min_max_achieved
plain-language theorem explainer
The declaration establishes that the positive integer pair (1,1) attains the value 1 under the max function. Researchers closing the zero-parameter ledger to hierarchical structure in Recognition Science cite it to anchor the base rung before invoking uniform scaling. The term-mode proof reduces the equality by direct simplification of the max expression.
Claim. For positive integers $a$ and $b$, the pair $(1,1)$ satisfies $a=1$, $b=1$ and therefore achieves the minimal value of the maximum function equal to 1.
background
The module closes Gap 2 of the axiom-closure plan by deriving hierarchical structure from the nontrivial zero-parameter ledger. It defines ScalePerturbed as an explicit perturbation shifting levels above a chosen position by exp(t) and proves that uniform inter-level ratios are forced once the zero-parameter condition holds. The sibling result additive_composition_is_minimal states that among positive-integer pairs the pair (1,1) uniquely minimises max(a,b) by pure arithmetic.
proof idea
One-line wrapper that applies simp to reduce the max expression directly to the constant 1.
why it matters
The result supplies the arithmetic base case for the minimal-coefficients claim inside the hierarchy-forcing sequence. It precedes the derivation of uniform_scaling_forced and hierarchy_forced_gives_phi, which together force the self-similar phi-ladder from the Recognition Composition Law. No open scaffolding remains at this leaf.
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