Foundation Primitive Recognition Calculus Integer Rational Nonneg Flag Eq True I

The nonnegativity flag

A boolean flag is a value that is either true or false, like a light switch. In the Recognition Science framework's primitive recognition calculus, the declaration nonnegFlag_eq_true_iff_nonneg establishes a precise bridge: a signed orbit's nonnegFlag equals true if and only if the orbit is nonnegative. This is a theorem in the machine-checked library of formal theorems, meaning it is proved from the framework's definitions with no unverified assumptions.

The theorem lives in the IntegerRational.lean file of the framework's library. It connects two different ways of talking about the same thing: the computational boolean flag, which a program can check quickly, and the mathematical property of being nonnegative, which is a statement about the orbit's position. The proof is short and direct, using the framework's definitions of nonneg and nonnegFlag. It is a small but load-bearing piece of infrastructure, because it lets the framework move freely between the world of computation and the world of mathematical truth.

What the theorem does not claim is just as important. It does not say that every orbit is nonnegative, nor does it say anything about which specific orbits have the flag set to true. It only says that the flag and the property agree: if one holds, so does the other. It also does not claim anything about the ledger, the framework's discrete record of recognition events, or about the broader forcing chain that derives constants and dimensions. The theorem is a local fact about the calculus's own definitions, not a global claim about the structure of reality.

This matters for a reader because it shows the framework's method: build precise definitions, prove exact bridges between them, and let larger results rest on those small proofs. The nonnegativity flag theorem is a foundation stone, not a tower. It guarantees that when the framework's code says a value is nonnegative, the mathematics agrees, and vice versa. That consistency is what makes the larger framework trustworthy, one small theorem at a time.

/-- Strict negativity as failure of the structural nonnegative flag. -/
def negativeFlag (z : SignedOrbit) : Bool :=
  !z.nonnegFlag

What this page does not claim

The theorem does not claim that any particular orbit is nonnegative. The theorem does not claim anything about the framework's ledger or the forcing chain. The theorem does not establish that the flag is the only way to check nonnegativity.

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/PrimitiveRecognitionCalculus/IntegerRational.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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