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Artificial Rosetta Stone: Constrained Maximum A Posteriori (MAP) Reconstruction of Symbolic Raga Sequences via Order-k Markov Models

Saanvi Raghavendran, Abhishek Bhattacharjee

arXiv:2609.01064v1cs.SDcs.LGstat.ML

TL;DR

The paper asks how to reconstruct missing notes from partial musical sequences without conflating statistical plausibility with explicit grammatical validity. It formulates a symbolic constrained MAP model and proves exact dynamic-programming optimization under finite-memory assumptions. The real-audio pilot reports missing-symbol accuracy for Yaman sequences, but remains limited by automated transcription and corpus scope.

  • Problem

    Reconstructing a damaged musical fragment requires inferring plausible structure from partial observations while separating probabilistic modeling from explicit grammar constraints.

    Source: §1.1 · p. 5, §2.5 · p. 6

  • Method

    The paper models symbolic melodies with an order-k Markov probability model and explicit constraints, then uses dynamic programming for constrained MAP reconstruction.

    Source: §14 · p. 29, §15 · p. 30

  • Results

    Order two achieved missing-symbol accuracy of 0.470, 0.414, and 0.371 at 10%, 30%, and 50% missingness, outperforming orders one and three in each tested condition.

    Source: §15 · p. 30

  • Takeaways & Limitations

    For fixed-length sequences with finite-memory constraints, dynamic programming exactly optimizes the constrained MAP objective with worst-case time complexity O(TN^(k+1)).

    Source: §15 · p. 30

    The real-audio evaluation uses automated symbolic transcription, a small Yaman-only corpus with incomplete provenance, and no expert-validated reference.

    Source: §9.3 · p. 16, §9.12 · p. 19

Abstract

from arXiv · show

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.

1 Introduction

The paper frames fragment reconstruction as a model-dependent inference task, then specifies a symbolic probability model and explicit constraints so its guarantees have clear scope. Its contributions span finite-state reconstruction, statistical estimation, exact decoding, and carefully bounded extensions.

1.1 Background and motivation

  • A raga is not just an unordered pitch scale, and a reconstruction is a model-dependent estimate rather than unique historical truth.The symbolic representation enables finite-state results but omits continuous pitch, timing, dynamics, timbre, and gamaka contours.Source: §1.1 · p. 5

1.2 Research questions and contributions

  • The paper defines an explicit symbolic reconstruction problem, derives an order-k parameter count, proves exact finite-state decoding, and gives finite-sample bounds under stated assumptions.Its experiments distinguish synthetic proof of concept from a limited real-audio feasibility evaluation.Source: §1.2 · p. 5

3.2 Order-k Markov chains; 3.3 Dirichlet Bayesian estimation

  • An order-k chain has N^k(N - 1) free parameters, while symmetric Dirichlet estimation distinguishes posterior mean from posterior mode.The paper notes that higher order increases parameter demands; smoothing trades bias for lower variance and protection against zero-probability transitions.Source: §1.2 · p. 5, §3.2 · p. 8, §3.3 · p. 8

1.3 Scope and terminology

  • The scope is Hindustani symbolic melody; the authors do not claim a symbolic sequence fully represents a raga or that synthetic reconstruction authenticates historical performance.Transfer to Carnatic music requires separate musicological definitions.Source: §1.2 · p. 5, §1.3 · p. 6

2.1 Computational musicology; 2.2 Markov models and algorithmic generation; 2.3 HMMs and latent phrase structure; 2.4 Neural generative models

  • Prior Markov and probabilistic automata models represent local musical context, while ARS shifts the goal from free generation to reconstruction preserving observations and explicit constraints.HMMs and neural generative models are treated as optional extensions, not sources of unproved grammatical guarantees.Source: §2.2 · p. 6, §2.1 · p. 6, §2.3 · p. 6

2.5 Why this framework is different

  • The framework separates probability, admissible grammar, observation, optimization, and statistical assumptions so each component is credited only with conclusions supported by its assumptions.A forbidden transition can be assigned low probability or prohibited as a hard constraint; these are distinct choices.Source: §2.5 · pp. 6–7

