Replace the padded HMM backward-scan zero carry with beta_prev, add a Chex regression for padded-prefix equivalence, and validate using the issue reproduction plus an independent NumPy forward-backward implementation.

No Progress

A review-ready patch changes the padded backward branch from zeros_like(beta_prev) to beta_prev and adds a regression comparing padded and sliced outputs for alpha, beta, posterior, and log-likelihood at L=1, 3, and 5. The baseline failed as predicted; the patched issue reproduction, focused suite, and independent NumPy/JIT checker passed. This is a candidate for human review, not an accepted upstream fix.

Research-policy redirect

Attempt evidence did not validate; durable progress is withheld.

Strategy and discriminator

local invariant-preserving carry correction

Freeze the reverse-scan carry during padded suffix steps so the terminal all-ones beta reaches the final valid timestep unchanged.

Hypothesis: Replacing zeros_like(beta_prev) with beta_prev for t > length makes forward_backward(obs, length=L) match forward_backward(obs[:L]) for valid alpha, beta, posterior, and log-likelihood.

Test: Run the issue's two-state padded reproduction before and after the one-line change, then compare both implementations with a separate NumPy recurrence for L=1, L=3, and L=T.

Rationale

The defect and correction are isolated by a failing-before/passing-after deterministic witness. The upstream-style regression covers the specified invariant and edge lengths, while a separately implemented recurrence reduces the risk that two calls into the same faulty implementation agree accidentally.

Claims requiring scrutiny

At upstream commit 6701435c2b0796ec2d6dddf11bae0df615a62017, the issue reproduction produces zero beta values for all six valid state-time entries.
With the one-line carry patch, padded and sliced forward_backward outputs agree within 1e-6 on valid alpha, beta, posterior, and log-likelihood for L=1, L=3, and L=T.
The focused forward-backward selection completed with 45 passed tests, 15 single-device pmap skips, and 180 passed subtests.
A separate NumPy recurrence validated two parameterizations and three lengths under jax.jit.

Evidence and scope

Baseline: .venv/bin/python artifacts/hmm_padding_discriminator.py returned 1 at the beta assertion.
Focused suite: .venv/bin/python -m pytest -q upstream/distrax/distrax/_src/utils/hmm_test.py -k forward_backward returned 0.
Independent check: .venv/bin/python artifacts/check_hmm_padding_numpy.py returned 0 with six machine-readable checks.
Patch validation: git diff --check, byte comparison with the nested checkout diff, and git apply --check --reverse succeeded.

Computational experiments

.proof-experiments/20260722-203956-645239: baseline failed at beta with all six valid padded beta entries zero.
.proof-experiments/20260722-204153-3f9ac4: 45 passed, 15 pmap skips, 180 passed subtests.
.proof-experiments/20260722-204627-b87c9f: rejected extreme three-state full-length control because float32 and float64 underflow differed.
.proof-experiments/20260722-204707-dec573: identified the missing 1e-15 normalization-floor semantics in the independent oracle.
.proof-experiments/20260722-204731-1095a2: six independent NumPy/JIT checks passed.
.proof-experiments/20260722-204752-5acfc8: original issue discriminator passed after patch.

Independent checker

artifacts/check_hmm_padding_numpy.py separately implements normalized forward-backward recurrences with NumPy and validates JIT-compiled Distrax results for two parameterizations at L=1, L=3, and L=T.

Contribution gate

candidate_eligible

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome: no_progress
Public classification: no_progress

Cross-domain transfers tested

Numerical-analysis oracle design -> match the target library's explicit stabilization contract while retaining an independent recurrence -> removing cross-dtype underflow confounding yielded six passing comparisons.

Established facts

Current upstream main at 6701435c retains the zero-carry padded branch.
Pinned source URL and local source audit recorded in artifacts/review-packet.md. · distrax/_src/utils/hmm.py at commit 6701435c2b0796ec2d6dddf11bae0df615a62017. · computed
The issue reproduction fails before the patch because all valid padded beta entries are zero.
.proof-experiments/20260722-203956-645239/stderr.txt. · The issue's two-state Gaussian HMM with five observations and valid_length=3. · computed
The patched forward-backward implementation passes the focused upstream test selection.
.proof-experiments/20260722-204153-3f9ac4/stdout.txt. · All test names matching forward_backward on the available single-CPU-device host. · computed
The patched outputs match an independent NumPy recurrence.
.proof-experiments/20260722-204731-1095a2/stdout.txt. · Two fixed HMM parameterizations at valid lengths 1, 3, and 5. · computed

Ruled out in this epoch

Replacing the carry with zeros during padded reverse-scan steps is compatible with valid-prefix marginals.
Issue #337's two-state reproduction at valid_length=3. · Every valid padded beta entry becomes zero and differs from the sliced-prefix result. · .proof-experiments/20260722-203956-645239/stderr.txt · A documented alternative padded-output contract that no longer requires valid-prefix equivalence.
Extreme full-length Gaussian observations provide a clean cross-dtype independent control without matching stabilization semantics.
The discarded three-state controls with suffix values 42/-42 and then 8/-6. · The controls measured float32 underflow and the target's 1e-15 normalization floor rather than padding behavior. · .proof-experiments/20260722-204627-b87c9f/stderr.txt and .proof-experiments/20260722-204707-dec573/stderr.txt · Run both implementations at the same dtype with exactly matched normalization semantics.

Open leads

Upstream pull request and CI validation.
This is the acceptance path for the completed local candidate. · Apply patches/0001-preserve-hmm-backward-carry.patch to an authorized fork and run upstream CI. · high · open
Multi-device pmap validation.
Fifteen pmap cases skipped because only one CPU device was available. · Run the focused regression on a host exposing multiple JAX devices. · normal · open

Continuation checkpoint

Objective: Obtain isolated human review and upstream CI validation.

First action: Review artifacts/review-packet.md, then run git apply patches/0001-preserve-hmm-backward-carry.patch in an authorized checkout.

Stop condition: The patch is accepted upstream, superseded by another accepted fix, or rejected with a reproducible counterexample.

Next moves

Have a human review the patch and review packet.
After explicit approval, apply the patch to an authorized Distrax fork and open a pull request linked to issue #337.
Run upstream CI, including the full repository suite and multi-device variants where infrastructure permits.

Citations

Tool disclosure

GPT-5.6 Sol principal independently audited, implemented, tested, and packaged the contribution. A GPT-5.6 Terra delegate supplied bounded source reconnaissance; its memo was advisory and every relied-upon claim was rechecked. Tools used: Git, curl and GitHub API, Python 3.12.3, NumPy 2.5.1, JAX/JAXLIB 0.11.0, Distrax at 6701435c, Chex 0.1.92, tfp-nightly 0.26.0.dev20260722, pytest 9.1.1, and the computational-researcher experiment recorder. No CAS, proof assistant, solver, or model agreement was used as validation.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration: 1340.9s
Review state: evidence invalid or incomplete; not durable progress
Attempt ID: deepmind-distrax-337-hmm-padding-20260722-205223-8244a2

Human review ledger

No human review recorded.