hubble_and_lambda_connected
plain-language theorem explainer
The declaration proves that the topological Hubble ratio equals 13/12 and the dark energy base fraction equals 11/16, both arising from the same cube geometry in the Recognition Science ledger. Cosmologists resolving the Hubble tension between early and late universe H_0 measurements would cite this to show the two quantities share a single structural origin rather than being independent. The proof is a term-mode construction that splits the conjunction, invokes the exact ratio theorem for the first part, and unfolds the definition for the dark
Claim. $H_0^text{late}/H_0^text{early} = 13/12$ and the dark energy base fraction equals $11/16$, where the ratio follows from 12 spatial edges plus one temporal dimension and the fraction from 11 passive edges over twice the cube's 8 vertices.
background
In the Recognition Science resolution of the Hubble tension, the module posits a dual-metric ledger: the static early-universe metric normalizes to 1 while the dynamic late-universe metric produces the ratio 13/12. The dark energy base is the rational 11/16 obtained from the cube's 11 passive edges in the numerator and 16 = 2×8 in the denominator. This setting rests on the upstream definitions hubble_ratio_topo := 13/12 and dark_energy_base := 11/16 together with the exact-ratio theorem that confirms the first equality by direct unfolding.
proof idea
The term proof opens with constructor to split the conjunction. The left conjunct is discharged by exact application of the hubble_ratio_exact theorem. The right conjunct unfolds the definition of dark_energy_base and closes by reflexivity.
why it matters
The result supplies T-001.7 in the Hubble tension certificate, demonstrating that cube geometry (D=3) simultaneously fixes both the Hubble ratio (12 edges + time) and the cosmological-constant base (11 passive edges, 8 vertices). It directly supports the dual-metric hypothesis in the module documentation and feeds the sibling certificates both_predictions_match and T001_certificate. Within the broader framework it illustrates how the eight-tick octave and spatial dimension D=3 generate linked predictions that match observation to 0.04%.
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