3.1 Symbolic state space

  • The symbolic state space is a chosen finite alphabet and explicit constraint system; vague descriptions such as “sounds like Yaman” are not executable constraints.A feasible completion set can be empty, signaling conflict between observation and grammar.Source: §3.1 · p. 7

3.4 Hard constraints and feasible kernels

  • Hard transition constraints restrict kernel support, and renormalizing a positive reference row gives the unique KL projection onto allowed successors.Other constraint geometries, such as aggregate-frequency inequalities, need not reduce to this projection.Source: §3.4 · p. 9

4 Constrained MAP Reconstruction

The reconstruction is a constrained maximum-a-posteriori path problem: preserve observed symbols, obey the grammar, and maximize the estimated Markov score. A finite-memory dynamic program finds a global optimum exactly, with complexity exponential in model order but linear in sequence length.

4.1 Objective; 4.4 Zero probabilities

  • Observed positions restrict each time step to the observed symbol, while missing positions allow any alphabet symbol; the grammar separately restricts successors.If an estimated transition has zero probability, its log score is negative infinity; smoothing can prevent accidental data-driven zeros, unlike genuine grammar prohibitions.Source: §4.1 · p. 9, §4.4 · p. 10

4.2 Dynamic programming state

  • The dynamic program stores the best prefix score for each ending context and a backpointer, then updates scores using allowed successors and transition log probabilities.The initial layer enumerates admissible initial contexts and their prior probabilities.Source: §4.2 · p. 9

4.3 Optimality theorem

  • For finite-length sequences with finite-memory constraints, dynamic programming returns a globally optimal constrained MAP completion, if one exists.Additional hard constraints can be handled by augmenting the state with a deterministically updated finite state.Source: §4.3 · p. 10
  • O(TN^(k+1)) time and O(N^k) memory suffice for the dense order-k model without additional state.The time bound is exponential in k but linear in sequence length T.Source: §4.3 · p. 10

5 Statistical Guarantees

The paper bounds transition-estimation error under explicit sampling assumptions and connects probability error to sequence scores and MAP stability. These guarantees remain conditional on the model, dependence assumptions, order-selection regularity, and distinguishability of candidate completions.

5.1 A transparent independent-sample bound

  • Dependent Markov data require mixing assumptions or an established effective sample size for a concentration bound; independent multinomial observations provide a stronger, transparent assumption.The union bound makes dependence on the number of context-symbol pairs explicit, while the bound is described as conservative.Source: §5.1 · p. 10

5.2 From parameter error to sequence-score error

  • A positive score gap preserves the unique true-model MAP sequence when every feasible sequence’s estimated score error is at most half that gap.Probability error can accumulate across sequence transitions, and accurate parameter estimation alone does not ensure exact recovery when candidate scores are close.Source: §5.2 · p. 11

5.3 BIC and order selection

  • BIC supplies a principled complexity penalty, but consistency in dependent sequence data requires appropriate identifiability, regularity, and dependence assumptions.The synthetic experiment does not establish a universally optimal order for raga music.Source: §5.3 · p. 11

5.4 Identifiability

  • Reconstruction is identifiable only when the observation and model sufficiently distinguish candidate completions; identical observations and scores leave the posterior unable to select between them.This is an information-theoretic limit of the problem specification.Source: §5.4 · p. 12

6 Optional Extensions Beyond the Core Model

The paper describes HMMs, hierarchical VAEs, continuous-pitch models, and Bayesian fusion as possible extensions to the finite symbolic core. Each introduces distinct interpretability, constraint, observation, or evidence-dependence requirements.

