A State-Space Model of Figured-Bass Realization: Local Constraints, Coupled Voices, and Polynomial-Time Solvability

Constraint-following music generation asks a score to satisfy several user-specified properties at once, each checkable programmatically (key, meter, length, range, final note, rhythm, motion and form), yet no existing benchmark isolates this capability. We construct MusicConstraintBench, 2,180 items over eight constraint families, on which current models fail once a few constraints are combined. The natural remedy is reinforcement learning with these verifiers as reward, yet we observe that a reward paid only when every property holds leaves most training groups without a learning signal: over the first 50 updates, 0.550 of rollout groups score identically and receive no gradient, even though a failing score typically misses only one requested property. Under the joint criterion, rollouts for a prompt tend to fail together, so a binary reward cannot separate a nearly correct score from a malformed one. We therefore introduce MusicRLVR, which pays graded per-property credit behind a hard validation gate that rejects malformed outputs, plus a joint-satisfaction bonus, requiring no human annotation, learned reward model, or music-domain fine-tuning. On MusicConstraintBench, MusicRLVR lifts Qwen3-4B-Instruct from 0.160 to 0.807 on mixed constraints and leads every zero-shot baseline including Llama-3.1-70B at 0.380. It also generalises to property combinations unseen in training and to out-of-range parameter values, showing that verifiable rewards need not presuppose a target output.

Reconstructing a damaged musical fragment is an inverse problem: the observed sequence contains partial information, while a raga encodes constraints limiting allowable completions. This paper formalizes a mathematical framework for this, proposing the Artificial Rosetta Stone (ARS). We separate three claims often conflated: a symbolic sequence can be reconstructed probabilistically; a sequence can be consistent with an explicit grammar; and a historical performance can be authenticated. We only support the first two. We model a raga via a finite alphabet and constraint system, using an order-k Markov model for melodic probabilities. A symmetric Dirichlet prior yields a tractable posterior. We pose missing-note reconstruction as a constrained MAP problem. For fixed-length sequences and finite-order constraints, optimization admits an exact dynamic-programming solution with worst-case time complexity $O(TN^{k+1})$. We derive the parameter count $N^k(N - 1)$, prove a concentration bound under explicit mixing assumptions, and analyze estimation error propagation. A reproducible synthetic experiment uses six raga-inspired alphabets, orders $k \in \{1, 2, 3\}$, and masking rates up to 50%. This is a proof of concept, not historical reconstruction. A real-audio feasibility pilot evaluates 30 usable sequences from 42 Yaman clips via automated pitch extraction, segmentation, and quantization. Lacking documented provenance and relying on automated transcription, this is not expert-validated archival reconstruction. Claims are tied to stated conditions, not universal properties of Hindustani music. Code: https://github.com/mathacker23/ArtificialRosettaStone.

Large language models can produce superficially legal twelve-tone scores that collapse into degenerate textures. We introduce a neuro-symbolic harness that wraps a language-model proposer in a generate-verify-repair-trace loop with symbolic verification. The complete pipeline improves event-local consistency without claiming whole-piece legality. Across 40 controlled tasks and four paired models, constraint-checked delivery rises from 13.3% under raw generation to 48.1% with the harness; it abstains on the remaining 51.9% of runs. The pass rate of a narrower collision and serialisation-consistency check rises from 33.5% to 58.3%, while degeneracy remains near 0.05, including under adversarial prompting. A blinded evaluation by five experts also shows a descriptive aggregate preference for harness candidates over raw generation in adherence, perceived legality, coherence, and overall quality.