Source Rigidity in Bandwidth-One Weil Certificates: Stein Normalization, Cubic Phase Skew, and a Two-Deck Pricing Barrier

This paper develops a source-rigidity theory for a finite-rank common-depth model arising in bandwidth-one Weil-certificate analysis. The main objective is to distinguish constraints that follow from abstract Stein/multiplier geometry from those forced by the actual weighted-exponential provenance of a legal source. For a common-depth source of the form Q = B - GBG, G = M(ys), an orbit-frame argument yields the linear depth-capacity bound n/α ≤ y+1, where n is the magnitude of a negative eigenvalue and α = λₘₐₓ(Q₊). Thus large negative-to-positive spectral gain forces large multiplier depth. The paper then shows that this spectral and multiplier information alone is insufficient: an explicit finite-dimensional relaxed Stein construction realizes a near-rank-one negative spike together with an almost flat positive spectrum while retaining strict positivity of the associated Stein source. This identifies source provenance, rather than spectral data alone, as the essential remaining constraint. For the actual weighted-exponential source, the Stein-normalized channel satisfies the exact depth-stripping identity Φy(Q) = 2 E E^*, where ( Ec)(s) = √p₀(s) ∑ⱼ cⱼ e-ixⱼ s. Thus Stein normalization recovers a rank-q finite weighted-exponential synthesis with a difference-kernel Gram structure. A Ky Fan argument gives corresponding spectral ceilings for the normalized base Gram. Using endpoint synthesis matrices and writing A = U + iV, C = R^*L, the paper proves the exact cubic identity trQ³ = 16trC³ + 48tr(CU²) - 48tr(CVTV). Consequently, if α₁/n₁ ≤ a, qa³ < 1, then the source must pay the quantitative cubic phase-skew cost tr(CVTV) ≥ 1-qa³/48 n₁³, and VF² ≥ 1-qa³/12a n₁². The phase-skew term also admits the positive same-source representation tr(CVTV) = 12 EVTHS². At the same time, the paper derives an exact two-deck formula for VF² in terms of the pair-difference intensity PX(r)². This exposes a structural distinction between quadratic pair-difference information and the genuine three-center geometry appearing in the cubic phase-skew functional. A fixed-center endpoint analysis then proves a quadratic-pricing barrier. For endpoint-cancelling configurations with PX(1) = 0, PX'(1) ≠ 0, one has, for endpoint-regular quadratic modes, M₍X,y) e²ʸy⁴, whereas VF² e²ʸy². Hence no fixed nonnegative linear combination of finitely many such quadratic modes can provide a depth-uniform upper bound for the phase-skew energy over the unrestricted common-depth center class. A numerical specialization at $q=5147$ is included only as a conditional orientation to a separate ongoing bandwidth-one certificate analysis; the arithmetic derivation of that parameter is not a theorem of the present paper. In particular, this work does not claim a new global zero-proportion bound or a proof exceeding the benchmark 0.672500703679412…. The principal contribution is therefore structural: spectral extremality forces extreme depth, abstract Stein relaxation can survive that demand, legal weighted-exponential provenance restores additional rigidity, and this rigidity produces a cubic same-source phase-skew tax that cannot in general be priced by unrestricted quadratic two-deck data alone.