IndisputableMonolith.Foundation.RelationalQM3FromJCost
Foundation module that packages domain cost, a positive canonical threshold, and a certificate linking the unique J-cost to three-dimensional relational quantum mechanics. Foundation workers tracing the forcing chain from J-uniqueness to D=3 would open it. The file is mostly definitions plus nonnegativity and inhabitation lemmas, not a single deep existence proof.
claimFrom the unique cost $J(x)=(x+x^{-1})/2-1$, the module defines a domain cost, a strictly positive canonical threshold, and a certificate that relational quantum mechanics in three spatial dimensions is forced by that cost structure (with supporting nonnegativity and inhabitation facts).
background
Recognition Science forces a unique nonnegative cost on positive reals, $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), via the Recognition Composition Law and the T5 uniqueness step. The Cost import supplies that functional; Constants supplies the RS-native tick $\tau_0=1$.
This module sits in the Foundation layer and turns that scalar cost into a domain-level cost and a numerical threshold used as a gate for relational structure. Sibling names indicate equalities at evaluation points, nonnegativity of the domain cost, and positivity of the canonical threshold.
The intended physical reading is the passage toward relational quantum mechanics in $D=3$ (T8), with the eight-tick octave (T7) and $\phi$-fixed point (T6) as ambient landmarks rather than objects proved here.
proof idea
Definition-and-certificate module, not a monolithic theorem. It introduces a domain cost built from $J$, records evaluation identities and nonnegativity, defines a canonical threshold and proves it is positive, then packages a Relational QM-in-3D certificate type with an inhabitation witness. Argument structure is: define cost and threshold, discharge elementary sign lemmas, inhabit the certificate record. No deep tactic script is implied by the sibling list.
why it matters in Recognition Science
Closes a Foundation packaging step between raw J-cost and a citable relational-QM-in-three-dimensions certificate. Downstream edges are empty in the graph snapshot, so the module is a leaf exporter rather than an intermediate lemma factory. It is the natural citation point when arguing that T5 J-uniqueness plus the cost calculus already supply the quantitative gate for the T8 claim $D=3$ in a relational reading of QM. Open work is whatever still sits behind the certificate fields themselves if any remain axiomatic outside this file.
scope and limits
- Does not prove J-uniqueness; that is upstream T5.
- Does not derive the eight-tick octave or $\phi$ fixed point here.
- Does not construct full Hilbert-space QM or measurement postulates.
- Does not numerically fix $\alpha$, masses, or $G$; only cost-threshold structure.
- Does not claim downstream consumers; used_by is empty in this snapshot.