Boundedness transport point-consistency estimates #
This file contains the point-consistency consequences of the helper-agreement off-diagonal decomposition and the natural-error transports through the final projective fields.
References #
references/ldt-paper/self_improvement.texlines 435 and 747--755blueprint/src/chapter/ch07_self_improvement.tex
The tensor product with the identity on the left register is the right tensor placement.
The point measurement is complete and polynomial evaluation preserves the right-register total, so the tensor total has the same expectation as the right placement of the original total.
Left and right tensor placements of the point-measurement and evaluation totals reduce to the right-register total expectation.
The helper-stage consistency defect is exactly the averaged off-diagonal
mass appearing in the point-consistency add-in-u calculation.
This is the same algebraic identity as
helper_boundedness_slack_average_ev_eq_off_diagonal_avg, read as a
ConsRel defect for the point measurement against the polynomial-evaluation
family of H.
Helper-stage point consistency from the point-consistency add-in-u
transfer hypothesis.
The transfer bound controls the off-diagonal mass
E_u ∑_h ∑_{a ≠ h(u)} ⟨ψ, A^u_a ⊗ Hhat_h ψ⟩. The preceding algebraic
identity identifies this mass with the ConsRel defect for the point
measurement and the polynomial-evaluation family of Hhat.
Helper-stage point consistency obtained directly from the selected
add-in-u chain.
The hypotheses are the two analytic inputs used by the selected four-step
chain: bipartite self-consistency for the point measurement, and the
global-variance sum bound for the polynomial submeasurement T. The helper
submeasurement is the averaged sandwiched polynomial submeasurement built from
T, exactly as in the helper-stage application.
Natural-error transport of point consistency from the helper output to the projective output, with the submeasurement total-overlap displacement stated explicitly.
The measurement-valued right-register triangle lemma has no total-overlap term:
both right-register totals are the identity. In the present application
polynomialEvaluationFamily params Hhat and
polynomialEvaluationFamily params H.toSubMeas are only submeasurements, so
the total-overlap term
⟨ψ, A^u_{\mathrm{tot}} ⊗ H^u_{\mathrm{tot}} ψ⟩ must also be transported.
This theorem separates that displacement as the parameter η; the remaining
contribution is exactly the square root of the data-processing SDD error. The
note docs/paper-gaps/issue-1093-submeasurement-triangle-total-overlap.tex
records the corresponding discrepancy with the measurement-valued paper step.
Natural-error point-consistency transport when the total-overlap displacement is supplied as a single right-register total difference.
Since the point measurement is complete and
polynomialEvaluationFamily params H has the same total as H, the averaged
total-overlap term in final_fields_point_consistency_totalGap_natural is
independent of the point u. This theorem records the corresponding reduction
of the issue #1226 obstruction to the scalar difference between the totals of
the two right-register submeasurements.
The data-processing SDD comparison controls the total-overlap displacement which appears in the submeasurement form of the point-consistency triangle.
For each point u, the right-register total difference is the sum of the
field-answer differences in the two postprocessed polynomial families. The
finite-outcome Cauchy--Schwarz estimate
subMeas_total_ev_gap_abs_le_sqrt_card_qSDD, averaged over u, bounds this
single scalar by
sqrt (#F_q * ε). The factor #F_q records the cost of passing from the
outcomewise state-dependent distance to the total operator.
Natural-error point-consistency transport with the total-overlap displacement bounded internally from the data-processing SDD estimate.
The price of eliminating the explicit total-difference hypothesis is the
additional term sqrt (#F_q * selfImprovementDataProcessingError).