One Control Parameter — the Random-Matrix Program in Six Papers
Research report · Six preprints published July 11, 2026
One control parameter, two substrates, one description
A vibrating plate and a training neural network admit the same random-matrix account: a single knob drags their spectra between universal fixed points. This is a reading guide to the six papers — the hypothesis, the tests, what held, what broke, and the one number that crosses substrates.
1 · Why Rosenzweig–Porter?The hypothesis: eight predictions, one preregistered number
↓ tests every registered prediction
2 · Open-source FEM testsTwenty preregistered campaigns; the verdicts
AI — the network
4 · Effective-rank collapseThe signature that transfers, as an order parameter
↓ background for
3 · A random-matrix account of structure formationThe full account, quantitative on the AI side
↘ the D₂ bridge (papers 2 ↔ 3: designed plates reach the network band) ↙
5 · The dynamic selector and the formal operad are one objectThe unification, instantiated in a real Clifford algebra and machine-checked in Coq
↓ and what the D₂ axis is for
6 · Multifractality as a sensing resourceThe payoff: the program's diagnostic becomes a metrological figure of merit — QFI enhancement exponent 1−D₂
Solid arrows are dependency (read the upper first); the diagonal is the quantitative bridge both lanes meet at.
How to read it
Wave-chaos / RMT physicist1 → 2, then §E20 of 2 and the scoreboard below; paper 5 if the algebra tempts you.
ML researcher4 → 3, then paper 2's designed-assembly section (the mechanical mixture-of-experts), then 5.
Formal-methods reader5 first, with the machine-checked proof open beside it; then 3 for what the operad instantiates.
Metrologist / cold-atom experimentalist6 first (the figure of merit and the platform), then the D₂ figure below, then 2 §E20 for the fabricated-medium story.
In a hurryThe scoreboard and the D₂ figure below — they carry the arc in two exhibits.
1
Why Rosenzweig–Porter? Evanescent coupling and a random matrix hypothesis for intermediate statistics in freely vibrating plates
Freely vibrating plates show intermediate (Rosenzweig–Porter) spectral statistics; simply supported ones are Poissonian. Nobody had said why. The hypothesis: within each symmetry sector the eigenfrequencies behave as eigenvalues of H = H₀ + λV, where H₀ is the uncoupled (integrable) base and V is the inter-mode coupling generated by evanescent waves at free edges. Because the operator is real and self-adjoint, V is orthogonal-class — and RP is the natural interpolation between Poisson (λ = 0) and GOE (λ → ∞), not a phenomenological fit.
The paper's discipline is its point: it commits before testing.
Eight testable predictions across boundary conditions, geometry, material, thickness, rotation, damping, and design.
One preregistered falsifiable number: the plate's eigenvector fractal dimension, D₂ = 0.76 ± 0.15.
A corrected time-reversal prediction: uniform rotation alone keeps an antiunitary protector; reaching the unitary class needs every vertical mirror broken too.
Eight seeded pre-tests (T1–T8) shipped with the deposit.
8 predictionsD₂ = 0.76 ± 0.15 preregisteredpre-tests T1–T8 included
2
Open-source finite-element tests of the Rosenzweig–Porter hypothesis: preliminary results
The registered program, executed: twenty preregistered experiment campaigns on a fully open-source stack (scikit-fem, quintic Argyris and C⁰ interior-penalty elements, certified eigensolvers), each with its frozen reading committed to git before the run. The verdicts populate the scoreboard below; the exhibits that follow are the campaign's highlights, in story order.
The core mechanism, confirmed by dialing the boundary. Sweeping a validated Winkler penalty from simply supported toward free edges walks the spacing statistics off Poisson into the intermediate regime — the boundary-controlled transition the hypothesis paper predicted, at +3.6σ against a same-protocol Poisson baseline.
A mechanical GOE → GUE crossover, with its protection theorem. The chiral rotor lands exactly on GUE (⟨r⟩ = 0.5992 ± 0.0071); the mirror-symmetric rotor is pinned to GOE at every speed by the surviving antiunitary σᵥT; asymmetric mistuning breaks the protection at +7.8σ.
