Conclusion

A proof certificate and its surrounding proof field answer different mathematical questions. The field need not consist only of viable derivations. It can retain failed strategies and genuine local refutations without allowing either to masquerade as global unprovability. A checked contravariant system of local field models gives an explicit bridge to Komorebi Space. Whole-support coverage identifies a constraint that pairwise dependencies can miss. The semantic-cover criterion now determines exactly when visible rules recover a missing constraint. In the linear finite case, protected observations admit exact success or failure certificates and a measurable information gain. A separate review model distinguishes proof correctness from correlated social interpretation.

A final proof is indispensable, but it is not always the whole mathematical object. The theory distinguishes the certificate from the declared field in which it is recorded. It gives that field typed nodes, alternative justifications, evidence records, task-relative compressions, bridge graphs, version histories, and explicit audit models.

The central distinction is between derivability within a declared inference system and the wider information retained by its archive. Certified closure records which claims follow from admitted leaves and accepted rules. Its interpretation as mathematical validity depends on the soundness of those leaves and rules. Discovery information serves different tasks: explaining a definition, locating a failed method, preserving a comparison, or opening a route to another problem.

Compression can preserve the present derivation while losing those other capabilities. Proof Path Theory makes that difference explicit through task families, recovery maps, extension problems, and versioned evidence. The certificate and the surrounding field should therefore be maintained together, with their logical roles kept distinct.

The final thesis is therefore:

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