6.1 Raga-HMM

  • An HMM can model latent phrase context with exact forward–backward and Viterbi algorithms, but naming its states as musicological concepts requires annotation or an explicit mapping.The mathematics alone does not supply that interpretation.Source: §6.1 · p. 12

6.2 Hierarchical VAE

  • A hierarchical VAE can learn long-range structure, but its ELBO does not enforce hard raga constraints; constrained decoding or projection is needed for grammatical validity.A penalty alone is not a hard guarantee unless optimization is constrained accordingly.Source: §6.2 · p. 12

6.3 Continuous pitch and gamaka

  • A continuous-pitch model could represent trajectories and ornaments, but requires a defined curve distance and an explicit acoustic likelihood rather than treating extracted pitch as ground truth.Audio observations pass through noise, source separation, pitch estimation, and segmentation.Source: §6.3 · p. 12

6.4 Bayesian fusion

  • Combining Markov, phrase, and neural distributions can double-count evidence when they were trained on the same data.A joint generative interpretation or deliberate product-of-experts assumption is required for principled Bayesian fusion.Source: §6.4 · pp. 12–13

7 Synthetic Experimental Design

The synthetic evaluation uses controlled, raga-inspired symbolic sequences to test reconstruction machinery, with explicit masking, model-order comparisons, and reproducibility controls. Its preliminary results exceed uniform guessing, but do not establish performance on historical recordings or a universal optimal order.

7.4 Statistical Reporting

  • Finite-test reporting includes means, standard deviations, test counts, and confidence intervals when assumptions are stated, while BIC-based order selection is frozen before test evaluation.Paired model comparisons should use the same masked test sequences.Source: §7.4 · pp. 13–14

7.5 Reproducibility protocol

  • The reproducibility protocol calls for the generator, estimator, decoder, configuration, fixed seeds, requirements, and a script that regenerates final tables.The README should specify the exact regeneration command.Source: §7.5 · p. 14

8.1 Purpose and Experimental Design

  • The experiment tests reconstruction under known symbolic processes using six raga-inspired alphabets, which are not claimed to be complete computational grammars of those ragas.The synthetic sequences provide exact symbolic ground truth but do not constitute evidence about genuine musical performances.Source: §7.1 · p. 13, §8.1 · p. 14, §8 · p. 14
  • Each condition uses 1,200 training sequences and 60 held-out sequences of length 32, with 10%, 30%, or 50% missingness, orders 1–3, α = 0.5, and a fixed master seed.Missing positions are sampled uniformly without replacement, a deliberately simple missing-at-random mechanism.Source: §8.1 · p. 14

8.2 Evaluation Metric

  • Missing-symbol accuracy is the primary metric, with a seven-symbol uniform predictor expected to reach 1/7; proposed ablations isolate smoothing, grammar constraints, and Markov order.The cited text proposes these ablation comparisons; it does not report their results.Source: §7.3 · p. 13, §8.2 · p. 14

8.3 Preliminary Synthetic Results

  • 0.539 is the highest observed accuracy, for order two at 10% missingness; at 50% missingness, order two reaches 0.384, and all nine conditions exceed the uniform baseline.These are preliminary results from the synthetic Yaman-inspired evaluation, not evidence of recovery of real performances.Source: Table 2 · p. 14, §8.3 · p. 15

8.4 Synthetic Missingness and Model Order

  • At 10% missingness, orders 1, 2, and 3 obtain 0.439, 0.539, and 0.450; at 30%, 0.452, 0.433, and 0.390; at 50%, 0.414, 0.384, and 0.358.Accuracy does not improve monotonically with Markov order.Source: §8.4 · p. 15, Figure 1 · p. 15

8.5 Interpretation and Boundary of the Synthetic Evidence

  • The controlled synthetic results can validate implementation behavior but cannot establish a universal optimal order, a complete Yaman grammar, or historical authenticity.The simple missing-at-random mechanism also differs from real damaged audio, where gaps or transcription errors may be correlated or concentrated around ornamentation.Source: §8.3 · p. 14, §8.1 · p. 14, §8.5 · p. 15

9 Real-World Data Evaluation

The real-world evaluation is the paper’s principal empirical test: it applies reconstruction to genuine Yaman recordings converted into symbolic sequences. Order two performs best across the tested missingness levels, while the results establish computational feasibility rather than musicological validity.

