We determine exactly the random robustness of the quantum switch along its control-dephasing trajectory. In the five-qubit process-matrix convention W_v = Q_A + Q_B + vD, 0 <= v <= 1, with white process noise N = I_32/8, a rational parametrization gives the optimum over the entire physical interval. A rank-six Gram dual certificate provides a lower bound, and explicit causal decompositions into positive 32 x 32 matrices attain the same value. Exact Sturm and Bernstein certificates establish positivity on the whole interval; floating-point optimization is not used in the proof. As v tends to zero, corresponding to complete control dephasing, R_r(W_v) = (3/2)v^2 + (1/16)v^4 + (1/96)v^6 + O(v^8). At full visibility, R_r(W_1) = 1.5760266987417078... . The result gives the exact white-noise separability boundary for the trajectory introduced by Branciard. The accompanying archive contains the LaTeX sources, explicit certificates, verification scripts, an ordered-cone correspondence check, and completed computational verification records. The research workflow and substantive contributions of ChatGPT are disclosed in Section 7.1 of the manuscript.
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state
A, published as a 1,472-byte witness, with certified
Ic>0 at the exact rational
p=16239/250000=0.064956 (per-Pauli convention; total error rate
3p=0.194868) -- beyond both the best value printed in the Agarwal et al. paper (
0.064657) and the strongest state in the same authors' public repository (
≈0.064911), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in
p on
[0,1/4], so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of
A and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than
1/25000 -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.
The sine Loewner operator is a spatial compression of a whole-line Fourier exchange whose spectrum is contained in {−1, 0, 1}. We determine two effects of truncation: a logarithmic transition in the outer spectral bands and algebraically interacting states in the central gaps. An exact reduction of the infinite Bloch fiber gives a trace law with a remainder bounded uniformly by the first two derivative norms. We evaluate its Schatten and determinant coefficients and transfer it to finite matrices with an error independent of dimension. Every fixed nonzero central-gap level occurs at exactly one half-line boundary phase modulo one and is simple. An explicit Mellin equation determines the physical tail exponent and the full leading interaction matrix, including the complementary-resolvent correction. Exact interval certificates isolate a phase-zero mode in (0.405, 0.428) and prove its normalized tail amplitude has magnitude greater than 1/100, giving a nonzero even–odd splitting. Further consequences include phase-dependent third terms, parity determinant asymptotics and odd logarithmic moments generated by additive frequency triples. This record contains the complete 88-page manuscript, with three figures, and a self-contained source archive. All mathematical proofs are included in the paper. The archive contains LaTeX sources, verification programs, exact rational witnesses and pinned dependencies. Floating-point diagnostics are distinguished from the rigorous interval certificates.
We present an explicit rational certificate for a positive-partial-transpose (PPT) state on C^15625 tensor C^15625 outside the cone C2 + partial_transpose(C2), where C2 denotes the cone of positive semidefinite operators of Schmidt number at most two. A 5x5 grid-state seed has realignment norm bounded below by 1028/909, proved by a 25x25 integer matrix certificate. Six tensor copies amplify this to (1028/909)^6 > 2, while the realignment norm is at most twice the trace on C2 + partial_transpose(C2). An explicit Choi witness consequently defines a linear map that is simultaneously 2-positive and 2-copositive but nondecomposable. The accompanying archive contains the mathematical preprint, LaTeX source, exact-arithmetic verification programs, and computational checks. This is an unrefereed proof manuscript. Independent expert verification and exhaustive prior-art assessment remain outstanding. The construction does not resolve the PPT-cubed conjecture or establish an improved quantum teleportation protocol.