Ginibre Eigenvalue Braid Dynamics

This repository collects conjectures, exact identities, and reproducible finite checks connecting:

• complex and elliptic Ginibre matrix dynamics;
• nonnormal eigenvector overlaps and eigenvalue motion;
• braid-valued observables of collision-free spectral loops;
• local B3 grammar in finite spectral pockets; and
• knot and link observables of closed eigenvalue braids.

The project is exploratory. It is not affiliated with the authors cited in the literature notes, and it does not claim that its open conjectures are new theorems.

Central Question

What probability law on braid words and braid closures is induced by a collision-free non-Hermitian random-matrix path, and which parts of that law are controlled by local eigenvalue gaps, eigenvector overlaps, endpoint conditioning, and the Hermitian-to-Ginibre crossover?

Model Boundary

The phrase "Ginibre dynamics" can refer to distinct stochastic objects. This repository keeps them separate.

Object	Definition	Role here
Matrix Ginibre Brownian/OU process	Matrix entries diffuse; the complex Ginibre law is a one-time or stationary matrix law	Primary source of eigenvalue paths
Elliptic Ginibre matrix process	Correlated off-diagonal entries interpolate between non-Hermitian and Hermitian regimes	Crossover model
Ginibre interacting Brownian motion	Planar Coulomb particles with the Ginibre point field as equilibrium	Literature comparator and possible limiting model
GOE/GUE plus local nonnormal perturbation	Hermitian background with an embedded complex 3x3 path	Finite-N local-pocket experiment
Nonlinear moving-well diffusion	Planar Brownian particles confined around controlled moving centers	Tractable non-Gaussian braid and synchronization model

The eigenvalues of a nonnormal matrix diffusion are not generally an autonomous Dyson gas. Their increments are coupled to left/right eigenvector overlaps. That coupling is central to the conjectures below.

Exact Topological Bridge

Let A(t) be a closed complex matrix path with simple spectrum. Its unordered eigenvalues form a loop in the unordered configuration space UConf_n(C), whose fundamental group is the braid group B_n.

For the spectral discriminant

Disc(A(t)) = product_{i<j} (lambda_i(t) - lambda_j(t))^2,

the winding number of Disc(A(t)) around zero equals the exponent sum of the spectral braid. This is the braid abelianization, not a complete braid invariant. The repository includes a proof note and canonical half-twist regression fixtures.

Evidence Levels

Label	Meaning
proved_exact	A complete argument is included.
proved_exact_from_standard_results	A complete deduction is included, but its substantive inputs are cited classical theorems.
standard_theorem	A published or classical theorem is cited; no authorship is claimed.
verified_bounded_numeric	A numerical identity check passed on an explicit finite fixture set.
proved_bounded	Exhaustive discrete computation proves a statement only within an explicit bound.
empirically_supported	Existing finite experiments support the claim.
open	No proof or decisive experiment is available.
falsified	A counterexample or failed prediction is recorded.

Numerical agreement, automated testing, and bounded enumeration are never promoted to asymptotic theorems.