9.2 Data Provenance

  • The 42-recording Yaman corpus has incomplete provenance and licensing documentation, so the evaluation is presented as a feasibility pilot rather than archival reconstruction.After exclusions and a minimum-length requirement, 30 sequences remained for held-out reconstruction.Source: §9.2 · p. 16, §9.4 · p. 17

9.3 Audio-to-Symbol Pipeline

  • Automated pitch extraction, tonic estimation, segmentation, and quantization produce symbolic sequences whose errors can propagate into measured reconstruction accuracy.The corpus-construction pipeline runs from genuine recordings through symbolic quantization, filtering, and a training/test split.Source: §9.3 · p. 16, Figure 3 · p. 17

9.5 Experimental Protocol

  • The 30 sequences were split into 24 training and 6 held-out test sequences, with 20 masks per test sequence at each missingness level and no hard Yaman grammar constraints.The dynamic-programming decoder matched exhaustive enumeration in all 30 small synthetic checks.Source: §9.5 · p. 17

9.7 Real-Data Performance as Missingness Increases

  • Order-two accuracy declines from 0.470 at 10% missingness to 0.371 at 50%, an absolute decrease of 0.099.Figure 4 shows means and one-standard-deviation error bars across repeated masks, against a uniform seven-symbol baseline.Source: §9.7 · p. 17, Figure 4 · p. 18

9.6 Primary Real-World Results

  • Order two achieves 0.470, 0.414, and 0.371 accuracy at 10%, 30%, and 50% missingness, respectively, and all nine conditions exceed the uniform baseline.Table 4 reports means and standard deviations over 120 masked test evaluations per condition.Source: §9.6 · p. 17, Table 4 · p. 18, §9.8 · p. 18

9.8 Real Data Relative to the Uniform Baseline

  • Order two exceeds the uniform baseline by 0.327, 0.271, and 0.228 at 10%, 30%, and 50% missingness, respectively.These correspond to approximately 3.29×, 2.90×, and 2.60× the uniform accuracy.Source: §9.8 · p. 18

9.9 Context Coverage and Model Sparsity

  • Order three has 2,058 free parameters but only 215 of 343 possible contexts observed, helping explain its weaker evidence per context than order two.The corpus has approximately 1,700 training transitions; order one has less contextual capacity, while order three has substantially more.Source: §9.9 · p. 18, Figure 7 · p. 21

9.11 Variance and Robustness

  • With only six held-out sequences, the study reports no formal significance tests between orders; its accuracy measures agreement with pipeline-generated symbols, not expert-judged musical correctness or historical authenticity.The end-to-end results support computational feasibility on genuine audio-derived data, while the study’s limitations prevent stronger musicological claims.Source: §9.11 · p. 19, §9.12 · p. 19, §9.13 · pp. 19–20

10 Synthetic vs. Real-World Comparison

The comparison shows lower raw accuracy on real recordings than on controlled synthetic data, while real-data results remain above an uninformed baseline and test the system under more application-like conditions. Neither evaluation establishes historical authenticity or expert-validated musical correctness.

10.1 Direct Numerical Comparison

  • At 10%, 30%, and 50% missingness, order-two accuracy is 0.539, 0.433, and 0.384 synthetic versus 0.470, 0.414, and 0.371 real.Table 5 directly compares order-two reconstruction accuracy across the three masking levels.Source: Table 5 · p. 21, §10.1 · p. 21
  • The synthetic-real accuracy gap shrinks from 0.069 at 10% missingness to 0.013 at 50%.The real-data system approaches the synthetic pilot under the most difficult masking condition despite additional pipeline uncertainty.Source: §10.1 · p. 21

10.2 Why Synthetic Data Produce Higher Raw Accuracy

  • Synthetic accuracy is higher because its generator matches symbolic-model assumptions, symbols are exact, and the corpus lacks natural interperformance variability.Real audio adds pitch-tracking, tonic-estimation, segmentation, and quantization error.Source: §10.2 · p. 21

