# Claim ledger — v2.0.0

“Derived” means proved for the stated model, not independently reviewed or historically certified.

| ID | Result | Essential hypotheses | Evidence and scope |
|---|---|---|---|
| C01 | Active dimension = rank R | Positive-semidefinite density, quotient by its kernel | Definition and elementary quotient proof, Chapter 3 |
| C02 | Readout descends iff F B^-1 ker R = 0 | Invertible B; quotient-valued equilibrium | Exact equivalence, Chapter 3 |
| C03 | Fixed static shell and rank sum | Completed positive form; compensated visible block | Congruence and generalized Schur proof; 12 tests |
| C04 | Explicit polynomial programmability | Finite expanded monomial list, coefficient-programmed acyclic graph | Constructive proof and exact resource count; 12 numerical cases plus symbolic checks; no general efficient arithmetic-circuit claim |
| C05 | Dynamic slope H*H | Unit Euclidean mobility, positive R, zero-state response | Resolvent proof; 10 tests; mobility generalization in Appendix A |
| C06 | Rank-existence cusp | Fixed B,E; positive R approaching singular R0 | Interchanged-limit proof; 8 tests |
| C07 | Single-death rate rho/||B^-1 e_j||² | Exactly one zero residual; all other densities positive | Simple eigenvalue perturbation proof; tested in cusp group |
| C08 | Unique optimal weighted density | Real inverse-nonnegative B, positive costs, fixed weighted budget | Two exact certificates; 20 cases with 80 random allocations each; 6 chain cases |
| C09 | Semidefinite mirror price | Invertible complex B, positive costs | Standard strong duality specialized to this model, Chapter 8 |
| C10 | Unbounded unique certificate rank | Explicit complete frame, alpha>0, unit costs | Analytic primal/dual proof and uniqueness; ranks 2–7 tested |
| C11 | Four-mode rank-two witness | Tetrahedral frame construction | Exact uniqueness proof; 1 numerical test; no numerical value asserted for the strict rank-one gap |
| C12 | Finite-collapse freeze | Scalar finite integrated relaxation law | Exact ODE solution, generated raw data and figure |
| C13 | Recurrent energy/norm ledger | Joint unitary scattering; declared port normalization | Telescoping identity; 8 tests; no free energy |
| C14 | Monotone density feedback | Smooth simple least eigenvalue; positive optimizer; global smooth flow for convergence | Variance and relative-entropy proof; 5 derivative checks, not a physical implementation |
| C15 | Full parameter-work identity | Differentiable controls and specified gradient law | Chain-rule proof; no ignored density-control work |
| C16 | Multi-death echo projector | Independent released residual coordinates, invertible B | Exact projector formula in Appendix A |

## Not established

Historical world-first priority; independent theorem validation; Maxwell/plasma realization; performance advantage over a GPU; unlimited hidden dimensions in a finite device; energy creation; efficient compilation of all arithmetic circuits; universal autonomous computing with no programming or readout resources.