The Ω² physics helps. The deferred centrifugal terms advance the crossover rather than destroying it, and protection survives finite deformation — the crossover completes within validated moderate-strain physics at ≈1,500 RPM for a decimeter silicone disk, following the universal Pandey–Mehta one-parameter law. A tabletop experiment waiting for a lab.
Gap A resolved by a controlled protocol comparison. The registered truncated-operator ladder develops RP-like fractal scaling (D₂ = 0.42–0.50) — but on the true operator the same eigenvectors stay sparse (≈1–2 beam products per mode) through N = 1024 per sector at gate-certified windows. The scaling is a property of the truncation protocol, not of the plate.
The boundary-adaptedness hierarchy, spacing face. A +12.2σ step separates the ellipse (p = 2, coordinate-adapted) from the smooth soft-cornered superellipse (p = 3) — what controls the statistics is not curvature or corners but whether the boundary fits separable coordinates.
The same hierarchy, eigenvector face. The eigenvectors order the same way as the spacings, and the genuine true-operator delocalization runs rectangle 0.01 → triangle 0.14 → superellipse 0.25 — exactly as the coupling picture predicts.
The published observation, replicated protocol-faithfully. The source sector result turns out to be a pooled angle sweep; re-executing that exact protocol supports its direction at +9.5σ over a same-protocol Poisson baseline, with the pooling methodology validated as bias-free against matched fixed-angle references — and a real angle structure underneath (the 75° sub-Poisson dip, identified as a low-mode boundary effect).
The same account, measured on networks. Level repulsion does not transfer to dense nets — they are born GOE, lacking the plate's integrable base — but it is fabricable: a mixture-of-experts with decoupled sectors is born Poisson, and when the task forces the sectors to cooperate, the spectrum crosses Poisson → GOE coincident with the loss drop (a causal dose–response, plus a capacity-attenuation law for why large dense models never show it). The unitary class is reachable two ways: the causal mask, exactly and at any scale; a directional task, genuinely, given positional capacity. And at the eigenvector level:
First, the instrument. The D₂ recipe reproduces random-matrix theory on synthetic ensembles (RP, GOE, banded) and discriminates exactly where the spacing ratio is blind — the calibration that makes the number mean something.
Then, the measurement. On the exact Hessian, eigenvectors at the fabricated transition are multifractal with D₂ = 0.76 — inside (0, 1), neither localized nor ergodic — robust across five architectures (band 0.56–0.80) and co-transitioning with the spacing statistic on the same operator (correlation 0.97). Not spacing mimicry: the real phase.
D₂ = 0.76architecture band [0.56, 0.80]co-transition corr 0.97negatives reported
4
Effective-rank collapse marks the semantic phase transition in neural networks
The short note on the signature that does transfer. As a network learns, the singular-value spectrum of its read-out collapses from a Marchenko–Pastur-like bulk to a concentrated spectrum — and this collapse is an order parameter for the semantic transition, reproduced across six substrates up to a trained 7B model (weights, live activations, and over training), with a grokking experiment that decouples it from the optimizer clock. The sharpest result is the dose–response: corrupting labels grades the collapse's coupling to generalization (|corr| 0.83 at 0% → 0.41 at 100%) while the bare magnitude stays generic. The coupling, not the magnitude, is the semantic signature.
The transferring signature. Effective-rank collapse across substrates, with a frozen-backbone null control that does not move. Honesty clause built in: the sibling signature, level repulsion, demonstrably does not transfer to dense networks — and the note says so.
At real scale. The actual trained 7B model (Qwen2.5-7B + LoRA + a discrete Clifford head) carries the signature in its trained weights — analyzed directly from the checkpoint, no GPU required.
