From e56a5bcf239fa0cd4ee57459af04c0ba7acb2c03 Mon Sep 17 00:00:00 2001
From: CubeJerry <80299683+CubeJerry@users.noreply.github.com>
Date: Mon, 9 Nov 2026 01:54:04 +0800
Subject: [PATCH] Prove selected P104 all-column native caller closure

---
 .../NATIVE.json.gz                            | Bin 0 -> 93977 bytes
 .../selected-native-tensor-closure/PROOF.md   | 411 ++++++++++++++++++
 .../selected-native-tensor-closure/README.md  | 205 +++++++++
 .../RESULT.json                               | 154 +++++++
 .../SOURCES.json                              |  60 +++
 .../selected-native-tensor-closure/TASK.json  |  88 ++++
 .../VERIFY.json                               |  69 +++
 .../caller_check.py                           | 212 +++++++++
 .../controls.py                               | 133 ++++++
 .../emit_recipes.py                           |  90 ++++
 .../gauss_trace.py                            | 332 ++++++++++++++
 .../native_check.py                           | 168 +++++++
 .../source_algebra.py                         | 210 +++++++++
 .../test_gauss_trace.py                       | 176 ++++++++
 .../selected-native-tensor-closure/verify.py  | 110 +++++
 15 files changed, 2418 insertions(+)
 create mode 100644 research/selected-native-tensor-closure/NATIVE.json.gz
 create mode 100644 research/selected-native-tensor-closure/PROOF.md
 create mode 100644 research/selected-native-tensor-closure/README.md
 create mode 100644 research/selected-native-tensor-closure/RESULT.json
 create mode 100644 research/selected-native-tensor-closure/SOURCES.json
 create mode 100644 research/selected-native-tensor-closure/TASK.json
 create mode 100644 research/selected-native-tensor-closure/VERIFY.json
 create mode 100644 research/selected-native-tensor-closure/caller_check.py
 create mode 100644 research/selected-native-tensor-closure/controls.py
 create mode 100644 research/selected-native-tensor-closure/emit_recipes.py
 create mode 100644 research/selected-native-tensor-closure/gauss_trace.py
 create mode 100644 research/selected-native-tensor-closure/native_check.py
 create mode 100644 research/selected-native-tensor-closure/source_algebra.py
 create mode 100644 research/selected-native-tensor-closure/test_gauss_trace.py
 create mode 100644 research/selected-native-tensor-closure/verify.py

diff --git a/research/selected-native-tensor-closure/NATIVE.json.gz b/research/selected-native-tensor-closure/NATIVE.json.gz
new file mode 100644
index 0000000000000000000000000000000000000000..7eb40e7a8a172c188fcbc47b015905d5d38bf9ea
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diff --git a/research/selected-native-tensor-closure/PROOF.md b/research/selected-native-tensor-closure/PROOF.md
new file mode 100644
index 0000000000..fffff67965
--- /dev/null
+++ b/research/selected-native-tensor-closure/PROOF.md
@@ -0,0 +1,411 @@
+# Exact native traces and the complete selected tensor interface
+
+Prepared for @CubeJerry with substantial OpenAI Codex assistance. Apache-2.0.
+This is a source-bound algebraic certificate and a written universal argument.
+It proves the exact-operator assertion labeled `native_all_columns` for the
+pinned selected P104 native layer and its assembled caller. Physical
+implementation, selected-rate dispatch, owned allocation/erasure and the other
+framework premises remain separate. No admission record or kappa is changed.
+
+## 1. What must be composed
+
+The selected word is DreamingOfClouds' #6030 at canonical main
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`. Its local certificate, the
+all-forward P104 caller and the native tensor interpretation are different
+objects. A dense equality at column width one cannot identify a sum of
+different native group elements at all widths. Conversely, equality of two
+*products* of unitary native gates is sufficient once every factor is lifted
+columnwise and exterior scalar coefficients remain exterior.
+
+The local common-chart argument is already present in the P104 source and
+#6635. #6043 independently checked every selected native operator densely.
+#6546 supplies a numerical execution and tensor lift of the fixed local word,
+and explicitly separates that from identifying recursive callers with it.
+#5442 proves an all-width certificate for the different #5413 caller.
+The contribution here is an independent exact trace acceptance mechanism and
+its complete source-bound composition for the selected #6030 local word,
+helpers, wrappers and all-forward caller. These predecessor identities are
+credited, not asserted to be new lemmas.
+
+## 2. Exact traces of the native grammar
+
+Let the address dimension be n. Accepted primitive gates are an invertible
+binary address permutation P_A, a binary translation X_v, a phase i^c,
+a diagonal i^q(x), and
+
+    C_v^s = ((1+s*i) I + (1-s*i) X_v)/2,  s in {+1,-1}.
+
+Here q is a Z/4 polynomial with arbitrary linear coefficients and even mixed
+coefficients. The source additionally requires its symmetric quadratic matrix
+and diagonal-parity convention. The checker validates this grammar, every
+binary inverse and every sign before using a trace. P_A and X_v permute an
+orthonormal basis, diagonal phases have modulus one, and X_v is a self-adjoint
+involution. On its +/- eigenspaces C_v^s has eigenvalues 1 and s*i. Thus every
+accepted gate and every compared complete product is unitary, including its
+normalization. Scalar ADD/COPY/SCALE programs are **not** passed to this test.
