[zxv]

Cross-column routing, changing representations and approximate consumers sit outside many existing native-control obstructions. This contribution gives a usable finite-memory physical endpoint contract: a fixed transfer table either has an exact certificate for every positive width or a literal counterexample at bounded width. It also proves exponential transfer-dimension requirements for two routing families, including operator-norm approximation.

The proof and research guide develop the scope, constructions and follow-ups. This is a scoped written theorem with author-checked exact tools, awaiting independent mathematical review. The automata and matrix-product foundations are classical; no new general equivalence algorithm, global literature priority, paid-query minimum or κ is claimed.

Result

Let a complete residual endpoint have entries

$$\left[F\,T_{(y_1,x_1)}\cdots T_{(y_f,x_f)}\,B\right]_{ \text{output role},\,\text{input role}},$$

with fixed $D$-state transitions and boundaries, including every independent input and required dirty return. Then it vanishes at all positive widths iff it vanishes at every width $1,\ldots,D$. A $D$-cycle first fails exactly at width $D$, so the bound is sharp. The proof refines it to $D-r_1+1$ when the positive seed space has rank $r_1\gt 0$.

The tool avoids constructing those tensor matrices. It emits a finite invariant-row certificate

$$F T_s=E_sR,\qquad R T_s=C_sR,\qquad RB=0 \quad\text{for every physical column pair }s.$$

Alternatively, it emits a physical input/output column, role pair and exact nonzero discrepancy. A separately transcribed checker performs only exact equations or literal witness replay; it imports no producer or elimination code.

Consequence	Scope and practical use
Concrete all-width semantics	Physical column entries already contain native relations and absolute phases. Dense/non-Clifford maps and finite-memory column mixing are allowed. All entries and return roles matter; sampled widths/columns are insufficient.
Compositional contracts	Sums add transfer dimensions; physical compositions multiply them, with global owner sums handled separately. Supplied recurrence coefficients $aG^fb$ and finite schedule/width filters are included in the state bound.
A strict cross-column extension	Prefix XOR has minimum transfer dimension two and a certified inverse. It does not commute with column permutations, so identical one-column tensor-power expansions cannot represent it. This extends the semantic scope beyond #5436.
A pinned native adapter	The nine-axis source word from #5369 is checked with both dirty roles, all four owner/input blocks and all 512 local addresses. Its prefix-encoded shear has eight transfer states and a two-row invariant.
Exact and robust routing exclusions	Swapping two $k$-column halves of local dimension $d$ requires $D\ge d^{2k}$; their complex phase $(-1)^{a\cdot b}$ requires $D\ge 2^k$. Operator-norm error $\varepsilon\lt 1$ still requires $D\ge(1-\varepsilon^2)d^{2k}$ and $D\ge(1-\varepsilon^2)2^k$, respectively. These are cut-rank bounds in the declared column order.
Sharp limits on testing	Bounded bond separately at each width does not imply a fixed-table family. Small early-width errors do not imply a uniform approximation bound, even for fixed unitary families with a two-state comparison.

Significance for researchers

This opens a verifiable class of changed column interfaces and closes bounded-transfer-state searches for specific nonlocal routing demands. A proposal can yield either a complete all-width certificate, a concrete semantic defect, or a finite cut-rank obstruction before expensive physical synthesis. For #5419's changed consumers and #5427's observation questions, the next task becomes precise: extract the actual physical tables with source versions and returns. This package has not supplied a missing observation or decided those callers.

Risks and tradeoffs

Boundary	Opportunity and remaining obligation
Transfer dimension is a representation cost	Small $D$ can make exact checking cheap. It does not price owned fields, physical memory, Gaussian calls, column access or tape movement. A virtual two-state map is not a free native primitive.
Extraction can expand	Closure checking is polynomial in the explicit table. Extracting that table from a succinct program can grow exponentially; existing formal/native frontends may be much smaller.
Uniformity is essential	Arbitrary width-dependent boundaries, recipes or position schedules invalidate the cutoff unless a fixed finite controller is explicitly included. Regular restrictions prove only their declared domain.
Inverses and dirty state	A supplied inverse must have a valid representation and complete endpoint checks. Bounded representation of an arbitrary inverse is not assumed. Returning zero cannot replace restoration of an independent dirty input.
Routing and approximation	The bounds depend on the cut and column order; legal reordering or side information may change them. The signed-phase/norm bounds use complex semantics, not $\mathbb{F}_2$/BIT. Exact certificates do not supply recovery-safe precision bounds.
Source and proof status	The source instance is a nine-axis identity, not the full provider or selected caller. Independent review, physical compilation, pricing, precision, stock and analytic transfer remain open.

