A fleet that caught its own circular logic
Noogram · federated sister-work · 2026-06-18
Emmanuel Sérié
Applied mathematics, then quantitative research — physics PhD, then years inside quantitative research.
Teaching and research today — École Polytechnique, CMAP, the MaQI master.
One question today: can a fleet of agents do real mathematics — and notice when its own reasoning runs in a circle?
Noogram — an instrument built in the open
A research programme, not a product — federative agentic AI, human-amplified, never replaced.
Live today — the public core is shipping, not promised.
One instrument, many crafts
Cosmon — the engine, under your control
Four principles, formally pinned — self-reference, transport-and-cognition, intentions-not-ownership, minimum-action; TLA+-validated.
Built by the method it runs — agents propose, a chief decides.
La Formule 1, pas le pilote — one mission it just ran.
The question, verbatim
A half-formed line from a colleague — Étienne asked: blur an exponential-family law with Gaussian noise; does it stay in the family?
Same shape, new knobs \(\tilde\theta\) — that line was all the fleet got.
Not one agent — a fleet
One chatbot is one mathematician — fast, fluent, alone in the room with its own blind spots.
This is a research group — many agents split across the three questions (general, quadratic, Hamilton–Jacobi), with referees built in: a panel whose only job is to attack the others' reasoning.
The plan lays itself down
A plan written in one breath — the opening science plan, before a single worker moves. Four branches: the proofs, the paper, the Lean formalisation, the narrative report.
It caught its own circular reasoning
The fleet found a loop in its own logic — the "classical" proof of the characterisation secretly assumes what it sets out to prove. A referee flagged it; the fleet wrote the loop down before escaping it. the circular notes ↗
The result, stated formally
Blur preserves the family ⟺ closure under Cole–Hopf:
One operator decides it — and it caps polynomials at degree two.
Q2 — the tractable case, in closed form
Quadratic statistic \(\Rightarrow\) the new knobs are explicit:
A Riccati flow — smoothing a Gaussian-family law is just matrix inversion.
It is machine-checked — and honest at the edge
A machine read the quadratic proof and agreed — Lean 4, kernel-checked, no charisma gets past it.
The frontier stays visible — the Hamilton–Jacobi half needs a Laplacian absent from Mathlib. Documented, not faked.
See it yourself
exp-families.noogram-labs.dev — proofs, paper, Lean, the circular notes: all live, all fleet-made.
The trust is in the gate — and in the loops the fleet was honest enough to write down.
Appendix — backup slides
Everything past this point is past the end — reachable during Q&A with the arrow keys or slide overview.
A · Q3 — the Hamilton–Jacobi equivalence
Regular stability ⟺ a differentiable path solves an EDP — there exists \(t\mapsto\theta_t\) with
Cole–Hopf is the bridge — \(q=e^{-g}\) turns the heat equation \(\partial_t q=\tfrac12\Delta q\) into \(\partial_t g=-\mathcal{L}g\); uniqueness (Widder/Tychonoff) pins the path.
B · The two loops the fleet caught
Loop 1 — stability \(\Rightarrow\) closure — the textbook argument builds a path \(\theta_t\) assuming stability holds on a whole neighbourhood, not just at the starting \(\theta\). Hidden over-assumption, written up in circular-q1a.
Loop 2 — the Q1 ⟺ Q3 "circle" — read naïvely, the easy direction assumes the very differentiable path it claims to produce. Resolved as a glissement d'hypothèses — a sliding of hypotheses, not a vicious circle — in circular-q3.
C · Why polynomial ⇒ quadratic
Plug a polynomial \(\varphi\) into \(\mathcal{L}\) — the term \(\tfrac12\|\nabla(\theta^\top\varphi)\|^2\) doubles the top degree.
Closure caps it — for \(\mathcal{L}(\theta^\top\varphi)\) to land back in \(\mathrm{Span}\{\varphi\}\oplus\mathbb{R}\) for all \(\theta\), the doubled degree must already be in range — forcing \(\deg\varphi_i\le 2\).
So the quadratic case is not a toy — it is the whole polynomial world.