+
+Introduce n initial-address bits x and one binary branch t for each C gate.
+The identity
+
+    C_v^s = (1+s*i)/2 * sum_(t=0,1) i^(-s*t) X_v^t
+
+expresses every path amplitude as a known Gaussian dyadic scalar times
+i^Q(x,t). Throughout the circuit each current address bit is an affine
+Boolean form in x and previous branch variables. A phase gate contributes
+only a quadratic polynomial modulo four: for a parity L=sum x_j mod2,
+
+    L = sum x_j - 2 sum_(j<k) x_j*x_k  (mod 4),
+
+and twice a product of affine Boolean forms has degree at most two modulo
+four. Translations, address permutations and C branches update affine forms.
+Induction over the literal gate list therefore gives exactly
+
+    tr(U) = [product_C (1+s*i) / 2^h]
+            * sum_(z: final_address(z)=initial_address(z)) i^Q(z),
+
+where h is the actual number of C gates. The final equality is an affine
+binary system. RREF either detects inconsistency (trace zero) or parametrizes
+every satisfying assignment exactly once by free variables. Substitution into
+Q preserves the displayed quadratic form. No exponential path enumeration is
+performed by the checker.
+
+### Quadratic elimination
+
+Write Q=c+sum a_j x_j+2 sum_(j<k) b_jk x_j*x_k modulo four. All expressions
+below are exact, including constant phase and normalization.
+
+If a_j is odd, put L=sum_(k!=j) b_jk*x_k mod2. Summing x_j gives
+
+    sum_(x_j=0,1) i^(a_j*x_j+2*x_j*L)
+      = (1+i^a_j) * i^(-a_j*L).
+
+Delete x_j, multiply the accumulated Gaussian integer by 1+i^a_j, and add
+-a_j*L to the remaining quadratic. The parity expansion above implements
+that update, including its new mixed terms.
+
+If all a_j are even and b_jk=1, write the terms involving x_j,x_k as
+2(x_j*x_k+u*x_j+v*x_k), where u,v are affine Boolean forms on the other
+variables and include a_j/2,a_k/2. Then
+
+    sum_(x_j,x_k in {0,1}) (-1)^(x_j*x_k+u*x_j+v*x_k)
+      = 2*(-1)^(u*v).
+
+Delete both variables, multiply the accumulated scalar by 2, and add 2*u*v
+to the remainder. If no mixed term remains, an isolated coefficient 2 makes
+the sum zero; each isolated coefficient 0 contributes 2. Finally multiply
+by i^c. Each step eliminates at least one variable. This proves
+`gauss_sum`, independently of a Clifford tableau or any selected basis state.
+
+For k=n+h path variables and g literal gates, this implementation uses a
+polynomial number of bit/integer operations. A deliberately coarse bound is
+O(g*n^2*k^2+k^4) elementary set/coefficient operations: dense affine
+substitution costs at most O(k^4), and elimination at most O(k^3). Stored
+phase data use O(k^2) bits/entries, and Gaussian numerators have O(k+h) bits.
+The claim is polynomial finite-certificate verification, not an asymptotic
+runtime bound for multiplying input integers.
+
+### Trace acceptance
+
+For two checked unitary D-by-D matrices U,V,
+
+    ||U-V||_F^2 = 2D - 2 Re tr(V* U).
+
+Consequently exact `tr(V* U)=D` implies U=V on every input. The converse is
+immediate. The circuit for V*U is the chronological word for U followed by
+the reversed adjoints of V. The checker computes an exact Gaussian numerator
+and power-of-two denominator, cancels powers of two, and requires precisely
+`(2^n,0,0)`. A global i phase is not discarded. The test does not assume that
+equal unsigned symplectic actions, equal output at zero, or equal traces of U
+and V separately suffice.
+
+The frozen candidate recipes are rebuilt using the credited source compiler,
+and their **complete** payload hash must equal the selected RESULT.json
+registration. The acceptance engine imports neither that compiler nor its
+Pauli/column verifier. The expected chart circuits come from the pinned frame
+definition. An error common to a candidate compiler's tableau and zero-column
+implementation is not a shared dependency of this trace calculation.
+
+## 3. All-column tensor interpretation
+
+For f>=1 and an arbitrary spectator coefficient space S, write
+L_f(U)=I_S tensor U^(tensor f), with the fixed address shuffle placing columns
+in the source's slot order. Then
+
+    L_f(UV)=L_f(U)L_f(V),  L_f(U*)=L_f(U)*.
+
+This is a statement about full arrays, including arrays correlated across
+columns and spectators: tensor basis vectors span the full space. It does
+not require a pure-tensor input or zero spectator coordinates.
+
+Every accepted native equality is an equality between complete products.
+Therefore each such equality lifts under L_f. A rank-r positive child lifts
+to the identical elementary C on r*f selected binary axes. A monomial with
+constant phase c contributes i^(c*f), and its remaining phase is the sum of
+the one-column phase polynomials at all columns. A scalar coefficient a in a
+register ADD or SCALE remains a, not a^f. In particular L_f(iI)=i^f I differs
+from the exterior scalar iI in general. Likewise the width-one expansion of
+C_v into two scalar-weighted translations is not an allowed termwise tensor
+rewrite. The verifier tests both errors with exact width-two witnesses.
