MinimalEngine
plain-language theorem explainer
A minimal visibility engine is a cost map from naturals to reals equipped with stability and K-theta floor properties that hold for every non-identity reciprocal. Researchers closing Erdos-Straus residuals via phase budgets cite this structure as the minimal interface supplying only the required RS-physical inputs. The declaration is a direct structure definition that packages the two hypotheses and omits any visibility field.
Claim. A minimal engine consists of a map $c : ℕ → ℕ → ℝ$ such that for every non-identity reciprocal $n$, the budget $c(n)$ satisfies the stable integer ledger condition and the K-theta failure floor hypothesis.
background
Recognition Science models recognition via reciprocal events whose ratios invert under the reciprocal automorphism. Non-identity reciprocals are the natural numbers to which this automorphism applies non-trivially. The stable budget field ensures the assigned cost maintains a stable integer ledger, while the floor field enforces the uniform K-theta failure threshold drawn from bounded phase visibility.
proof idea
The declaration is a structure definition that directly introduces the costOf field together with the two universal fields encoding the stable budget and K-theta floor conditions. No tactics or lemmas are applied; the form is the interface itself.
why it matters
This structure supplies the minimal data required by the admissible-gate theorem and the phase-budget construction in the same module. It enables the theorem that closes the Erdos-Straus residual class from the phase budget, using the Recognition Composition Law. It aligns with the J-uniqueness and eight-tick octave steps of the forcing chain by providing the budget and floor inputs needed to derive visibility.
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