Flow-Matching Gaussians · the fleet

The fleet that watched itself work

Three sub-fleets — cognition (science + words), formal (the machine-checked proof), instrumentation (this self-report) — turned one question into a proof, a paper, notebooks, a site and a deck. Below: the role graph, then the whole run as it actually happened, then the closed citable set. Every bar traces to a recorded timestamp; hover anything for detail.

1 · Architecture — who does what

The FLEET STATE panel (left) is the shared blackboard every role reads and writes: briefs, the source-ledger, the proofs, Lean status, faults, the chronicle. A dashed ENTRY is the mission; the dashed green EXIT is the artifacts.

Fleet architecture three sub-fleets · a question in, the artifacts out · hover any role FLEET STATE briefs — Per-molecule briefing + frame — the sealed contract each worker runs under. briefs source-ledger — The closed citable set — tier-scored, DOI-verified references. source-ledger proofs — theorem.md + the LaTeX proof — both directions of the iff. proofs lean status — Mathlib build state — what is machine-checked vs sorry-stubbed. lean status faults — Collapses (redundant probes) + stuck gates (operator safety holds). faults chronicle — The append-only narrative — why each decision was taken. chronicle ENTRY — the mission / the question. MISSION.md: when is flow matching optimal transport? ▸ ENTRY — the question / mission COGNITION — science + words sourcer — Curates the closed citable set — every DOI verified, no fabrication. Owns the source-ledger. sourcercogn concept-writer — Writes the conceptual wiki notes — definitions, the stochastic-interpolant frame. concept-writercogn coder — Builds the notebooks: the three-Dirac OT contrast, the Σₜ-vs-Bures geodesic experiments. codercogn proofsmith — Drafts the theorem and its proof in prose + LaTeX, both directions of the iff. proofsmithcogn skeptic — Adversarial reader — attacks every claim before the formal branch trusts it. skepticcogn synthesizer — Folds the deliberation panels into one position the writers can act on. synthesizercogn writer — Composes the paper from wiki + proofs, citing only the source-ledger. writercogn editor — Consolidation pass — paper.tex must reflect the patched theorem.md, no drift. editorcogn reviewer — Final gate on the prose artifacts before delivery staging. reviewercogn FORMAL — the machine-checked proof lean-engineer — Formalises the theorem in Lean 4 / Mathlib — the machine-checked core. lean-engineerform probe-engineer — Writes the numerical probes that stress the commuting-covariance dichotomy. probe-engineerform red-team — The referee sub-fleet: attacks the proof, found the invalid step + the d≥3 gap. red-teamform INSTRUMENTATION — the fleet watches itself collector — Snapshots cosmon state read-only → the tidy report/data tables. collectorinst dataviz — Renders the fleet life-cycle — these figures — from the data, never decorated. datavizinst narrator — Writes the self-report: the honest scorecard of what was proved vs gated. narratorinst EXIT — the artifacts: the Lean proof, the paper, the notebooks, the site, the deck, this self-report. ▸ EXIT — Lean proof · paper · notebooks · site · deck · self-report
Figure 1 · The role graph. Rounded boxes are roles, one colour per sub-fleet. The referee — red-team — is a role whose only job is to attack the others’ work; it found the invalid proof step and the d≥3 gap. Hover any box for its charge.

2 · Swimlane — the whole run

Outcome is read off the fill: solid = completed, red hatch = collapsed (redundant / junk-probe), amber outline = stuck (an operator safety-gate), faint dashed = pending (not yet unblocked). Ticks are step completions; small squares are worker dispatches.