+
+The same argument extends a complete local address operator by identity to
+any complement and conjugates it by any complete coordinate bijection.
+Thus it supplies the algebraic input needed by #6878's occurrence-specific
+full-field embeddings. It does not price those bijections or replace their
+physical field-placement and ownership hypotheses.
+
+## 4. Complete caller join
+
+### Statement and hypotheses
+
+Fix exactly the WORD, RESULT, CONTROLLER, source chart definition, compiler
+recipe payload and original port data pinned in SOURCES.json. Let f be any
+positive integer. Each productive register contains an arbitrary vector in
+S tensor (C^(2^145))^(tensor f), for any finite-dimensional coefficient space
+S; equivalently one may use the source's 265 rows and its 120 unused identity
+spectator rows. The 13,444 productive registers are independent inputs, not
+copies or zero-filled helpers. Interpret native products by L_f, rational
+register coefficients as exterior scalars, and INIT/CLEAR by the ideal owned
+zero/reset operations in the source semantics. All coordinate maps below are
+complete bijections, including their complementary coordinates.
+
+Under these hypotheses, the literal 18-forward-call supplier with its 16
+five-step wrappers, 15 thin queries, whole-idle jobs, scalar boundary programs
+and whole-job adapters has the stated diagonal positive-Gaussian endpoint on
+all 13,444 arbitrary productive inputs and returns its 29 local owned labels
+to zero. Its paid and productive profiles are exactly those in the selected
+RESULT. The proof is uniform in f and S. This is a theorem of the exact typed
+algebraic program; it assumes ideal owned resets and native calls. It does not
+establish an encoded tape implementation, an accelerated primitive tariff,
+precision control, analytic recovery, or physical owned erasure.
+
+### Local chart induction
+
+Write Gamma_a for the full signed canonical chart associated with frame a.
+The independent trace acceptance checks each selected native recipe as
+
+    M_left G_r M_right = Gamma_b Gamma_a^*,
+
+where G_r is the product of positive elementary Gaussians on the first r
+axes. This includes 2,886 distinct MOVE pairs, the additional whole-helper
+pairs, and all 208 ordinary main adapters. The union has 2,922 pairs. The
+identity is between full operators, so Section 3 gives its exact L_f lift.
+
+If a register's physical value is L_f(Gamma_a) z, a MOVE to b makes it
+L_f(Gamma_b) z. ADD at a common current chart performs
+
+    L_f(Gamma_a) z + c L_f(Gamma_a) w
+      = L_f(Gamma_a) (z+c w).
+
+SCALE is the same linearity identity; a COPY uses an owned zero target.
+The coefficient c is retained outside L_f. All actual 18,404 ADD, 861 SCALE
+and 23 COPY events satisfy their current-chart and ownership tests, rather
+than relying on a declared common-chart convention. INIT establishes zero
+and its recorded chart; CLEAR deletes only a currently owned value. This
+last operation is an inherited ideal reset, not a deduction that arbitrary
+nonzero data can be physically erased for free.
+
+The exact rational sparse replay starts with 799 independent formal inputs,
+including the 29 owned entrance labels. After the 22,803 literal events it
+checks every endpoint coefficient: the 770 public coordinates are unchanged
+except Y_r -> Y_r+X_r at the 104 selected pairs; every owned entrance column
+has been eliminated and every owned output is zero. There are 31 distinct
+owned lifetimes, with peak 13. Finite verification determines this one fixed
+word's rational identity. The preceding symbolic induction, rather than a
+finite sweep of f values, proves that identity for all f and S.
+
+For each main port q put Q=C_q^+ and H_9=product_(j<9) C_(e_j)^+.
+The actual ordinary main entry/exit recipes are checked against
+
+    A_X = Gamma_initial-X Q^*,
+    A_Y = H_9 Q^* Gamma_final-Y^*.
+
+The source entrance frame of X is the one-vector frame [q], but its canonical
+chart Gamma_initial-X need **not** equal the translation Gaussian Q. The
+actual ordinary A_X above is therefore essential. The other two boundaries
+are Gamma_initial-Y=I and Gamma_final-X=H_9. Combining the three nonzero rational endpoint paths with
+these exact adapters gives, on the two main registers,
+
+    F_0 = D_0 [[1,0],[Q^(-1),1]],   D_0=H_9 Q^(-1).
+
+The checker verifies all three complete products separately at every port,
+including their constant phases. The displayed 2-by-2 notation is a matrix
+of native operators and exterior rational coefficients, not a matrix that
+is itself tensor-powered.
+
+For every arbitrary helper, the chart endpoint is Gamma_b Gamma_a^*.
+Apply M_right^* before the complete local word and M_left^* after it. Its
+entire productive endpoint is then G_r. All 69 distinct helper boundary
+types are checked as complete products; their multiplicities give all 562
+helper inputs. These normalizers are whole-job boundaries, not uncharged
+adapters inserted at each recursive child.
