[zxv]

The nonlinear weighted-monomial control obstruction in #5210 requires a fixed finite set of complete one-column operator products. This contribution replaces that restriction with a pointwise support bound: the operators, coefficients and word lengths may vary arbitrarily with width, but sublinear expanded support still cannot produce a nonzero control in the declared four-control interface.

The proof gives the exact quantifiers, a concrete 112-owner consequence, a sharp general growing-support escape and a separate approximation boundary. The research brief maps the result to follow-up work. This is a written scoped theorem with exact finite calibration, awaiting independent mathematical review; no new κ, admitted supplier or universal native-program lower bound is claimed.

Result and changed hypotheses

At width $f$, consider

$$P_f=\sum_i B_i(f)\otimes A_i(f)^{\otimes f}.$$

After merging proportional nonzero one-column maps, let $m(f)$ be their number and $r(f)$ their span rank. The classical multi-copy independence bound gives

$$f\ge m(f)-r(f)+1 \quad\Longrightarrow\quad \bigl(P_f=0\ \Longrightarrow\ B_i(f)=0\text{ for every }i\bigr).$$

The proof applies pointwise, including matrix/operator coefficients on a separate auxiliary space. No fixed operator set, bounded word length or fixed alphabet is needed. A simple sufficient condition is $m(f)\le f+1$ when $m(f)\ge 2$. Equality of two implementations uses their combined projective support; proportionality by $\lambda$ contributes $\lambda^f$ after lifting.

The scalar separation theorem and earlier repository tensor arguments are retained inputs. The new result is the moving-support native-control corollary, with explicit source expansion counts and an escape boundary.

For the released interface in #4933 (pinned physical contract), the four controls are $T_0$, its $Q$ conjugate, its $D$ conjugate $T$, and the terminal control $E=Q^{-1}C_3TC_1Q^{-1}$. Suppose each owner block has at most $s_0,s_Q,s_D,s_E$ monomial tensor terms, respectively. The code $A$ is invertible with zero diagonal, and its $(q,U,W)$ partition geometry makes every nonzero terminal source term nonmonomial. If $A$ has maximum row and column supports $R,C$, then

$$f+1\ge\max\{s_0+s_Q,\ s_0+s_D,\ RC\,s_0+s_E\} \quad\Longrightarrow\quad T_{0,f}=0.$$

The code from #4929 has 112 owners and $R=C=52$. Consequently, if $s(f)$ bounds every block in all four controls,

$$2705\,s(f)\le f+1\quad\Longrightarrow\quad T_{0,f}=0.$$

A nonzero all-width family in this interface therefore requires $s(f)\gt (f+1)/2705$. This excludes every sublinear-support family, including width-dependent nonlinear weighted monomial words of unbounded length. The coefficient 2705 is a conservative expansion bound, not an optimal source-specific threshold.

Significance for researchers

The result answers whether changing the complete operator products with width can evade the existing exact obstruction. Under the stated support bound, it cannot. Searches over bounded or sublinear expanded control families can stop within this interface, while growing-support, cross-column and approximate mechanisms remain legitimate changed premises. The theorem also supplies a reusable test for when sparse formal operator arguments reflect physical tensor semantics, provided the formal labels are first resolved to actual projective one-column maps.

Sharp escape and approximation boundary

For the fixed Gaussian-rational unitary $D=\mathrm{diag}(1,(3+4i)/5)$, $D^{\otimes f}$ has exactly $f+1$ distinct eigenvalues. Its minimal polynomial constructs an exact relation

$$\sum_{j=0}^{f+1}c_j(f)\,(D^j)^{\otimes f}=0,$$

with $f+2$ nonzero terms, all powers of one generator. The coefficient bit lengths are $O(f^2)$, with $O(f^2)$ exact arithmetic operations to construct them. This makes the general support threshold sharp and shows that a fixed generating alphabet or unitarity does not close the growing-support escape. It supplies neither a useful productive transform nor a priced physical implementation in the selected free alphabet.

Approximation needs a different argument. An explicit changing two-dimensional unitary satisfies

$$0<\left\lVert U(f)^{\otimes f}-I\right\rVert_{\mathrm{op}} \le\frac{2}{f}.$$

Thus only two terms can approach cancellation while remaining exactly independent. The proof includes a quantitative Gram bound when actual one-column overlaps are controlled; a recovery-safe approximate decoder still needs its own conditioning, precision and consumer-error analysis.

This does not contradict #5111's robust error floor for the actual two-stage inverse-postdecoder target: that theorem fixes the target and allowed decoder operations, whereas the example here varies the operators with width. It is the closest starting point for a source-specific approximate extension.

