feat(interval): retain bounded branch trees

[kim-em]

Summary

• add the first function-independent retained runtime branch-tree manager
• store exact root and child CheckerInputs in an append-only tree
• support replaceable depth-first and breadth-first frontier scheduling
• bound processed leaves, splits, leaf count, depth, scopes, and per-leaf policy fuel
• retain blocked, unfinished, and child-start-failure states without treating them as proof closure
• run a live authenticated exponential split through two exact child sessions
• keep proof-tree emission explicitly separate

Reconciliation

• rebased onto exact current main 055eaeb92a1b25105261b65d018dbf37da50aaa6
• preserve merged refutation, ReLU, PNT-log, Machin, and endpoint-sine content
• seed sealed branch state from the exact root session and use the current depth-owning branch-start API
• refresh the narrow exact lakefile factor-freshness exemption for the isolated branch-tree registration; the factorization executable dependency graph is unchanged

Validation

• combined branch-tree, ReLU, PNT-log, Machin, and endpoint-sine lake build (2245 jobs)
• PNT inventory validation
• copyright, line-count, DAG, released trust-surface, Phase 4, factor-sweep freshness, diff, and banned-proof checks

No native_decide, axiom, or sorry is introduced.