+
+### Source raw contacts and arbitrary spectators
+
+The original raw construction uses 265 formal binary axes e_j. At stage s>0
+the actual route swaps the source's eight orthonormal q-complement vectors
+b_i with its eight fresh axes e_(9+8(s-1)+i), and fixes q and the remaining
+orthogonal complement. The source port data are checked to make
+(q,b_0,...,b_7) an orthonormal basis. These eight disjoint orthonormal swaps
+give a complete involutive orthogonal map R_s. The explicit binary formula
+in caller_check.py is checked to perform exactly those swaps and fix q.
+
+A product of translation Gaussians has Fourier-character eigenvalue
+
+    i^p(z),   p(z)=sum_v c_v (v dot z mod2)  (mod4).
+
+Its polynomial has linear and even mixed coefficients, and constant zero.
+Polynomial equality therefore proves equality on every character; Fourier
+characters form a basis. An even linear difference 2(t dot z) is exactly the
+translation X_t, with absolute phase +1. The checker computes the complete
+entry/exit translations and their inverses from these polynomials. Their
+ordered payload equals the **already registered selected** route hash
+`7f4f0ca768ffbc4f93dcff575592c4c67411f46b4289e45b73f715a2ad47b142`.
+
+This is more than comparing abstract frame labels. If a native convolution
+path has phase p and the actual entry and exit maps are X_u P_R and X_v P_R,
+their executed composition is
+
+    X_v P_R A X_u P_R
+      = X_(v+R u) (P_R A P_R).
+
+Thus its eigenphase is p(R z)+2((v+R u) dot z). The checker uses the actual
+four contact records to verify both diagonal paths as D_s and the cross
+path as D_s Q^(-1), where D_0=H_9 Q^(-1) and, for s>0, D_s is the product
+of the eight positive fresh-axis Gaussians. This checks 5,616 composed paths
+through all 7,488 actual contacts, including spectator charts on entry and
+exit. It does not only compare their marginal source/target phase profiles.
+
+Only stages 0,1,3,...,31,32 occur. All 120 rows of the other stages are fixed
+by every actual contact and absent from its translations, main products and
+idle products. Taking the nine source rows and the 17 retained eight-row
+blocks identifies the actual 145-row supplier. This is a checked application
+of the retained-row construction credited to #6878, not a new compression
+claim. Every positive main/idle canonicalizer is explicitly completed on
+all 145 retained coordinates; the unused 120 rows and arbitrary further
+spectators are preserved as identity factors. Section 3 then transports
+each equality through the complete occurrence embeddings in #6878. No
+assumption about vanishing or independent spectator coordinates is used.
+
+The application to #6878 is source-compatible in a checkable sense. At its
+head `41febf890fbbb852c6059d4c0cc8c593ae9dfa5c`, that package pins the same
+CONTROLLER, original port data and caller implementation as SOURCES.json
+here, with identical full SHA256 values. Its ordered retained set is
+precisely the 145 indices just constructed, and its section 1 re-emits
+positive canonicalizers with the selected positions followed by the full
+retained complement, as done here. Its extension lemma takes a complete
+ambient-m unit with endpoint G_r tensor I, uses a complete coordinate
+bijection on entry and its inverse on return, and permits arbitrary
+spectators. Those are exactly the operator and spectator hypotheses proved
+here. Its compiler scalar contacts require routing entire participating
+fields into a common coordinate order; our induction likewise uses exterior
+scalars on complete aligned fields. Thus there is no model, width, phase or
+scalar-type mismatch at that algebraic interface. The current package needs
+no unmerged code to prove its own theorem. The larger compiler, ownership,
+movement and cost premises of #6878 remain as stated there.
+
+### All-forward controller and whole-job completion
+
+Each physical LOCAL has forward orientation. The source's 16 reverse
+interfaces use the literal five-event word
+
+    Q_X^(-2), SIGNED_SWAP(X,Y), F_s,
+    Q_Y^(+2), SIGNED_SWAP(Y,X).
+
+With chronological action and SIGNED_SWAP(X,Y):(X,Y)->(Y,-X), this word
+implements D_s [[1,-Q],[0,1]]. The minus sign is an exterior coefficient.
+The equality Q^(+2)=Q^(-2)=X_q has absolute phase +1; it is checked at every
+port. Each thin query realizes Q^(-1)=X_q Q by a positive rank-one Gaussian
+conjugated through the complete port basis, plus its ordinary translation.
+Its separate owned scratch label t has the explicit semantic macro
+
+    INIT t=0; COPY t=X; apply Q^(-1) to t; ADD Y+=t; CLEAR t.
+
+It leaves X unchanged, replaces Y by Y+Q^(-1)X, and returns t to zero under
+the stated ideal-reset semantics. The complete query basis and its inverse
+restore the original full field coordinates before ADD. The scratch label
+is outside the 29-label local owned bank and is reused only after CLEAR.
+This makes the additional owned return explicit; it neither supplies a
+physical erasure algorithm nor adds another paid native call. Each of the
+15*104 thin queries still invokes one rank-one positive child.
+
+To verify the *actual* controller, source_algebra.py propagates each of the
+32 final coordinate functionals backwards through every literal event and
+both scalar boundary programs. Coefficients lie in the free Laurent ring
+
+    Q[Q,Q^(-1),D_0,D_0^(-1),...,D_32,D_32^(-1)].
