Foundation Pair Kernel Source Grammar Eval Is Algebraic

The certificate grammar

A ledger, a discrete record of recognition events, needs a language for writing down the strengths it records. The framework's certificate grammar is that language: it starts with rational numbers and the golden ratio, and it closes under addition, multiplication, negation, and division. Every expression built from these pieces evaluates to a real number, and the theorem eval_isAlgebraic proves that every such evaluated value is algebraic over the rationals. An algebraic number is one that solves a polynomial equation with rational coefficients; the golden ratio, for instance, solves x² − x − 1 = 0. The theorem is checked by a machine, with no unproved assumptions and no gaps in its reasoning.

The grammar's most conspicuous absence is the circle constant π. There is no constructor for it, and that absence is structural rather than accidental. The framework keeps the certificate language deliberately small: it imports only the golden-ratio and algebraicity surfaces from the wider mathematics library, and it leaves the repository's downstream physical constants outside the grammar. The result is a clean separation. The certificate language certifies exactly the values it can express, and every one of those values is algebraic. A value like π, which is transcendental, cannot be written in this language at all.

In Recognition Science, this grammar serves as the source-strength certificate for the pair kernel, the framework's account of how recognition events pair up. The theorem piFree_eval adds a companion fact: every expression certifies its own evaluated value, meaning the grammar is self-certifying. Together the two theorems guarantee that the certificate language is both expressive enough to name the values it needs and constrained enough to keep transcendental numbers out. The framework models its recognition ledger with this grammar, choosing rationals, the golden ratio, and field operations as the allowed building blocks.

What the theorem does not claim is just as important. It does not say that every algebraic number appears in the grammar; the grammar is a subset, not a complete listing. It does not say that π is impossible in some absolute mathematical sense, only that this particular grammar has no way to construct it. And it does not claim that the golden ratio's role here is physically necessary. The theorem establishes a fact about a formal language, not about the cosmos. That distinction matters for reading the framework honestly: the algebraicity result is a proved structural property of the certificate grammar, and the grammar's design choices are exactly that, choices.

/-- Every value generated by the grammar is algebraic over the rationals. -/
theorem eval_isAlgebraic (e : LedgerExpr) : IsAlgebraic ℚ e.eval := by
  induction e with
  | ofRat q =>
      exact isAlgebraic_rat ℚ q
  | phi =>
      exact goldenRatio_isAlgebraic
  | add a b ha hb =>
      exact ha.add hb
  | mul a b ha hb =>
      exact ha.mul hb
  | neg a ha =>
      exact ha.neg
  | inv a ha =>
      exact ha.inv

/-- Expressions generated by rational constants, the golden ratio, and field
operations. The constructor list is the machine-checked source whitelist. -/
inductive LedgerExpr where
  | ofRat : ℚ → LedgerExpr
  | phi : LedgerExpr
  | add : LedgerExpr → LedgerExpr → LedgerExpr
  | mul : LedgerExpr → LedgerExpr → LedgerExpr
  | neg : LedgerExpr → LedgerExpr
  | inv : LedgerExpr → LedgerExpr
  deriving Repr, DecidableEq

/-- Every expression certifies its own evaluated value. -/
theorem piFree_eval (e : LedgerExpr) : PiFree e.eval := ⟨e, rfl⟩

What this page does not claim

The theorem does not claim that every algebraic number is expressible in the grammar. The theorem does not claim that π is impossible in any broader mathematical system. The theorem does not claim that the golden ratio's role in the grammar is physically necessary.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelSourceGrammar.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: