phantom_shadow_uniqueness
plain-language theorem explainer
Any phantom balance structure fixes a unique integer that cancels its accumulated signal sum to zero. Analysts of mock modular forms in the Recognition Science setting cite this to confirm the shadow is the only possible completion for an unclosed 8-tick window. The proof reduces immediately to the uniqueness lemma for integer cancellation applied to the accumulation.
Claim. Let $pb$ be a phantom balance with accumulated signal sum $a$. There exists a unique integer $d$ such that $a + d = 0$.
background
The module develops the analogy between Ramanujan's mock theta functions and unclosed 8-tick windows. A phantom balance is the structure holding an accumulated integer signal sum, a remaining tick count at most 8, and the debt defined as the negation of the accumulation. This rests on the balanced ledger predicate and the J-cost structures that enforce neutrality for closed windows.
proof idea
The proof is a one-line term that applies the shadow uniqueness lemma directly to the accumulated field of the input phantom balance.
why it matters
This secures uniqueness of the shadow in the mock theta to phantom light correspondence. It supports the eight-tick octave completion and aligns with the forcing chain from J-uniqueness through periodicity, closing a step in the Ramanujan bridge without recorded downstream dependencies.
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