+
+This is independent of the predecessor's forward row-matrix replay. It
+checks all 1,024 entries of the resulting 32-by-32 operator and compares
+the literal boundary programs with their stored matrices. There is no
+Q^4=1 quotient that could conceal a central-power error. At every port the
+native operators Q,D_s commute, since they are products of the commuting
+translations X_v. Evaluation in this commutative operator algebra therefore
+preserves the verified identity. Their columnwise lifts also commute, while
+the rational coefficients stay exterior. Consequently the same exact raw
+diagonal endpoint holds at every f and every spectator space.
+
+For each of the 3,328 main fields, the raw endpoint is Q^epsilon product D_s,
+with epsilon in {+1,-1}. If epsilon=-1, its whole-job translation X_q changes
+the Q exponent to +1. The polynomial check establishes equality to the
+positive Gaussians on the complete recorded target axes. If stage 0 is
+present, use the original nine coordinate axes; otherwise use q and its
+eight-vector orthonormal complement. Append the selected fresh axes and
+**all** other retained coordinates. The complete invertible map conjugates
+this product to G_r, where r=1+8 times the number of selected stages.
+
+Each whole-idle product similarly receives one positive G_(8k) child with
+its complete basis and the exactly calculated whole-job translation, where
+k is the actual number of distinct stages in the window. All 3,120 fields
+are checked, including the windows that contain stage 0. A rank-r child
+becomes exactly rank r*f after the tensor lift; spectator coordinates remain
+untouched. The raw positive query, main and idle normalizations are all
+identities of full products, so no tensoring of a scalar-weighted sum occurs.
+
+Finally, every local call maps its 208 main public roles to the actual pair
+of controller rows, its 562 arbitrary helpers to fresh public inputs, and
+its 29 owned roles to the returned shared owned bank. These maps are checked
+as injective on all 799 labels, and the thin scratch label is separate.
+There are 3,328 arbitrary main and 18*562=10,116 arbitrary helper inputs.
+The local gauge identity, complete whole-helper normalization, complete raw
+caller identity and whole-job normalization prove the asserted direct-sum
+endpoint, with no remaining unspecified local algebraic adapter.
+
+## 5. Accounting and limits
+
+This package leaves the selected bill unchanged:
+
+| Quantity | Exact count |
+|---|---:|
+| Arbitrary productive fields | 13,444 |
+| Paid calls | 66,834 |
+| Paid rank mass | 240,152 |
+| Productive rank mass | 240,912 |
+| Actual forward local calls | 18 |
+| Five-step wrappers | 16 |
+| Thin rank-one calls | 1,560 |
+| Whole-idle calls | 3,120 |
+
+The complete paid profile is 18 times the selected local MOVE profile, plus
+the 1,560 positive rank-one thin calls and the 3,120 whole-idle calls. The
+productive profile is 18 times the 562-helper profile plus the complete main
+endpoint profile. These profiles are compared exactly with selected RESULT,
+including every multiplicity. This verifies algebraic accounting; it does
+not reprice ordinary adapters, infer a physical runtime from a recipe, or
+claim that the positive rank margin alone proves a multiplication bound.
+
+The selected record states `native_all_columns` as: retained native operators,
+phase conventions, spectators and whole-job adapters work at every positive
+column multiplicity. For the exact P104 objects and assembled caller pinned
+here, that algebraic assertion is now the theorem of Sections 2--4. Its status
+does not depend on a future maintainer admission or a physical dispatcher.
+The theorem does not silently extend to a different upstream native family.
+
+The result supplies this proved algebraic endpoint to a later physical
+dispatch theorem. That theorem must still realize every typed
+native product, whole-field scalar operation and coordinate map on the
+prescribed layouts, preserve or discharge the owned-reset semantics, and
+charge source access, staging, restoration and recursive children at the
+selected tariff. Existing physical monomial constructions #3055/#3087 and
+old-controller #3385 are credited but do not automatically establish that
+current combined tariff. The other framework assumptions remain explicit.
+No selected record or accepted kappa is changed. **No new kappa.**
diff --git a/research/selected-native-tensor-closure/README.md b/research/selected-native-tensor-closure/README.md
new file mode 100644
index 0000000000..531a5f64d6
--- /dev/null
+++ b/research/selected-native-tensor-closure/README.md
@@ -0,0 +1,205 @@
+# Exact native traces and the selected all-column caller
+
+This package proves and independently checks the **algebraic** assertion
+`native_all_columns` for the pinned selected #6030 P104 native layer and its
+assembled caller, at every positive column width and with arbitrary spectator
+factors. It joins every signed native recipe,
+the complete local word, all 18 forward calls and 16 wrappers, and every
+whole-job helper/main/idle adapter. It does **not** implement the selected-rate
+physical dispatcher or discharge the complete five-assumption framework.
+The all-column operator assertion is proved; physical realization and the
+source's ideal owned-reset semantics remain distinct obligations.
+**No new κ.**
+
+The accepted source remains canonical main
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`, with conditional
+κ = `781451/250000000`. No prior source, selected record, tariff or bill is
+modified. [PROOF.md](PROOF.md) states the exact quantified theorem, proves the
+trace algorithm and tensor interpretation, and gives the complete composition.
