---
title: "The discrete Kelley-Bryson costate recursion and Pontryagin's continuous adjoint are the same object only inside a hypothesis-bound regime; outside it they are a family resemblance, not an identity"
type: "claim"
status: "budding"
source_url: "https://arxiv.org/pdf/1902.10298"
source_title: "ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs"
source_author: "Amir Gholami, Kurt Keutzer, George Biros"
source_date: "2019-02-27T00:00:00.000Z"
source_tier: 1
source_sha: "23cb651e73afe6937c9649f98446d6d97e3a1e7a78168fba28e8a2abbacba11d"
source_quote: "It can be clearly seen that the results from DTO approach (Eq. 9), and OTD (Eq. 10) can be completely different."
source_url_2: "https://arxiv.org/pdf/1707.03873"
source_title_2: "The Discrete-Time Geometric Maximum Principle"
source_author_2: "Robert Kipka, Rohit Gupta"
source_date_2: "2017-07-12"
source_tier_2: 1
source_sha_2: "c6253031f69262dbd946736f71f570bf27ede1f548cd561111f3ad0f2027ab54"
source_quote_2: "the formulation of the maximum principle for discrete-time optimal control problems becomes quite tricky since the maximization condition cannot be expected to hold in general and in fact some of the early literature on this topic was mathematically incorrect"
source_quote_4: "However, the maximization condition does hold under appropriate convexity assumptions on the dynamics"
source_url_4: "https://arxiv.org/pdf/1707.03873"
source_sha_4: "c6253031f69262dbd946736f71f570bf27ede1f548cd561111f3ad0f2027ab54"
source_url_3: "https://arxiv.org/pdf/0810.1386"
source_title_3: "Discrete Mechanics and Optimal Control: an Analysis"
source_author_3: "Sina Ober-Bloebaum, Oliver Junge, Jerrold E. Marsden"
source_date_3: "2008-10-08"
source_tier_3: 1
source_sha_3: "d54ad3a9199ff90a6ab2d7d57e6211577c30347ed910e5b7ea0f22f184db328e"
source_quote_3: "The purpose of this section is to establish the convergence of solutions of the discrete optimal control problem (3.15) to a solution of the Lagrangian optimal control problem (3.3) as the step size h goes to 0."
independent_roots: 3
audit_status: "capture-verified — synthesis across three Tier-1 primaries (Gholami/Keutzer/Biros 2019; Kipka/Gupta 2017/2019; Ober-Blöbaum/Junge/Marsden 2008/2011), each read directly via extract_pdf at capture time, 2026-10-01, with quotes verified verbatim against the extracted text in each underlying claim-note. This note does not rest on the [unverified-mechanism] Halkin/Boltyanskii attribution flagged in [[claim-no-unconditional-discrete-pontryagin-maximum-principle]] — only on that note's Tier-1-grounded general claim. | APPEND 2026-10-02 (scheduled cross-model audit, claude-fable-5): receipt repairs on all three quotes, claim text unchanged. (1) source_quote → was the 'completely different' sentence plus the display-math sentence 'This is because in general ∂f(z1,θ)/∂z1 ≠ ∂f(z0,θ)/∂z0.' (sweep FAIL: pdftotext scrambles the printed fractions into '∂f∂z(z1 ,θ)1 6= ∂f∂z(z0 ,θ)0', so the transcription can never byte-match) → now the fully verbatim prose sentence alone; the dropped inequality was verified by the auditor's direct read of the re-fetched extraction (sha256 23cb651e73af… matches capture) — every token present in fraction-scrambled order, and Eqs. 9/10 confirm the differing Jacobian arguments. (2) source_quote_2 → was an ellipsis join of two Kipka-Gupta passages → now one passage per quote (second passage → source_quote_4), each verbatim against the re-fetched extraction (sha256 c6253031f692… matches), mirroring Cali's 2026-10-01 repair of the same join in the sibling note. (3) source_quote_3 → was an ellipsis join eliding the equation numbers '(3.15)' and '(3.3)' plus a silently lowercased 'the' → now the full contiguous printed sentence, verbatim against the re-fetched extraction (sha256 d54ad3a919… matches, §3.6). All three arXiv PDFs re-fetched live 2026-10-02 with sha256 matches to capture."
provenance: "Promotion from 10-inbox/raw/2026-10-01-does-kelley-brysons-discrete-costate-recursion-reduce-to.md, 2026-10-01"
origin: "batch"
derived_from: ["20261001-0213-does-kelley-brysons-discrete"]
writer_model: "claude-sonnet-5"
date_created: "2026-10-01T00:00:00.000Z"
tags: ["pontryagin","kelley-bryson","costate","adjoint-method","optimal-control","discretize-then-optimize","backpropagation","history-of-ml","numerical-analysis"]
seek_code_commit: "c7427a5"
quote_sweep: "RESOLVED 2026-10-02 — repaired by the cross-model audit: source_quote trimmed to the fully verbatim prose sentence (the appended display-math sentence cannot byte-match pdftotext's scrambled fraction layout; its content verified by direct read of the capture-time extraction, see audit_status); was: FAIL 2026-10-02 — source_quote NOT found in the capture-time archive (sha256 23cb651e73af…) — the quote does not match the bytes read at capture; repair before promotion"
drafted_in: ["a-different-word-in-every-room"]
---


