gauge_alignment_possible
plain-language theorem explainer
Positive gauge factors A_lepton, A_up, A_down exist for the lepton, up-quark, and down-quark sectors, permitting the phi-ladder mass formula to align to a shared structural scale. Recognition Science researchers modeling sector-consistent masses would reference this existence result. The proof supplies explicit positive constructions from each sector's B_B, the positive coherence energy E_coh, and phi to the sector r0 power, then chains positivity lemmas for the products and powers.
Claim. There exist positive real numbers $A_l$, $A_u$, $A_d$ (the sector gauge factors) such that the mass formula $m_i = A_B · ϕ^{r_i + f_B}$ aligns with the common structural scale $m_{struct}$ for leptons, up quarks, and down quarks.
background
The module formalizes sector-global gauge parameters (B_B and r0) that align the phi-ladder across different fermionic sectors. The phi-ladder is the discrete scaling of masses by powers of the golden ratio phi, the self-similar fixed point forced in the Recognition Science chain. Upstream results include E_coh_pos, which states that the coherence energy E_coh is positive, as derived from phi to the power -5 in the eight-tick period.
proof idea
The proof explicitly constructs the three positive reals using the expressions from lepton_gauge, up_quark_gauge, and down_quark_gauge. It introduces auxiliary facts for 0 < 2, phi > 0 from phi_pos, and E_coh > 0 from E_coh_pos. For each sector it applies simp to obtain B_B > 0 via zpow_pos, Real.rpow_pos_of_pos for the phi power, then mul_pos twice to conclude the product is positive.
why it matters
This declaration establishes the possibility of gauge alignment for fermionic sectors in the Recognition Science mass formula, filling the role described in the theorem's doc-comment for aligning to m_struct. It relies on the phi-ladder and E_coh from the forcing chain (T5 J-uniqueness through T8 D=3). With no downstream uses yet, it serves as a base for sector-specific mass calculations and touches the question of determining the specific B_B and r0 values.
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