Input & Operator Disclosure
The terms below (EOA, LCR, β-operator) are specific to this project and are not standard in symbolic dynamics.
Input: spoken English letter name. Exact EOA phonetic encoding disclosed under collaboration access (not reproduced in the public artifact).
LCR: limiting-ratio control parameter of the EOA construction. Currently set to 3.14. Formal mathematical definition is part of the EOA technical documentation, available to qualified collaborators under the stated agreement.
β-operator: generates the released sequence. Transformation rules and implementation available to qualified researchers under a signed collaboration agreement.
Full disclosure is included in the exported report.
Placeholder notice: the built-in sample sequence is a motif placeholder for UI demonstration. It is not generated by the β-operator. Paste a released EOA sequence to analyse a real one.
Access: the β-operator and the exact phonetic encoding are available on request via research.neurolabs.space .
Live Controls
Projection order drives Panel 5. Saturation order = smallest k at which the entropy ladder flattens.
Metrics by Epistemic Tier
p(k) = # distinct length-k subwords of released word; windows counted once; no circularization.
ρ(A_k) = spectral radius of the binary adjacency matrix A_k of the order-k factor-overlap graph G_k (edge weights are ignored for the spectral calculation). Distinct from ρ(M_kstride) shown in Panel 5.
h_k = ln ρ(A_k) is the topological entropy of the sofic shift presented by the empirical graph G_k — a well-defined number for that object. It is not asserted to equal the topological entropy of any underlying system, which is not defined in this tool.
Zero topological entropy • primitive substitution • morphic class • balanced system • fractal structure — none established by finite-sample analysis; each requires a proof over the full language.
1. Symbolic Turtle Rendering
Letters are mapped to geometric instructions via a deterministic rule. No rewriting system is applied in EOA mode. Vary Turn Strength to test whether fractal structure is intrinsic or induced by the map.
2. Chaos Game Representation
Observed density of symbol transitions. Empty regions mark transitions absent from the released word. Compare against Panel 12 before reading structure as order.
3. Recurrence Plot
Diagonal lines indicate repeated symbol positions; block structure indicates subword recurrence.
4. Rewrite Engine (explicit morphism only)
Applies a user-supplied rewriting rule σ(a)=w_a and renders the resulting word class. If no rule is supplied, only the raw symbol waveform is shown.
5. Empirical Eigenvector Projection
Prefix Parikh walk projected onto the empirical contracting plane. The reported ρ(M_kstride) is the spectral radius of the stride-k letter matrix — distinct from the binary-adjacency ρ(A_k) shown in the metrics card. Point cloud geometry is estimated from the released word, not proved for the language.
6. Subword Transition Graph
Empirical order-k factor-overlap graph G_k of the released word. Node degree and the binary-adjacency spectral radius ρ(A_k) are estimated. h_k = ln ρ(A_k) is the topological entropy of the sofic shift presented by G_k — a well-defined number for that empirical object. It is not asserted to equal the entropy of any underlying system, which is not defined in this tool.
7. Desubstitution Test
Tests whether the released word parses uniquely under a user-supplied morphism. Absence of a parse is not proof of non-morphic structure.
8. Factor Complexity p(k) vs k
Log-log slope of the observed complexity curve. Read the curve, not any single point.
9. Abelian Complexity a(k) vs k
Number of distinct Parikh vectors of length-k factors of the released word.
10. Finite-Sample Block-Growth Ladder
Sofic bounds from order-k binary factor-overlap graphs vs. the finite-sample block-growth estimate. Each h_k = ln ρ(A_k) is the topological entropy of the sofic shift presented by the empirical graph G_k. It is not asserted to be the topological entropy of any underlying system, which is not defined in this tool. Edge weights are ignored for the spectral calculation.
11. Prefix-Frequency Discrepancy
Maximum observed deviation of prefix symbol counts from the asymptotic frequency vector.
12. Surrogate Controls (CGR comparison)
Experiment: Compare the original CGR against a frequency-shuffle, a Markov order-1 surrogate, and a periodic control. Structural features present in the original but absent in all surrogates are evidence of long-range order. Features reproduced by the Markov surrogate are explained by lower-order statistics alone. Surrogates are seeded deterministically from the sequence, so this panel is reproducible across exports. A statistically rigorous surrogate study would require replicate ensembles per surrogate and multiple-testing control across statistics; that is out of scope here.