Nearest overlapping work

Relevant work	Retained contribution and distinction
#5235, #5262, #5352	Classical automata-based complete register certificates, true named inverses and bit bounds. Their alphabet is formal operator letters; here it is concrete physical input/output column pairs, with a fixed finite-memory family across widths.
#5369, #2250	Specialized affine-quadratic native lifting and the recipe lineage. The source fixture is retained exactly. Their compact specialized criterion remains preferable within its language.
#5436	Moving-support columnwise obstruction. Prefix XOR supplies a concrete map outside that representation; the current theorem requires fixed finite transfer tables instead of its pointwise support hypothesis.
#5419, #5427	Changed coupled consumers and frozen-word observation/reuse tests. They motivate source adapters; their availability, price and physical nonmembership claims are not promoted through this theorem.
#5425	Standalone paid-fanout optimum in its stated grammar. Transfer dimension and cut rank do not extend that result to incremental costs at a caller cut.

Classical inputs retain credit to weighted-automata equivalence, Tzeng's algorithm and the matrix-product/automata correspondence. Cut-rank and projection-error arguments are also classical. The new contribution is this concrete repository contract, source-bound frontend and explicit enabling/exclusion boundaries.

Validation

python3 -B research/finite-memory-lifting/verify.py
python3 -O -B research/finite-memory-lifting/verify.py
python3 -B research/finite-memory-lifting/audit_mutations.py
python3 -B research/finite-memory-lifting/verify.py --source-root .

• 24 small transfer tables, 138 independent path-sum comparisons, and sharp first-failure examples at widths 1 through 12.
• 126 literal prefix/inverse columns and 340 complete prefix matrix entries, with a nonzero cut minor and a five-state inverse residual certificate.
• All four dirty owner/input blocks, 512 independent native address checks, an eight-state encoded shear and a rejecting absolute-phase mutation.
• Exact half-swap unfoldings through width eight, Walsh cut/Gram controls through width twelve, and fixed-unitary examples with small early but large late errors.
• Ten invalid-certificate/input controls and four semantic code mutations in normal/optimized modes. Standalone checker runs without the producer, Git or PATH tools; portable read-only replay preserves source bytes.

RESULT.json, VALIDATION.json and MUTATIONS.json retain the outputs and actual environment. SOURCES.json pins seven canonical provenance files at 32890bc, priority revision 7; selected conditional $\kappa=0.003027288235$. OVERLAP.json states bounded coverage, including its historical/local limitations.

No full 112-owner/17-axis provider, selected cold replay, physical cross-column implementation, generic replay registration, Lean, Docker or hosted CI run is claimed. Universal conclusions rest on the written proof.

Credit

Retain zxv's register-certificate bridge, Krzysztof Wójt's inverse frontend, Ali Romman's bit bounds, Jose Manuel's native recipe implementation, and the full retained native/phase/compiler/source lineage. New prose and code are Apache-2.0 with substantial OpenAI Codex assistance. No independent human or proof-kernel acceptance is claimed.

Open next questions

Investigation	First useful result and principal blocker
One actual caller	Extract complete physical transfer tables for a named provider/consumer, with source versions and dirty returns. Table/state growth may defeat direct extraction.
Physical observation membership	Bind one candidate from #5427 to actual phases and entries. Formal-word nonmembership alone does not answer this.
A legal column-mixing adapter	Price prefix encoding, its inverse, access and restoration in an allowed native/tape contract. Virtual state count does not supply the bill.
Source-specific cut obstructions	Produce one exact coefficient minor for a real routing demand, after allowing legal reordering and side information. A convenient artificial cut is insufficient.
Compact semantic extraction	Combine the existing formal frontends with physical transfer tables and polynomial bit certificates. Concrete relations can cause state explosion.
Uniform approximation and recovery	Prove a conditioning/contractivity criterion for a real consumer, then account for precision and exact recovery. Early-width tolerance checks cannot certify it.