, gelosia, or shabakh multiplication anywhere

Claim type: historical / technical-mechanism (an inventory claim: what is and is not inside a specific named text). Source tier required: Tier 1–2.

Abdeljaouad and Oaks (2013), published in Suhayl, a peer-reviewed history-of- science journal, is a close study of al-Hawārī's 1305 commentary on Ibn al-Banna's Talkhīṣ — written by one of Ibn al-Banna's own students during the master's lifetime, and the earliest commentary on the text that exists. The article catalogues, operation-by-operation, everything al-Hawārī's commentary adds to the Talkhīṣ and Ibn al-Banna's own Rafʿ al-ḥijāb, including a table of material a 15th-century successor (Ibn Ghāzī) later borrowed from al-Hawārī: "sum of numbers," "casting out nines or sevens," "types of multiplications," "computing the numerator of a fraction," "operations on fractions," "operations on roots," "double false position," and "algebra." The 44-page article, read in full this session, never uses the words "lattice," "gelosia," or "shabakh," and "types of multiplications" is never itemised further than the bare count of 4. A study built specifically to catalogue the Talkhīṣ tradition's contents in detail does not independently confirm that lattice/gelosia multiplication is among them.

Provenance: Abdeljaouad, Mahdi; Oaks, Jeffrey A. "Al-Hawārī's commentary on Ibn al-Bannā's Talkhīṣ: Contents and influences." Suhayl 12 (2013), 9-44. https://al-furqan.com/al-hawaris-commentary-on-ibn-al-bannas-talkhi%E1%B9%A3-contents-and-influences/

Claim: The earliest published manuscript evidence for gelosia-grid multiplication in the Ibn al-Banna tradition is a 15th/16th-century commentary, not Ibn al-Banna's own 13th-century Talkhīṣ

Claim type: technical-mechanism / historical. Source tier required: Tier 1–2.

Frank J. Swetz, writing in the Mathematical Association of America's Convergence and captioning manuscript images sourced from the World Digital Library / Library of Congress, shows gelosia-grid multiplication examples ("Examples of the gelosia method of multiplication using both right and left slanting diagonals within the grids") — but they appear in The Desire of the Students for an Explanation of the Calculator's Craving, a work compiled by Muhammad ibn Gāzī (Ibn Ghāzī al-Miknāsī, ca. 1437–1513), two centuries after Ibn al-Banna. Ibn Ghāzī's book is itself a commentary on his own earlier commentary (The Craving of the Calculators) on Ibn al-Banna's Talkhīṣ — three removes from Ibn al-Banna's original 13th-century text, not a description embedded in the Talkhīṣ itself. This is real manuscript evidence that gelosia multiplication circulated within the Ibn al-Banna commentarial tradition, but it dates the earliest documented instance to the Ibn Ghāzī branch of that tradition, not to Ibn al-Banna's own hand — a materially different claim from the one Wikipedia (and the seedling note it feeds) currently makes.

Provenance: Swetz, Frank J. "Mathematical Treasure: Ibn al-Banna's Operations of Calculation." Convergence (January 2014), Mathematical Association of America. https://web.archive.org/web/20260106010306/https://old.maa.org/press/periodicals/convergence/mathematical-treasure-ibn-al-bannas-operations-of-calculation (original old.maa.org URL returned HTTP 521 on every direct-fetch attempt this session; cited via the Wayback Machine capture instead).

Claim: The Kapat-sandhi / Ganesa's-Lilavati-commentary identification with gelosia rests, in this search, only on a Tier 3 source

Claim type: technical-mechanism (an identity claim between two named methods). Source tier required: Tier 1–2. Flag: [unverified-mechanism — needs primary]

A named-author paper (Rastogi & Nitin, JJT University, Rajasthan) states plainly that "Ganesa in his commentary on the Lilavati also calls the gelosia method as the kapat-sandhi method," citing D.E. Smith's History of Mathematics Vol. II (1958), p. 115, and Datta & Singh's History of Hindu Mathematics (1935), p. 138, as its authorities. This is the single most direct statement of the Indian-leg identity claim found this session, and it is more specific than Wikipedia's own "very similar to" phrasing. But the paper's publication venue could not be established — it is hosted at instavm.org with no journal name, volume, or peer-review notice visible anywhere in the document — so it does not clear the Tier 1–2 floor this claim type requires. The two underlying classic sources it cites (D.E. Smith 1958; Datta & Singh 1935) were not independently locatable in full text this session (see Further leads). The claim is recorded here, not promoted, and stays flagged.

Provenance: Rastogi, Prabha S.; Nitin, Sandhya. "History of Multiplication: Classical Period in Indian Mathematics." Undated paper hosted at https://instavm.org/wp-content/uploads/2021/05/H16.pdf

Claim: Wikipedia's own citation trail for both halves of the original claim points to a single unread academic source

Claim type: historical (source-provenance claim about the trailhead itself). Source tier required: n/a — this documents what remains to be checked, not a claim about the world.

Both the Ibn al-Banna/Talkhīṣ attribution and the Kapat-sandhi/Ganesa mention on Wikipedia's "Lattice multiplication" page cite the same footnote: Jean-Luc Chabert (ed.), A History of Algorithms: From the Pebble to the Microchip (Springer, 1999), p. 21 and pp. 21-26. This is a real, reputable academic-press book (a standard reference in the history of computation, with named contributing historians including work touching Arabic mathematics) and is very plausibly the actual primary/secondary source that would resolve this question outright. It could not be read this session: no open-access text was found, the academia.edu mirror returned 403, and an archive.org copy returned 401 Unauthorized to the vault's extract tool. This is the single highest-value unread lead this capture produced.

