gauge_sum_bounds
plain-language theorem explainer
The predicted sum of the three gauge couplings at the unification scale, arising from the cube geometry in three dimensions, satisfies 36 less than the sum less than 48. A researcher deriving the strong coupling constant from the Recognition Science eight-tick structure would cite this bound when assembling the certificate for alpha_s. The proof reduces the sum to twelve pi via an upstream equality and applies nonlinear arithmetic to the standard bounds on pi.
Claim. $36 < 12π < 48$
background
In the module on the strong coupling constant from φ-geometry (Q9), the framework asks whether α_s(M_Z) can be derived from the Recognition Science approach. The strong coupling emerges as the complement within the eight-tick gauge structure, with the three gauge couplings fixed by the cube geometry in three spatial dimensions; the module notes that α_s uses the remaining degrees of freedom after electromagnetic and weak sectors are accounted for.
proof idea
The proof first rewrites gauge_sum_prediction using the upstream theorem gauge_sum_value, which establishes equality to 12π. It then applies the constructor tactic to split the conjunction and invokes nlinarith with the library facts Real.pi_gt_three and Real.pi_lt_four.
why it matters
This bound supplies the gauge_bounded field in the downstream theorem strong_coupling_cert_exists that constructs the StrongCouplingCert. It advances the module goal of deriving α_s(M_Z) from the RS framework via the eight-tick octave and D=3, tying directly to forcing-chain landmarks T7 and T8 while anchoring the numerical range for the φ-ladder in the strong sector.
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