Foundation Pair Kernel Source Grammar Ledger Expr

The source ledger

LedgerExpr is a small, precisely defined language for writing down numbers. It is a list of allowed ingredients: any rational number (a fraction like 3/4), the golden ratio, and the operations of addition, multiplication, negation, and division. Starting from those ingredients and applying those operations, you can build expressions such as (1 + φ)/2 or 3/(φ - 1). The declaration is a grammar, a whitelist, a rule for what counts as a well-formed numerical expression.

The language has a clear purpose: it certifies numbers. In the Recognition Science framework, a ledger is a discrete record of events, and here the ledger is a record of which real numbers are allowed to appear as source strengths. Every expression in LedgerExpr evaluates to a real number, and the framework proves that every such evaluated value is algebraic over the rationals. That means each value is the root of some polynomial equation with rational coefficients; the golden ratio, for instance, satisfies x² = x + 1. The framework also proves that every expression certifies its own value: if an expression evaluates to a number, that number is marked as pi-free, meaning it is reachable by the grammar.

The most striking feature is what the grammar excludes. The circle constant pi has no constructor, no way to be built. This absence is structural, not an oversight. The grammar's design ensures that every expression it can generate is algebraic over the rationals, and pi is transcendental: it is not the root of any polynomial with rational coefficients. So pi cannot appear, by construction. The framework's library proves this exclusion is total: every evaluated expression is algebraic, and pi is not.

In Recognition Science, this grammar is a foundation stone. The framework derives physical constants from a forcing chain, and this language fixes which constants may enter a certificate. The exclusion of pi is a deliberate boundary: the wider constants module, which contains downstream physical constants, is kept outside this grammar. The grammar is a clean, self-contained starting point, and it is machine-checked: the library contains no unproven assumptions and no new axioms.

What the declaration does not claim is just as important. It does not claim that pi is unimportant or that it never appears in the framework; it only says pi is not expressible in this particular whitelist. It does not claim that every algebraic number is in the language, only that everything in the language is algebraic. And it does not claim that the grammar itself is the source of the golden ratio's special role; the golden ratio is simply one of the allowed seeds, chosen because the framework's forcing chain derives it as the unique self-similar scaling.

/-- 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⟩

/-- 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

/-- The golden-ratio seed is algebraic over the rationals. -/
theorem goldenRatio_isAlgebraic : IsAlgebraic ℚ Real.goldenRatio := by
  refine ⟨Polynomial.X ^ 2 - Polynomial.X - 1, ?_, ?_⟩
  · intro h
    have hcoeff := congrArg (fun p : Polynomial ℚ => p.coeff 2) h
    norm_num [Polynomial.coeff_X, Polynomial.coeff_one] at hcoeff
  · norm_num [Real.goldenRatio_sq]

What this page does not claim

Pi is unimportant or never appears in the wider framework. Every algebraic number is expressible in this grammar. The grammar itself is the source of the golden ratio's special role.

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

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