ultramassiveBHCert
plain-language theorem explainer
ultramassiveBHCert constructs the verified certificate for ultramassive black holes by bundling non-negativity of J-cost, positivity of entropy and temperature, monotonicity of temperature with mass, and the Hamiltonian approximation bound. Researchers modeling black hole thermodynamics within Recognition Science would reference this to establish the formal status of results for objects like TON 618. The definition proceeds by direct assignment of the supporting lemmas to each field of the UltramassiveBHCert structure.
Claim. The ultramassive black hole certificate is the structure containing: for all $x > 0$, $J(x) = (x + x^{-1})/2 - 1$ satisfies $J(x) > 0$ except at $x=1$; for every RS black hole, entropy and Hawking temperature are positive; temperature decreases as mass increases; and for strain $|ε| ≤ 1/2$, $J(1+ε) = ε²/2 + c ε³$ with $|c| ≤ 2$.
background
In the Recognition Science treatment of ultramassive black holes (M ≳ 10¹⁰ M☉), the J-cost function J(x) = (x + x^{-1})/2 - 1 quantifies recognition cost for positive real configurations, with non-negativity following from the AM-GM inequality. The module formalizes four key results: no singularity (J-cost finite on (0,∞)), RS entropy S_BH = (ln φ) · A/(4ℓ₀²) with k_R = ln φ, RS Hawking temperature T_H = 1/(8π M) in native units, and the Hamiltonian approximation bound that Ĥ emerges from R̂ only for |ε| ≪ 1.
proof idea
The definition is a direct record construction for the UltramassiveBHCert structure. It assigns the no_singularity field to the lambda applying the Jcost_nonneg lemma, entropy_positive to rs_entropy_pos, temp_positive to rs_hawkingTemp_pos, temp_monotone to temp_decreases_with_mass, and hamiltonian_approx to hamiltonian_approximation_bound.
why it matters
This definition supplies the top-level certificate that verifies the key results for ultramassive black holes in Recognition Science. It incorporates the no singularity theorem from the EarlyUniverse module (initial state has zero total defect) and the local Hamiltonian approximation bound. The certificate realizes the four enumerated results in the module documentation, including the cold temperature limit T_H → 0 as M → ∞, and closes the formalization for this domain.
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