Fourth-Moment Strong Universality for Finite-Type Transpose-Correlated Random Matrices

1 Introduction

• Strong convergence controls normalized traces and operator norms of every fixed noncommutative *-polynomial, including spectral edges and singular values beyond empirical eigenvalue distributions.The distinction matters in nonnormal problems, where polynomial singular values are not captured by the limiting eigenvalue measure alone.Source: §1 · p. 1

Main contribution

• The full real covariance of unordered-pair atoms determines an explicit joint strong limit, including masked circular cross-type blocks and independent symmetric and antisymmetric semicircular sectors within each type.The result covers fixed matrix amplifications and fixed noncommutative *-polynomials, with deterministic companions jointly strongly convergent with the type projections.Source: Main contribution · p. 2, §1 · p. 2
• Nested corners converge almost surely on one common event, whereas fixed-law nonnested rows converge in probability for each fixed test; the fourth-moment threshold is sharp for the nested Wigner subfamily.At the threshold, fresh rows do not yield a coupling-invariant almost-sure upgrade of the in-probability conclusion.Source: Main contribution · p. 2
• The application gives strong limits for finite-separable continuous left/right profiles, including literal-transpose terms, but does not cover arbitrary continuous covariance kernels.The finite-type theorem is a transfer principle; the transpose is included as an additional color at the unordered-pair level.Source: Main contribution · p. 2

Relation to previous work

• The claimed extension combines exact-fourth-moment universality, finitely typed color/orientation covariance, jointly strongly convergent companions, and fixed-level strong convergence beyond earlier homogeneous models.Prior work supplies comparison tools and results for Wigner coordinates, Gaussian profiles, or weaker eigenvalue-distribution targets, but not this combination.Source: Relation to previous work · pp. 3–4

Proof strategy

• The proof uses companion-augmented linearization, covariance-matched Gaussian comparison, and radial truncation with sector-specific tail controls before upgrading pencil bounds to joint strong convergence.The nested and nonnested couplings require different convergence arguments: a common almost-sure event versus convergence in probability for each fixed test.Source: Proof strategy · p. 4

2.1 Endpoint types and unordered-pair atoms

• Independent unordered-pair atoms may have arbitrary within-atom coordinate dependence, subject to reversal consistency and same-type endpoint exchangeability; the diagonal contributes no limiting component.The diagonal’s norm vanishes almost surely in the nested model and in probability in the fixed-law nonnested model.Source: §2.1 · p. 5

2.2 Deterministic companions and fixed amplified tests

• The deterministic type projections and companions must converge jointly strongly, because separate strong limits do not determine mixed words.The tests use a matrix level and polynomial chosen independently of n.Source: §2.2 · p. 6

2.3 The covariance-matched free family

• Cross-type blocks use circular sources, while same-type endpoint exchangeability yields independent endpoint-symmetric and endpoint-antisymmetric semicircular sectors.The cross-type orientations share a circular source to retain orientation covariance.Source: §2.3 · pp. 6–7
• The covariance-matched Gaussian tuple has the same joint moments and scalar and matrix-amplified polynomial norms regardless of the real covariance factorizations chosen.Its Gaussian realization converges jointly strongly with the deterministic tuple.Source: §2.3 · p. 7

2.4 Main strong-universality theorems

• Nested corners converge strongly almost surely on one common event, while fixed-law nonnested arrays converge in probability for each fixed test.Both formulations use the covariance-matched family as their limit.Source: §2.4 · p. 7

2.5 Finite-separable profile application: preview

• The profile application covers fixed finite-separable continuous masks, including transpose coordinates that reuse the original free sources with an endpoint-reversal sign.It does not extend by uniform density alone to arbitrary covariance kernels without an additional operator-norm estimate.Source: §2.5 · p. 8

3.1 Self-adjoint coordinates and companion-augmented pencils

• Non-self-adjoint variables can be replaced by self-adjoint coordinates, and companions enter the deterministic part of affine pencils; arbitrary polynomial norms reduce to self-adjoint tests at size 2q.The deterministic and random coordinates are separated in the pencil.Source: §3.1 · p. 9

3.2 Fixed matrix amplification

• The fixed-amplification lemma extends strong convergence to each separately fixed matrix level, but gives no uniform estimate as the level grows.In particular, it yields no conclusion for q = q_n →∞.Source: §3.2 · pp. 9–10

