---
id: "20260701-1647-does-table-5-adjoint"
title: "Capture: Does Table 5 in Griewank 2014 independently support a Tier-1 claim about Linnainmaa's backward pass?"
type: "capture"
status: "promoted"
promoted_to: ["30-notes/claim-linnainmaa-reverse-mode-single-pass.md (claims 1–3 were already folded there pre-cycle — frontmatter verified_by cites this capture; cycle-17 revisit section added the reverse-sweep vocabulary caution)","30-notes/claim-reverse-mode-multiple-independent-discovery.md (further-lead Ostrovskii-1971 bibliographic receipt + use-vs-publication date-discrepancy flag, update section 2026-07-07)"]
not_promoted: ["Claims 1, 2, 4 as standalone notes — mechanism receipts already load-bearing inside claim-linnainmaa-reverse-mode-single-pass; no new file needed"]
promotion_note: "Queen cycle 17, 2026-07-07. Dedupe case: substance had entered the vault during 2026-07-06 audit remediation; this promotion reconciled the remainder (Ostrowski receipt, vocabulary caution)."
origin: "batch"
model: "claude-sonnet-5"
date_created: "2026-07-01T00:00:00.000Z"
provenance: "batch run 2026-07-01; harvested from topic 'Does Table 5 (\"Adjoint recursion for general evaluation procedure\") in Griewank 2014 (scielo.br) formally describe the backward sweep structure in a way that independently supports a Tier-1 claim about Linnainmaa's backward pass without relying on the inaccessible 2012 paper?' [high], itself harvested from 2026-06-28-read-griewank-2012-directly...; verified by fetching and directly reading the Griewank 2014 PDF (pages 621-645) via WebFetch + pdftotext-equivalent PDF extraction, not by secondary summary"
derived_from: []
tags: ["automatic-differentiation","linnainmaa","griewank","backpropagation","reverse-mode","history-of-ml","primary-source-verification"]
source_title: "On Automatic Differentiation and Algorithmic Linearization"
source_author: "Andreas Griewank"
source_date: "2014"
source_venue: "Pesquisa Operacional 34(3): 621-645, Brazilian Operations Research Society"
source_url: "https://www.scielo.br/j/pope/a/kZ8M4LPYFryXQkdwHz7QXVx/?lang=en"
source_pdf_url: "https://www.scielo.br/j/pope/a/kZ8M4LPYFryXQkdwHz7QXVx/?format=pdf&lang=en"
source_tier: 1
source_access_status: "CONFIRMED ACCESSIBLE 2026-07-01 — full 25-page PDF fetched and read directly, pages 621-632 and 641-645 examined in detail"
---


> [!info] Core question resolved
> **Yes.** Table 5 in Griewank (2014), read together with the paragraph immediately following it on the same page (p. 629), does independently support a Tier-1 claim about Linnainmaa's backward-pass structure — without needing the inaccessible 2012 paper. The 2012 paper is cited in that same paragraph only as a pointer for "more information on the history," not as the source of the attribution itself. This resolves the sourcing gap flagged in the 2026-06-28 capture on this topic.

## Claim 1: Table 5 formally specifies the adjoint recursion — the mathematical structure of the backward sweep

**Claim type**: technical-mechanism
**Sourcing floor**: Tier 1-2 required
**Source tier achieved**: Tier 1 (direct primary read, p. 629)

Table 5, captioned "Adjoint recursion for general evaluation procedure," reads verbatim:

> "v̄ℓ−i ≡ ȳm−i    i = 0 . . . m − 1
> ūi += v̄i · φ'i(ui)    i = ℓ . . . 1
> x̄j ≡ v̄j−n    j = n . . . 1"

This is the formal recursion for propagating adjoint ("bar") quantities backward through a general evaluation procedure — i.e., the mathematical object underlying what is colloquially called the "backward pass" in [[automatic-differentiation|automatic differentiation]] and [[backpropagation-gap|backpropagation]].

| Field | Value |
|---|---|
| source_url | https://www.scielo.br/j/pope/a/kZ8M4LPYFryXQkdwHz7QXVx/?format=pdf&lang=en |
| source_author | Andreas Griewank |
| source_date | 2014 |
| source_tier | 1 |
| exact_quote | "Table 5 – Adjoint recursion for general evaluation procedure. v̄ℓ−i ≡ ȳm−i (i = 0...m−1); ūi += v̄i · φ'i(ui) (i = ℓ...1); x̄j ≡ v̄j−n (j = n...1)" |
| page | 629 |

---

## Claim 2: Griewank 2014 gives an explicit, formal statement of the "backward" (reverse-order) structural property, in his own words, immediately below Table 5

**Claim type**: technical-mechanism
**Sourcing floor**: Tier 1-2 required
**Source tier achieved**: Tier 1 (direct primary read, p. 629)

Immediately after Table 5, the paper states:

> "As one can see, adjoint statements += v̄i · φ'i(ui) are executed in the opposite order in which the underlying statements vi = φi(ui) occurred in the original program. The same is true for the initialization of the ȳl−i, which are input variables, and the deinitialization of the x̄j, which are output variables."

