Scoped proof: exact 3-ADD minimum for PR1221's R439 frame-509 endpoint

[1337isnot1337]

Result

For the fixed R439 frame-509 Boolean endpoint on (a,b,h,c)=(33,38,468,77), the minimum in the scalar one-destination ADD model is exactly 3. The endpoint changes three distinct destinations, so fewer than three scalar ADDs cannot realize it. PR #1221's three-ADD word attains the lower bound. An exact four-register CNOT census has 1, 12, 97 and 626 distinct maps at lengths 0–3, with exactly three target words at length 3.

The four-ADD and three-ADD integer transfers are different by exactly 2 e_468 e_33^T; this is a Boolean endpoint result, not integer-map equality.

Relationship to concurrent work

Hanson Sun's open PR #1221 provides the witness and explicitly makes no minimum claim. Concurrent PR #1344 independently verifies the maps, full splice, and invoice, also without claiming a minimum. This package adds the local lower bound and exact shortest-word classification; it does not repeat PR #1344's full replay.

Provenance and scope

• Base commit: bce54fb3f895cbe5cd4c0eb477c3afeed100190f.
• Selected conditional result stays κ = 836330098538459/500000000000000000; the complex supplier remains binding.
• PR Independent bit construction: R439 frame509 at 20 ADDs with original H1-H5 (no new kappa) #1221 head 6deda901721316fcb61c6e2a6dd6b537de183452; parent circuit PR Finite bit construction: R439 frame 509 uses 21 ADDs (no new kappa) #973 head c05dc72734c25dd069077157224d4dd30c6e8c19; selected PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 head a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46 on PR Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555.
• Source pins and overlap notes are in research/r439-b509-local-minimality-seed06/.

This does not establish a shortest factorization for the full frame-509 block, reprove H1–H5, establish hosted supplier admission, change the paid recursive profile, or claim a new κ.

Reproduction and checks

python3 -I -O -B research/r439-b509-local-minimality-seed06/verify.py
python3 -I -O -B research/r439-b509-local-minimality-seed06/test_package.py -v
python3 scripts/research_package.py --package research/r439-b509-local-minimality-seed06 --check-only

The verifier prints PASS_R439_B509_LOCAL_MINIMALITY minimum=3 length3_words=3 maps=1,12,97,626 boolean_inputs=16 integer_delta=2e_h_e_a no_new_kappa. Four focused tests pass, and the package pin check prints PASS_PACKAGE_METADATA_AND_PINS. Local verification used Python 3.14.7. Python 3.12 is not installed locally, so I could not run the registered Python 3.12 replay in this session.

Credit and disclosure

Credit Hanson Sun (@bbMisk), PR #1221, for the three-ADD witness; Cinna Davis (@1337isnot1337), PR #973, for the 21-ADD parent; DreamingOfClouds, PR #589; and eumemic, PR #555, for the retained supplier. Prepared with substantial OpenAI Codex assistance; no human independent review is claimed. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 6a10c0eaa7e896bb5c6d297a0b9c6bbe46143498.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Sharp three-ADD minimum for the fixed four-register PR1221 endpoint, by destination counting and complete short CNOT census. This classifies the small endpoint, not the complete eleven-register frame-509 block.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.