Whole-run swimlane one row per molecule · bar length ∝ tackle→settle · ticks = steps · squares = worker dispatches · x = real wall-clock completed collapsed = redundant / junk-probe stuck = operator-gate pending = not-yet-unblocked running sub-fleet hue: cognition formal instrumentation 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00 01:00 02:00 03:00 04:00 05:00 06:00 07:00 08:00 09:00 delib-b9c7 delib-20260617-b9c7 · cognition · completed task-1889 task-20260617-1889 · cognition · completed task-bc1b task-20260617-bc1b · cognition · completed task-290c task-20260617-290c · cognition · completed task-453e task-20260617-453e · cognition · completed task-0515 task-20260617-0515 · cognition · completed task-1669 task-20260617-1669 · cognition · completed task-a4a6 task-20260617-a4a6 · cognition · completed delib-ddce delib-20260617-ddce · cognition · completed task-45e8 task-20260617-45e8 · cognition · completed task-595c task-20260617-595c · cognition · completed task-f8af task-20260617-f8af · cognition · completed · GATE: Gate: wait for verify-panel proof-repair molecules 8772 (rigidity qualifier), fbcd (delete invalid det-continuity argume task-51e0 task-20260617-51e0 · cognition · completed delib-da31 delib-20260617-da31 · cognition · completed task-bbb4 task-20260617-bbb4 · formal · completed edit-f4bd edit-20260617-f4bd · formal · completed task-8772 task-20260617-8772 · formal · completed task-fbcd task-20260617-fbcd · formal · completed task-c55d task-20260617-c55d · formal · completed task-32e5 task-20260617-32e5 · formal · completed task-7a09 task-20260617-7a09 · formal · completed task-cca6 task-20260617-cca6 · formal · completed task-6204 task-20260617-6204 · formal · completed task-×28eb task-20260618-28eb · formal · completed task-×131a task-20260618-131a · formal · completed task-×1dc9 task-20260618-1dc9 · formal · completed task-5292 task-20260617-5292 · formal · collapsed · junk probe molecule created while inspecting nucleate JSON shape; not a real child task-8e73 task-20260617-8e73 · formal · collapsed · junk probe to inspect nucleate JSON shape; not a real molecule task-×8447 task-20260618-8447 · formal · pending task-64d0 task-20260617-64d0 · instrumentation · completed task-5451 task-20260617-5451 · instrumentation · completed task-500e task-20260617-500e · instrumentation · completed task-a347 task-20260617-a347 · instrumentation · completed · GATE: SAFETY GATE (operator): no outward delivery until the verify-panel proof repairs (8772/fbcd/c55d/f4bd) merge and a clean verify-2f15 verify-20260617-2f15 · instrumentation · completed task-3063 task-20260617-3063 · instrumentation · completed verify-4cce verify-20260617-4cce · instrumentation · completed verify-b1cb verify-20260617-b1cb · instrumentation · completed verify-2238 verify-20260617-2238 · instrumentation · completed task-c612 task-20260617-c612 · instrumentation · completed task-5115 task-20260617-5115 · instrumentation · completed task-×62aa task-20260618-62aa · instrumentation · completed verify-×344d verify-20260618-344d · instrumentation · completed task-×53b4 task-20260618-53b4 · instrumentation · completed task-×a8b5 task-20260618-a8b5 · instrumentation · completed verify-×9401 verify-20260618-9401 · instrumentation · completed task-×f0fc task-20260618-f0fc · instrumentation · completed verify-×aea2 verify-20260618-aea2 · instrumentation · completed task-×1cb2 task-20260618-1cb2 · instrumentation · completed verify-×ac48 verify-20260618-ac48 · instrumentation · completed task-×db45 task-20260618-db45 · instrumentation · completed verify-×2340 verify-20260618-2340 · instrumentation · completed verify-×b5b8 verify-20260618-b5b8 · instrumentation · completed task-×ab7a task-20260618-ab7a · instrumentation · completed verify-×434e verify-20260618-434e · instrumentation · completed verify-×6254 verify-20260618-6254 · instrumentation · completed task-×23f1 task-20260618-23f1 · instrumentation · running task-118e task-20260617-118e · instrumentation · collapsed · duplicate of canonical site builder task-20260617-7a09 (pre-wired by mission controller); synthesis briefing injected into 7a09 instead task-97fb task-20260617-97fb · instrumentation · collapsed · duplicate of canonical deck builder task-20260617-cca6 (pre-wired by mission controller); synthesis briefing injected into cca6 instead task-×e42d task-20260618-e42d · instrumentation · pending Built from real .cosmon/state/events.jsonl + runtime-trace.jsonl + each molecule’s state.json. Sub-fleet is an inferred label (prompt heuristic); every bar edge is a recorded timestamp.
Figure 2 · One row per molecule. Bar length is real tackle→settle duration on a wall-clock axis. The collapsed/stuck/pending story is legible at a glance: a handful of redundant probes pruned, two molecules held at an operator gate until the proof repairs merged.

3 · References — the closed citable set

  1. 1Building Normalizing Flows with Stochastic Interpolants ↗Michael S. Albergo, Eric Vanden-Eijnden · 2023 · International Conference on Learning Representations (ICLR)
  2. 2Stochastic Interpolants: A Unifying Framework for Flows and Diffusions ↗Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden · 2023 · arXiv preprint (Journal of Machine Learning Research version)
  3. 3Flow Matching for Generative Modeling ↗Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, Matt Le · 2023 · International Conference on Learning Representations (ICLR)
  4. 4Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow ↗Xingchao Liu, Chengyue Gong, Qiang Liu · 2023 · International Conference on Learning Representations (ICLR)
  5. 5Score-Based Generative Modeling through Stochastic Differential Equations ↗Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon et al. · 2021 · International Conference on Learning Representations (ICLR)
  6. 6Flow Matching Guide and Code ↗Yaron Lipman, Marton Havasi, Peter Holderrieth, Neta Shaul, Matt Le et al. · 2024
  7. 7Computational Optimal Transport: With Applications to Data Science ↗Gabriel Peyr\'{e}, Marco Cuturi · 2019 · Foundations and Trends in Machine Learning
  8. 8Polar Factorization and Monotone Rearrangement of Vector-Valued FunctionsYann Brenier · 1991 · Communications on Pure and Applied Mathematics
  9. 9The distance between two random vectors with given dispersion matricesIngram Olkin, Friedrich Pukelsheim · 1982 · Linear Algebra and its Applications
  10. 10On the Bures--Wasserstein distance between positive definite matrices ↗Rajendra Bhatia, Tanvi Jain, Yongdo Lim · 2019 · Expositiones Mathematicae
  11. 11Wasserstein geometry of Gaussian measuresAsuka Takatsu · 2011 · Osaka Journal of Mathematics
  12. 12Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry ↗Klas Modin · 2017 · Journal of Geometric Mechanics
  13. 13A Convexity Principle for Interacting GasesRobert J. McCann · 1997 · Advances in Mathematics