3.3 From affine pencils to strong convergence

• Control of every fixed companion-augmented affine pencil implies joint strong convergence at every separately fixed amplification.Self-adjoint linearization, matrix-unit polarization, and a faithful-trace lower bound supply the deterministic upgrade from pencil estimates to polynomial norm and trace convergence.Source: §3.3 · pp. 10–11

4.1 Exact covariance factorizations

• Realification makes the Gaussian factorizations act on the full real covariance, including complex pseudocovariances.The Gaussian sources are driven by shared real vectors for the two orientations of each cross-type edge.Source: §4 · p. 11, §4.1 · p. 11
• Real covariance factorizations retain correlations among colors and orientations, while endpoint exchangeability makes the same-type symmetric and antisymmetric Gaussian sectors independent.The two same-type sectors yield free semicircular families; masked cross-type blocks yield free circular variables.Source: §4.1 · pp. 11–12, §4.3 · pp. 13–14
• The same-type polynomial model matches the off-diagonal law exactly, and its outer factor 2^-1/2 is compulsory.A separate diagonal correction handles prescribed comparator diagonals and is norm-negligible.Source: §4.1 · p. 12

4.2 Real Ginibre matrices as doubled GOE corners

• Doubled GOE matrices realize the Ginibre sources as polynomial corners, allowing scalar strong convergence to transfer from the GOE theorem.The diagonal covariance correction is norm-negligible and does not change that conclusion.Source: §4.1 · p. 12, §4.2 · p. 13

4.3 Compression and the explicit free limit

• The joint free limit is independent of the chosen real covariance factorizations, including at the level of amplified polynomial norms.Independent doubled-GOE sources give freeness across distinct type pairs and sectors.Source: §4.3 · p. 14
• For every fixed matrix amplification and polynomial, the comparator converges in normalized trace and operator norm to the explicit free limit.The convergence holds on one probability-one event; no uniformity in the amplification size is asserted.Source: §4.3 · p. 14

5.1 One self-adjoint summand per unordered pair

• Each unordered-pair vector must remain one self-adjoint summand, since splitting its orientations would discard dependence that belongs to the pair law.The Gaussian replacement matches the full real covariance, preserving correlations among colors and orientations.Source: §5 · p. 15, §5.1 · p. 16, §5.3 · p. 17

5.2 The three comparison parameters

• For fixed pencils, the comparison parameters remain bounded independently of n and are unchanged by adding an arbitrary deterministic self-adjoint shift.The bounded-support hypotheses apply to the pair atoms and, when retained, the diagonal atoms.Source: §5.2 · p. 16

5.3 Quantitative comparison with the Gaussian pencil

• Spectral comparison, expected-moment comparison, and concentration play distinct roles in transferring the Gaussian pencil limit.The concentration step requires uniformly bounded expected roots.Source: §5.3 · p. 18

5.4 Transfer of the Gaussian strong limit

• Bounded, centered, endpoint-consistent off-diagonal pair laws with zero diagonal yield joint strong convergence to the covariance-matched limit at every fixed matrix amplification.The convergence is almost sure on one common event in the nested model and in probability separately for each fixed matrix-valued polynomial in the nonnested model.Source: §5.4 · p. 19
• The transfer covers mixed polynomials in the deterministic companions, but neither argument gives uniformity for amplification sizes growing with n.For nonnested rows, the fixed-pencil limits hold in probability.Source: §5.4 · p. 19

6 Removal of boundedness at the fourth-moment threshold

• Truncation is applied to each entire unordered-pair atom so that correlations among colors and the endpoint-reversal law are preserved.Coordinatewise truncation need not preserve either property.Source: §6 · p. 19

6.1 A continuity estimate

• The continuity estimate controls changes in polynomial norms and normalized traces when two bounded tuples are close in operator norm.Its bound follows by telescoping the factors in each word.Source: §6.1 · p. 20

6.2 Centered radial truncation

• Centered truncation preserves the endpoint-swap identities for both truncated vectors and their residuals.The Euclidean truncation ball is invariant under the endpoint swap.Source: §6.2 · p. 20

6.3 Cross-type residual blocks

• Cross-type residual directions are completed in separate iid arrays, controlling their norms without imposing independence on the two correlated directions.The reversal relation gives a common law within each ordered block.Source: §6.3 · p. 20

6.4 Same-type residual blocks

• Same-type residuals are split into real symmetric and skew-symmetric arrays, whose norms are controlled using finite-fourth-moment bounds.The nested residual estimate follows after combining these bounds over finitely many colors and types.Source: §6.4 · p. 21

