[petalas]

Summary

Extend PR3549's common-span triple-reader floor to three distinct groups with unequal, different binary spans under an orthogonal block refinement. A group contains a complete Gram-I basis of a block or none of it, and every complete plane stays in one block. Bases may differ inside shared blocks.

n_k = binary dimension present in exactly k target groups
original D = n_1 + 2n_2 + 3n_3
VVV bill >= D + n_2
MMM bill >= D + n_1 + n_2
therefore every exact realization with at most four terms costs >= D

The proof covers every coefficient field and arbitrary ambient factors. The original three terms attain D. Residual bounds are sharp over allowed basis choices. General nonreducing spans, block-crossing planes, partial block bases, five terms and physical supplier costs remain open. No kappa or selected-frontier change is claimed.

Package: research/p112-orthogonal-block-triple-floor. It retains the original hypothesis, explicit narrowed deliverable and scope history. Credit and actual proof/runtime sources include PR3549, PR3538, PR3513, PR3505, Philippe Van Velzen/Arham Tahir's expanded-identity method, Tekalign/PR3370 and their retained lineage. The two pending proof texts are published as exact local bytes with origin commit/path/blob/SHA256; replay needs no foreign Git object.

Evidence

• Before: PR3549 requires two largest groups to span the same binary space. Its different-span handoff remains open in general.
  After: The written blockwise proof establishes the stated enlarged contract and explicit residual excess. The unequal nested 6/4/2 fixture has D=12, VVV minimum 14 and MMM minimum 16. A pairwise-overlap fixture with no equal spans has D=12 and both residual minima 18.
• Two separate implementations match all 11,856 assignments and complete four-charge streams across eight fixtures. Integer coefficient identities, 135 gauged finite-field entries, eight optimized actual CLI faults and two positives pass.
• CPython 3.12 registered replay and inherited complete seven-point classification checks pass. The actual final commit passes metadata/pins and check_research_commit.py --check-source-history.
• All five optimized scientific/control commands pass from the exact 22-file read-only, regular, inode-disjoint committed Git-free export. Source/export hashes are unchanged. Focused Ruff diagnostics and formatting pass.

python3.12 -Werror -B scripts/research_package.py --package research/p112-orthogonal-block-triple-floor
python3.12 -Werror -B scripts/check_research_commit.py --package research/p112-orthogonal-block-triple-floor --check-source-history

The 60 exploratory random general-span fixtures are corroboration only and do not enter acceptance. Independent human/formal proof review, hosted CI, the selected full cold replay and physical/compiler/supplier obligations are unclaimed. Investigation and original claim.

Merge Danger

Door: two-way

Blast Radius: package

Only a new isolated research package is added. Canonical inputs, frozen predecessors, shared code and the selected frontier are unchanged.