Preface
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A proof establishes an implication. The work surrounding that proof may answer other questions: why a definition was chosen, which hypothesis prevented an argument from failing, or how a change of representation opened a route that had not previously been visible. Publishing a certificate and preserving this larger mathematical record are related tasks, but they are not the same task.
Proof Path Theory studies that distinction through explicit finite objects. A proof field contains typed claims, justifications, evidence, and targets. Each justification requires all of its listed premises, while alternative justifications supply different routes to a conclusion. A certified derivation lies inside this field without converting its conjectures, examples, and unsuccessful attempts into additional proof steps.
The theory has two complementary directions. One selects or compresses a record while asking what remains recoverable. The other assembles local descriptions into coherent realizations. The first leads to certificate complexity, task sufficiency, information loss, and retention. The second gives a precise connection with Komorebi Space through local rule models, restriction maps, and gluing. A separate linear model yields positive realization certificates, negative obstruction witnesses, and exact observation counts.
Failed approaches remain part of the ambient record for a reason. They may isolate a missing hypothesis or reveal a limitation of one method. Their value does not turn a failed search into a proof of impossibility. A refutation must identify the assertion it refutes, the assumptions under which it applies, and the witness that establishes the failure. That scope is as important as the witness itself.
The later chapters examine how an archive changes and how its claims are evaluated. Their review and queue models are deliberately conditional. A covariance formula describes a specified dependence model, not an empirical diagnosis of a community. A record of social reception does not determine whether a proof is valid. Keeping those distinctions explicit allows the theory to study the transmission of mathematics without replacing mathematical standards with social judgments.
The mathematical ingredients come from proof theory, graph theory, category theory, linear algebra, information theory, and optimization. Their role here is to form one account of certified reasoning and the surrounding information that later inquiry may need. Familiar ingredients are not concealed, and a new notation is not presented as evidence of new mathematical truth.
This monograph preserves written proofs, finite examples, and the connection to the accompanying implementation. Its formal concordance identifies the selected Lean interfaces. The aim is a record that a reader can inspect at several levels: the statement, its argument, the hypotheses that make it valid, and the computational evidence supplied for its implementation.