PhantomBalance
plain-language theorem explainer
PhantomBalance encodes the RS analog of the mock modular defect for partially filled 8-tick windows via an accumulated integer signal and bounded remaining ticks. Researchers mapping Ramanujan's mock theta functions to harmonic Maass forms cite it to represent the required compensation as balance debt. The definition specifies fields directly and defines debt as the negation of the accumulated sum.
Claim. A structure consists of an integer accumulated signal sum $a$ and a natural number of remaining ticks $r$ with $r ≤ 8$. The debt of such a structure is defined as $-a$, and its phantom magnitude is the absolute value of the debt.
background
The module interprets mock theta functions as unclosed 8-tick windows in the Recognition Science framework, where true modular forms correspond to balanced windows with zero sum. LedgerForcing.balanced defines a ledger as balanced when its event list satisfies the balance condition. The PhantomBalance captures the pending debt when the window is incomplete, with the module stating that the mock modular defect equals the Phantom Magnitude as the pending future debt constraining the configuration space. Upstream, Actualization.A maps configurations to their actualized form via argmin of the J-cost, and IntegrationGap.A sets the active edge count to 1 per tick.
proof idea
PhantomBalance is defined as a structure with three fields: accumulated as an integer, remaining as a natural number, and a proof that remaining is at most 8. The debt function is introduced as the negation of accumulated, and phantomMagnitude as the real-valued absolute value of that debt. No external lemmas are invoked; the construction is direct.
why it matters
This structure supplies the central object for the RamanujanBridge module, directly supporting downstream results including balanced_phantom_zero, nonzero_accumulation_forces_phantom, and phantom_completes_to_balanced which establish that phantom debt completes an indebted window to balance. It implements the RS version of Zwegers' completion theorem, linking mock theta functions to the eight-tick octave from the forcing chain T7. The construction leaves open the full verification of the MockThetaPhantomCorrespondence hypothesis.
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