6.5 Diagonal entries

• Finite second moments suffice for diagonal entries: deleting or restoring the diagonal does not affect any fixed polynomial test.For nonnested rows, the corresponding conclusion holds in probability.Source: §6.5 · p. 22

6.6 Passage to the full covariance and proof of the main theorems

• Passing from truncated laws to the full covariance gives the strong-limit assertions, with almost-sure convergence for nested arrays and in-probability convergence for fixed nonnested tests.The nonnested result supplies neither a common almost-sure event nor uniformity in coefficient size.Source: §6.6 · p. 22

6.7 Why the coupling distinction is sharp

• The fourth moment is necessary for the nested almost-sure formulation on the Wigner subfamily, while fresh independent rows can have nonsummable spike probabilities.Such spikes can prevent almost-sure edge convergence even though the fixed-test in-probability conclusion remains valid.Source: §6.7 · p. 22

7 Finite-separable continuous profiles

• The corollary gives strong limits for fixed matrix-valued polynomial tests of finite-separable continuous profiles, almost surely in the nested model and in probability for fixed-law nonnested rows.The normalized traces converge in the same respective modes.Source: §7 · pp. 23–24
• Literal transposes are installed as additional colors at the unordered-pair level, so their covariance is retained rather than inferred from strong convergence of the original matrices.The application permits separately weighted transpose terms.Source: §7 · pp. 23–24
• The example uses distinct position-dependent weights for the direct and endpoint-reversed coordinates.The displayed profile combines the two orientations with different functions of the row and column locations.Source: §7 · p. 24
• For each fixed λ, the singular-value sets of Z_n − λI converge in Hausdorff distance to the spectrum of |z − λ|, in the same mode as the strong limits.This does not assert convergence of the eigenvalue support of the nonnormal matrix.Source: §7 · p. 24
• The profile class allows fixed finite sums of separable continuous masks generated from finitely many homogeneous latent colors, not arbitrary continuous covariance kernels or growing separable rank.Uniform kernel approximation alone does not provide the dimension-free Schur-multiplier bound needed to pass to the limit.Source: §7 · p. 25

8.1 Spectra and singular values of fixed polynomials

• Fixed self-adjoint polynomial spectra converge in the nested model and in probability for fixed-law rows; singular-value sets of fixed, possibly non-self-adjoint polynomials converge in the same modes.The result rules out asymptotic operator-norm outliers for covered self-adjoint polynomials, but does not establish eigenvalue-support or Brown-measure convergence for nonnormal polynomials.Source: §8.1 · p. 25

8.2 A covariance universality principle

• Under the hypotheses, identical full real covariance matrices for every type-pair law give the same joint strong limit with every admissible deterministic companion tuple.The covariance includes color, pseudocovariance, and endpoint-orientation correlations, while independence across distinct unordered pairs remains required.Source: §8.2 · p. 25
• The finite-fourth-moment threshold supports an operator-norm statement, not quantitative concentration: removing truncation under L4 yields qualitative convergence without an exponential-tail or finite-sample concentration claim.The passage distinguishes quantitative bounded comparison estimates from the unbounded model.Source: §8.2 · p. 25

8.3 Directions beyond the finite-type theorem

• Arbitrary continuous kernels, growing numbers of types, colors, separable terms or amplification levels, and continuum-varying endpoint laws are outside the claimed uniform regimes.These extensions require additional stability or uniformity estimates.Source: §8.3 · p. 26
• The fourth-moment threshold is sharp for nested almost-sure convergence already on the Wigner subfamily; arbitrary fresh rows do not permit a coupling-invariant almost-sure upgrade of the in-probability conclusion.The fixed-law nonnested conclusion remains convergence in probability.Source: §8.3 · p. 26

A.1 Real covariance and directed cross-type blocks

• The corner calculations preserve the full real covariance, including information not contained in the Hermitian covariance, and lose no type-proportion factor.A real covariance factorization yields the coefficients used to realize directed cross-type blocks; reversal consistency and distinct unordered edges give the required covariance and independence.Source: A Covariance and normalization… · p. 26, §A.1 · p. 26