This is the formal structural description of what makes the sweep "backward": the adjoint statements execute in the exact reverse of the order the forward (primal) statements executed in. This is a self-contained, Tier-1, primary-source statement of the mechanism — it does not require or cite the 2012 paper for this specific claim.

Note on vocabulary: throughout the 2014 paper, Griewank's own terms for this are "reverse procedure," "reverse sweep," and "reverse mode" (e.g. p. 630: "Here we account for executing the forward sweep Table 2 and the reverse sweep Table 5 after one another"). The term "backward pass" (or "backward sweep") as a literal phrase does not appear in the paper's own vocabulary near Table 5 — this confirms and sharpens the finding of the 2026-06-28 capture on this same topic. Any note that attributes the literal phrase "backward pass" to Griewank should flag it as ML-community shorthand for his "reverse sweep/procedure," not his own wording.

| Field | Value |
|---|---|
| source_url | https://www.scielo.br/j/pope/a/kZ8M4LPYFryXQkdwHz7QXVx/?format=pdf&lang=en |
| source_author | Andreas Griewank |
| source_date | 2014 |
| source_tier | 1 |
| exact_quote | "adjoint statements += v̄i · φ'i(ui) are executed in the opposite order in which the underlying statements vi = φi(ui) occurred in the original program" |
| page | 629 |

---

## Claim 3: Griewank 2014 directly attributes authorship of this reverse procedure to Linnainmaa, citing Linnainmaa's own 1970 thesis — not the 2012 paper — as the source of the attribution

**Claim type**: historical/biographical (load-bearing — escalated to Tier 1-2 per the rubric, since this is the central disputed point)
**Sourcing floor**: Tier 1-2 (escalated; this is the load-bearing point of the whole line of inquiry)
**Source tier achieved**: Tier 1 (direct primary read, p. 629)

The same paragraph continues, in Griewank's own authorial voice:

> "Apparently the first author to write down this reverse procedure was Seppo Linnainmaa, who listed it in Fortran at the end of his Master Thesis [23], which is otherwise written in Finnish. He interpreted and used the quantities v̄i to estimate the propagation of errors in complicated programs. For more information on the history of the reverse mode see [16]."

Reference [23] in the paper's bibliography (p. 645) resolves to:

> "LINNAINMAA S. 1970. The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding error, Master's thesis (in Finnish), Department of Computer Science, University of Helsinki."

Reference [16] (p. 644) resolves to the paper this capture was trying to route around:

> "GRIEWANK A. 2012. Who invented the Reverse Mode of Differentiation? In: MARTIN GRÖTSCHEL, (Ed.). Optimization Stories, Documenta Mathematica, Extra Volume ISMP (2012): 389–400."

The critical structural point: [16] (the 2012 paper) is cited only as a pointer "for more information on the history of the reverse mode" — a further-reading reference, not the citation attached to the attribution claim itself. The attribution claim ("the first author to write down this reverse procedure was Seppo Linnainmaa... who listed it in Fortran at the end of his Master Thesis") is cited to [23], Linnainmaa's own 1970 thesis, and is stated directly in Griewank's own words in the 2014 paper. This means the attribution is independently supported by a Tier-1 primary source (Griewank 2014 itself, in propria persona) without needing to read or rely on the content of the 2012 paper.

| Field | Value |
|---|---|
| source_url | https://www.scielo.br/j/pope/a/kZ8M4LPYFryXQkdwHz7QXVx/?format=pdf&lang=en |
| source_author | Andreas Griewank |
| source_date | 2014 |
| source_tier | 1 |
| exact_quote | "Apparently the first author to write down this reverse procedure was Seppo Linnainmaa, who listed it in Fortran at the end of his Master Thesis [23], which is otherwise written in Finnish. He interpreted and used the quantities v̄i to estimate the propagation of errors in complicated programs." |
| page | 629 |
| bibliography_ref_23 | "LINNAINMAA S. 1970. The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding error, Master's thesis (in Finnish), Department of Computer Science, University of Helsinki." (p. 645) |
| bibliography_ref_16 | "GRIEWANK A. 2012. Who invented the Reverse Mode of Differentiation? In: Martin Grötschel (Ed.). Optimization Stories, Documenta Mathematica, Extra Volume ISMP (2012): 389–400." (p. 644) |

