QCI — Quantitative Closure Interface

\[ \mathcal T \xrightarrow{\Phi} \mathcal A_{\mathrm{admissible}} \xrightarrow{Q} \mathcal R \]

Where \(\mathcal T\) is the original qualitative/structural argument, \(\Phi\) is the map that filters and rewrites it into an admissible form, \(\mathcal A_{\mathrm{admissible}}\) is the intermediate form that has passed the legitimacy check, \(Q\) is the quantification operator, and \(\mathcal R\) is the final, checkable quantitative result.

Any argument that claims to have achieved QCI closure must be able to account for the status of each of the following nine components:

\[ D^{\mathrm{QCI}} = ( D_{\mathrm{def}}, D_{\mathrm{adm}}, D_{\mathrm{quant}}, D_{\mathrm{uniform}}, D_{\mathrm{error}}, D_{\mathrm{tail}}, D_{\mathrm{global}}, D_{\mathrm{consistency}}, D_{\mathrm{formal}} ) \]

• D_def — whether the object has a clear definition (well-definedness);
• D_adm — whether it passes the admissibility check;
• D_quant — whether there is a clear quantitative bound;
• D_uniform — whether the bound is uniform across the relevant parameter range;
• D_error — whether error terms have been estimated and controlled (error budget);
• D_tail — whether tail behavior has been handled;
• D_global — whether local results genuinely glue into a global conclusion (local-to-global);
• D_consistency — whether it is consistent with known results and special cases;
• D_formal — whether it has (or needs) formal verification (formalization).

QCI
✓ Well-definedness
✓ Admissibility
△ Quantitative bound
✗ Uniformity
△ Error budget
N/A Tail
✗ Local-to-global
✓ Consistency
○ Formalization

The overall status is fixed to one of five values: CLOSED / PARTIAL / OPEN / FAILED / N/A.