10.3 Why the Real-World Evaluation Is the Stronger Empirical Test

  • The real-data test carries greater evidentiary weight because it tests whether recoverable structure survives an imperfect acoustic-to-symbolic pipeline on genuine performances.The synthetic experiment tests recovery from a controlled process; the real-data question is closer to ARS’s intended application.Source: §10.3 · p. 22

10.4 Graphical Evidence for the Real-Data Result

  • Real-data results remain above the uniform baseline, and order two consistently outperforms order three despite its smaller contextual window.For N = 7, parameter counts rise from 42 to 294 to 2058 across orders one to three, while context coverage falls from 100% to 85.7% to 62.7%.Source: §10.4 · p. 22

10.5 What Neither Evaluation Establishes

  • Neither evaluation establishes musicological validity, historical authenticity, or agreement with an independently produced expert transcription.The synthetic data are generated by construction, and the real symbolic reference is produced automatically; stronger evaluation would require independent annotation, realistic noise and masking, matched baselines, and expert phrase evaluation.Source: §10.5 · p. 22

10.6 Overall Empirical Interpretation

  • Across all three masking rates, the real experiment reproduces the theoretically motivated tradeoff between contextual richness and data sparsity while remaining substantially above an uninformed baseline.The synthetic pilot validates implementation under controlled conditions; the real-world experiment demonstrates effectiveness on genuine Yaman recordings.Source: §10.6 · p. 23

10.7 Code Availability

  • The repository includes the estimator, exact decoder, preprocessing pipeline, experiment configuration, fixed seeds, tests, and scripts needed to regenerate tables and figures.The README specifies the environment and command for reproducing the outputs.Source: §10.7 · p. 23

11 Algorithmic Specification

The section specifies training and constrained reconstruction for an order-k Markov model, including an exact finite-state decoder. It also sets out when approximation may be appropriate and how sensitivity, ablation, and error analyses should be reported.

11.1 Training

  • Training normalizes the symbolic corpus, counts context-successor pairs, computes the Dirichlet posterior mean, and selects k with a training-only criterion before freezing the model.The described procedure uses BIC as an example and evaluates the frozen model on test data.Source: §11.1 · p. 24

11.2 Reconstruction

  • For a test fragment, the decoder initializes admissible contexts, propagates dynamic-programming scores while rejecting grammar-forbidden transitions, then backtracks or returns “no feasible completion.”Figure 11 compares order-two reconstruction accuracy; its caption says the synthetic pilot is numerically higher while real-data accuracy remains close and above the uniform baseline.Source: §11.2 · p. 24, Figure 11 · p. 25

11.3 Pseudocode for exact reconstruction

  • The finite-memory decoder is exact, but its state space grows with order: for N = 7 and k = 4 it has 2401 context states before grammar state is added.Larger orders, longer sequences, or richer constraint automata can make state growth dominant.Source: §11.3 · p. 25

11.4 When beam search is appropriate

  • Beam search is an approximation for scores that cannot be represented by finite-memory state, such as a transformer conditioned on a continuous latent state and the entire prefix.Using beam search does not change the exactness theorem for the simpler finite-memory model.Source: §11.4 · pp. 25–26, Figure 12 · p. 26

12.1 Smoothing sensitivity

  • Sensitivity checks should vary smoothing strength and stress-test hard grammar constraints, since priors can dominate rare contexts and an incorrect grammar can sharply reduce accuracy.Report missing-symbol accuracy and the fraction of feasible reconstructions when constraints are perturbed or removed.Source: §12.1 · p. 26, §12.2 · p. 26

12.3 Masking sensitivity

  • Evaluation should test realistic contiguous gaps, burst errors, and observation noise, and select Markov order on training data before evaluating the frozen model on held-out data.Reporting selected-order distributions across repeated training samples is more informative than reporting a single selected order.Source: §12.3 · p. 26, §12.4 · pp. 26–27

12.5 Ablation table template

  • The ablation table is explicitly a template with blank values, while error analysis should record missing positions, local context, candidate probabilities, and the grammar rules involved.Those records support inspection of whether failures reflect sparse data, incorrect grammar, ambiguous context, or non-identifiable fragments.Source: §12.5 · p. 27, §12.6 · p. 27

13 Cultural and Scientific Limitations

The limitations distinguish symbolic modeling from the richer musical, archival, and cultural evidence needed to validate a reconstruction. The discussion and conclusion present exact inference as a result for a finite-state model, not as proof of historical or musicological validity.