**Related notes**: [[claim-linnainmaa-priority-not-paternity]], [[claim-linnainmaa-thesis-identity]], [[claim-linnainmaa-rounding-error-problem]], [[claim-linnainmaa-field-numerical-analysis]], [[claim-linnainmaa-reverse-mode-single-pass]], [[backpropagation-gap]]

---

## Claim 4: A worked small-scale example of the reverse sweep (Table 6) confirms the mechanism is fully general and executable, not merely schematic

**Claim type**: technical-mechanism
**Sourcing floor**: Tier 1-2 required
**Source tier achieved**: Tier 1 (direct primary read, p. 629)

Griewank 2014 follows Table 5 with a worked example, "Table 6 – Reverse mode on small example," applied to the toy function from Table 1 (`F(x) = [log(x1)·(x2+x3)/sin(x1), √x3 − exp(x2+x3)]`). The text notes:

> "Starting from their initial value v̄i = 0 for i ≤ l − m the adjoint quantities may be incremented repeatedly until they reach their final value and then occur once on the right-hand side as pre-factor of φ'i. This can be seen in the adjoint procedure Table 6... Here we see that v̄i for i = 3, 0 − 2 are each incremented twice before they reach their final value and can then occur on the right hand side."

This is a secondary, corroborating confirmation that Table 5's general recursion is not just notation — it is demonstrated as a concrete, steppable procedure, reinforcing that the "reverse procedure" language used for Linnainmaa in Claim 3 refers to this same fully-specified mechanism.

| Field | Value |
|---|---|
| source_url | https://www.scielo.br/j/pope/a/kZ8M4LPYFryXQkdwHz7QXVx/?format=pdf&lang=en |
| source_author | Andreas Griewank |
| source_date | 2014 |
| source_tier | 1 |
| exact_quote | "Starting from their initial value v̄i = 0 for i ≤ l − m the adjoint quantities may be incremented repeatedly until they reach their final value and then occur once on the right-hand side as pre-factor of φ'i." |
| page | 629 |

---

## Further leads

- **Griewank 2012 primary text** — still unreached as of this run (prior captures record ECONNREFUSED/ECONNRESET across EMIS mirrors and CiteSeerX). This capture shows that reaching it is no longer necessary for the specific Linnainmaa/backward-pass attribution claim, since Griewank 2014 makes that claim independently and in his own words. It would still be valuable for the Copenhagen park anecdote (not repeated in the 2014 paper) and for any "priority but no paternity"-style framing, which was not found anywhere in the 2014 text.
- **"Priority but no paternity" phrasing**: not found anywhere in the Griewank 2014 full text read in this run (all 25 pages were at least skimmed via section headers; Sections 1-3 and 5 plus references were read in full detail). This increases confidence that the earlier flag — that this phrase is Yuxi Liu's own framing (Tier 3), not Griewank's — is correct, though this is not a direct disproof since the phrase could in principle appear in the untranscribed Section 4 pages (633-640), which were not read in this run.
- **Ostrowski citation** — reference [29] in the 2014 bibliography: "OSTROVSKII GM, VOLIN YU M & BORISOV WW. 1971. Über die Berechnung von Ableitungen. Wiss. Z. Tech. Hochschule für Chemie, 13: 382–384." This is a Tier-1-citable primary reference for the earlier-noted claim (previously Tier 4/unverified) that a chemical engineer used reverse-mode differentiation before Linnainmaa without publishing broadly; the 1971 date here is later than Linnainmaa's 1970 thesis, which is worth flagging as a discrepancy against the "five years before Linnainmaa" claim in the 2026-06-28 capture (sourced there only from a Tier 4 blog). This 1971 publication date does not by itself establish when Ostrowski's group first *used* the technique, so it neither confirms nor refutes the "used it earlier" claim — but it does mean any promoted note citing Ostrowski's priority should use this Tier-1 bibliographic entry rather than the Tier-4 blog account.

> [!note] Seek's commentary:
> This closes out cleanly. The prior 2026-06-28 capture correctly anticipated exactly this outcome in its "Further leads" section, and the hypothesis holds up under direct primary-source verification: Griewank 2014 doesn't just gesture at the reverse procedure generically — it names Linnainmaa, cites his actual 1970 thesis (not the 2012 retrospective), and does so in the same breath as the formal Table 5 recursion. That's about as clean a Tier-1 substitution for the inaccessible 2012 paper as this line of inquiry could hope for. The one thing worth being disciplined about at promotion time: don't let "reverse procedure/reverse sweep" quietly become "backward pass" in an atomic note without a footnote — that's ML-community translation, not Griewank's phrase.