Lean
The elaborator, then the kernel.
A term or a tactic script that says which lemma it hopes is already in the library.
Founding draft · Bologna
Language, distilled into proof.
A companion to LogicArtificialis and LogiBo. It asks how a language model may propose, and how Lean, Rocq, Isabelle, and Agda may refuse — until what remains is short enough for a kernel to sign.
The model proposes. The still condenses. The kernel signs.
Four movements of the still
- 01
Propose
A model drafts in ordinary mathematical English, and in the surface syntax of a proof assistant. The draft is allowed to be fluent. Fluency is not acceptance.
- 02
Translate
The same statement has to be able to move: into Lean, into Rocq, into Isabelle, into Agda, and between them. A change of keyword is not a translation.
- 03
Repair
The kernel’s refusal is the lesson. An open goal is isolated, the failed tactic is named, and the model is asked again — about that goal, not about the whole story.
- 04
Distill
A checked argument is shortened until a person can read it aloud and a machine can replay it. What boils off is length, hedge, and any step that was only local habit.
Two distillations
Not a smaller model. A shorter proof.
One use of the word is engineering: train a large prover, then compress it. That work can matter. It is not this note.
The other use is the old one. A mixture is heated, and only the part that survives is kept. Here the mixture is a draft — informal mathematics, or a tactic script a model found plausible. The heat is a proof kernel. The distillate is a statement whose argument a person can read and a checker can replay.
Length is not understanding. A proposal is not a proof. The checker, not the chat, is the authority — the sentence LogicArtificialis already uses, applied to the moment a model speaks.
Ex vapore, gutta certa.
From the vapor, a certain drop.
The emblem
The same animals. A new instrument.
LogicArtificialis keeps a sapphire, a turtle, and two companions, and sets a gold turnstile in its monogram. Distilled Proof uses that family and adds the still: the place where a draft is boiled down to something the turnstile is allowed to mark.
- 01
The corgi. Proposes, the way a model proposes. On the LogiBo seal he is the President. Here he is off duty, and not allowed to conclude.
- 02
The capybara. Stays. Someone has to read the distillate later, under a portico, without having been in the room where the draft was made.
- 03
The turtle. The kernel. Slow on purpose. The flask sits on the shell because nothing is signed until the turtle has crossed.
- 04
The alembic. Distillation, not compression for its own sake. A mixture goes in. Heat is a checker. What survives is smaller and named.
- 05
The sapphire. LogicArtificialis says the sapphire is the mind. Here the mind is a drop: one claim, faceted, held up to the light.
- 06
The turnstile. ⊢ in gold, the same mark set into the LogicArtificialis monogram. It is the signature. It does not appear until the refusal has been answered.
Instruments
Four flasks, one refusal.
The still is not loyal to a single assistant. A distillate that exists in only one grammar has not yet travelled. The authority in each column is the checker, not the interface the model was shown.
Rocq
The vernacular, then the kernel.
A proof script that does not invent an axiom by changing the keyword.
Isabelle
The proof object behind the Isar.
A structured argument. A sledgehammer call is a search, not the note.
Agda
The type checker. There is nowhere to hide a tactic.
A construction a reader can walk, recursion and all.
Kinship
Between the institute and the portico.
Distilled Proof does not rename either neighbour. It is the bench between a prospectus and a society: where a model is allowed to speak, and not allowed to conclude.
Institute
LogicArtificialis
Artificial logic, made checkable. Meaning guides. Syntax decides. The sapphire, the turtle, and the turnstile come from there.
logicartificialis.org
Society
LogiBo
Logicians and the logic-curious, under the porticoes of Bologna since 2025. The room where a distillate would actually be read aloud.
logibo.org
In the flask
A specimen, not a prover.
Advance the still. Three claims — a picture, a smuggled axiom, one fact in four grammars — show the loop this note is for. They are written out. Nothing on this page calls a model or a kernel.
A first shape of work
Proposed. Not scheduled.
A programme for the bench. No seminars, repositories, or appointments are claimed as already underway.
- 01
A public bench
Small theorems taken from a blackboard to a kernel while the room is still there. The interesting part is where the formalization resists the talk.
- 02
A repair log
What the model said, what the checker refused, and what was kept. Published as a log, not as a leaderboard.
- 03
A translation shelf
The same statement in Lean, Rocq, Isabelle, and Agda, short enough to diff, explicit about the lemmas each one calls.
- 04
Notes for the portico
Brief enough to be read aloud at LogiBo: what was formalized, in which system, under which axioms, and what was left informal on purpose.
Correspondence
Write beside the prospectus.
If you have a statement worth the heat — a lemma from a seminar, a translation that failed at the keyword, a proof a model made too long — set it down. This draft keeps the note locally. The letter, if you send one, goes with LogicArtificialis.
Bologna, with the neighbours.
distilledproof.org