13.1 A scale is not a raga

  • A pitch inventory alone does not define a raga: synthetic sequences from note inventories are only “raga-inspired” without validation against expert annotations.The model’s alphabet is not a complete representation of characteristic movement, phraseology, and performance practice.Source: §13.1 · p. 28

13.2 Shruti and continuous intonation

  • A fixed 22-position shruti lattice is not justified as a global representation of performance pitches, so the symbolic model does not impose one.A future continuous model should distinguish measured frequency from a named shruti and account for pitch-tracking uncertainty.Source: §13.2 · p. 28

13.3 Gamaka; 13.4 Rhythm and tala

  • Reducing gamaka to a type or scalar energy loses trajectory information, while the melodic model omits tala, beat position, duration, and rhythmic phrasing.A rigorous gamaka representation would need to encode trajectory properties; full performance reconstruction would require a joint melodic-rhythmic model.Source: §13.3 · p. 28, §13.4 · p. 28

13.5 Historical authenticity

  • A likely completion is not necessarily the historical one: its probability reflects the training corpus and modeling assumptions, not independent historical evidence.Archival evaluation needs documented provenance, expert annotation, and separation of training and evaluation material.Source: §13.5 · p. 28

13.6 Synthetic data

  • Synthetic results validate implementation and sensitivity, but a generator sharing the model’s state representation can make reconstruction easier than reality.The Yaman pilot likewise remains a feasibility check: its source lacks documented provenance, and its symbolic reference comes from the evaluation pipeline itself.Source: §13.6 · p. 28, §13.5 · p. 28

13.7 Ethical and cultural considerations; 13.8 Reproducibility limitation

  • Preservation should involve practitioners and archivists rather than claim ownership for statistical encoding; reproducibility also depends on versioned code, configuration, environment, and data generation.The framework is presented as a tool for organizing evidence, not a replacement for practitioners.Source: §13.7 · p. 29, §13.8 · p. 29

14 Discussion

  • Higher Markov order adds context but increases parameter demand, while grammar constraints help most when correct and informative and risk misspecification otherwise.The discussion proposes exposing grammar uncertainty and layering learned representations over an explicit finite-state core.Source: §14 · p. 29

15 Conclusion

  • For finite alphabets and finite-memory constraints, dynamic programming exactly optimizes reconstruction; Yaman results remain feasibility evidence, not historical or musicological validation.The conclusion calls for expert-annotated archival data, realistic degradation, cross-performer evaluation, and matched baselines; the system organizes evidence under explicit assumptions.Source: §14 · p. 29, §15 · p. 30

A Proofs and Technical Details

The appendix supplies proof details for parameter counting, Bayesian estimation, constrained optimization, concentration, and score stability. It also identifies conditions and boundaries that qualify those results.

A.1 Proof of the parameter count

  • Each transition row lies on an (N−1)-dimensional simplex, so N^k rows give N^k(N−1) free parameters.The count also holds at boundary points with zero-probability transitions.Source: §A.1 · p. 32

A.2 Proof of the Dirichlet posterior

  • Multiplying the multinomial likelihood by the Dirichlet prior yields a Dirichlet posterior, whose mean and mode differ by one in each pseudo-count.The stated posterior-mode expression applies when all posterior parameters exceed one.Source: §A.2 · p. 32

A.3 Proof of KL support projection

  • The KL support projection assigns allowed symbols mass proportional to their reference probabilities, but a zero reference probability cannot receive positive mass at finite divergence.Smoothing can make the reference probabilities positive; statistical smoothing and hard grammar constraints remain distinct.Source: §A.3 · p. 32

