IndisputableMonolith.Cost.CauchyAuxiliary
The CauchyAuxiliary module defines the auxiliary map φ(t) = H(t) + √(H(t)² - 1) that converts d'Alembert solutions H into exponential form. Researchers in Recognition Science cost theory cite it when moving from hyperbolic to multiplicative structures. The module supplies the explicit construction together with elementary identities such as φ(0) = 1 and H_from_phi.
claim$φ(t) := H(t) + √(H(t)^2 - 1)$ for solutions $H$ of the d'Alembert equation $H(t+u) + H(t-u) = 2 H(t) H(u)$ with $H(0)=1$ and $H(t) ≥ 1$.
background
The module sits inside the Cost domain and imports the Aczél theorem, whose doc-comment states: 'Every continuous solution of H(t+u) + H(t-u) = 2·H(t)·H(u) with H(0) = 1 is C^∞. The complete Aczél classification (1966, Ch. 3): 1. H(t) = 1 (trivially C^∞) 2. H(t) = cosh(λt) (C^∞)'. It introduces the auxiliary φ together with sibling definitions phi_at_zero, H_from_phi, H_PhiMultiplicative and H_CauchyToExponential that translate between the additive and multiplicative presentations of the same solution.
proof idea
This is a definition module, no proofs. It supplies the explicit algebraic construction of φ from H and records the immediate consequences φ(0)=1 and the recovery map H_from_phi.
why it matters in Recognition Science
The module supplies the φ function that realizes the self-similar fixed point required by the Recognition framework (T5 J-uniqueness and T6 phi forced). It feeds the multiplicative structure used in the Recognition Composition Law and the phi-ladder mass formula, providing the concrete bridge from the Aczél-classified H to the exponential yardstick.
scope and limits
- Does not prove the Aczél smoothness theorem.
- Does not classify solutions when H(t) < 1.
- Does not fix the numerical value of λ from physical constants.
- Does not address discrete or non-continuous solutions.