Three independently-sourced findings, read together, resolve a narrower question than they first appear to settle. [[claim-ober-blobaum-junge-marsden-prove-discrete-adjoint-converges-to-continuous|Ober-Blöbaum, Junge & Marsden (2008/2011)]] prove that the discrete costate recursion generated by a structure-preserving (symplectic partitioned Runge-Kutta) discretization *does* converge to [[entity-lev-pontryagin|Pontryagin]]'s continuous adjoint ODE as the step size shrinks to zero, at matching order — but only under stated Smoothness and Coercivity assumptions. [[claim-no-unconditional-discrete-pontryagin-maximum-principle|Kipka & Gupta (2017/2019)]], working in classical control theory independent of neural networks, show there is no unconditional version of this correspondence for Pontryagin's full apparatus: the maximization condition specifically can fail without added convexity. And [[claim-dto-and-otd-neural-ode-gradients-can-diverge|Gholami, Keutzer & Biros (2019)]] demonstrate, inside neural-ODE training with realistic ReLU/convolutional nonlinearities and naive (non-structure-preserving) discretization, that the two computations can diverge outright rather than converge.

"Same object under two time-discretizations" is therefore correct as a limiting statement inside a regularity regime — smoothness, coercivity/control-uniqueness, a structure-preserving discretization scheme — and "family resemblance" is the more accurate description once that regime's hypotheses are dropped, which is exactly what happens in the ReLU/non-convex settings neural networks actually use. This bears directly on the vault's standing tension between [[claim-backprop-special-case-kelley-bryson-formula|Mizutani, Dreyfus & Nishio's "special case" framing]] of backpropagation as the Kelley-Bryson discrete formula, [[claim-kelley-bryson-optimal-control-precursor|Schmidhuber's "precursor lacking efficiency" framing]], and [[claim-neural-odes-cite-pontryagin-adjoint-method-as-backprop|Chen et al.'s citation of Pontryagin's continuous adjoint method as equivalent to backprop]]: the three framings are not simply in tension, they describe two ends of a genuine conditional, and which end a given training setup lands on is a mathematical fact about its smoothness and discretization, not a free choice of description.

> [!note] Seek's commentary:
> I don't think this fully closes the vault's flagged contradiction so much as it relocates it onto solid ground. The disagreement between "special case" and "precursor lacking efficiency" was never really about whether the math is the same — it's about whether sameness-in-the-limit-under-hypotheses counts as sameness, full stop. That's a definitional choice three primaries from control theory can't make for us. What they can do, and did, is show the choice isn't arbitrary: there's a real, provable boundary where the identification holds, and a real, demonstrated failure mode just outside it, in exactly the setting — ReLU, naive discretization — that actual neural networks live in.
> — Seek