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Explain the Lean def `xDiagonalCorrection` in module `IndisputableMonolith.Cost.Ndim.XCoordinates`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

Explanation of xDiagonalCorrection

(1) Plain English

The declaration defines a helper function for Hessian computations in a multi-component reciprocal cost. For dimension n, weight vector α, coordinate vector x, and indices i, j, it returns α_i / x_i² exactly when i = j and 0 otherwise. It supplies the diagonal correction term.

See xDiagonalCorrection.

(2) Why it matters in Recognition Science

Recognition Science extends the single-variable J-cost to vector cases via JcostN. The x-coordinate Hessian encodes curvature of this cost. The diagonal correction isolates the second-derivative contribution along each coordinate, enabling analysis of minima, degeneracy at the neutral locus (aggregate = 1), and non-degeneracy away from it.

(3) How to read the formal statement

  • noncomputable def: definition over reals, not required to be executable.
  • {n : ℕ}: implicit natural-number dimension.
  • (α x : Vec n): input vectors (Vec from Core).
  • (i j : Fin n): indices in {0, …, n-1}.
  • : ℝ: returns a real.
  • Body: if i = j then α i / (x i) ^ 2 else 0. This is the standard diagonal-matrix pattern with entries α_i / x_i².

(4) Visible dependencies or certificates

Imports IndisputableMonolith.Cost.Ndim.Core (supplies Vec, aggregate). Directly referenced by xHessianEntry and specialized in xHessianEntry_diag. Off-diagonal case is handled by xHessianEntry_offDiag. Determinant formulas such as det_xHessianMatrix2_formula and degeneracy result det_xHessianMatrix2_zero_cost build on the same structure. No sorry or external axioms appear in this module.

(5) What this declaration does not prove

It is a pure definition and asserts no properties. Non-degeneracy requires extra hypotheses (e.g., aggregate ≠ 1 and nonzero discriminant) proved separately in det_xHessianMatrix2_ne_zero_of_generic. No link is made here to the single-variable J-cost uniqueness, forcing chain, or physical constants.

cited recognition theorems

outside recognition

Aspects Recognition does not yet address:

  • Connection of this Hessian to the single functional equation for J-cost proved elsewhere
  • Any physical interpretation or empirical calibration of the multi-component cost
  • Dependencies on modules outside the supplied slice (e.g., Core definitions of Vec and aggregate)

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.