IndisputableMonolith.Foundation.RecognitionHilbertSpace3
Foundation module packaging the three-dimensional recognition Hilbert-space layer: a domain cost functional, a canonical positive threshold, and an inhabitation certificate bundle. Researchers tying the D=3 forcing step to Hilbert-space structure would land here. Content is mostly definitions plus elementary nonnegativity and positivity lemmas, not a deep existence proof.
claimIn the Recognition foundation layer one introduces a domain cost functional $C$ on the three-dimensional recognition Hilbert space, a canonical threshold $\theta>0$, and a certificate type asserting that the three-dimensional recognition Hilbert-space data are inhabited and meet the cost/threshold constraints.
background
Recognition Science forces spatial dimension $D=3$ at step T8 of the unified forcing chain. This module sits in the Foundation domain and supplies the Hilbert-space-side bookkeeping that matches that dimension count.
It imports the RS constants layer (including the native tick $\tau_0=1$) and the Cost module that carries the $J$-cost calculus ($J(x)=(x+x^{-1})/2-1$ and the Recognition Composition Law). Sibling names indicate a domain cost map, evaluation and nonnegativity facts for that cost, a canonical threshold with a positivity lemma, and a certificate record RecogHilbert3Cert with an inhabitation witness.
No separate module doc-comment is supplied; the theoretical setting is the post-forcing Hilbert packaging of three-dimensional recognition geometry rather than a new derivation of $D=3$ itself.
proof idea
This is primarily a definition and certificate module, not a long derivation. It introduces the domain cost and canonical threshold, then records elementary analytic facts (cost nonnegativity, threshold positivity) and packages them into a certificate type with an inhabitation instance. Expect short lemma proofs by unfolding definitions and applying Cost/Constants facts, not a multi-step forcing argument.
why it matters in Recognition Science
The module earns its place as the Foundation-side Hilbert packaging for the forced three-dimensional geometry (primer landmark T8). Downstream used_by edges are empty in the supplied graph, so it currently acts as a leaf certificate bundle rather than a widely cited lemma source.
Anyone assembling a full Recognition Hilbert-space story after the forcing chain would import this for the domain-cost and threshold interface and for the inhabitation certificate that marks the three-dimensional data as live. It does not itself re-prove $D=3$; it assumes that dimensional outcome and organizes the matching Hilbert-space cost structure.
scope and limits
- Does not re-prove spatial dimension $D=3$; that is upstream forcing (T8).
- Does not derive the $J$-cost or RCL; those live in Cost.
- Does not supply spectral theorems or operator algebras beyond the named cost/threshold interface.
- Does not claim downstream physics applications; used_by is empty here.
- Does not fix numerical mass or alpha values; only Hilbert-domain cost scaffolding.