balanced_phantom_zero
plain-language theorem explainer
A PhantomBalance structure with zero accumulated signal sum has zero phantom magnitude. Researchers modeling mock theta functions as unclosed 8-tick windows in Recognition Science would cite this to confirm closure eliminates residual defect. The proof is a direct simplification that substitutes the definitions of phantomMagnitude and debt under the zero-accumulation hypothesis.
Claim. Let $pb$ be a PhantomBalance structure recording an accumulated signal sum and remaining ticks in an 8-tick window. If the accumulated sum equals zero, then the phantom magnitude of $pb$ equals zero.
background
The module treats mock theta functions as partition functions of phase debt in partially filled 8-tick windows. A true modular form corresponds to a closed window whose sum is neutral; a mock theta function records an open window whose unfilled ticks impose a future compensation requirement. The PhantomBalance structure holds the current accumulated signal sum (an integer), the number of remaining ticks (a natural number at most 8), and defines debt as the negation of the accumulated sum.
proof idea
The proof is a one-line simp that unfolds PhantomBalance.phantomMagnitude and PhantomBalance.debt, then substitutes the hypothesis that accumulated equals zero.
why it matters
This result closes the zero case for phantom magnitude, confirming that a fully balanced 8-tick window carries no mock modular defect. It supports the RS reading of Zwegers' shadow as phantom light that restores symmetry once the window closes, consistent with the eight-tick octave and the distinction between closed windows (true modular forms) and open ones (mock theta functions). No downstream theorems are recorded yet.
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