IndisputableMonolith.Cosmology.BBNHeliiumExact3FromJCost
Module packages a BBN helium certificate that ties a domain cost built from the RS J-functional to a fixed canonical threshold, aiming at an exact factor-of-three relation in early-universe nucleosynthesis. Cosmologists working in the RS ladder would cite the certificate and the nonnegativity/positivity lemmas. The argument is definitional plus elementary inequalities on J, not a full BBN network solve.
claimDefine a domain cost $C$ from the RS cost $J$, a positive canonical threshold $\theta$, and a certificate asserting that the BBN helium exact-3 relation holds when $C$ meets $\theta$. The module also records $C\ge 0$, $\theta>0$, and inhabitation of the certificate type.
background
Recognition Science measures mismatch with the cost $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), forced unique by the T5 step of the unified forcing chain and obeying the Recognition Composition Law. Cosmology modules import the RS constants (including the native tick $\tau_0$) and the Cost API so that dimensionless ratios on the $\phi$-ladder can be scored by $J$.
Big Bang nucleosynthesis fixes light-element yields from expansion rate, baryon density, and neutron-proton freeze-out. In the RS reading, a domain cost aggregates $J$-penalties on the relevant scale ratios; a canonical threshold marks the recognition boundary at which an exact combinatorial factor (here three) is selected rather than a continuum fit.
This module sits in the Cosmology domain and only depends on Constants and Cost. It introduces the local objects domain cost, canonical threshold, and the BBN helium exact-3 certificate, together with elementary positivity facts needed by downstream cert consumers.
proof idea
Definition-heavy module. Domain cost is introduced as a $J$-derived functional on the BBN domain, with an evaluation identity and a nonnegativity proof inherited from $J\ge 0$ on the positive reals. Canonical threshold is a positive constant (positivity lemma separate). The certificate is a Prop-level bundle packing the exact-3 claim against that threshold; inhabitation is a one-line witness that the bundle type is nonempty once the inequalities are in place. No full reaction-network or Boltzmann integration appears here.
why it matters in Recognition Science
Gives the Cosmology layer a named, importable certificate that helium’s exact factor-of-three structure is scored by $J$-cost rather than by an external nuclear fit. Downstream used_by edges are empty in the current graph, so the module is a leaf cert provider: parent developments that assemble multi-element BBN or link yields to the $\phi$-ladder and eight-tick timing are expected to import cert and BBNHeliiumExact3Cert. It touches the RS program of replacing continuum cosmological parameters by discrete recognition thresholds forced from $J$, without yet closing the full abundance pipeline or the $\alpha$-band linkage.
scope and limits
- Does not solve the full BBN differential network or predict absolute mass fractions.
- Does not derive neutron lifetime, baryon-to-photon ratio, or expansion history from first principles.
- Does not prove uniqueness of the threshold beyond the local positivity lemmas.
- Does not connect helium yields to T8 spatial dimension or eight-tick timing inside this file.
- Does not supply observational error bars or likelihood comparisons to measured Y_p.