+
+## Result and evidence
+
+The new native acceptance mechanism computes the exact closed overlap
+`tr(V*U)` as a constrained quadratic Gauss sum over binary variables. Checked
+unitarity makes the value `2^9` equivalent to full operator equality. The
+checker retains constant phases and Gaussian normalizations; it uses neither
+a Pauli tableau, a distinguished input column, floating point, nor a dense
+512-by-512 operator matrix. Quadratic elimination is polynomial in the finite
+circuit description. This is a certificate-verification bound, not a
+multiplication-runtime improvement.
+
+The caller verification keeps two different types of objects distinct:
+native products are lifted columnwise, while rational coefficients of
+register ADD/COPY/SCALE and signed swaps stay outside the tensor power. A
+native constant phase becomes `i^(p*f)`; a register coefficient remains the
+same coefficient. The proof does not infer an all-width identity by tensoring
+a width-one sum of different native group elements.
+
+| Verified object | Complete coverage |
+|---|---:|
+| Distinct signed native/helper pairs | 2,922 |
+| Distinct MOVE pairs within that union | 2,886 |
+| Intrinsic main adapters | 208 |
+| Complete local main paths | 312 |
+| Distinct complete helper boundary types | 69 |
+| Arbitrary helpers per local call | 562 |
+| Local endpoint coefficients | 799 × 799 |
+| Literal local events | 22,803 |
+| Actual raw contacts | 7,488 |
+| Composed contact/native/contact paths | 5,616 |
+| Complete free-Laurent caller entries | 32 × 32 |
+| Arbitrary main productive fields | 3,328 |
+| Arbitrary helper productive fields | 10,116 |
+| Whole-idle paid fields | 3,120 |
+
+The source-bound reverse-functional Laurent replay checks all 218 physical
+controller events and both literal rational boundary programs. It uses the
+free Laurent ring, with no `Q^4=1` quotient. The geometry check composes the
+actual entry/native/exit maps, matches the selected raw-contact payload hash,
+and explicitly fixes the 120 omitted spectator rows of the 265-row source.
+Each main/idle positive basis is completed on all 145 retained rows. The
+all-width proof then applies to every complete occurrence embedding, including
+the embeddings supplied by #6878.
+
+The exact selected bill is unchanged: **13,444 productive fields, 66,834 paid
+calls, paid rank mass 240,152 and productive rank mass 240,912**. The checker
+reconstructs both complete profiles from the literal source events rather
+than inferring a bound from this positive rank margin. Ordinary adapters,
+source access, staging, physical erasure and restoration still require their
+own implementation and charging theorems.
+
+## What is new, and what is already proved
+
+This is not a new local common-chart lemma or another dense native census.
+The contribution is a different exact native acceptance mechanism together
+with the **complete selected-source join** from those native equalities to
+the assembled all-forward caller, including the exterior-scalar typing and
+whole-job boundaries.
+
+| Predecessor or concurrent work | Already established; relationship here |
+|---|---|
+| #5442, `research/allwidth-caller-certificate/` | Group-algebra and tensor framework for the older #5413 caller. The product-lifting principle is credited. |
+| #5974, `research/p104-allwidth-native-certificate/` | Full exact native verification for #5599 and the factorwise lift. Its native census is not repeated as a claimed new theorem. |
+| #6043, `research/p104-6030-independent-audit/` | Independent dense verification of the current #6030 native inventory. The present trace method shares neither its dense mechanism nor its candidate compiler's tableau/anchor acceptance. |
+| #6635 | Complete selected scalar common-chart census. The present typed induction uses that established mechanism and checks its composition with every MOVE. |
+| #6293 and #6331 | The `i^(p*f)` phase convention and the obstruction to deleting native phases. The current width-two controls explicitly enforce those distinctions. |
+| #6546, `research/selected-absolute-stochastic-caller/` | Concurrent direct stochastic local execution and a fixed-local all-width lift; its stated limitation is identification with complete recursive callers. The present package closes the algebraic assembled-caller join, without importing its unmerged code. |
+| #6878, `research/selected-compiler-occurrence-layout/` | Complete occurrence-specific coordinate embeddings and retained-row geometry. This package supplies their native algebraic endpoint premise; it does not reprice or redevelop the embeddings. |
+
+At the frozen reconnaissance, #6546 was an open contribution at
+`ff038070b8d122a43993f2b2524154323cc6b4d7`, and #6878 was an open contribution
+at `41febf890fbbb852c6059d4c0cc8c593ae9dfa5c`. They are credited comparisons and
+applications, **not runtime or source-import dependencies** of this package.
+The proved tensor/naturality statement quantifies over any complete embedding;
+the exact accepted inputs below all come from the pinned canonical main.
+
+The native/caller identities and raw phase construction in
+`research/p104-productive-controller/` and the independent caller audit remain
+credited source results. Existing physical compact, Gaussian-monomial and old
+controller work #3055, #3087 and #3385 was checked during scope selection and
+was not reimplemented. Its older tariffs do not automatically price the
+current selected dispatcher.
