[mihir-s-05]

The sparse-union tape bound assumes a p-bit payload plus o(p) metadata. This contribution supplies a concrete per-microbox header contract for that assumption, taking the field-capacity part of #4327's open question 2 and the handoff also left by #4376.

The seven-file package research/union-header-encoding/ stores coordinates, membership flags, boundary ordinals, count and active tag in exactly H = 2gh + g + 2 ceil(log2(g+1)) + 2 bits. At #4100's actual core side L = 2^ceil(400 log2 p) and beta = 1 - 3/10^30, the written proof gives H <= 810 p^beta log2 p = o(p) for downward group rounding. It also covers upward/power-of-two rounding with twice that bound. An exact symbolic cutoff proves H < p/16 for either rounding once p >= 2^(40*10^30); this is only a sufficient header-width cutoff.

Validation on Windows / CPython 3.12.0, standard library only:

• Registered normal and optimized replays pass, with 2,040 exact finite width cases, 746 codec round trips, 688 active-header checks and 14 rejection controls.
• Six executions of Audit literal head travel of #4100's sparse-union schedule: section 2.3 holds with constant 17 (no new κ) #4327's unchanged simulator use serialized headers: 287 records, 705 visits, 355 lines and 28,976 header decodes. Exact operator outputs, visit identities and movement bounds with the actual header width pass.
• The committed-byte preflight checks all 13 package/input files. The checked export runs in a fresh directory without Git metadata in both modes. Source corruption, receipt corruption and a wrong-tag code mutation are rejected; all 14 exported files, including its inventory, remain byte-identical afterward.
• git diff --check passes. Hosted CI is not claimed.

Reproduce:

python -B scripts/research_package.py --package research/union-header-encoding
python -B scripts/check_research_commit.py --package research/union-header-encoding

The source simulator is reused evidence, not an independently implemented tape schedule. The new codec is a sufficient reference format, not a claim that #4100 already serializes its records this way. Actual carving/rank emission, bit-level parsing travel, global counters/stacks, coarse-context gather/page delivery and the remaining multiplication premises are outside scope. A concrete collision fixture shows that identical local headers cannot identify distinct coarse contexts without preserved boundaries or an added key. The optional full-history preflight was not run in this shallow checkout; all pins resolve at the current base and no foreign-head dependency is introduced. The orientation baseline stopped on sparse-omitted selected packages; no selected cold replay was run.

Base and final canonical-main refresh: f626db5c3a3d781d8f7695038776c621651f75e1, priority revision 7. Contribution commit: 8d259a89fcff2a347de640a2f91cbe629c2e7a5c. Final overlap refresh covered 358 open PR bodies and 3,493 Discussion records with complete reported pagination; local notes were sparse. The central ledger labels #4100/#4327/#4376 partially reviewed. No new kappa, selected-frontier change or scientific acceptance is asserted.

Coordination and next steps: Discussion #4576. The README and TASK.json leave concrete handoffs for an actual field emitter and the two-context caller boundary.

Credit Cinna Davis, Jeremy Kothe, Vlado Vojdanovski, Ryanson Jonathan, unpaidmoon, rtyu193-rgb and LeverRank, with all original source notices retained. Prepared for Mihir Srivastava / mihir-s-05 with substantial OpenAI Codex assistance; no subagents or independent human/formal-kernel review. Apache-2.0.