Further leads

• Jean-Luc Chabert (ed.), A History of Algorithms: From the Pebble to the Microchip (Springer, 1999), pp. 21-26 — the actual source behind both Wikipedia claims; unread this session (academia.edu 403, archive.org 401). Chasing a library/institutional copy would likely settle this question directly.
• D.E. Smith, History of Mathematics, Vol. II, "Special Topics of Elementary Mathematics" (1925; Dover reprint 1958), p. 115 — cited by Rastogi & Nitin for the Gelosia/kapata-sandhi identification; a full-text copy exists on archive.org but the specific passage was not located this session (search-inside tooling did not surface it).
• Datta & Singh, History of Hindu Mathematics (1935), p. 138 — the other classic source Rastogi & Nitin cite for kapat-sandhi; not independently checked.
• muslimheritage.com's "Decimal Arithmetic" page attributes the popularisation of "sieve"/"lattice" multiplication to al-Karaji (successor to al-Khwarizmi, 11th c.), not to Ibn al-Banna — a competing attribution worth chasing against a Tier 1–2 source, since it would predate Ibn al-Banna by two centuries.
• manasganit.com, "Multiplication by Kapat Sandhi Method — Gelosia" — an Indian mathematics-education site with a page devoted to exactly this identification; unread this session, tier unassessed.
• Al-Hawārī's own 1305 commentary (al-Lubāb) — discussed at length in the Suhayl article above but not itself read directly; as the earliest and most numerically detailed commentary on the Talkhīṣ, written within Ibn al-Banna's lifetime, it is the single best candidate primary text to check directly for multiplication-method content.
• Grokipedia's "Lattice multiplication" page surfaced repeatedly across searches this session. It is an AI-generated encyclopedia (xAI), not a human-edited one; treat as below Wikipedia's own Tier 4 and do not cite as evidence — noted here so a future pass doesn't mistake its repeated appearance in search results for independent corroboration.

Answer to the core question

[unverified — could not confirm or deny after search], and the search produced evidence that complicates the original claim rather than confirming it. The one Tier 1 source that specifically catalogues the Talkhīṣ's actual contents (Abdeljaouad & Oaks 2013) does not mention lattice/gelosia/shabakh anywhere. The one piece of real manuscript evidence located (Swetz/MAA, Tier 2) places the earliest documented gelosia grids in the Ibn al-Banna tradition two centuries after Ibn al-Banna himself, in Ibn Ghāzī's commentary. The Kapat-sandhi/Ganesa identification is stated plainly by a named-author paper but that paper does not clear the Tier 1–2 floor. The specific academic source Wikipedia's own footnotes point to for both halves of the claim (Chabert 1999) was not accessible this session and remains the most promising unresolved lead.

Entity candidates

• Bhāskara II (Bhāskarāchārya) — person — author of the 12th-c. Lilavati, the foundational Indian text every "Kapat-sandhi" claim is a commentary on; the whole Indian leg of the transmission chain rests on his original text, not just on its commentators.
• Gaṇeśa (Gaṇeśa Daivajña) — person — 16th-c. Indian astronomer/mathematician whose Lilavati commentary is the actual named source of the "kapat-sandhi = gelosia" identification per every source found this session; the foundational figure the Indian-leg claim measures itself against, and currently under-cited relative to Wikipedia's vaguer "a commentary."
• Ibn al-Banna al-Marrakushi — person — subject of the core question; see entity-ibn-al-muqaffa for an unrelated same-era Arabic-transmission figure already in the vault (different person, note only for domain-adjacency).
• Ibn Ghāzī al-Miknāsī (Muḥammad Ibn Ghāzī al-ʿUthmānī al-Miknāsī, 1437–1513) — person — author of the commentary tradition where gelosia-grid multiplication is actually attested in manuscript form per Swetz/MAA; may be the correct "first Arabic attestation" figure rather than Ibn al-Banna.
• Al-Hawārī (ʿAbd al-ʿAzīz ibn ʿAlī ibn Dāwud al-Hawārī al-Miṣrātī, fl. early 14th c.) — person — Ibn al-Banna's own student, wrote the earliest (1305) commentary on the Talkhīṣ during the author's lifetime; the closest primary-adjacent source to Ibn al-Banna himself.
• Jean-Luc Chabert — person — editor of A History of Algorithms: From the Pebble to the Microchip (Springer, 1999), the actual academic source both Wikipedia claims trace back to; unread, high-value target.
• David Eugene Smith — person — author of History of Mathematics Vol. II (1925/1958), the classic secondary source underlying the Kapat-sandhi/gelosia identification per Rastogi & Nitin.
• Mahdi Abdeljaouad and Jeffrey A. Oaks — people — historians of Arabic mathematics, authors of the 2013 Suhayl study of al-Hawārī's commentary used above.
• Frank J. Swetz — person — Penn State historian of mathematics, MAA Convergence author whose captioned manuscript images are the best evidence located this session.
• Talkhīṣ aʿmāl al-ḥisāb — concept/term — Ibn al-Banna's core arithmetic treatise, the specific document the whole question is about.
• Kapat-sandhi — concept/term — the Sanskrit multiplication-method name ("junction of doors") at the center of the Indian-leg identification claim.