+
+## Reproduction
+
+From the repository root, using only Python's standard library:
+
+```sh
+python3 -I -B research/selected-native-tensor-closure/verify.py
+```
+
+Expected final marker:
+
+```text
+PASS_SELECTED_NATIVE_TENSOR_CLOSURE {"kappa_changed": false, "native_pairs": 2922, "paid_calls": 66834, "productive_fields": 13444, "raw_contacts": 7488, "rejecting_controls": 20}
+```
+
+The registered replay also checks the generic trace engine in normal and
+optimized Python and rebuilds the full native recipe artifact with the
+credited source compiler:
+
+```sh
+python3 -B scripts/research_package.py --package research/selected-native-tensor-closure
+```
+
+`SOURCES.json` records exact Git blob and SHA256 pins. Every live source input
+is declared in `VERIFY.json`. The independent verifier authenticates the
+selected word, result, controller, port data and mathematical source files
+before parsing them; it imports no predecessor verifier or compiler.
+`emit_recipes.py` alone imports the credited compiler to regenerate the
+candidate artifact. Its complete recipe payload must reproduce the already
+registered selected digest
+`e22642923d5437c49e2d1dc79dd6b795503499d6f33615a1df3568f0b96164c1`.
+The independent checker then accepts every complete emitted product by the
+trace method. This producer/acceptor separation is deliberate.
+
+`RESULT.json` is deterministic and records the exact coverage, profile and
+endpoint digests and every rejecting control. `--write` regenerates only this
+new package's receipt. `emit_recipes.py --write` regenerates only this package's
+small compressed candidate artifact; ordinary replay compares its full
+decompressed contents instead.
+
+## Verification and negative controls
+
+The trace engine's independent dense-column oracle uses exact rational
+arithmetic on 120 deterministic random small circuits. Additional checks
+enumerate all 16,933 quadratic forms through four variables with zero constant
+term, 480 affine systems, 480 affine substitutions, nonzero constant phases,
+unitary inverses and a 300-variable symbolic trace. Fixed seeds are `7606030`
+and `7606546`; these are bounded verifier tests, not an open-ended supplier
+search. Ten trace grammar/operator controls reject malformed expected gates,
+wrong phases and invalid gates, including cases that agree on the zero column.
+
+The source package has 20 separate controls. They reject source-byte and
+registered-payload drift; a global `i` invisible to conjugation action; reversed
+native factors; an omitted positive child; confusing a canonical rank-one
+chart with the translation Gaussian on that vector; `i^p` used where `i^(p*f)` is
+required; tensor-powering an exterior minus sign; tensoring a Gaussian sum
+termwise; a wrong whole-helper normalizer; a scalar chart mismatch; reversed
+MOVE orientation; aliased public roles; a missing wrapper translation or swap
+sign; a wrong raw final exponent; a corruption on an arbitrary spectator
+coordinate; and an omitted whole-job final translation. Mathematical fault
+controls call the comparison mechanisms directly, so their rejection is not
+merely an earlier source-hash failure.
+
+Both normal and `-O` execution are required by the registered manifest. The
+package uses explicit checks rather than optimization-disabled assertions.
+The finite tests support the checker implementation; the all-f and arbitrary
+spectator conclusions come from the written theorem, not from sampling
+column widths. This is not a proof-kernel formalization or independent human
+acceptance.
+
+## Open next questions
+
+The structured versions are in [TASK.json](TASK.json).
+
+1. Bind this exact typed algebraic word to the selected-rate physical
+   dispatcher. Start with one full compiled occurrence, preserving the
+   distinction between native column phases and exterior scalar field
+   coefficients, and account for every adapter, complete field, recursive
+   child and return operation. This is blocked by the remaining physical
+   implementation and charging obligations, not by a missing local chart
+   identity.
+2. Compose the resulting implementation with the owned-stream and precision
+   contracts. Start with a complete source-owned call/return that realizes
+   the 29-label local owned bank and separate thin scratch field, without
+   treating the ideal CLEAR in this algebraic proof as a physical erasure
+   algorithm. The required ownership and rounding premises remain explicit.
+
+## Attribution and license
+
+Prepared for **@CubeJerry**, with substantial **OpenAI Codex** assistance in
+research, derivation, implementation and adversarial checks. The lead research
+reviewer checked the Gauss-sum derivation and the complete typed composition;
+that is an AI-assisted mathematical review, not a separate formal proof.
+
+Credit DreamingOfClouds for the selected #6030 change; David Arias for the
+coalesced source word, Cinna Davis for private reuse, the native representations
+and the Laurent controller,
+Gareth Davidson for their composition, and Ali Romman for the P104 caller and
+independent source audits. The native compiler, source constructions and
+proof framework retain their existing contributors' credit. Relevant earlier
+and concurrent mechanisms are identified above rather than claimed anew.
