Births outside the light cone: entropy deficit in a growing quantum register

We study a quantum register whose Hilbert space grows: fresh qudits are born in a pure state, uncorrelated with the rest, and the register is scrambled by local dynamics. The observable is the deficit G(A) = log q · min(|A|, |Ā|) − S(A) between the maximal and the actual entropy of a region. A velocity criterion in the spirit of Frautschi (relaxation versus expansion) predicts that the deficit closes when scrambling is fast enough; simulations refute it. We show that the controlling variable is causal, not kinetic. (i) For every unitary circuit, on any graph and with arbitrary births, S(A) ≤ log q · |Ā ∩ Inf(A)|, where Inf(A) collects the qudits whose world line meets the causal past of A (Theorem A); a Lieb–Robinson version holds for local Hamiltonians (Theorem B). (ii) For an isotropic front R ∝ t^α, the deficit is extensive for every local dynamics if α ≥ 1, and can be made sub-extensive if α < 1 (Theorem C); in one dimension the causal bound is attained. (iii) For gates drawn from a 2-design, the averaged purity of a prefix reduces exactly to the walk of a single domain wall absorbed by the line of births. This yields E[G] = Θ(m^(2−1/α)) for 1/2 < α < 1 (Theorem D) and, for one birth every k layers, the exact annealed Rényi-2 density θ_k with (2q/(q²+1)) cosh θ_k = exp(−θ_k/k) (Theorem E). (iv) Stabilizer simulations up to 6400 qubits and Haar state-vector simulations up to 22 qubits agree with these statements; predictions were written before the computations, and the ones that failed are reported. The deficit is robust to the choice of birth state and to local correlations with the past. In a free lattice field with a vacuum, however, uncorrelated births cost an energy of the order of the cutoff, so the cosmological reading of these results remains a constrained analogy. AI-use statement: this work was produced by an AI system (Claude, Anthropic) in an extended interactive session directed by the author, who posed the research questions and made editorial decisions. The derivations, proofs, Coq formalisation, code, numerical experiments and text were generated by the AI system. The author has not independently verified the mathematical content, and the work has so far been checked only by automated reviewers. Please report errors through the repository's issue tracker. Code, data, Coq proofs and LaTeX source: https://github.com/GofKingnose/births-outside-the-light-cone (release v1.0.0).