shadow_fringe_exists
plain-language theorem explainer
Recognition Science predicts a non-zero phase fringe density at black hole event horizons due to the eight-tick cycle. Researchers modeling ILG corrections to black hole shadows, such as for M87*, would cite this when deriving fringe-induced shifts in shadow diameter. The proof is a direct construction that exhibits the explicit sine density and evaluates it at t = 2 tau0 to obtain the value 1.
Claim. Let $tau_0 > 0$ be the fundamental tick duration. There exists a function $rho : mathbb{R} to mathbb{R}$ such that $rho(t) = sin(2 pi t / (8 tau_0))$ for all real $t$, and there exists $t'$ with $rho(t') neq 0$.
background
PhaseFringeDensity is the sine function $sin(2 pi t / (8 tau_0))$ that encodes interference fringe density at the event horizon boundary. The module formalizes ILG-corrected lensing predictions for black hole shadows, with the explicit goal of showing that the 8-tick cycle produces a detectable phase fringe. Upstream results supply tau0 as the fundamental time unit (derived from hbar, G, c in Constants.Derivation) together with ledger factorization and nucleosynthesis tier structures that calibrate discrete phi-ladders in the Recognition framework.
proof idea
The tactic proof exhibits PhaseFringeDensity tau0 as the witness rho. It then picks the concrete point t' = 2 tau0, unfolds the sine definition, and applies field_simp followed by ring to reduce the argument to pi/2. The final simp step invokes sin(pi/2) = 1 to witness the required non-zero value.
why it matters
This theorem supplies the existence claim for the phase fringe inside the black hole shadow predictions module. It directly implements the eight-tick octave (T7) of the forcing chain, guaranteeing that the periodic density is non-trivial at the horizon. The accompanying doc comment links the result to the shadow diameter correction epsilon_fringe ~ lambda_rec / R_s, which is negligible for supermassive holes but potentially detectable for primordial black holes.
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