[JoyboyBrian]

Four ordered graph-frame images need similarity data that three-frame dimensions cannot capture. Settle the complementary-graph first case of Open next question 2 in the pinned three-frame shared-map package (#4919): for frames A, B, graph(L), graph(M), a common invertible map exists exactly when L^-1 M and Lprime^-1 Mprime are similar. Give the all-field cyclic-module construction, exact finite-field centralizer count including nilpotent/repeated/mixed-primary blocks, and the retained unused-complement factor q^(2rs)|GL_s(q)|.

Binary irreducible cubic companions give identical incidence dimensions but incompatible quadruples. A stronger nonsemisimple example has the same characteristic polynomial and pairwise complementary frames, but primary partitions (2) and (1,1) and self-centralizer counts 12 and 180. The native nine-dimensional compatible cubic self-case has 308,281,344 ambient completions.

The bounded reference is independently checked against complete literal GL2(F2), GL2(F3), and GL3(F2) actions: 609 operators, 32 similarity orbits, and every realizing intertwiner. Eight primary fixtures also use an independent all-matrix bit-vector oracle; literal geometry verifies all four images, nonidentity graph normalizations, supplied shear/complement action, singular fourth graphs, directed ternary images and zero active space. Sixteen malformed/boundary cases reject.

Validation on macOS 27.0 arm64 / CPython 3.12.7:

• Registered warning-strict normal and optimized commands pass.
• Committed-byte/source-history check and official 12-file export/integrity check pass.
• Git-free, read-only export passes normal/optimized direct checks with an empty PATH, preserving every exported byte; occupied receipt writes reject.
• Nine actual mutation types reject in both modes, including graph-I omission, reversed intertwining, characteristic-only matching, discarded radical factors, missing complement freedom, source/proof drift and Boolean/integer receipt substitution. Algorithm mutants update their local code hash so hash mismatch cannot substitute for the mathematical control.
• A separate Astra agent independently derived and inspected the written construction, radical/count proof and executable interfaces, finding no correctness blocker. This is agent review, not human or proof-kernel acceptance.

All source pins are retained. The actual emitted source operator has not been shown to impose four independent graph contacts; this is an algebraic interface result. The executable reference is restricted to prime fields 2/3/5, dimension at most four, ambient dimension at most nine and at most 4096 intertwiner coefficient choices. The written all-field proof is distinct from these finite experiments. No native optimization, MILP, historical scientific suite, Lean, Docker, physical savings, new kappa or supplier admission is claimed.

Exact scope, construction, pins, receipt and open questions are in research/four-frame-shared-map/. Canonical claim: #1166 (comment).

Retain Nick Petalas's #4919 three-frame result, Cao Yifan's PR4610 native-map/helper work and their credited supplier/frame/compiler/recursion lineage. Similarity, Smith normal form and primary-module algebra are classical background. JoyboyBrian, with substantial OpenAI Codex assistance. Apache-2.0.

Submission refresh: canonical main 4c0a359d88f22f05189d51f48387750b801a45d4; governing guidance, selected record and this contribution's source inputs remain unchanged from the scientific base. Fresh overlap covers 309 open PR bodies, 4,156 Discussion records and all 6,760 local notes with complete reported pagination/inventory. New public deltas were checked; this remains advisory text coverage, excluding arbitrary diffs and unpublished work.