Can Finite Experiments Certify Actual Infinity? Dimension Witnesses, Tsirelson's Problem, and Holographic Resource Bounds

Whether physical reality contains an actual infinity is usually discussed philosophically. We ask a narrower, quantitative question: what would it cost a finite experiment to certify it? We combine sharp results from quantum information with physical resource bounds. (1) Certification budget. Resolving a game value to ε requires N ≳ ln(1/4δ) w(1−w)/ε² rounds (Bretagnolle–Huber), so the cost of certifying a Hilbert-space dimension D is governed by the gap ε_G(D) between the optimal value and the best dimension-D value. Take the unspecified constants in the known bounds to be one. Then the best known two-player dimension witness needs at least 10⁸⁶ rounds to certify a dimension beyond the Bekenstein bound of a 1 kg, 1 m laboratory (2.6×10⁴³ qubits). Such a laboratory can perform only 2.4×10⁶⁸ operations in the age of the universe. Beyond the de Sitter entropy (4.8×10¹²² qubits), the witness needs at least 10²⁴⁵ rounds, against the 10¹²⁰ operations of the universe. Spatial parallelisation does not help, because the Margolus–Levitin bound limits the total rate summed over all subsystems that share the energy. For the I₃₃₂₂ inequality the requirement is doubly exponential. (2) Embezzlement asymptotics. The optimal fidelity of van Dam–Hayden embezzlement of one EPR pair is F_n = (Hn/2 + σn/2)/H_n. Hence 1 − F_n = (ln 2 − σ)/H_n + O(1/(n ln n)) with σ = 0.2992041. This holds for all n. It is confirmed to 10⁻⁶ for dimensions up to 2²⁶, and to 6×10⁻⁸ for odd n = 10⁷ + 1. The infinite-dimensional resource behind these witnesses is approached only logarithmically. (3) The Tsirelson loophole. MIP*=RE implies that commuting-operator correlations can exceed every finite-dimensional value by a finite gap, which finite statistics could in principle detect. However, the set of true bounds ω*(G) ≤ t is Π⁰₁ but not computably enumerable. No consistent theory therefore proves all of them, and this route is limited by provability rather than by statistics. In practice, no separating game with small question and answer sets and a computed gap is known. (4) Thermodynamic infinities. Kaufman's exact formula for the two-dimensional Ising model on a finite torus shows that a 1 cm sample rounds the critical singularity at |K − K_c|/K_c ~ 4×10⁻⁹. The logarithmic amplitude of the specific-heat peak equals that of the bulk singularity. The orbital helium lambda-point experiment, which reached |t| ≈ 10⁻⁹, stays about four orders of magnitude above the finite-size scale of its sample. These results do not show that nature is finite. They show the following: apart from one route whose cost is logical rather than statistical, finiteness bounded by holography cannot be refuted from inside a universe that respects that bound. Code, data and figures: https://github.com/Ruqing1963/certify-infinity