[pinterbanget]

Summary

This adds research/pr4541-packed-io-check/, an independent check of the packed IO codec in Dugongue's #4541 (head 46ca24a8, §4 and PACKED-IO.md). The O(n/(log n)^κ) corollary depends on this codec to avoid a Θ(n) bit-per-word IO term. Together with #4552 (§1–§3), it completes my coverage of #4541's construction. All checked claims are confirmed. No new κ.

The codec was written from the vendored spec text (blob 675e520c, SHA256-pinned). Nothing is imported from #4541's packed_codec.py.

Checks

For (r, c) ∈ {1,2,4,8}×{2} ∪ {1,2,4}×{4}, with 131,620 decoder rows built in full and 3,372 cases compared with direct binary:

• Tables. E, Er and the decoder (length, value, valid) keys and entries all fit in cr bits.
• Input expansion. LSB-first and MSB-first expansion, including the partial top digit Er[x] ≫ c(r−ℓ), gives the exact coded stream followed by genuine blanks.
• Reservoir. It holds fewer than 2r cells, with at most one flush per append.
• Output. The coded prefix is decoded under caps m ∈ {n, n+1, n+r, n+3} with dirty cells after the terminator, giving exactly ⌈m/r⌉ LSB digits of the right integer.
  • A full cap does not read the terminator.
  • Leading zeros are kept.
  • An invalid cell before the blank rejects.
  • A zero cap emits nothing.
• Shared tape. A continuous reservoir across a delimiter is exact. Separately padded concatenation inserts blanks in all 37 runs where n₁ ≢ 0 (mod r), confirming Packed Word RAM transfer: conditional O(n / log^κ n) multiplication (no new TM κ) #4541's warning.
• Controls. 25 of 25 decoder mutations are detected. 7 spec statements are bound verbatim.

python3.12 -B scripts/research_package.py --package research/pr4541-packed-io-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4541-packed-io-check outcome=completed   (~1 s, stdlib only)

Limitations:

• The 100-per-digit and 160-per-digit operation constants are model envelopes and are not re-derived here.
• The corollary's TM antecedent is still inherited.
• These are finite tests that corroborate the written invariants.

Head: cf88c6f65b668c3e848f8ac9f7818152cf7b9860. Base: canonical main f626db5c.

Credit: Dugongue (#4541). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code