Repository Map

• GINIBRE_MODEL_BOUNDARY.md: precise model distinctions.
• CONTINUOUS_BRAID_GENERATOR_AND_MONODROMY.md: diffusion generators, deck laws, KZ holonomy, and new conjectures.
• NUMBER_THEORY_AND_PRIME_KNOTS.md: the B3 modular representation, determinant traces, and prime-knot cautions.
• BLOCK_PAIR_WINDING_THEOREMS.md: exact local-block reduction, pair-winding coordinates, amplitude threshold, and the weak-coupling resolvent identity.
• FINITE_QUOTIENT_RANDOMNESS.md: finite arithmetic projections, IID expander baselines, entropy boundaries, and source-dependent randomness conjectures.
• CONTROLLED_BRAID_SANDBOX.md: target-braid synthesis, clearance-tube assurance, Brownian-bridge fidelity, and the classical-versus-anyonic computation boundary.
• BRAID_CONTROL_ENERGY_AND_LARGE_DEVIATIONS.md: exact canonical compiler energy, optimal timing, and bridge tube-exit action.
• NONLINEAR_MOVING_WELL_BRAIDS.md: nonlinear co-moving wells, phase-offset clock synchronization, central full twists, and the finite-quotient bottleneck.
• BROWNIAN_WINDING_SKELLAM_BOUNDARY.md: correction of the angular-mode/winding-sector conflation, the discrete-Gaussian circle-bridge law, twisted periodic generators, and the constant-rate proxy boundary.
• NONLINEAR_BROWNIAN_WINDING_PERTURBATIONS.md: Yor's heavy-tailed Brownian-lace baseline, exact Gibbs sector reweighting, uniform perturbation bounds, and the quartic-OU Girsanov score.
• NONLINEAR_WINDING_NOVELTY_AUDIT.md: claim-by-claim review separating classical inputs, elementary deductions, and the open radial-confinement and quartic-score targets.
• MCSHANE_PROBABILITY_AND_MATRIX_MODELS.md: the conditional McShane measure, its exact Weil-Petersson-annealed length law, trace/Markoff moment threshold, and the matrix-loop-equation bridge.
• MCSHANE_NOVELTY_AUDIT.md: specialist-facing literature audit separating known results, immediate corollaries, elementary observations, and unresolved research questions.
• FORMALIZATION_BLUEPRINT.md: proof-assistant targets and the external supervision checklist.
• CONJECTURES.md: statements, evidence, and falsifiers.
• conjectures.csv: machine-readable conjecture registry.
• PROOF_STATUS.md: proof and evidence ledger.
• proofs/README.md: exact arguments and automation limits.
• results/README.md: generated finite-check results.
• results/CONTROLLED_BRAID_SANDBOX_RESULTS.md: generated finite control and Brownian-bridge results.
• results/CONTROL_ENERGY_RESULTS.md: generated exact control-energy and bridge-action checks.
• results/MOVING_WELL_BRAID_RESULTS.md: generated clock topology, quartic-well, splay-stability, energy, and quotient checks.
• results/BROWNIAN_WINDING_BRIDGE_RESULTS.md: generated angular-mode proxy, circle-winding, and twisted-generator checks.
• results/NONLINEAR_WINDING_PERTURBATION_RESULTS.md: deterministic checks of nonlinear reweighting bounds, Cauchy-cell tails, and the quartic-OU score.
• REVIEW.md: external proof-review checklist.
• src/braid_stats.py: exact word operations and canonical spectral-loop fixtures.
• src/controlled_braids.py: finite target synthesis and bridge perturbations.
• scripts/check_finite_claims.py: bounded proof and regression checks.
• scripts/run_controlled_braid_sandbox.py: bounded control experiment.
• scripts/check_control_energy_claims.py: exact rational control checks.
• scripts/check_moving_well_claims.py: nonlinear-well and clock-braid symbolic fixture checks.
• scripts/check_winding_bridge_claims.py: Cotler angular-mode Skellam proxy, circle discrete-Gaussian winding, and twisted-generator checks.
• scripts/check_nonlinear_winding_perturbations.py: bounded Gibbs, Cauchy-cell tail, and quartic-OU regression fixtures.
• tests/: deterministic unit tests.

Reproduce

Only the Python standard library is required.

python -m unittest discover -s tests -v
python scripts/check_finite_claims.py --max-word-length 8 --write-results
python scripts/run_controlled_braid_sandbox.py --trials 500 --write-results
python scripts/check_control_energy_claims.py --max-word-length 7 --write-results
python scripts/check_moving_well_claims.py --write-results
python scripts/check_winding_bridge_claims.py --write-results
python scripts/check_nonlinear_winding_perturbations.py

What Is Actually New Here?

The configuration-space and discriminant-winding facts are classical. The elementary word lemmas are also standard. The plausible research contribution is the specific stochastic observable:

Ginibre or locally nonnormal matrix path
    -> tracked local eigenvalue triple
    -> signed B3 word
    -> grammar and closure spectrum,

especially when conditioned jointly on spectral-pocket geometry and eigenvector-overlap data. The present repository organizes that question; it does not yet prove a new asymptotic law.

The control branch adds elementary but useful guarantees: every Artin word has an explicit collision-free realization; fixed-endpoint tracking within half the reference clearance preserves the full braid; and Brownian endpoint pinning has a quantitative, non-unit fidelity bound. These are classical control/topology sandbox results, not claims of fault-tolerant topological quantum computation.

The follow-up energy layer shows that this simplest compiler is exactly order-blind: at equal spacing, every length-L word executed in time T costs pi^2 L^2/(2T). It also identifies Delta^2/(2T) as the exact Cameron-Martin action for exiting the sufficient half-clearance bridge tube. Packet-order energy and actual braid-change large deviations remain open only after this baseline is removed.

