Kernel Positivity from Physical Principles: Macroscopic Locality Closes the Gap, Information Causality Faces Seven Benchmarks (Part 6)

Part 6 of this record supersedes Parts 1-5. Earlier parts conjectured that Information Causality (IC) alone enforces the positive-semidefinite (PSD) structure of translation-invariant correlation kernels; the Sole-Origin and Eigenvector versions (Parts 1, 4, 5) claimed constructive proofs. This paper withdraws those proofs and replaces them with what can be proven. Two structure theorems show the withdrawal is structural, not incidental. The Barrier Lemma: any protocol whose measurement settings are independent of the encoded data satisfies S ≤ E(0)²‖u‖₂² ≤ 1, for every Bob strategy and every resource model (verified by exhaustive maximization over all Bob functions at N=4). The Pyramid Factorization Theorem: the Pawłowski concatenation satisfies S2ⁿ = Π levels ≤ (sup S₂)ⁿ, so nesting can never rescue a kernel that a single level cannot detect. Two positive results take their place. A visibility dichotomy: at sub-unit visibility, explicitly non-PSD kernels are quantum-realizable via a Tsirelson factorization with sign-flipped harmonics, so no correlation principle can enforce PSD there; at unit visibility (E(0) = −1), Cauchy-Schwarz rigidity forces single-family Gram structure. The main theorem: Macroscopic Locality implies kernel positivity at unit visibility (via the Q¹ characterization of ML plus the rigidity argument). The axiomatic gap open since Pawłowski et al. (2009) is thereby closed, by Macroscopic Locality. Corollary: every unit-visibility non-PSD kernel lies outside the Almost-Quantum set, confirmed numerically by robust NPA-1+AB semidefinite programming (N = 7 settings, Clarabel, PSD control at 10⁻⁹). Whether the strictly weaker principle IC suffices remains open, and is sharpened by seven benchmark kernels, unit visibility, non-PSD, provably outside Almost Quantum, that resist every IC protocol tested: optimized N=2/3 functionals, general random-access codes up to four bits, the full concatenation pyramid with free angles, linear parity encodings, entropic and mutual-information functionals, and continuous-angle protocols. Either a fundamentally new protocol detects them (IC ⇒ PSD), or one of them satisfies IC (IC strictly weaker than ML here). Both outcomes are discoveries; the benchmark coefficients are given in full and machine-readable. An open verification suite (6 self-contained Python scripts, MIT license, ~90 s runtime) is included and reproduces every numerical claim: kernel validity, the Barrier Lemma (three exact checks), the Pyramid Factorization, the Tsirelson factorization of the dichotomy, the NPA-1+AB distances for all seven benchmarks, and the H² bound. Following the documentation practice of this series, all withdrawn claims and their precise failure modes are recorded, including one hypothesis raised and refuted during review of this work.