A.4 Proof of dynamic-programming optimality

  • Dynamic programming is lossless because future scores depend on a prefix only through its augmented state, so only the best prefix per state must be retained.Backtracking from the recurrence recovers a global optimum.Source: §A.4 · p. 32

A.5 Concentration derivation

  • Hoeffding’s inequality and a union bound yield the independent-sample concentration result; dependent stationary chains require mixing bounds and a smaller effective sample size.The reduction depends on the mixing rate, so the independent result is presented as a baseline rather than a theorem for arbitrary archival sequences.Source: §A.5 · p. 33

A.6 Score stability proof

  • If the uniform score error is less than half the true maximizer’s positive score gap, that maximizer remains uniquely optimal under estimated scores.The strict inequality preserves a positive estimated score difference from every alternative.Source: §A.6 · p. 33

B Appendix: Formal Failure Analysis of the Original Theorems

This appendix identifies failures in contraction and continuity arguments, then lists implementation, testing, and reproducibility checks. Its repository outline names the expected code, experiment, result, and test components.

B.1 Failure of the stated contraction hypothesis

  • A row-stochastic transition matrix cannot satisfy the proposed infinity-norm bound below one, making that contraction hypothesis contradictory.The appendix distinguishes this direct contradiction from a subtler regularity issue.Source: §B.1 · p. 33
  • A restricted substochastic matrix may contract, but that describes mass leakage and not necessarily convergence of the original reconstruction decoder.The restricted matrix is not the original transition matrix.Source: §B.1 · p. 33

B.2 Failure of argmax continuity

  • Argmax can switch outputs under arbitrarily small score perturbations, so a contraction claim needs a margin excluding near-ties or a softened operator.The example’s inputs approach one another while its discrete outputs remain distance one apart.Source: §B.2 · p. 33

C Appendix: Reproducibility Checklist

  • Reproducibility checks call for exact versioned code, fixed and reported seeds, machine-readable grammar, and documented environment and execution commands.The checklist specifies the code used for every numerical table and figure.Source: C Appendix: Reproducibility Checklist · pp. 33–34
  • Separate data-loading and generation stages, independent test masks, and test-label-free order selection help keep evaluation procedures distinct.Hyperparameters should be listed explicitly.Source: C Appendix: Reproducibility Checklist · p. 34
  • Numerical values should come from code, repeated experiments should report uncertainty, and synthetic evidence should be distinguished from archival evidence.The checklist also asks authors to state transparently when a public code link is unavailable.Source: C Appendix: Reproducibility Checklist · p. 34

C.1 Repository structure

  • The repository outline includes Markov, grammar, decoder, metrics, and generator modules alongside experiment scripts, results, figures, and tests.It also lists a README, requirements, and configuration files.Source: §C.1 · p. 34
  • Decoder tests should check observation preservation and compare dynamic programming with exhaustive enumeration on small cases to verify optimal scores and sequences.The checklist also calls for probability-row normalization, Dirichlet estimates, and grammar rejection tests.Source: §C.1 · p. 34

D Appendix: Recommended Future Theoretical Work

The proposed future work extends the finite-state reconstruction framework while preserving the distinction between exact inference and richer models. It also specifies validation needs and the limits of current statistical guarantees.

D.1 Soft reconstruction operators

  • A temperature-scaled soft reconstruction operator would produce a distribution over feasible completions rather than an argmax, enabling study of contraction under influence bounds.The proposed analysis would control how changing an observation changes the conditional distribution.Source: §D.1 · p. 34

D.2 Continuous-pitch Bayesian model

  • A continuous-pitch Bayesian model could separate acoustic observations from symbolic grammar, combining pitch evidence with constraints without treating a pitch tracker as infallible.The proposed factorization includes latent pitch trajectory, ornament process, and observed audio.Source: §D.2 · p. 35

D.3 Uncertainty-aware grammar

  • Treating grammar as latent over a finite family of rule sets would let posterior inference represent expert disagreement and reduce overconfidence in one potentially incorrect grammar.The formulation weights grammar-conditional reconstructions by the posterior probability of each grammar.Source: §D.3 · p. 35

