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Abstract

We prove the complete lower bound q_H[h] ≥ −1.5 × 10⁻¹¹ ‖h‖² on the original constrained minimal odd domain in its reciprocal-theta norm. The certificate retains the full joint covariance of 4080 phase columns and five exact polar/null rows. Shorting that certified 4085-dimensional receiving Gram gives a directed 4080-dimensional exact-row Schur certificate. Its proof includes the complete arithmetic remainder, infinite frequency and physical tails, source conversion errors and the original-domain return. A real Bessel suppression argument and complete spatial-endpoint control sharpen the approximation without changing the frame. Three complementary compact-support certificates, in the distinct ordinary physical L² norm, give A₈ ⪰ 10⁻⁴⁰I + 4.75 × 10⁻¹¹Pₑ₇,₈, A₉ ⪰ 10⁻⁵⁰I + 4 × 10⁻¹²Pₑ₈,₉ and A₁₀ ⪰ 10⁻⁶⁰I + 10⁻¹³Pₑ₉,₁₀, with explicit sharp directional upper trials and joint eleven-, thirteen- and fifteen-feature receivers. An exact ground transformation preserves signed arithmetic pairing through closure even where the two positive edge energies separately diverge. It yields a norm-convergent paired tail, a sharper localized sign bound and exact compatibility with polar repair. Further results give fixed-null-prefix funding, all-support gamma approximation, whole complementary-space bounds and an explicit fixed-cutoff exterior obstruction. These are certified original-domain and physical receiving results with their respective norms and hypotheses.