deflection_ratio
plain-language theorem explainer
The ratio of successive deflection angles on the phi-ladder equals phi exactly. Cosmologists certifying lensing regimes in Recognition Science cite this to confirm uniform scaling across the five regimes. The proof is a direct algebraic reduction that unfolds the power definition, invokes positivity and the division equivalence, then applies the successor exponent rule followed by ring simplification.
Claim. Let $θ(k) := φ^k$ be the deflection angle at ladder rung $k$. Then $θ(k+1)/θ(k) = φ$ holds for every natural number $k$.
background
The Gravitational Lensing from RS module defines five canonical regimes (weak, strong, microlensing, cluster, time-delay) each carrying a characteristic deflection angle on the phi-ladder. The upstream definition sets the deflection angle at rung $k$ to $φ^k$, encoding the self-similar scaling required by the Recognition Composition Law and the T6 fixed-point condition. The module works entirely inside RS-native units with $c=1$ and no additional axioms.
proof idea
The term proof unfolds the deflection-angle definition to obtain two powers of phi. Positivity of $φ^k$ permits rewriting the ratio equation via the division equivalence. The successor exponent identity is applied, after which ring normalizes the resulting polynomial identity.
why it matters
This supplies the phi_ratio field inside gravitationalLensingCert, which assembles the five-regime certification. It realizes the geometric progression forced by the phi fixed point (T6) and the eight-tick octave structure. The result closes the scaling property needed for the lensing certificate without introducing new hypotheses.
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