Section 11 commutativity: the phase-six reverse insertion of the evaluated-slice paper chain

The first reverse eq:add-an-a bound used after the paper line-87 phase-five removal: the counterpart of Commutativity/ScalarApproximation/PaperChainPhaseSix.lean under MIPRE/Background/LIDT/MIPStarRE/LDT/ in the port of planning/c6b-plan.md (milestone M7, section "Port conventions").

evaluatedSlice_phaseSix_first_reverse_bound drops the vendored hypothesis hnorm : strategy.state.IsNormalized, a theorem of the model, so its arguments are params strategy zeta family hcombined_fst; hcombined_fst is stated with evaluatedPointFamilyLeft strategy.state and Preliminaries.totalSandwichFamily strategy.state, which take the model explicitly in the port. The vendored proof obtains the self-adjointness of the two families from Matrix.PosSemidef; here it is IsSelfAdjoint.of_nonneg, and the absorption of the inserted total into the first factor is one opTensor_mul.

Not ported

Every declaration of the vendored file has a counterpart here.

References

In LionSR/MIPStarRE at commit 507e8122, not in this repository:

• references/ldt-paper/commutativity-G.tex
• blueprint/src/chapter/ch08_commutativity.tex

theorem MIPRE.LIDT.Co.Commutativity.evaluatedSlice_phaseSix_first_reverse_bound {𝔓 : Type u_1} [CStarAlgebra 𝔓] [PartialOrder 𝔓] [StarOrderedRing 𝔓] {K : Type u_2} [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K] (params : MIPStarRE.LDT.Parameters) [MIPStarRE.LDT.FieldModel params.q] (strategy : SymStrat params.next 𝔓 K) (zeta : ℝ) (family : IdxPolyFamily params 𝔓) (hcombined_fst : strategy.state.SDDRel (MIPStarRE.LDT.uniformDistribution (MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params)) (fun (q : MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params) => evaluatedPointFamilyLeft strategy.state params family q.1) (fun (q : MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params) => Preliminaries.totalSandwichFamily strategy.state (evaluatedPointFamily params family) (evaluatedSlicePointMeas params strategy) q.1) (4 * zeta)) : have 𝒟 := MIPStarRE.LDT.uniformDistribution (MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params); have phase5PaperRemoved := evaluatedSlicePhaseFivePaperRemoved params strategy family; have phase6FirstRemoved := evaluatedSlicePhaseSixFirstRemoved params strategy family; |MIPStarRE.LDT.avgOver 𝒟 phase5PaperRemoved - MIPStarRE.LDT.avgOver 𝒟 phase6FirstRemoved| ≤ 2 * √zeta

Paper lines 99--102: reverse the first eq:add-an-a insertion after eq:gcom10.