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

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

evaluatedSlice_phaseSeven_second_reverse_bound drops the vendored hypothesis hnorm : strategy.state.IsNormalized, a theorem of the model, so its arguments are params strategy zeta family hcombined_snd; hcombined_snd is stated with evaluatedPointFamilyLeft strategy.state and Preliminaries.totalSandwichFamily strategy.state, which take the model explicitly in the port. The vendored proof absorbs the inserted total with simp [opTensor_mul, …]; here it is one opTensor_mul and Preliminaries.projSubMeas_outcome_mul_total_eq_outcome.

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_phaseSeven_second_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_snd : strategy.state.SDDRel (MIPStarRE.LDT.uniformDistribution (MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params)) (fun (q : MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params) => evaluatedPointFamilyLeft strategy.state params family q.2) (fun (q : MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params) => Preliminaries.totalSandwichFamily strategy.state (evaluatedPointFamily params family) (evaluatedSlicePointMeas params strategy) q.2) (4 * zeta)) : have 𝒟 := MIPStarRE.LDT.uniformDistribution (MIPStarRE.LDT.Commutativity.EvaluatedSliceQuestion params); have phase6FirstRemoved := evaluatedSlicePhaseSixFirstRemoved params strategy family; have phase7GonnaCite := evaluatedSlicePhaseSevenGonnaCite params strategy family; |MIPStarRE.LDT.avgOver 𝒟 phase6FirstRemoved - MIPStarRE.LDT.avgOver 𝒟 phase7GonnaCite| ≤ 2 * √zeta

Paper lines 103--104: reverse the second eq:add-an-a insertion.