[thu-nmrc]

For every even n>=4 and every coupled width q, any three commuting compatible full-stage projectors in a shortest n+1-query tuple on nq independent fields, satisfying their actual chronological-prefix conditions, have total stage rank at least7q. Individual ranks may differ and internal port matrices need not commute. An explicit unimodular ordinary frame attains(2q,3q,2q) with literal endpoint S diag(ABC,AB,B,C,1,...,1) at every n,q. The field has characteristic different from two.

In particular, two minimum-rank2q full stages force every other full stage to have rank at least3q. A coupled q2 rank13 Gram configuration cannot become a complete independent-stage caller by increasing the even field count while retaining the shortest-query contract.

The uniform proof has three cases:

• An even interior site has zero-paired image/row closure of dimensions2q and2q, so its stage already needs rank4q.
• Each odd interior site contains a forced rank2q projector. Two such forced projectors have commutator rank2q; making the actual stages commute needs at leastq extra dimensions between them.
• With only one odd interior site and both endpoints, a far-endpoint leakage bound gives rank(Q)>=4q-2e, where e is the odd stage's extra rank. This imposes the same total extra-rank debtq.

These cases exhaust every triple at every even n. The attainment retains every original independent input, including the extra central-only outputs. A genuine compatible q2 rank13 shortest tuple is included as a failed-prefix scope control, not a priced supplier.

Credit: Ali Romman's #6398 already gives the scalar all-even obstruction, image/row closure method and forced odd projector. The complete coupled normal form and the all-length fixed alternating-interior/curvature and generic low-rank correction tools are also retained predecessor results, following Cinna Davis's query/controller lineage. Original source bytes/notices remain exact. The new contribution is the uniform arbitrary-three mixed-rank threshold from quantitative odd-pair and endpoint-leakage arguments, plus the explicit all-n attaining family. Targeted repository review found no matching coupled mixed-rank statement; no world priority is claimed.

Validation with Python3.12 and the standard library:

• Metadata and exact source pins passed; every generic Git pin is reachable in the submitted history. The separate fork's original proof remains authenticated by its original Git blob/SHA256 through a packaged exact copy, requiring no network fetch.
• All four registered checks passed locally in18.92s: main normal/-O and a separately written independent implementation normal/-O.
• A separate Git-free export of all 20 committed package files passed those four checks in18.089s, with every input byte unchanged.
• The registered upstream workflow passed on head 3c9072b8738323dda9f9c7dfb421d578ed3fc17a, including both replay implementations in normal/optimized mode and the committed-input cleanliness guard. CI success establishes reproducibility; scientific admission remains pending.
• Complete multivariate stage endpoints retain all independent inputs. Uniform prefix/commutator/far-closure formulas, exact attaining words and necessary failure controls are rebuilt.
• The written proof supplies unbounded n,q and field scope. Finite controls corroborate conventions. Reviews and both implementations are author-side AI-assisted evidence.

python3.12 -B scripts/research_package.py --package research/p104-coupled-three-stage-rank-budget

No selected-frontier file changes. This is a stage-dimension theorem under a precise algebraic contract; it claims no newκ, physical native supplier, paid-call invoice or multiplication theorem, and does not discharge the inherited framework assumptions. Prepared for thu-nmrc with substantial OpenAI Codex assistance, including separate proof/overlap/arithmetic review. Apache-2.0. Scientific admission remains pending.