Discharge the selected BIT fractional-moment premise in Lean

[bitplane]

PR #2289 deliberately left the selected BIT fractional moment as a formal hypothesis, supported by an external Python certificate. This proves that actual Real.rpow inequality in Lean and composes it with the unchanged root-uniform recurrence theorem. The resulting theorem has no supplied numerical moment premise; the physical recurrence, leaf estimates and one local-charge constant shared across root contexts remain explicit assumptions.

The proof rounds tau=9983/10000 down to 130849/131072 and checks 136 rational square inequalities across the eight selected ranks. A general Real-power lemma justifies each step. The complete weighted moment, including the fallback allowance, is proved at most 11290320473664709188758299/11291503906250000000000000, strictly below 9999/10000. The numerical point and original inequality belong to Jose Manuel's PR #2234; this closes the numerical-to-Real proof gap rather than improving the point.

Validation: all five local Lean modules built fresh, with 22 axiom reports using only propext, Classical.choice and Quot.sound. Two corrupted square certificates failed on false arithmetic obligations, and reversed exponent rounding produced a type error. Independent exact integer powers checked eight actual-exponent upper bounds and 4,774 dyadic calibration cases. Eleven failure-control groups and registered Python 3.12.3 replay passed. Both lightweight commands also passed on Python 3.14.4 from a Git-free, read-only export with all 22 files unchanged. The scoped workflow performs the checksum-pinned Lean installation and kernel build; hosted success is not yet claimed.

Two recurrence modules are preserved byte-for-byte from PR #2289 at 32ef80a. Source snapshots bind the selected PR589 profile and assembly. The Python generator proposes certificates and lies outside the formal trust base; Lean checks its claims. Exact dependency heads and tracked source cleanliness are checked; Mathlib proof objects use the official cache, without a full source-library rebuild.

Existing Lean PRs #585/#605/#842/#885 cover different historical complex expressions. PR #2284's generic aggregation retains per-term Real bounds. Prior repeated-square-root numerical methods in independent-queue-ceilings and idle-stability-radical-audit are credited; no numerical-method invention is claimed. Physical packets, frozen-ancestor access/preparation, parking, the uniform local charge, full supplier admission and the multiplication theorem remain open. No selected frontier or kappa changes.

Canonical claim: #2209 (comment)

Single random draw de346470d4c367b36f566f9208c8fab2, independent-verification / independent-moment, refined without rerolling. Canonical main remains 0fdd7d8; the patch applies cleanly. Final live overlap covered 512 open PRs and 1,003 Discussion records, with complete pagination and no warnings. TASK.json records exact freshness, attribution and the next physical proof obligation.

Gareth Davidson with substantial OpenAI Codex assistance; no subagents or independent human review. Apache-2.0; predecessor credit retained.