[codex] add rootfile Hamiltonian proof machinery

[Mopati123]

Summary

• Corrects the Hamiltonian canon against the active rootfile architecture and real evidence-chain wording.
• Adds deterministic v1 Ramanujan path-family signatures for liquidity, time, entry, risk, and topology.
• Replaces the hardcoded entropy value with measured posterior uncertainty and information gain passed into scheduler collapse.
• Adds post-outcome falsification scoring for closed MT5 and Deriv demo settlements without placing broker orders.

Validation

• Scoped diff hygiene passed for intentional files; global git diff --check remains blocked by unrelated .github/instructions/kluster-code-verify.instructions.md whitespace drift left out of this PR.
• python -m pytest tests/rootfile/test_hamiltonian_missing_pieces.py -q -> 5 passed
• python -m pytest tests/unit/test_geometry -q -> 64 passed
• python -m pytest tests/rootfile -q -> 92 passed
• python -m pytest tests -q -> 185 passed, 9 warnings

Notes

• No MT5 or Deriv canary was run in this pass.
• No real-money execution or sailing escalation is enabled.
• kluster review could not produce findings because the kluster.ai trial has ended.