B.4 Theorem and evidence-status ledger
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| Statement | Status | Qualification |
|---|---|---|
| Invariant-basis simulation | Proved | Exact gatewise subspace invariance. |
| Polynomial packet simulation | Proved | Requires efficient discovery, updates, measurement, and polynomial representation size. |
| Fixed \(k\) number-preserving circuits lie in \(P\) | Proved corollary | \(k\) fixed and local restricted gates efficiently applicable. |
| Approximate circuit leakage bounded by sum of gate leakages | Proved | Operator-norm state-error certificate. |
| Hamiltonian leakage bounded by \(|t|\|[H,P]\|\) | Proved | Bounded self-adjoint Hamiltonian. |
| Permutation-symmetric qubits lie in \(P\) | Derived theorem | Gate matrices and measurement efficient in Dicke coordinates. |
| Fixed-\(d\) symmetric qudits lie in \(P\) | Derived theorem | Fixed local dimension and efficient occupation updates. |
| Polynomial-orbit quantum walks lie in \(P\) | Derived theorem | Quotient data computable from succinct input. |
| Fixed-particle boson and fermion circuits lie in \(P\) | Derived theorem | Particle number fixed and updates efficient. |
| Compressed-packet theorem | Proved abstraction | Representation class closed under gates with polynomial size. |
| Weight-two runtime and memory gains | Recorded computation | Implementation-specific archival benchmark. |
| Half-filling remains exponential | Proved dimension statement | Runtime constants remain implementation dependent. |
| Parity gives only a factor-two dimension reduction | Proved dimension statement | Does not preclude further internal compression. |
| Quantum Latin order-six product realization absent, entangled realization present | External theorem | Contextual existence defect, not a simulation theorem. |
| Adaptive discarded-state error certificate | Proved | Unnormalized trajectories, explicitly accounted discarded norms. |
| Universal polynomial packetization | Open conjecture | Would imply \(P=BQP\). |
| Representation-independent interference lower bound | Open | Required for a full hard-side complexity conclusion. |
B.4.1 Publication discipline
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Every extracted theorem paper should declare the circuit model, input encoding, gate precision, measurement, symmetry discovery method, packet representation, error norm, and complexity parameter. Benchmarks should be accompanied by code and frozen data. Open conjectures should not be phrased as consequences of finite experiments.