The nonlinear moving-well layer pursues the blog's earlier proposal to replace Gaussian bridges by interacting measures on the circle. A feed-forward quartic well has an exact autonomous co-moving Gibbs diffusion. Regular phase-offset wells rotating like a clock realize the central full twist, with every labelled pair winding together. This gives exact nontrivial topology but also an exact negative result for random generation: a perfect B3 clock reaches only I and -I in odd-prime Burau quotients. Rich nonabelian output therefore requires phase slips, sequential dwell structure, or explicit symmetry breaking. Controlled moving centers can compile any signed Artin word exactly, but that T35 controllability result is not a stationary random-support theorem; the autonomous claims remain C36-C40.

The Brownian-winding layer now corrects a foundational model boundary. Cotler's normalized integer-order Bessel coefficients form a symmetric Skellam law over angular Fourier modes, not over Brownian-loop winding sectors. Yor's true point-puncture Brownian-lace winding law is a Cauchy mixture with inverse-square tails and infinite variance. A Brownian bridge confined to a circle instead has discrete-Gaussian deck-sector weights. A periodic nonlinear well is governed by a twisted heat kernel, not automatically by either closed form.

The nonlinear perturbation layer answers the blog's 2015 suggestion symbolically. Exact conditional partition-function ratios describe the perturbed winding law; bounded path energies give explicit likelihood, KL, TV, and Taylor bounds; and a quartic OU drift has an explicit Girsanov radial occupation score. These are exact deductions from standard machinery, not claimed literature-new theorems. C42-C43 isolate the real open asymptotic questions.

The McShane layer gives a second exact scalar law without simulation. On a fixed punctured torus, normalized McShane summands define a probability measure on simple geodesics. If the torus is first sampled from normalized Weil-Petersson measure, the selected geodesic length has density (12/pi^2) l/(1+exp(l)), with explicit zeta-valued moments. This is a probability reformulation of McShane-Mirzakhani unfolding, not a claimed new proof. Mirzakhani's generalized pants kernel becomes, after integration and Laplace transform, the matrix-model recursion identified by Eynard and Orantin. The claim-by-claim review in MCSHANE_NOVELTY_AUDIT.md finds no defensible literature-new theorem in this branch. T43 is retained only as an apparently unrecorded elementary corollary pending specialist review, and C41 is now a research question rather than a stated conjecture.

The holonomy branch replaces the informal phrase "continuous braid generator" with three precise objects: a configuration-space diffusion generator, its lift to a cover with braid-group deck labels, and KZ/Chen holonomy. A bounded fixture verifies that second-level pair-log iterated integrals distinguish two ordered pure-braid packets whose first-level pair winding agrees.

The arithmetic branch adds a second exact shadow. The reduced Burau representation at t=-1 maps B3 into SL(2,Z). For a knot closure, its determinant is |2-trace(M)|; powers of a braid packet therefore obey an exact Cayley-Hamilton trace recurrence. This is a useful number-theoretic observable, but a prime determinant is not a proof that the closure is a prime knot.

The symbolic block branch adds a third exact layer. A perturbation embedded strictly through selected eigenvectors reduces to an autonomous 3 x 3 spectral block; pure pair winding is an additive Z^3 shadow but provably forgets packet order; and winding is impossible when twice the perturbation norm is smaller than the minimum active gap. Genuine influence from the outside spectrum begins only after an off-block coupling is introduced, where the Schur complement exposes an exact resolvent self-energy. A standard finite-range-dependent CLT also settles the abstract additive pair-winding limit under explicit stationarity, moment, and dependence assumptions; proving that Ginibre-derived episodes satisfy such assumptions remains open.

The finite-quotient branch treats a braid as a random object to be compressed, not necessarily closed and named as a knot. Reducing the Burau matrix modulo a prime maps onto all of SL(2,F_p). The lazy IID signed-letter baseline mixes rapidly by congruence expansion, while the Ginibre-induced transition law remains open. The exact amplitude threshold is stronger than a no-winding claim: below it the complete braid is trivial, forcing an identity atom in every quotient. Deterministic projection can expose or discard dependence but cannot create entropy.

Research Discipline

• A matrix Ginibre process is not silently replaced by Ginibre interacting Brownian motion.
• Eigenvalue gaps are not assumed to determine nonnormal dynamics without overlap information.
• Braid exponent sum is not treated as a complete braid or knot invariant.
• Motif enrichment is not treated as a knot invariant.
• Pocket selection must be evaluated out of sample.
• Candidate topology labels remain separate from confirmed names.