phi_pow_51_lo_pos
plain-language theorem explainer
Lemma establishing positivity of the lower bound for the rational interval approximating φ^51. Researchers bounding successive powers of the golden ratio in Recognition Science numerics cite it to justify interval multiplications for φ^102 and beyond. The argument reduces to unfolding the explicit interval definition and normalizing the resulting numerical inequality.
Claim. Let $I_{51}$ be the closed rational interval with lower endpoint $45537548334$ and upper endpoint $45537549354$. Then the lower endpoint satisfies $0 < 45537548334$.
background
The module computes numerical bounds for the galactic time ratio τ★ / τ₀ ≈ 5.75e29. An Interval is a structure with rational endpoints lo and hi satisfying lo ≤ hi. The interval for φ^51 is defined with explicit bounds lo := 45537548334 and hi := 45537549354 that match prior PhiBounds results. Upstream, UniversalForcingSelfReference supplies the meta-realization axioms while Certification.Interval records the corresponding real closed-interval structure.
proof idea
One-line wrapper that unfolds the definition of the φ^51 interval and applies norm_num to discharge the strict positivity of the lower rational endpoint.
why it matters
The lemma is invoked to construct the φ^102 interval via multiplication and to prove the φ^140 containment theorem. It supplies a concrete step on the phi-ladder used for mass formulas and the eight-tick octave in the Recognition framework. The result closes a numerical link between the self-reference axioms and explicit galactic-scale bounds.
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