A.2 Exchangeable endpoints and the factor 2−1/2

• Endpoint exchangeability makes the symmetric and antisymmetric Gaussian sectors independent, and the same-type edge construction requires the outer factor 2^-1/2.Without that factor, every real covariance in the block would be doubled.Source: §A.2 · p. 27
• The diagonal entries need not match the comparator covariance, so the proof couples off-diagonal parts and subtracts diagonal Gaussian vectors to form a correction.The stated variance bound controls this correction through a Gaussian tail bound and union bound.Source: §A.2 · p. 27

A.3 The doubled corner and its trace

• The GOE block construction makes every entry of the off-diagonal block an independent Gaussian entry, including its diagonal positions.The associated unordered global index pairs are distinct.Source: §A.3 · p. 27
• The limiting corner projection has trace 1/2, so its normalized corner trace is twice the ambient trace.The finite-dimensional calculation records the corresponding normalized trace factor.Source: §A.3 · p. 27
• In the normalized corner, c is standard circular with variance one, not variance 1/2.The calculation gives the circular covariance identity and vanishing higher cumulants.Source: §A.3 · p. 27

A.4 Type proportions are already encoded by compression

• The limiting fractions of admissible column indices appear automatically as τB(pa) = πa and τB(pb) = πb, without extra square-root factors.These proportions are encoded by compression.Source: §A.4 · p. 28
• The displayed limits retain the star-covariance and pseudocovariance, with normalization dictated by the n−1/2 entry scale.Mixed covariances between distinct type-pair components vanish by freeness.Source: §A.4 · p. 28

B Amplification and target-independent linearization

• The matrix levels and all polynomials are fixed before the matrix dimension tends to infinity.Source: B Amplification and target-independent… · p. 28

• Strong convergence bounds the coordinate sequences, enabling a polynomial-algebra map into the C*-ultraproduct.The proof begins by fixing a free ultrafilter and representing each coordinate by its sequence class.Source: §B.1 · p. 28
• Vanishing polynomial evaluations make the map independent of the chosen polynomial representative, and the resulting *-homomorphism is injective.These steps establish a well-defined embedding of the limiting polynomial algebra.Source: §B.1 · p. 28
• Complete isometry transfers the norm identity to every fixed matrix amplification.The argument uses the canonical identification of the fixed matrix level with its matrix ultraproduct.Source: §B.1 · p. 28
• The resulting convergence is for each fixed matrix level; the argument provides no uniform control as the level varies.The arbitrary-ultrafilter argument yields ordinary convergence, and scalar trace convergence gives the corresponding trace statement.Source: §B.1 · p. 29

B.2 Quotient spectra and the polynomial upper bound

• Quotient spectra and self-adjoint linearization transfer resolvent bounds to polynomial evaluations, yielding an upper bound for the polynomial norm.The linearization coefficients depend only on the polynomial and spectral parameter, not on the target's structure.Source: §B.2 · p. 29

B.3 Polarization and the faithful-trace lower bound

• Polarization recovers traces of ordered words from self-adjoint pencil convergence, and faithfulness of the limiting trace supplies the norm lower bound.Fourier extraction of a cyclic coefficient recovers each ordered word's trace.Source: §B.3 · p. 30

C Precise forms of the external probabilistic inputs

• The cited probabilistic inputs divide the proof among bounded-summand matrix comparison, Gaussian strong-limit identification, and edge control of fourth-moment residuals.The comparison retains full real and imaginary entry-coordinate covariance within an unordered-pair summand.Source: C Precise forms of the external… · p. 30, §C.1 · p. 31

C.1 Brailovskaya–van Handel comparison

• The comparison framework permits dense deterministic companion pencils when random terms attached to each unordered pair are grouped first.The comparator uses the same mean and full covariance as the self-adjoint matrix model.Source: §C.1 · p. 31

C.2 Fan–Sun–Wang and the doubled GOE model

• Doubled GOE matrices provide real Ginibre corners, and the resulting Gaussian theorem applies at scalar matrix level before separate fixed-level amplification.No uniformity for a growing amplification is asserted.Source: §C.2 · pp. 31–32

C.3 Bai–Yin for cross-type residuals

• For cross-type residuals, auxiliary iid-array completions and orthogonal compression retain the original pair atoms' prescribed orientation correlations.The two directed orientations are completed in separate arrays; their dependence inside each original atom is not replaced by independence.Source: §C.3 · p. 32

C.4 Anderson for same-type symmetric and skew residuals

• The same-type residual argument treats real symmetric and skew coordinates one at a time, allowing arbitrary dependence among the original atom's four real coordinates.The skew-coordinate construction uses a twisted model and a two-sign argument.Source: §C.4 · p. 32