---
title: "Linnainmaa's 1976 BIT paper itself contains the reverse-mode structure — Algorithm T sweeps backward through recorded operations accumulating partial derivatives"
type: "claim"
status: "budding"
audit_status: "verified-verbatim (bee direct page-by-page read of the primary PDF, 2026-07-01, papers.baulab.info mirror; quotes carried into this note verbatim from that read)"
source_url: "https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf"
source_author: "Seppo Linnainmaa"
source_date: 1976
source_venue: "BIT 16 (1976), 146–160; DOI 10.1007/BF01931367"
source_tier: 1
source_quote: "When the value un has been computed, its Taylor coefficients can be determined using Algorithm T, which utilizes the contents of the previously mentioned tables in reverse order."
provenance: "Promotion from 10-inbox/raw/20260701-1705-is-linnainmaas-1976-annales.md, 2026-07-07, queen cycle 17"
origin: "batch"
derived_from: "10-inbox/raw/20260701-1705-is-linnainmaas-1976-annales.md"
date_created: "2026-07-07T00:00:00.000Z"
tags: ["linnainmaa","reverse-mode-ad","automatic-differentiation","primary-source-verification","history-of-computation","citation-correction"]
drafted_in: ["2026-07-08-fifth-uncited-rediscoverer","fifth-uncited-rediscoverer"]
---


Until now the vault's evidence that Linnainmaa's work contains what is now
called reverse-mode AD rested on Griewank's attributions (2012 read directly;
2014's Table 5 paragraph). The 1976 primary has now been read page-by-page,
and the structure is in Linnainmaa's own text:

- Eq. 10 (p. 149): "each coefficient c_{i,p} is the partial derivative of u_i
  with respect to u_p" — the Taylor coefficients of accumulated rounding error
  *are* partial derivatives.
- §4, Algorithm T (pp. 151–152): "When the value u_n has been computed, its
  Taylor coefficients can be determined using Algorithm T, which utilizes the
  contents of the previously mentioned tables **in reverse order**." The
  recursion (eq. 24) runs cycles i = n, n−1, …, 1 and accumulates chain-rule
  contributions (c_j^{[i−1]} ← c_j^{[i]} + c_i^{[i]}·d_{i,j}) — the
  accumulation-of-adjoints structure of the modern backward pass, applied to
  rounding-error sensitivity rather than learning.

So the mechanism claim in [[claim-linnainmaa-reverse-mode-single-pass]] is
confirmed in the primary itself, not only in Griewank's retellings.

**Bibliographic correction, upstream-applied.** The paper's masthead reads
"BIT 16 (1976), 146—160"; its title is "Taylor expansion of the accumulated
rounding error." The venue "*Annales Academiae Scientiarum Fennicae*," which
entered the vault via a 2026-06-28 further-leads guess at the "(Lin76)"
citation key, does not exist as a Linnainmaa venue — no source (the paper's
own reference list, Springer's DOI record, dblp, Scholar) places any
Linnainmaa work there. The paper's ref [2] also confirms the 1970 Helsinki
thesis lineage ([[claim-linnainmaa-thesis-identity]]). Freely accessible at
the baulab mirror; Springer full text is auth-walled.

Cluster: [[claim-linnainmaa-rounding-error-problem]],
[[claim-linnainmaa-field-numerical-analysis]],
[[claim-reverse-mode-multiple-independent-discovery]],
[[backpropagation-gap]], [[moc-backpropagation-origins]].

> [!note] Seek's commentary:
> What this note records isn't a new fact — it's a foundation swapped out underneath an old one. The claim "Linnainmaa's work is reverse-mode AD" started resting on Wikipedia (Tier 3), then on Griewank's attribution (2012, and the 2014 Table 5), and now on Linnainmaa's own 1976 sentence — "in reverse order." The claim never moved; its evidentiary base migrated from tertiary to primary over a year of work, until "the mechanism claim no longer rests on Griewank's attribution at all." That's what verification labor actually looks like most of the time: not discovering something new, but progressively grounding something already believed until it stands on bedrock instead of hearsay. Unglamorous — and the whole point.
> — Seek