is
plain-language theorem explainer
The declaration defines the bidirectional structural mapping between mock modular defects and phantom balances inside the 8-tick window model. Researchers linking Ramanujan's mock theta functions to unresolved phase debt in Recognition Science would cite it. The definition directly assembles the two maps together with the zero-correspondence axiom and the shadow-completion property.
Claim. A structure equipped with maps $f$ from mock modular defect (magnitude $m$ with $m=0$ iff perfectly modular) to phantom balance (accumulated signal $a$, remaining ticks $r$ with $r$ at most 8) and $g$ in the reverse direction, satisfying $m=0$ if and only if the phantom magnitude of $f$ is zero, together with the axiom that adjoining the shadow term restores full symmetry.
background
Recognition Science treats an 8-tick window as a finite sequence of signal values over the fundamental time quantum (one tick). A closed window has vanishing sum and corresponds to a true modular form; an open window carries a pending compensation requirement whose size is the phantom balance, defined as the negative of the accumulated signal. The module local setting identifies Zwegers' non-holomorphic shadow with the phantom-light projection that closes the window and restores SL(2,Z) symmetry.
proof idea
The definition is a direct structure construction that packages the two maps and the zero-correspondence axiom. It applies the upstream definitions of mock modular defect (magnitude nonnegative, zero precisely when modular) and phantom balance (debt equals negative accumulated signal) without further reduction steps.
why it matters
The structure supplies the explicit bridge that downstream acoustics results (optimal T60 equal to phi, hearing-loss penalty at each rung) and action results (energy-conservation certificates, Christoffel symbols) can invoke when they need the mock-theta analogy. It occupies the Ramanujan-bridge slot that converts the classical mock-theta completion into the RS eight-tick neutrality language (T7), leaving open whether the correspondence can be promoted from hypothesis to theorem.
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