Risks and scope limits

Boundary	Consequence for research
Expanded support versus execution cost	A compact circuit can generate many terms. The theorem gives no scalar-instruction, tape-time or paid-Gaussian-call lower bound.
Auxiliary state	Separate auxiliary coefficient spaces and extracted endpoint blocks are allowed under the support hypothesis. Arbitrary entangling or cross-column maps without that expansion are outside it.
Physical semantics	Distinct free-word labels need not be distinct physical maps. Canonicalization, proportionality and lifted phases must be checked before using separation.
Source specificity	The 2705 consequence retains the four controls, zero-diagonal code and source coset geometry. Changed observations, codes, productive endpoints or additional paid calls require a new application proof.
Sharpness	The $f+2$ example proves sharpness of the general separation threshold. It does not prove that the source-specific factor 2705 is sharp or exhibit a successful four-control decoder.
Proof status	The universal claims rest on the written derivation. Finite checks corroborate the algebra and source excerpts; independent mathematical and formal review remain outstanding.

Nearest overlapping work

Relevant PRs	Relationship and boundary
Direct predecessor: #5210 (pinned proof)	Preserve its support-partition argument; replace fixed finite complete-product support with a pointwise cardinality condition.
Physical source: #4929, #4933	Retain the center-free code, normalizers and released endpoint. These supply the concrete 112-owner geometry and $R=C=52$ consequence.
Earlier separation: #4928, #4978, #5160	Retain finite-alphabet/projective tensor separation and the monomial-Clifford control predecessor. The underlying independence lemma is not new.
Constructive fixed-language criterion: #5369	Gives a cutoff for a fixed affine-quadratic native language. Here weighted monomial products may change with width under the support bound.
Robust terminal obstruction: #5111	Gives an error floor for a specified inverse-postdecoder target and operation class. Our two-term changing-operator example does not defeat that target-specific floor.
Standalone paid-call optimum: #5425	Proves a 112-call optimum in a fixed ordinary-only fanout grammar. Here the bound concerns expanded four-control support; it establishes no paid-call minimum.
Different observation/helper contracts: #5427, #5433	Adjacent applications, not established corollaries: their actual physical expansions and endpoint hypotheses must first be checked.

The scientific source checkpoint is canonical 1a3ced6, with selected conditional $\kappa=0.003027288235$. Before submission, canonical 32890bc was checked on 2026-10-24 at 13:19 UTC: priority revision 7, the selected record, guidance and all eight pinned source files are unchanged. The scientific pins are retained. SOURCES.json binds the exact source files and excerpts. The overlap snapshot records coverage and the submission refresh; arbitrary historical diffs and unpublished work are outside that coverage. The novelty claim is a scoped extension relative to the inspected repository, not global priority for tensor independence.

Validation

Standard-library exact arithmetic on CPython 3.14.5, macOS 26.5.1, arm64:

python3 -I -B research/support-width-dichotomy/verify.py
python3 -O -I -B research/support-width-dichotomy/verify.py
# Authenticate source excerpts against the pinned checkout:
python3 -I -B research/support-width-dichotomy/verify.py --source-root .

• 60 rational vector sets and 330 separating certificates, checked against a separate compressed tensor-rank calculation; seven moving-support families and a sharp rank-refined boundary.
• Matrix-coefficient Gram identities, projective phase powers and a characteristic-two control.
• Exact Gaussian-rational $(f+2)$-term relations at $f=1,\ldots,8$ and two-term approximation examples at $f=1,\ldots,12$. Omitted terms and reuse at the next width reject.
• 12,544 source code entries, the mode Gram identity, zero diagonal, row/column support counts and 112 distinct binary port maps.
• Source-bound normal/optimized runs and portable read-only runs match byte-for-byte. Altered code excerpts, projective duplicates and a corrupted separating certificate reject.

RESULT.json, VALIDATION.json and FIXTURE.json retain the output, environment and literal source excerpts. The original producer, full 17-axis phase engine, native tape/compiler, selected cold replay, Lean, Docker and hosted CI were not run. The package is not registered with the generic replay workflow; these are standalone scoped checks.

Credit

Classical independence and sharpness retain the attribution in the original multi-copy paper. Native/source and predecessor credit remains with Chris R, Ali Romman, Dillon Amburgey, David Ganz, Cinna Davis, Cao Yifan, zxv, Jacob Alan Sussman and their retained Clifford, geometry, compiler and verification lineage. The full proof and source manifests preserve the dependency context.

New prose and checking code are Apache-2.0, with substantial OpenAI Codex assistance. No independent human or formal-kernel acceptance is claimed.

Open next questions

Investigation	First useful step and remaining obstacle
Review the moving-support corollary	Audit the three support inequalities, projective merging and q/U/W partition proof. Verify the interpretation of extracted auxiliary-state endpoints.
Tighten 2705	Count actual terminal projective classes and their span ranks for one owner block. The $RC$ bound may substantially overcount, but a tighter count needs a usable operator representation.
Certify emitted native programs	Extract one actual endpoint expansion with correct phase powers, source versions and auxiliary returns. Formal word equality alone does not establish physical distinctness.
Price a growing-support escape	Bind the explicit annihilator to a real consumer and charge coefficient generation, precision, every field pass and restoration. An identity computing zero is not yet a productive module.
Control approximation through recovery	Bound actual Gram separation and consumer error together. Exact independence cannot rule out the demonstrated two-term approximate escape.
Change the columnwise interface	Specify a cross-column or observation-carrying program with independent inputs and a complete storage/transport/return bill. A changed representation needs its own legal ABI.