Stein-Normalized Weighted-Exponential Rigidity in a Bandwidth-One Weil-Certificate Source Model: Extreme-Depth Gain, 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. Version v1.0. 주요 수정사항 및 보강 내역 삼차 흔적 항 등식 및 부등식 교정: trQ³ 삼차 항 식에서 누락되었던 마이너스 연산자(-) 및 등호(=)를 복원하여 trQ³ = 16trC³ + 48tr(CU²) - 48tr(CVTV)로 교정했습니다. 부등식 tr(CVTV) ≥ 1-qa³/48 n₁³ 및 VF² ≥ 1-qa³/12a n₁²의 관계식 연산자를 올바르게 정돈했습니다. 누락된 등호(=) 복원: $Q = B - GBG$, G = M(ys), Φy(Q) = 2 E E^*, ( Ec)(s) =, $A = U + iV$, C = R^*L, tr(CVTV) = 12 EVTHS², PX(1) = 0 수식에서 누락되어 있던 등호(=)를 복원했습니다. MathJax/LaTeX 파싱 에러 및 이스케이프 교정: 전치 행렬 표기 중 VT로 오기되어 위첨자 이스케이프 파싱 에러가 발생하던 부분을 VT로 올바르게 정돈했습니다. 프로베니우스 노름 표기 중 V_F^2 및 EV^THS^2 구문을 표준 MathJax 노름 표기인 VF² 및 EVTHS²로 교정했습니다. 디스플레이 및 인라인 수식 모드 정돈: 주요 디스플레이 수식을 $$ ... $$ 구문으로 분리하여 Zenodo MathJax 렌더링 안정성을 확보했습니다. 본문 속 Q, G, n, α, q, PX(1)=0, PX'(1)≠0, $q=5147$ 등 수식 요소를 표준 인라인 LaTeX($...$) 모드로 정돈했습니다.