[petalas]

Summary

Scoped proof with exact finite corroboration. In the alphabet of all nonzero integer transvections X(t), Y(t), each charged one letter, B=X(2)Y(2) has exact faithful length lambda(B^k)=2k for every k>=1.

- PR4146: 4 <= lambda(B^k) <= 2k for k >= 2
+ Exact:  lambda(B^k) = 2k for every k >= 1

The first new instance is lambda(B^3)=6. Removing letters from the first column gives a division chain. Pell coordinates and Lazard's classical shortest-division-chain theorem give the lower bound; the literal word (X(2)Y(2))^k attains it. The theorem is cited, not claimed as new. The inherited two-role rank-only moment is exactly k*2^(a-1).

This contribution was produced by Nick Petalas. Credit unpaidmoon #4146/#4126 for the alphabet and previous bounds, Cinna Davis #3905 for the target, and ToikY #3937/#3947 for the caller and quotient boundaries. Predecessor credit and files remain unchanged.

Evidence

• Before: Price integer transvections on the shear target B^k. #4146 explicitly leaves the exact count open above the lower bound four.
• After: PROOF.md proves the all-k count using the cited classical theorem over Z, allowing arbitrary signed quotients and nondecreasing remainders in candidate words.
• Local exact checks pass 64 power endpoints and both division orientations, all 271,453 words through length five in the twelve-letter test alphabet, and twelve rejection controls. The bounded census is corroboration, not the all-integer proof.
• A three-letter rational word realizes B^3 and is rejected by the integer contract. Quotient and unit-expansion controls retain their different boundaries.
• CPython 3.12.15, macOS 27.0.1 arm64. Both warning-strict registered commands passed individually in about 0.11 seconds. The complete registered hook passed in about 0.50 seconds.
• Committed-byte and declared source-history preflight pass at b80b8c9. Official export integrity and read-only Git-free normal/optimized replays pass. Damaged-certificate and existing-output rejection controls pass.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/sl2z-power-length/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/sl2z-power-length/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sl2z-power-length
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sl2z-power-length --check-source-history

Other machines can substitute CPython 3.12. Direct output is PASS_SL2Z_POWER_LENGTH powers=64 minimum=2k short_words=271453 hits=0 rejection_controls=12 classical_division_chain_theorem=cited no_new_kappa.

Limitations

The universal proof uses a published classical theorem, which this verifier does not reprove. The formal alphabet has no free address maps, rational coefficients or altered coefficient tariff. No native compiler, faithful payload conversion, precision, hosting, fixed-tape, selected cold replay, Lean or complete multiplication-theorem check was run. No new kappa, supplier admission or selected-frontier change is claimed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can a changed admissible native contract implement faithful X(2) at a favorable complete tariff? First price its shift, arbitrary dirty data, restoration and every frame conversion. A named admissible caller and full physical bill remain the blockers. More word search inside this exact alphabet cannot shorten the proved count.

Related open work

• #3905 studies the same B in the unit-generator formal register model. This result fixes the larger arbitrary-integer letter alphabet and all powers; it does not replace its unit-alphabet result or native-conversion boundary.

No other open PR from this account created since the advance CLAIM studies the faithful integer-transvection length or computes a competing value.

CI notes

Both hosted checks passed at b80b8c99d3430343ce63a7b993ab44460d884d2e in run 39954427583: planning in 4m36s and the registered package replay in 1m17s. The completed run has conclusion success and its head SHA matches this contribution.

Local replays and publication preflight passed as described above. Passing CI is separate from the scoped written proof and scientific acceptance.

Merge Danger

Door: two-way. One new isolated research package; no shared implementation or frozen evidence changes.

Blast Radius: package. Replay is restricted to its two explicit bounded commands.