refactor(Schonhage): namespace native product register aliases #1617

Available after #1615; bounded follow-up to its outside review, within #1394.

Public register abbreviations in QueriedProduct collide with the finite Schonhage.tiles, leaves and bands names. Product cost/correctness proofs consequently spell out the finite namespace, and the upcoming caller adapter needs both APIs together. In Product/Program.lean, place register aliases such as tiles, leaves, bands, output, index and value in a small QueriedProduct.Reg namespace, or adopt a consistent register prefix after inspecting existing native conventions. Keep the general finite vocabulary unchanged.

Migrate all native Product modules, tests and the real caller-adapter clients together; remove superseded aliases under the repository compatibility rule. Show a checked client using both finite tile/leaf definitions and native registers without full namespace disambiguation. Preserve the fixed program, register numbers, execution/resource statements and source credit; verify native token/body identity apart from renamed references. Update affected module/PR documentation, assess abstracts, complete required validation and obtain API review of the public rename. No mathematical or runtime claim changes are intended.

This is a concrete optional API cleanup, not a new blocker on the reviewed #1615 implementation. Check ownership and current PR heads before claiming; no worker assigned yet. Root coordinator retains the global/static audit queue.