IndisputableMonolith.Cost.Ndim.Calibration
The Calibration module supplies lemmas establishing uniform weight properties for the N-dimensional reciprocal cost. It proves that uniform weights summing to one equal 1/n for n>0. Researchers simplifying multi-component costs under isotropy cite these results to reduce the weighted log aggregate from Core. The arguments consist of direct summation and norm verifications on the listed sibling definitions.
claimIf weights $w_1,…,w_n$ ($n>0$) are uniform and satisfy $∑_{i=1}^n w_i=1$, then $w_i=1/n$ for each $i$.
background
The upstream Core module defines the N-dimensional reciprocal cost by lifting the scalar kernel through a weighted log aggregate. This calibration module introduces auxiliary objects weightSum, sqNorm and the UniformWeights predicate, then records their values under uniformity via lemmas such as weightSum_uniform and uniform_weight_of_sum_one.
proof idea
This is a definition module with supporting lemmas; it contains direct algebraic verifications of sum and norm identities under the uniform predicate, using only basic arithmetic from Core.
why it matters in Recognition Science
Supplies the uniform calibration identities required by the downstream Symmetry module on permutation symmetry of coefficient weights. It fills the isotropic case needed before symmetry arguments are applied to the N-dimensional cost model.
scope and limits
- Does not derive the general weighted cost expression.
- Does not treat non-uniform or arbitrary weight vectors.
- Does not address permutation symmetry itself.
- Does not connect to the J-function or phi-ladder.