[zhaobenny]

For matrices with absolute row cap rho and column cap m*rho, column-budget control has a different Euclidean critical boundary than the induced 1-norm results in #6788/#6806. This isolated package proves the sharp dimension-growth order of the worst ||(I-E)^-1||_2 for integer m>=2 and fixed 0<rho<1:

• rhosqrt(m)<1: limit exactly 1/(1-rhosqrt(m)).
• rho*sqrt(m)=1: Theta(log n).
• rhosqrt(m)>1: Theta(n^(log_m(rhosqrt(m)))).

A finite Neumann-series upper bound covers arbitrary complex matrices. Nilpotent, zero-diagonal m-ary trees give all lower bounds. The critical lower bound uses coherent input across levels; a single inverse column would miss the logarithmic order. Exact finite-dimensional maxima are unclaimed.

Only six new files under research/column-cap-spectral-growth/ are added. The original uniform-transfer-audit card is retained with explicitly changed/partial scope, an abstract conditioning side result. Its actual-caller acceptance gate remains unmet. No emitted Gaussian matrix, physical precision bill, supplier, selected assumption or new kappa is admitted. No selected or predecessor frozen files change.

Validation at head 867ecd59bce6e6a3ae0e894e5aa4f4d0d66e55dd, Linux x86_64, CPython 3.12.9, standard library only:

• Registered warning-strict normal and optimized replays pass with identical counts: 15 dense trees, 960 capped real matrices, 12 complex-phase realifications, 63 virtual radial cases through height 64, six controls. Dense inverses are checked in both multiplication orders, against path formulas, and with rational positive-semidefinite Euclidean upper certificates.
• Metadata/pins, exact committed bytes, executable modes and declared source history pass. Official export integrity and both declared inputs pass. Both direct modes from the final read-only Git-free export with empty PATH/PYTHONPATH pass with matching receipts; all nine exported files including the manifest remain unchanged.
• git diff --check passes. Selected baseline source bindings passed; no selected cold replay, native suite, predecessor scientific suite, Lean or Docker run. Hosted checks and independent scientific review are not claimed.

Reproduce:

python3.12 -B scripts/research_package.py --package research/column-cap-spectral-growth
python3.12 -B scripts/check_research_commit.py --package research/column-cap-spectral-growth --check-source-history --check-modes

Canonical main was refreshed at entry and before submission, unchanged at 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. Final overlap snapshot covers 762 open PR bodies/heads and 6,887 Discussion records with complete remote pagination, plus all tracked local-note paths and the new notes. #6788/#6806 remain at their cited heads. Their induced 1-norm optimum is distinct from this Euclidean growth result. Body/note search does not cover every diff or unpublished work. Advance coordination claim; this PR is the canonical investigation artifact.

Credit Nick Petalas #6788, ToikY #6806, Ali Romman #5255/#5306, Cinna Davis #4100 and James Reed #4282, with retained conditioning/Gaussian lineage. Classical Neumann, norm interpolation and regular-tree radial methods retain their credit; general literature priority is unclaimed. Prepared for zhaobenny, Apache-2.0.

Produced with OpenAI GPT-6.1 Sol at high reasoning effort through Codex.

The same assistant wrote the proof and verifier after reading related sources. No subagents, independent human review or proof-kernel acceptance. Open next questions cover actual source binding and exact finite optima. All five selected framework assumptions remain; no merge or frontier change.