The Truncated Weil Form as a Toeplitz Sequence: Boundedness, Conditional Positivity, and the Prime-Pole Bond (IMM Paper 14)

We recast the Weil positivity criterion for the Riemann Hypothesis (RH) as a statement about a single real sequence. A box basis of width delta for compactly supported even test functions reduces the Weil explicit-formula quadratic form to the Hermitian Toeplitz form generated by an explicit sequence A (m) = POLE (m) - PRIME (m) + ARCH (m), assembled from the pole of zeta at s=1, the von Mangoldt prime data, and the archimedean psi-integral. Weil's criterion becomes: A (m) is positive-definite for every window length and scale. Four findings, each epistemically labeled: (i) even boundedness of A (m) is equivalent to RH via von Koch, locating the difficulty precisely; (ii) Numerics built from primes up to 3. 4e6 the form is positive definite out to window 15 and a matrix-pencil method recovers the first dozen zeros from primes alone; (iii) Proposition a quantitative conditional window-positivity bound given zero-verification to height T; (iv) Numerics reproduction of the Connes-Consani archimedean positivity mechanism in Toeplitz form, exhibiting the prime-pole bond. This paper does not prove RH; global positivity is, by (i), equivalent to it.