[aliromman]

Result and exact scope

Proves an exact additive tariff for heterogeneous GL2 targets on disjoint address-coordinate pairs. The six block labels e,a,b,ab,ba,aba cost 0,1,1,2,2,3 calls. Sum these costs over all blocks and target registers. The formula remains exact in every containing GL_n(F2), with arbitrary cross-pair coordinate CNOTs, free scalar register mixing and any finite number of restored dirty registers.

Proof and scope, exact receipt, checker, and source pins.

This is a scoped formal-model theorem. Each paid call has a scalar rank-one idempotent and costs one. The complete arbitrary-input endpoint must restore every unused register. The selected kappa stays 60403/20000000. There is no native supplier or complete multiplication-exponent improvement.

Method, novelty and credit

The mixed tariff at address dimension two already follows from Ross Richardson's PR #1989 and GL2 bridge in PR #2415; see the inversion-sum proof and bridge source. This PR does not claim that dimension-two result as new.

The extension uses James Reed's averaged row-order method in PR #4950, with its retained proof. One coordinate CNOT charges only the cut of its written pair. Its averaged charge is at most one unit, including cross-pair reads. The six target charges are exactly the listed tariffs. Summing disjoint cuts gives the lower bound; direct diagonal formal words attain it.

Cinna Davis's fixed-three-register heterogeneous target and question supplies another predecessor. The target diag(a,b,aba) costs five even with larger address spaces. The source notices also retain the earlier register/cut lineage. Prepared for Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No predecessor executable is imported.

Base: f259dabc7d70e0db8d6d36bfda888f008bbfecc1, priority revision 7. The changed task premise is the ambient coordinate alphabet and disjoint-block sum, not another fixed-register census. A final overlap refresh covered 318 open PRs, 4,130 Discussion records and 83 of 6,710 local notes. Its focused matches were this claim and package; sparse omitted notes, arbitrary diffs and unpublished work remain limits.

Verification

CPython 3.12, standard library:

python3.12 -B scripts/research_package.py --package research/mixed-gl2-block-tariff

Expected markers:

PASS_MIXED_GL2_BLOCK_TARIFF costs=0,1,1,2,2,3 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/mixed-gl2-block-tariff outcome=completed

Normal and optimized replay passed. The checker compares all six cut counts with independent group word lengths, checks 80 actual coordinate-CNOT cases over 31,992 relative orders, and verifies 7,128 complete diagonal formal endpoints and 39,852 call omissions. Nine boundary controls include a literal overlapping-pair product that drops from nine separate calls to three. Four actual code mutations are rejected with assertions disabled.

Exact source pins, base ancestry, committed bytes and a Git-free read-only optimized export replay passed. The full-history wrapper was not run in this shallow checkout. No independent human or proof-kernel review, native/compiler admission, selected cold replay or archive-wide sweep ran.

Open next questions

Overlapping coordinate blocks need a different cost law: a list of their separate tariffs need not add. Unequal prices, rank-two calls charged as one, free address maps and restricted clean-input endpoints are outside this theorem.