Principal nonsingularity of the Fourier matrices of orders \(70\) and \(143\)

Abstract: We give computer-assisted proofs that every principal minor of each of the (70\times70) and (143\times143) Fourier matrices is nonzero. A lifting theorem of Caragea, Lee, Malikiosis, and Pfander reduces the two assertions to the nonvanishing of all principal minors of the Fourier matrix of order (10) in characteristic (7), and of order (11) in characteristic (13), respectively. We realize primitive roots in (\mathbb F_{7^4}) and (\mathbb F_{13^{10}}) and evaluate all (2^{10}) and (2^{11}) principal determinants by exact, division-free arithmetic. None vanishes. The lifting theorem in fact yields the stronger conclusions that every (10)-principal minor of the order-(70) matrix and every (11)-principal minor of the order-(143) matrix is nonzero. Self-contained standard-library verifiers for the finite-field calculations accompany the paper.

• The paper proves that every principal minor of the Fourier matrices $F_{70}$ and $F_{143}$ is nonzero, settling unresolved cases confirmt he conjecture linking principal nonsingularity with square-free orders.
• The proof involves finite-characteristic lifting and field determinant computations, evaluating only $2^{10}$ and $2^{11}$ determinants instead of the full $2^{70}$ or $2^{143}$ subsets.
• The findings have direct implications on applications involving Fourier matrices, confirm the square-free conjecture on a relevant case, improve calculus of helical properties and design of performance search through exact circulant.

Overview

This paper proves that every principal minor of the Fourier matrices $F_{70}$ and $F_{143}$ is nonzero, thereby settling the two smallest unresolved cases of the conjecture that a Fourier matrix has no zero principal minor if and only if its order is square-free. The proof is computer-assisted but exact: it combines the lifting theorem of Caragea, Lee, Malikiosis, and Pfander with finite-field determinant computations over $\mathbb F_{7^4}$ and $\mathbb F_{13^{10}}$, performed in division-free integer-free arithmetic. The key structural point is that only $2^{10}=1024$ and $2^{11}=2048$ determinants are evaluated, rather than the $2^{70}$ or $2^{143}$ principal submatrices one would naively need to inspect.

Background: from Chebotarëv to square-free orders

Chebotarëv's theorem states that when $N=p$ is prime, every square submatrix of $F_p=(\omega_N^{ij})$ is nonsingular. This property characterizes prime order: for composite $F_{143}$0, suitable $F_{143}$1 Fourier submatrices are singular. The theorem underlies the sharp finite uncertainty principle $F_{143}$2 on $F_{143}$3 [(Caragea et al., 30 May 2025) context; 0312398; math/0409506-related literature cited as Tao2005].

Restricting attention to principal minors changes the arithmetic boundary. Caragea and Lee proved that for $F_{143}$4, square-freeness is exactly the condition for nonvanishing of all $F_{143}$5 and $F_{143}$6 principal minors, while nonsquare-free orders admit zero principal minors of every intermediate size (Caragea et al., 2024). Motivated partly by woven Riesz basis constructions, where Cabrelli, Molter, and Negreira isolated the role of principal Fourier minors, this led to the conjecture

$F_{143}$7

Caragea–Lee–Malikiosis–Pfander proved the conjecture for families $F_{143}$8 with $F_{143}$9 via a lifting theorem for congruence-balanced minors, identifying $\mathbb F_{7^4}$0 and $\mathbb F_{7^4}$1 as the smallest three-prime and two-prime orders left open (Caragea et al., 30 May 2025). This paper closes both cases.

Reduction to characteristics 7 and 13

A subset-indexed minor $\mathbb F_{7^4}$2 is called $\mathbb F_{7^4}$3-principal when row and column residue-class counts modulo $\mathbb F_{7^4}$4 agree; ordinary principal minors ($\mathbb F_{7^4}$5) are $\mathbb F_{7^4}$6-principal for every divisor $\mathbb F_{7^4}$7. The lifting machinery used is:

Proposition (finite-characteristic lifting). If $\mathbb F_{7^4}$8 is square-free with $\mathbb F_{7^4}$9 prime and $\mathbb F_{13^{10}}$0 for every $\mathbb F_{13^{10}}$1, where $\mathbb F_{13^{10}}$2, then every $\mathbb F_{13^{10}}$3-principal minor of the complex $\mathbb F_{13^{10}}$4 is nonzero.

