IndisputableMonolith.Cost.Ndim.RicciScalar
This module defines the Ricci scalar for the N-dimensional reciprocal cost in (q,r)-coordinates using sinh and cosh forms. Researchers extending the cost geometry to higher dimensions cite these expressions. The module supplies auxiliary definitions and equivalence lemmas that rewrite the scalar to match Eq. (4.26).
claimThe Ricci scalar $R(q,r)$ in $(q,r)$-coordinates, Eq. (4.26), expressed via $\sinh$ and $\cosh$.
background
The module imports IndisputableMonolith.Cost.Ndim.Core, whose doc states: 'This module defines the multi-component reciprocal cost by lifting the scalar kernel through a weighted log aggregate.' It introduces component expressions (ricciQ, ricciZexp, ricciW) and equivalence results for the Ricci scalar in the N-dimensional setting. The local context is the extension of the reciprocal cost kernel to N dimensions within the Recognition Science framework.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the explicit Ricci scalar form that supports higher-level N-dimensional cost calculations in the Recognition Science framework. It fills the geometric component referenced by Eq. (4.26) and connects to the core lifting construction from the imported Core module.
scope and limits
- Does not derive the Ricci scalar from an underlying metric tensor.
- Does not treat coordinate systems other than (q,r).
- Does not include numerical evaluation or approximation schemes.
- Does not connect to the one-dimensional scalar case.