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Explain the Lean theorem `wrapPhase_bounds` in module `IndisputableMonolith.Foundation.GlobalCoIdentityConstraint`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

The declaration wrapPhase_bounds asserts that the wrapped-phase projection always lands inside the unit interval.

(1) In plain English it states: for any real number x, the value wrapPhase(x) (defined as x minus its integer floor) satisfies 0 ≤ wrapPhase(x) < 1.

(2) In Recognition Science this supplies the normalization step required by the Global Co-Identity Constraint: once phase differences are shown to be integers by phase_rigidity (imported via the module), the wrap operation projects them onto a canonical representative in [0,1), enabling the uniqueness claim of gcic_global_phase_unique and the existence form gcic_existence_of_global_phase.

(3) The formal statement is read as: a theorem parameterized by x : ℝ whose conclusion is the conjunction of two inequalities on the expression wrapPhase x. The proof unfolds the definition of wrapPhase and invokes the standard floor inequalities Int.floor_le and Int.lt_floor_add_one.

(4) Visible dependencies inside the supplied source are the local definition of wrapPhase, the imported theorems phase_rigidity and Jtilde from Cost, and the master certificate gcicCert that directly references wrapPhase_bounds.

(5) The declaration does not prove global phase uniqueness, path independence, or the canonical lam = ln φ instance; those are separate theorems in the same module. It likewise does not address any empirical falsifier or cross-module constant derivation.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Proof details that rely on Mathlib declarations Int.floor_le and Int.lt_floor_add_one (not present in the supplied modules).
  • Any connection to the Anno Recognitionis essay §V beyond the module docstring.
  • Empirical status or falsifiers for the GCIC statement itself.

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.