simple_geometric_ratio
plain-language theorem explainer
Recognition Science derives the base Weinberg angle ratio of exactly 1/4 from the eight-tick geometry with phase counts 3, 1 and 8. Model builders would cite this result as the uncorrected geometric starting point before applying the phi correction. The proof reduces directly to arithmetic evaluation after unfolding the mixing definition.
Claim. The geometric mixing ratio for the eight-tick geometry with three SU(2) phases, one U(1) phase and eight total ticks equals $1/4$.
background
The module derives the Weinberg angle from the eight-tick phase geometry in Recognition Science. EightTickGeometry encodes the discrete phase counts for SU(2) and U(1) sectors within the 8-tick octave structure. The geometric mixing ratio is defined as the U(1) phase count divided by the sum of SU(2) and U(1) phase counts, yielding the base sin²(θ_W) value. This base ratio is the starting point before the phi correction from quantum channel capacity is applied, as noted in the module documentation. The upstream correction definition supplies a positive factor 1/(phi * N) for finite N, which adjusts the ratio toward the observed value around 0.23. The local setting is the RS derivation of electroweak mixing parameters from information-theoretic principles, targeting a PRL paper.
proof idea
The proof unfolds the geometric mixing definition to expose the phase ratio expression and applies norm_num to simplify the concrete numbers 1 over 4 to the value 1/4.
why it matters
This result establishes the exact geometric base for the Weinberg angle in the Recognition Science framework, directly implementing the 8-tick geometry constraint (T7). It feeds the subsequent phi-corrected predictions listed in the module, such as the best prediction close to observed. The module positions it within the target of deriving sin²(θ_W) from phi optimization for electroweak unification.
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