largeScaleStructureCert
plain-language theorem explainer
This definition constructs the large-scale structure certificate in Recognition Science cosmology by supplying the exact count of five regimes, the golden-ratio scaling between consecutive comoving scales, and strict positivity of all scales. Cosmologists modeling structure formation within the RS framework would cite it to confirm that the five canonical regimes obey the phi-ladder. The construction is a direct record instantiation that assembles three prior component results.
Claim. A large-scale structure certificate consists of the record whose fields assert that the number of regimes is exactly five, that consecutive scales satisfy scale(k+1)/scale(k) = phi for every natural number k, and that scale(k) > 0 for every natural number k.
background
The module identifies five canonical large-scale structure regimes corresponding to configDim D = 5: the CMB acoustic scale, baryon acoustic oscillation, galaxy clusters, filamentary structure, and cosmic voids. Each regime sits one rung higher on the phi-ladder in comoving length, consistent with the self-similar scaling forced by the Recognition functional equation. The referenced structure LargeScaleStructureCert packages three required properties: the cardinality of the regime type equals five, the ratio of successive scales equals phi, and every scale is positive. The upstream theorem scale_ratio proves the ratio identity by unfolding the scale definition and applying the ring tactic after establishing positivity; scale_pos follows from iterated application of pow_pos.
proof idea
The definition is a one-line record constructor that directly assigns the five_regimes field to the decide-based cardinality theorem, the phi_ratio field to the scale-ratio theorem, and the scale_always_pos field to the positivity theorem. No additional tactics or reductions are performed; the body simply supplies the three component results already established in the same module.
why it matters
This definition supplies the concrete certificate that ties the five LSS regimes to the phi-ladder scaling required by the Recognition framework, specifically instantiating the self-similar fixed point at T6. It completes the local cosmology module whose module documentation states that Lean status is zero sorry and zero axiom. Although no downstream uses are recorded yet, the certificate stands ready to feed larger derivations that connect the eight-tick octave and spatial dimension D = 3 to observable cosmic structure.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.