The reduction exploits inertness of the relevant primes. For the order-70 case: $\mathbb F_{13^{10}}$5, so $\mathbb F_{13^{10}}$6 stays irreducible over $\mathbb F_{13^{10}}$7, giving $\mathbb F_{13^{10}}$8 in which the class $\mathbb F_{13^{10}}$9 has exact order 10. Since 7 is inert in $2^{10}=1024$0 with $2^{10}=1024$1 the unique prime above it, nonvanishing of $2^{10}=1024$2 over $2^{10}=1024$3 is equivalent to $2^{10}=1024$4 — precisely the lifting hypothesis. Choice of identification among Frobenius conjugates $2^{10}=1024$5 is immaterial since $2^{10}=1024$6.

The order-143 case is parallel: $2^{10}=1024$7, so $2^{10}=1024$8 is irreducible over $2^{10}=1024$9, yielding $2^{11}=2048$0 with a primitive 11th root $2^{11}=2048$1, and 13 inert in $2^{11}=2048$2.

Consequently, proving all principal minors of the $2^{11}=2048$3 matrix over $2^{11}=2048$4 nonzero suffices for all $2^{11}=2048$5-principal (hence all ordinary principal) minors of $2^{11}=2048$6; likewise over $2^{11}=2048$7 for $2^{11}=2048$8. This is the paper's central quantitative claim: two computations involving at most 2048 exact field determinants replace exhaustive verification over $2^{11}=2048$9 and $2^{70}$0 subsets.

The exact finite-field certificates

Field elements are represented uniquely as polynomials in $2^{70}$1 (resp. $2^{70}$2) of degree below 4 (resp. 10) with coefficients in the base prime field, with multiplication reduced via $2^{70}$3 and $2^{70}$4. Each verifier first re-establishes the field structure using the Rabin irreducibility criterion and checks that the root has the correct exact order.

Determinants are computed by a division-free Laplace recurrence expanding along the last row, so every operation occurs exactly in the finite field — no floating-point arithmetic is involved anywhere. All masks are visited, including the empty set (determinant 1). The rankwise outputs are summarized below.

Order	Field	Subsets checked	Zero determinants	Product of all determinants
$2^{70}$5 reduction	$2^{70}$6	$2^{70}$7	0	$2^{70}$8
$2^{70}$9 reduction	$2^{143}$0	$2^{143}$1	0	$2^{143}$2

Because the ambient rings are fields, the nonzero aggregate product alone certifies that no individual factor vanishes. The characteristic-13 computation carries an independent audit: all 2048 determinants were also computed in $2^{143}$3 before reduction, with norms evaluated by both multiplication-matrix determinants and Sylvester resultants, agreeing throughout, and with total norm

$2^{143}$4

in which 13 does not appear — independently confirming the finite-field nonvanishing.

Combining the certificates with the reductions yields the main theorem: for $2^{143}$5 and every $2^{143}$6, $2^{143}$7. A remark notes the conclusions extend verbatim to normalized Fourier matrices and to either sign convention in the root of unity.

Reproducibility

The paper ships self-contained Python verifiers using only the standard library (Python 3.10+): two direct finite-field checkers and an independent norm-audit script for the order-143 case. Each script re-verifies irreducibility, root orders, binomial counts, and per-size outputs, and publishes SHA-256 digests of source files and canonical output records so results can be pinned against tampering. Notably, the ancillary programs do not implement the lifting proposition itself; that mathematical step is invoked from (Caragea et al., 30 May 2025), so the computational evidence and the theoretical reduction are cleanly separated. The acknowledgments state the results were obtained with assistance from GPT-5.6 Sol and that the finite-field computations were rerun through independent exact implementations.

Limitations and open questions

The proofs are conditional on two external dependencies that the paper states plainly. First, the lifting theorem is cited rather than reproved, so correctness rests on the published result of Caragea, Lee, Malikiosis, and Pfander. Second, the finite-field certificates are machine-checked rather than human-verified line-by-line; their assurance derives from the division-free design, the aggregate product checks, the independent norm audit (for $2^{143}$8 only — no analogous integer-norm audit is reported for $2^{143}$9), and the digest-pinned scripts, not from formal verification in a proof assistant. The method also does not extend automatically: each new order requires a prime $N=p$0 inert in the relevant cyclotomic field, and the paper leaves open whether the full square-free conjecture admits a uniform proof covering all remaining orders without case-by-case computation.

Conclusion

The paper establishes principal nonsingularity of the Fourier matrices of orders 70 and 143 by reducing, via finite-characteristic lifting, to exhaustive exact determinant computations in $N=p$1 and $N=p$2 involving only 1024 and 2048 cases respectively. No computed minor vanishes, and an independent integer-norm audit corroborates the larger computation. With these two cases closed, the smallest open instances of the square-free-order conjecture move beyond $N=p$3.