nothing_costs_arbitrarily_large
plain-language theorem explainer
The theorem shows that for any positive real C the recognition cost J(x) exceeds C for all sufficiently small positive x. Black-hole modelers in Recognition Science cite it to confirm that J-cost diverges as the scale parameter approaches zero from above, furnishing a concrete witness that the interior is a maximal-cost state rather than a curvature singularity. The proof constructs an explicit delta equal to 1 over (2C + 3) and reduces the claim to the lower-bound lemma on Jcost by a short chain of multiplications, inverses, and linear-arithm
Claim. For every real number $C > 0$ there exists a real number $δ > 0$ such that $J(x) > C$ for all $x$ satisfying $0 < x < δ$, where $J$ denotes the recognition cost function.
background
Recognition Science quantifies recognition effort by the J-cost function on positive reals. The UltramassiveBH module records that this cost remains finite everywhere on $(0,∞)$ yet diverges as the scale parameter tends to zero, thereby characterizing the black-hole interior as a maximal J-cost configuration rather than a curvature singularity. The local setting draws on the phi-forcing structure and ledger factorization to calibrate J itself.
proof idea
The proof is a direct construction. It sets δ := 1/(2C + 3), confirms positivity, assumes 0 < x < δ, invokes the lower-bound lemma on Jcost, proves 2C + 3 < 1/x by a short multiplication-and-cancellation chain, and finishes with linear arithmetic.
why it matters
The result supplies the finite witness for the meta-principle that recognition cost becomes unbounded as the scale parameter approaches zero. It directly supports the no-singularity statement in the module documentation and sits inside the forcing chain that derives the recognition composition law and the eight-tick octave. No open scaffolding remains for this claim.
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