Explanation of additive_decomposition
(1) In plain English, the declaration states that for vectors α and ε in any dimension n, the additive quadratic approximation of ε equals the multiplicative quadratic approximation (weighted by α) plus the residual compensatory quadratic term. This is a direct algebraic decomposition.
(2) In Recognition Science this identity bridges additive and multiplicative quadratic costs within the N-dimensional cost module, supporting decomposition of cost structures that appear in recognition and ledger mechanisms.
(3) The formal statement reads:
theorem additive_decomposition {n : ℕ} (α ε : Vec n) :
additiveQuadratic ε
= multiplicativeQuadratic α ε + compensatoryQuadratic α ε := by
unfold compensatoryQuadratic
ring
It asserts equality of three locally defined functions; the proof simply unfolds the compensatory definition and applies the ring tactic.
(4) Visible dependencies are the three definitions additiveQuadratic, multiplicativeQuadratic and compensatoryQuadratic inside the same module. The proof contains no sorry, no axioms, and no external lemmas beyond the ring tactic. Related declarations present in the supplied source are dot_sq_le_sqNorm_mul, multiplicative_le_additive_of_sqNorm_le_one and compensatory_nonneg_of_sqNorm_le_one.
(5) The declaration does not prove any inequalities, non-negativity results, bounds involving the norm of α, or any connection to J-cost, φ, or the forcing chain. It also does not address the definition of Vec or any content from the imported Core module.