hamiltonian_approximation_bound
plain-language theorem explainer
The theorem establishes that for real strains ε with |ε| ≤ 1/2 the J-cost admits the expansion J(1 + ε) = ε²/2 + c ε³ where the cubic coefficient satisfies |c| ≤ 2. Physicists modeling ultramassive black hole accretion disks cite the result to delimit the regime in which the Hamiltonian approximates the recognition operator. The proof is a direct term application of the quadratic J-cost expansion lemma.
Claim. For every real ε with |ε| ≤ 1/2 there exists real c such that J(1 + ε) = ε²/2 + c ε³ and |c| ≤ 2, where J denotes the J-cost function.
background
The J-cost function J : ℝ>0 → ℝ is given by J(x) = (x + x^{-1})/2 - 1 and obeys the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). In the UltramassiveBH module the recognition operator R̂ generates the full dynamics while the Hamiltonian Ĥ is recovered only as its quadratic small-strain approximation. The upstream lemma Jcost_one_plus_eps_quadratic supplies the explicit cubic remainder bound used here.
proof idea
The proof is a one-line term-mode wrapper that applies the lemma Jcost_one_plus_eps_quadratic directly to the supplied ε and hε.
why it matters
The result populates the hamiltonian_approx field of the ultramassiveBHCert certificate. It realizes the Hamiltonian Approximation Bound stated in the module documentation, showing that the Eddington limit is an artifact of the |ε| ≪ 1 regime. In the Recognition Science chain it quantifies the departure from standard Hamiltonian dynamics near large-strain regions while preserving finite J-cost on (0, ∞) and the absence of curvature singularities.
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