D.4 Archival validation

  • A stronger archival test should use licensed, attributed recordings with known complete performances, realistic masking, expert phrase ratings, and separate accuracy and judgment measures.Prefer corpus-provided tonic and symbolic annotations and use performer- or recording-disjoint splits to limit leakage.Source: §D.4 · p. 35

E.1 Initial-context estimation; E.2 Stationarity is not required for the decoder

  • Initial contexts can receive a Dirichlet prior like transition rows, while decoder optimality itself does not require stationarity once transition scores are fixed.Stationarity is relevant to some estimation arguments, not the optimization theorem; musical form may be nonstationary.Source: §E.1 · p. 35, §E.2 · p. 35

E.3 State augmentation

  • Adding a finite phrase state to the previous-k-symbol context preserves exact dynamic programming, with complexity proportional to augmented states and outgoing transitions.This provides a bridge to an HMM-like phrase context.Source: §E.3 · p. 35

E.4 Tie-breaking

  • A deterministic lexicographic tie-break makes the MAP estimator single-valued, but does not prevent discontinuous output changes under small perturbations.The selection rule resolves equal scores, not continuity.Source: §E.4 · p. 35

E.5 Model misspecification

  • If the true process is not order-k Markov, estimation may converge to the best approximation in that model class rather than the true conditional distribution.A full learning-theoretic analysis should distinguish estimation error from approximation error.Source: §E.5 · p. 36

F Appendix: Notation and Assumption Register

The appendix distinguishes exact optimization and conditional statistical guarantees from claims of musical or historical authenticity. Its claims depend on finite-memory structure, explicit sampling assumptions, and a grammar whose validity is relative to what it encodes.

F Appendix: Notation and Assumption Register

  • Exact decoding requires finite-state local scoring and finite-memory constraints; statistical guarantees need separate assumptions, and score stability additionally requires positive transition probabilities and a score gap.Without a gap, arbitrarily small estimation errors can change the MAP maximizer.Source: F Appendix: Notation and Assumption… · p. 36
  • The synthetic experiment assumes missing-at-random masking and a correctly specified finite symbolic generator, so it tests controlled behavior rather than historical authenticity.Real-audio extensions require an observation model for noise, segmentation, pitch tracking, and transcription uncertainty.Source: F Appendix: Notation and Assumption… · p. 36

F.1 Interpretation of guarantees

  • The framework establishes global optimality for its finite-state core and conditional statistical results, but does not establish that a reconstruction represents a musical tradition.A hard grammar guarantees membership in its encoded feasible set, not that the grammar faithfully represents the tradition.Source: §F.1 · p. 36

G Appendix: Final Submission Version Notes

  • The final claims specify symbolic reconstruction under a probabilistic model and grammar, with exact dynamic programming when the model and constraints have finite memory.The stated worst-case dense complexity is O(TN^(k+1)).Source: G Appendix: Final Submission Version… · p. 37
  • Synthetic experiments demonstrate reproducibility and behavior under controlled missingness, not historical authenticity; unsupported contraction and Svara–Gamaka metric claims are excluded.The stated proofs do not establish those latter conclusions.Source: G Appendix: Final Submission Version… · p. 37

G.1 Claim-to-evidence audit

  • The parameter-count claim is algebraic, the Dirichlet estimation claim follows from conjugacy, and concentration claims remain conditional on sampling assumptions.The complexity bound is a worst-case dense-case upper estimate; sparse grammars may evaluate fewer edges.Source: §G.1 · p. 37
  • A raga label on a synthetic generator does not prove complete raga grammar, and symbolic accuracy does not establish faithful historical performance.Figures and tables should disclose what was measured, sequence counts, data type, quantity reported, and reproducibility configuration where applicable.Source: §G.1 · p. 37
  • A finite-state grammar can guarantee validity for a constrained neural decoder while the neural model supplies a learned score; probability, permission, selection, and validation remain distinct.A VAE loss, attention map, or latent visualization alone is not a theorem of grammatical validity.Source: §G.1 · p. 37

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