MockModularDefect

formal source

 108    In the Zwegers formalism, a mock theta function f(τ) fails to be
 109    modular by a specific "error" E(τ, τ̄) that depends on both τ and τ̄
 110    (hence non-holomorphic). -/
 111structure MockModularDefect where
 112  /-- The defect magnitude (how far from perfect modularity) -/
 113  magnitude : ℝ
 114  /-- Non-negative -/
 115  magnitude_nonneg : 0 ≤ magnitude
 116  /-- Zero iff perfectly modular -/
 117  zero_iff_modular : magnitude = 0 ↔ True  -- Simplified; full version needs modular form definition
 118
 119/-- The Phantom Balance: RS analog of the mock modular defect.
 120
 121    When an 8-tick window is partially filled, the unfilled portion
 122    must compensate. The magnitude of this compensation requirement
 123    is the Phantom Balance. -/
 124structure PhantomBalance where
 125  /-- Current accumulated signal sum -/
 126  accumulated : ℤ
 127  /-- Remaining ticks in the window -/
 128  remaining : ℕ
 129  /-- Remaining ≤ 8 -/
 130  remaining_le : remaining ≤ 8
 131
 132/-- The balance debt: what must be compensated to achieve neutrality. -/
 133def PhantomBalance.debt (pb : PhantomBalance) : ℤ := -pb.accumulated
 134
 135/-- The phantom magnitude: |debt| measures required compensation. -/
 136def PhantomBalance.phantomMagnitude (pb : PhantomBalance) : ℝ := (|pb.debt| : ℝ)
 137
 138/-- A fully balanced window has zero phantom magnitude. -/
 139theorem balanced_phantom_zero (pb : PhantomBalance) (h : pb.accumulated = 0) :
 140    pb.phantomMagnitude = 0 := by
 141  simp [PhantomBalance.phantomMagnitude, PhantomBalance.debt, h]