r0_constraints_true_iff
plain-language theorem explainer
For any integer four-sector r0 assignment (lepton, up, down, electroweak), the Boolean structural filter returns true exactly when the Prop-level principle constraints hold. Yardstick choice-set verification cites this bridge when moving between decidable filters and mathematical statements. The proof unfolds both definitions and closes by simplification plus tautology.
Claim. For every assignment $a$ of integer $r_0$ values to the lepton, up, down, and electroweak sectors, the Boolean structural constraints evaluate to true if and only if $a$ satisfies the principle constraints: $a_{\mathrm{down}}<0$, $a_{\mathrm{lepton}}>a_{\mathrm{ew}}$, $a_{\mathrm{ew}}>a_{\mathrm{up}}$, and $a_{\mathrm{lepton}}+a_{\mathrm{up}}+a_{\mathrm{down}}+a_{\mathrm{ew}}$ equals the $r_0$ sum target.
background
This module treats the O1 yardstick discussion as a finite combinatorial search: four candidate $r_0$ values are assigned to the four sectors (all permutations), then filtered by structural constraints. Valid choice sets collapse to a singleton under those filters.
An $r_0$ assignment is a four-tuple of integers (lepton, up, down, electroweak). The structural filter is the Boolean conjunction of four decidable checks: down negative, lepton strictly above electroweak, electroweak strictly above up, and the four components summing to the fixed $r_0$ sum target. The principle constraints are the same four conditions stated as a Prop.
The theorem is the equivalence bridge between those two presentations of the same filter.
proof idea
Unfold the Boolean structural filter and the Prop-level principle constraints to expose identical atomic conditions (three strict inequalities plus the sum identity). simp reduces the decide/&& Boolean packaging to the underlying propositions; tauto closes the resulting propositional equivalence. No external lemmas are required.
why it matters
The equivalence lets later results switch freely between decidable Boolean filters (used in enumeration) and Prop-level hypotheses (used in uniqueness arguments). Downstream, the canonical-forcing theorem converts a principle-constraint hypothesis into a true Boolean filter via the reverse direction, then places the assignment in the valid set and concludes it equals the canonical $r_0$ assignment. The Boolean-form theorems under affine sector roles and the depth-gap hypothesis also route through this bridge when stating forced filters or reducing the filter to a pure sum check. In the broader Recognition verification stack this is bookkeeping for the yardstick rung assignment that feeds the mass ladder, not a new physical law.
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