# Proof index

The numbering is from the distributed manuscript.

| Central result | Location | Proof mechanism |
|---|---|---|
| Unrestricted linear minimax recovery | Chapter 3 | Pseudoinverse upper bound and opposite-state lower witness |
| Exact common-matrix convexification | Chapter 5 | Y=t(CCᵀ)⁻¹; reverse determinant interpolation |
| Target capacity and budget regimes | Chapter 6 | Floored MAXDET feasibility and spectral threshold |
| Complete noncommuting frontier | Chapter 7 | Reflection averaging, scalar determinant envelope, analytic roots |
| Joint affine continuous design | Chapter 8 | Convex joint hypograph |
| Common-envelope dual certificate | Chapter 9 | Log-determinant conjugacy and PSD complementarity |
| Two-scenario closed form | Chapter 9 | Whitening and sign averaging |
| Tensor envelope composition | Chapter 10 | Product primal feasibility and tensorized KKT multipliers |
| Direct-sum capacity | Chapter 11 | Block sign symmetry and log-determinant additivity |
| Future-universal summary criterion | Chapter 12 | Equality of feasible regions; singleton future constraint witness |
| Dimension-dependent memory bound | Chapter 12 | First-order trace inequality and scaling |
| Exact coordinate schedule realization | Chapter 13 | Largest-residual induction |
| Finite ellipsoid minimax | Chapter 14 | Two-resource LP/SDP duality and quadratic completion |
| Infinite Sobolev-ellipsoid minimax | Chapter 15 | Explicit shrinkage rule and two-coordinate witness |
| Local nonlinear stability | Chapter 16 | Integrated Jacobian perturbation bound |
| Integrability obstruction | Chapter 16 | Mixed partial incompatibility |
| Passive dilation | Chapter 19 | Singular value decomposition |
| Linear-head task certificate | Chapter 20 | Pairwise margin and Cauchy–Schwarz |

Fourteen theorems, thirteen propositions, three lemmas and two corollaries appear in the main text. Thirty-one explicit proof environments accompany the results; one compatibility proposition is justified in the immediately following prose. These are manuscript proofs, not formal proof-assistant objects.