modelTransitionFrequency
plain-language theorem explainer
Converts a horizon area gap ΔA at surface gravity κ into the transition frequency ω = κ·ΔA/(8π·ℓ_P²), the first-law map of black-hole thermodynamics in ℏ = c = 1 units (where ℓ_P² = G). Cited by anyone auditing the φ-horizon absorption-comb preflight as the explicit MODEL conversion step. Pure algebraic definition: no proof content, hypothesis status kept visible by construction.
Claim. The model transition frequency for surface gravity $\kappa$, Planck area $\ell_P^2$, and area gap $\Delta A$ is $\omega(\kappa,\ell_P^2,\Delta A) = \kappa\,\Delta A/(8\pi\,\ell_P^2)$. This is the first-law conversion $\Delta M = \kappa\,\Delta A/(8\pi G)$ with $\omega = \Delta M$ and $\ell_P^2 = G$ in $\hbar = c = 1$ units.
background
Module setting is the Seven Gaps Pillar 3 fallback: a falsifier-gated preflight of a MODEL φ-horizon absorption comb. Nothing in the file is a prediction. The candidate mechanism posits area quantization with gap $\Delta A = 4\ln\varphi,\ell_P^2$, then converts that gap via black-hole thermodynamics into a repeated absorption comb (headline Schwarzschild value $GM\omega_* = \ln\varphi/(8\pi)\approx 0.019147$).
The conversion itself is classical first-law algebra. From $\Delta M = \kappa,\Delta A/(8\pi G)$ and the identification $\omega = \Delta M$, together with $\ell_P^2 = G$ in $\hbar = c = 1$ units, one obtains $\omega = \kappa,\Delta A/(8\pi,\ell_P^2)$. Existing RS capital does not force the quantization: sealed horizon area is the continuous real $A = 4\pi R_s^2$, ledger capacity is a real bound $A/\ell_0^2$, and boundaryCost on a recognition ledger is real-valued with no spectrum.
This definition is deliberately not a theorem: stating the map as a def keeps the MODEL status of the area-gap input explicit before any algebraic consequence is derived.
proof idea
Definitional one-liner. The body is the explicit formula $\kappa\cdot\Delta A/(8\pi,\ell_P^2)$; no lemmas, rewrites, or tactics. Downstream theorems simply unfold this def and cancel factors.
why it matters
Load-bearing conversion step for the absorption-comb preflight. Two immediate parents use it: (1) feeding the MODEL gap $\Delta A = 4\ln\varphi,\ell_P^2$ through this map yields exactly the Kerr comb offset $\kappa\ln\varphi/(2\pi)$; (2) specializing to Schwarzschild $\kappa = 1/(4GM)$ recovers the headline observable $GM\cdot\omega_* = \ln\varphi/(8\pi)$ from the inserted MODEL hypotheses alone.
Pillar 3 stays OPEN. The kinematic algebra is real; the area quantization is not forced by RS capital, and the P1 scaling falsifier already blocks a uniform ledger gap under continuous horizon-area scaling. This is categorically an absorption/level-structure claim, not a revival of the dead $\varphi$-rung echo-train route (damping $1/\varphi\approx 0.618$), which remains killed against O3/O4 bounds.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.