IndisputableMonolith.Foundation.MultiChannelJCost
The module defines multi-channel J-cost as the sum of per-channel J-costs to aggregate costs across independent substrates. Researchers validating Recognition Science predictions in language models, photonic systems, and plasma would cite it for shared J-cost behavior. It supplies basic properties such as non-negativity and symmetry through direct definitions and short lemmas.
claimThe multi-channel J-cost is $J_n(x_1, x_2, ..., x_n) = J(x_1) + J(x_2) + ... + J(x_n)$, where each $J(x_i)$ satisfies the single-channel functional equation from the Cost module.
background
J-cost is introduced in the upstream Cost module via the Recognition Composition Law: $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$. The present module extends this to multiple channels by defining Jcost_n and its variants (Jcost_n_nonneg, Jcost_n_symm, Jcost_n_zero_iff). These support simultaneous application across distinct physical or computational substrates without altering the underlying J function.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the aggregate cost object required by the downstream ThreeSubstrateValidationCert, which records RS Exp B results across language models (96.4% of MLP layers), photonic qubits (code rate 7/8), and magnetized plasma (convergence to x = 1.036). It thereby enables the claim that all three substrates obey the same J-cost predictions.
scope and limits
- Does not derive the explicit form of J from first principles.
- Does not specify numerical channel counts or substrate mappings.
- Does not include experimental data or convergence proofs.
- Does not address interactions between channels.