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luminous-hypercontinuum-foundation-v3

# 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.

	# 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.