IndisputableMonolith.Cosmology.HubbleConstantPrecise2FromJCost
Packages a J-cost-based certificate for a second precise form of the Hubble constant in RS-native units. Cosmologists tracing H0 back to the Recognition cost would cite the inhabited cert and the domain-cost lemmas. The module is mostly definitional: it builds a nonnegative domain cost, a positive canonical threshold, and a certificate record with an inhabitation proof.
claimFrom the Recognition cost $J$, the module defines a nonnegative domain cost, a strictly positive canonical threshold, and an inhabited certificate that a precise Hubble constant (second form) is fixed by that threshold in RS-native units ($c=1$, tick $\tau_0=1$).
background
Recognition Science measures mismatch with the J-cost $J(x)=(x+x^{-1})/2-1$, forced uniquely by the Recognition Composition Law and the T5 step of the unified forcing chain. Cosmology in this stack is written in RS-native units where $c=1$ and the fundamental time quantum is one tick ($\tau_0=1$), imported from the Constants module.
This module sits in the Cosmology domain and treats the Hubble scale as a threshold phenomenon on a domain cost built from $J$. Sibling definitions introduce that domain cost, prove it is nonnegative, fix a canonical threshold and its positivity, then package the claim as a certificate type with an inhabitation witness.
Upstream material is thin: only Constants (tick convention) and Cost (J-cost infrastructure). No external Hubble data enter; the precise form is an internal RS relation.
proof idea
Definition-and-certificate module rather than a deep derivation. Domain cost is defined from $J$; equality-at-point and nonnegativity are recorded as short lemmas. The canonical threshold is defined and shown positive. The main object is a certificate structure for the second precise Hubble form, together with an inhabitation proof that assembles the threshold facts. No long tactic script; the argument is packaging and positivity bookkeeping.
why it matters in Recognition Science
Gives Cosmology a named, inhabitable certificate that a precise Hubble constant (variant 2) is pinned by J-cost structure rather than by external fit. Downstream use is not yet wired in this graph (no used_by edges), so the module is a leaf certificate package for later H0 or expansion-history theorems.
It ties the Hubble scale to the same cost that forces $\phi$, the eight-tick octave, and $D=3$ in the T5–T8 chain, keeping cosmological constants inside the single-functional-equation program. Until parent theorems consume the cert, its role is to freeze the precise-2 claim in a form Lean can reuse.
scope and limits
- Does not derive numerical H0 in SI units or fit observational supernova data.
- Does not prove uniqueness of this precise form among all possible Hubble normalizations.
- Does not connect the certificate to FLRW dynamics or dark-energy equations of state.
- Does not discharge broader cosmology forcing beyond the packaged threshold claim.
- Does not depend on measured alpha or mass-ladder rungs.