The Cost of Conservation: Coordination-Memory Laws for Exact-Support Generation

A parallel sampler can choose at random which coordinates to generate together. For a target $p$ and the output law $q_U$ of a schedule $U$ drawn afresh for each output, we compare the schedule-averaged divergence $\mathbb{E}_U D_{\mathrm{KL}}(p\|q_U)$ with the output-law divergence $D_{\mathrm{KL}}(p\|\mathbb{E}_U q_U)$, in nats. Rounds commit independent draws from exact conditional marginals frozen at the round's start, each output's schedule is drawn before sampling from a finite mixture of schedules, and every scheduled round is charged. For the uniform distribution on binary strings of even length $n$ with $n/2$ ones, the least expected number of rounds at a fixed tolerance $\varepsilon>0$ is $Θ_\varepsilon((\log n)^2)$ under the first criterion but $(e^{-\varepsilon}+o(1))\log_2\ln n$ under the second, and the double-logarithmic order is attainable with the same round count on every run. For this target, a count-state recursion gives the exact least output divergence at every round cap over all causal randomized policies. The separation extends to weighted fixed-count targets with bounded weight ratios and quota fractions bounded away from zero and one, without permutation symmetry, and to the full-support Beta$(1,1)$--Bernoulli target, where the sharp expected cost under the output-law criterion is $(e^{-\varepsilon}/2+o(1))\log_2\log_2n$, attained with rare $n$-round runs. Prior work implies the order of the schedule-averaged cost on the balanced target and its $O_\varepsilon((\log n)^2)$ upper bounds on both extensions.

Finding the shortest program that generates a sequence is uncomputable, and for six decades that fact has been mistaken for a wall around finding any generating program. It is not a wall but a price, and this paper measures it. For every algorithm that learns about a candidate program only through its score, a class spanning Levin search, evolutionary methods, simulated annealing, and the cross-entropy method, we define the coupling width of a search problem and prove an unconditional worst-case lower bound, exponential in that width with base one less than the domain size. From it follows a conservation law: structural knowledge injected into a search trades one for one against the search it removes, and their sum can never fall below the length of the program sought. Levin's 1973 upper bound and the lower bound proved here are the two ends of one conserved quantity, closing on each other as the instruction set grows. The only escape is to read a candidate's structure rather than its score, and its price, which we prove for generic targets, is incompleteness. A deterministic engine built on this theory recovers a generating program, certified by compressing its data and predicting an unseen continuation, for 2,383 of 3,914 sequences across four independent populations, including 244 of the 256 elementary cellular automata, with measured discovery cost rising along program length more than an order of magnitude inside the score-oracle worst case.

We prove inference-time quantum coordination advantages for specified AI state-tracking tasks. A solver compresses semantic history into a future-accessible boundary state and later answers a query. We count communication $B$, persistent instance-dependent memory $M$, and local work $D$; classical recurrence, caches, tools, and recomputation are allowed and charged. The central result is a boundary-preserving semantic-compilation theorem. It maps a finite one-way, streaming, or adaptive causal task into a semantic AI interface while preserving event order and access to past input. Classical boundary-state lower bounds and quantum-memory upper bounds transfer up to explicit compiler overhead, independently of the finite-precision recurrent architecture. Two applications have classical semantics. Matched-entity synopsis QA inherits the hidden-matching separation between $O(\log N)$ qubits and $Ω(\sqrt{N})$ classical boundary bits. Continual requirements auditing inherits a Max-$k$SAT streaming separation: a recurrent solver uses $O(\log^5 n\log(1/δ))$ qubits and polylogarithmic classical workspace to obtain a $0.7172$-approximation, whereas every classical one-pass finite-information solver attaining that ratio requires $Ω(\sqrt{n})$ coordination width. As a quantum-native compiler test, a stabilizer latent-state dialogue uses $n$ qubits, while every exact finite-state classical causal online realization satisfies $B+M \ge \frac{1}{2}n^2+(\frac{3}{2}-\log_2 3)n+O(1)$. The source protocols, streaming algorithms, and stabilizer witness are imported; the new result is their architecture-independent semantic transfer. These are memory and coordination separations, not runtime or empirical advantages for present-day language models. The stabilizer result assumes exact simulation and ideal noiseless quantum memory.