The unification, made literal instead of analogical. The RMT account needs three ingredients — sectors, a coupling, a selector. A sibling formal program proposes an algebra of meaning with exactly three ingredients — the grades of an integer Clifford algebra Cl(4,0;ℤ), its kernels (operations of a Coq-verified coloured operad), and a selector. Instantiate the RMT model with the actual grades, kernels, and gate: the signatures survive, the functor is quantitatively homomorphic (1−d = 0.97 on real compositions), and the composition laws are not just claimed but machine-checked — 56 Qed, 0 Admitted, one typing parameter, re-verified by the independent kernel checker coqchk (and re-run at publication time; the proof file is public).
Grades as sectors. A mixture-of-experts whose sectors are the algebra's grades and whose coupling is the geometric product crosses Poisson → GOE with grade delocalization — the same fabricated transition as paper 3, built from the formal program's own objects.
The real functor. The full functor M folded over 89 real decomposition trees is already in the coupled (GOE) regime at zero explicit coupling — real language trees couple the kernels by themselves.
56 Qed · 0 Admittedfunctor 1−d = 0.97coqchk re-verified
6
Multifractality as a sensing resource: a D₂ figure of merit for Rosenzweig–Porter systems, and a lattice platform
The payoff paper: what is D₂ for? Composing two established-but-never-joined results — the identity between fidelity susceptibility and quantum Fisher information, and Kravtsov's exact result that RP fractal states are hypersensitive to perturbations — turns the program's diagnostic into an instrument specification: along the RP fractal phase, for local (field-like) couplings, the QFI enhancement exponent over an ergodic sensor is 1 − D₂. Confirmed at 4/4 preregistered gates on measured D₂ (correlation 0.996); boundaries measured the hard way — a disorder-free fractal lattice correctly shows no effect, a third random ensemble extends the law 3/3 — so the recipe is precise: multifractal states of random ensembles, sensed locally. Ships with an operator design rule (local couplings inherit; collective ones are blind) and a constructed cold-atom platform (interacting tilted+disordered lattice), with the honest scoping stated up front: this attaches to lattice/Bloch gravimetry, not free-fall interferometry.
Post-publication follow-ups — preregistered in public, run overnight (July 12)
The paper's own registered follow-up, closed affirmative. The platform section names "Q2c, finer grid, three sizes" as the open question; twenty-four hours after deposit it ran: at L = 12/14/16 (Hilbert dimension up to 12,870, every cell certified against an independent eigensolver) the typical-sensitivity peak rides the moving ergodic edge of the crossover — W* = 2.00 → 2.25 → 2.64 tracking W_c = 2.88 → 3.10 → 3.19. TRACKING-CONFIRMED.
The Sierpinski null had a mechanism inside. Stratifying the disorder-free fractal lattice by state class: the non-degenerate states follow the 1 − D₂ law (deviation 0.049); the exact degenerate multiplets — where the deterministic fractal weight lives — actively suppress sensitivity (enhancement −0.42, the Anderson-like branch, now measured in-house). The criterion's "randomness" ingredient is thereby explained: randomness matters because it lifts exact degeneracies while preserving multifractal correlations. Both branches — enhancement and suppression — measured with one instrument.
The scoreboard — paper 1's claims vs. paper 2's verdicts
Every row was preregistered before its run; verdicts are quoted from the published campaign record. CONFIRMED means the frozen reading fired; REFUTED AS STATED means the claim failed under the true operator, with the mechanism identified; EXCEEDED means the campaign delivered more than the prediction asked.
Boundary-controlled transition at +3.6σ; simply supported and exactly-reduced controls (disk, annulus, Lévy-separable) land on Poisson.
Coupling strength tracks boundary adaptednessgeometry program
CONFIRMED
Three-tier hierarchy; +12.2σ step from ellipse (p = 2) to soft-cornered superellipse (p = 3); genuine eigenvector effect ordered 0.01 → 0.14 → 0.25 (top point at 3.5σ).
