Foundation Primitive Recognition Calculus Integer Order Negative Flag Mul Of Non

Signs under multiplication

The declaration negativeFlag_mul_of_nonnegFlag_of_not_balanced_zero_of_negativeFlag is a theorem in the framework's machine-checked library of formal theorems. It states a sign rule for multiplication in a specific algebraic structure called SignedOrbit, which is a discrete record of events that can be counted forward or backward. The theorem says: if you multiply a negative number by a non-negative number, and the non-negative number is not balanced with zero (meaning it is not exactly zero), then the result is negative. In symbols, if a has a negative flag and b has a non-negative flag, and b is not balanced with zero, then the product a * b has a negative flag.

This is a formalization of a rule that ordinary arithmetic takes for granted. The framework proves it from its own definitions of SignedOrbit, its flags, and its multiplication operation. The theorem is not about the physical world directly; it is about the internal consistency of the framework's counting system. It ensures that the sign of a product behaves as expected, which is a necessary step for building more complex statements about ratios and orders within the framework.

In Recognition Science, this theorem is part of a larger effort to show that the framework's basic objects obey the laws of ordinary arithmetic. The framework models reality as a ledger of recognition events, and SignedOrbit is one of its primitive counting tools. This particular theorem does not claim anything about the physical universe, nor does it derive any physical constants. It is a purely formal result about the framework's own definitions, a building block that must be in place before the framework can make any claims about the world.

The theorem also does not claim that a negative times a non-negative is always negative when the non-negative is zero. That case is excluded by the condition that the non-negative is not balanced with zero. The theorem's scope is narrow: it covers the sign of a product under specific conditions, and it says nothing about other operations like addition or division. It is a precise, limited statement that contributes to the framework's internal arithmetic.

theorem nonnegFlag_negate_ofOrbit_eq_true_iff_zero (n : DistinctionNat) :
    (SignedOrbit.negate (SignedOrbit.ofOrbit n)).nonnegFlag = true ↔
      n = DistinctionNat.zero := by
  constructor
  · intro h
    rw [SignedOrbit.nonnegFlag_eq_true_iff, SignedOrbit.negate_toInt,
      SignedOrbit.ofOrbit_toInt] at h
    apply DistinctionNat.toNat_inj
    rw [DistinctionNat.toNat_zero]
    omega
  · intro h
    rw [h]
    rw [SignedOrbit.nonnegFlag_eq_true_iff, SignedOrbit.negate_toInt,
      SignedOrbit.ofOrbit_toInt, DistinctionNat.toNat_zero]
    norm_num

What this page does not claim

This theorem does not derive any physical constant or make a claim about the physical universe. This theorem does not cover the case where the non-negative number is balanced with zero. This theorem does not state a rule for addition or division, only for multiplication.

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

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