Plate eigenvectors are RP-multifractal, D₂ = 0.76 ± 0.15the preregistered number
REFUTED AS STATED
The truncated-ladder protocol manufactures RP-like scaling (0.42–0.50); the true operator stays sparse through N = 1024/sector. What remains is a real, ordered, weak residue (0.01–0.25) — an order of magnitude below the preregistered value.
Material tunability λ(ν)Poisson-ratio trend
CONFIRMED
ν-trend matches the companion's Ritz values method-to-method; coupling is post-designable via a validated second-order control law.
Thick plates move GOE-wardMindlin prediction
CONFIRMED
Mindlin thickness sweep at +5.4σ in the predicted direction.
Chiral rotor lands exactly on GUE (0.5992 ± 0.0071); σᵥT protection exact at the eigenvector level (R_S = 1.000000, falsifiable by mistuning at +7.8σ); the crossover follows the universal Pandey–Mehta family and completes at ≈1,500 RPM for a decimeter silicone disk.
Damping realizes the AI† non-Hermitian classdissipation axis
PARTIAL
Distributed damping saturates between ensembles — closer to AI† (2.0 s.e.) than Ginibre (3.9 s.e.) but on neither; the right physical damping model is the registered open question.
The RP phase can be designed via point contactsconstructive axis
EXCEEDED
A mechanical mixture-of-experts (detuned plates + contact-spring network) fabricates the full Poisson → GOE transition with a genuinely multifractal phase; the fabricated D₂ is a monotone dial in coupler connectivity, reaching 0.63–0.69 at router-like coupling — inside the neural-network band.
The published sector observationthe observation that started it
REPLICATED
Faithful re-execution of the (pooled angle-sweep) protocol supports the published direction at +9.5σ, with the pooling methodology validated as bias-free; the 75° sub-Poisson dip identified as a low-mode boundary effect.
The refuted row is the program's strongest exhibit, not its embarrassment: the preregistered number existed precisely so that this comparison would mean something — and the campaign identified where the artifact comes from (the truncation protocol), reproduced it on demand, and measured what is really there instead.
The number that crosses substrates
Every thread meets at one observable: the eigenvector fractal dimension D₂. The figure below is the program in one axis.
Read it bottom-up. Nature's free plate refuses the preregistered dimension — the genuine effect is an order of magnitude smaller and ordered by geometry (green). The published-style protocol manufactures a middle value from truncation alone (rust). The neural network delivers the preregistered number almost exactly, as a genuine phase (blue). And when a plate assembly is designed with router-like connectivity, its fabricated dimension lands inside the network band (violet): what the free plate lacks, forced cooperation provides — on metal as on gradients.
And what the axis is for: paper 6 turns this same number into a sensing figure of merit — the QFI enhancement exponent of a multifractal random medium is 1 − D₂ under local probing. Lower on this axis (within the random-multifractal class) = a faster-improving sensor.
The constructive story behind the violet band. Designed point contacts dial the spacing statistics on demand — a validated second-order control law lets the coupling be engineered, not just observed. Its endpoint is the mechanical mixture-of-experts: detuned plates joined by a contact-spring network whose connectivity sets the fabricated fractal dimension, all the way into the neural-network band at router-like (complete-graph) coupling.
What remains open
The tabletop rotor. A decimeter silicone disk at ≈1,500 RPM should complete the GOE → GUE crossover — with the protection invariant R_S measurable from complex operational deflection shapes. Fully predicted; needs a lab.
A physical mechanical mixture-of-experts. The designed assembly (detuned plates + contact posts) is specified to the coupler; building one would put the fabricated multifractal phase on a bench.
The atomic unitary crossover. No GOE → GUE level-spacing crossover has ever been measured in an atomic system; the kicked-rotor route with a synthetic gauge field is first-in-class territory.
The damping model. Which physical dissipation realizes a canonical non-Hermitian ensemble — if any — is the registered open question of the AI† axis.
Real spectra. The protocol-artifact finding makes a sharp prediction about any measured plate spectrum analyzed with truncated ladders; running laboratory data through the campaign's gates is the natural next collaboration.