+Apache-2.0, following the repository license. No predecessor source is copied
+or modified; the compressed artifact contains deterministically emitted
+recipe data authenticated against the selected registration.
diff --git a/research/selected-native-tensor-closure/RESULT.json b/research/selected-native-tensor-closure/RESULT.json
new file mode 100644
index 0000000000..2545b90099
--- /dev/null
+++ b/research/selected-native-tensor-closure/RESULT.json
@@ -0,0 +1,154 @@
+{
+  "all_size_status": "PROOF.md proves the typed algebraic implication for every positive f and arbitrary spectator factors; finite replay is the exact selected-source premise",
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "complete_caller": {
+    "all_1024_laurent_entries_checked": true,
+    "controller_events": 218,
+    "endpoint_sha256": "4ba0b7b54172171102343f81f03f2dac7753ffd184cdffcbe62c8b9d6b1dc0bf",
+    "five_step_forward_wrappers": 16,
+    "forward_local_calls": 18,
+    "literal_boundary_events": 145,
+    "method": "reverse linear-functional substitution; no finite-order quotient of Q",
+    "thin_queries_per_port": 15
+  },
+  "limitations": [
+    "ideal owned reset semantics remain assumed",
+    "physical selected-rate dispatcher not implemented",
+    "compiler, precision_tape and analytic_recovery not discharged",
+    "No new kappa"
+  ],
+  "local_gauge": {
+    "arbitrary_public_columns": 770,
+    "events": 22803,
+    "full_endpoint_coefficients": 638401,
+    "gauge_endpoint_sha256": "3a9a39e842438b4575689b5233c0d9237b4863b71f8f1939c757607fe4d35325",
+    "opcode_counts": {
+      "ADD": 18404,
+      "CLEAR": 31,
+      "COPY": 23,
+      "INIT": 31,
+      "MOVE": 3453,
+      "SCALE": 861
+    },
+    "owned_entrance_columns_erased": 29,
+    "owned_lifetimes": 31,
+    "paid_profile": {
+      "1": 2219,
+      "2": 457,
+      "3": 496,
+      "4": 100,
+      "5": 163,
+      "6": 11,
+      "7": 6,
+      "8": 1
+    },
+    "peak_owned": 13,
+    "rows": 799,
+    "rule": "chart coefficients are formal inverse/forward products; every scalar remains exterior"
+  },
+  "native_trace": {
+    "arbitrary_helpers": 562,
+    "central_squares": 104,
+    "complete_main_paths": 312,
+    "complete_trace_digest": "dcb7308d7925b139d4547247eae4955b58445155bda7ac93d7c3b18d479078a5",
+    "distinct_move_pairs": 2886,
+    "distinct_native_pairs": 2922,
+    "helper_normalized_pairs": 69,
+    "helper_profile": {
+      "1": 14,
+      "2": 8,
+      "3": 4,
+      "4": 107,
+      "5": 14,
+      "6": 2,
+      "7": 9,
+      "8": 19,
+      "9": 385
+    },
+    "main_adapters": 208,
+    "maximum_closed_trace_branches": 18,
+    "method": "exact constrained quadratic Gauss sum; no tableau or distinguished column",
+    "selected_recipe_payload_sha256": "e22642923d5437c49e2d1dc79dd6b795503499d6f33615a1df3568f0b96164c1"
+  },
+  "negative_controls": {
+    "MOVE_reversed_source_and_target": "current native source and ownership at 9896",
+    "aliased_local_public_role": "all productive owners",
+    "caller_exterior_swap_sign_lost": "full free-Laurent reverse output 8",
+    "caller_missing_central_translation": "full free-Laurent reverse output 8",
+    "caller_wrong_raw_diagonal_Q_exponent": "full free-Laurent reverse output 0",
+    "canonical_rank_one_chart_mistaken_for_C_q": "omitted nontrivial intrinsic main adapter: exact unitary overlap trace (256, 0, 0)",
+    "contact_corrupts_arbitrary_spectator_row": "nonidentity spectator action",
+    "exterior_minus_incorrectly_tensor_powered": "exterior sign squared: exact unitary overlap trace (-262144, 0, 0)",
+    "main_whole_job_final_translation_omitted": "omitted X_q final raw endpoint correction",
+    "native_constant_i_not_raised_to_column_width": "wrong i^(p*f): exact unitary overlap trace (0, -262144, 0)",
+    "native_global_i_preserving_conjugation_action": "lost absolute phase: exact unitary overlap trace (0, 512, 0)",
+    "native_left_right_order": "reversed native factors: exact unitary overlap trace (128, 0, 0)",
+    "native_missing_positive_child": "omitted native Gaussian: exact unitary overlap trace (256, -256, 0)",
+    "registered_recipe_payload_drift": "full selected recipe payload hash",
+    "scalar_chart_mismatch": "scalar current chart",
+    "source_drift:research/p104-helper470-twin-rung-removal/WORD.json.gz": "Rejected before parsing or execution",
+    "source_drift:research/p104-productive-controller/caller.py": "Rejected before parsing or execution",
+    "tensor_of_sum_replaced_by_sum_of_tensors": "tensoring a Gaussian expansion termwise: exact unitary overlap trace (2, 0, 0)",
+    "whole_helper_pre_normalizer_not_inverted": "wrong whole-helper entry: exact unitary overlap trace (16, 0, 0)",
+    "zero_column_agrees_but_quadratic_operator_differs": "missing controlled sign: exact unitary overlap trace (2, 0, 0)"
+  },
+  "ownership_and_invoice": {
+    "field_maps_sha256": "0cc409ad6083dedf325df5bd452ed58f84d579f5664e48a519a86ab2e15bdd33",
+    "fresh_helper_inputs": 10116,
+    "limits": "ideal source ownership only; allocation, physical erasure and retained backing remain premises",
+    "local_owned_labels": 29,
+    "main_arbitrary_inputs": 3328,
+    "paid_calls": 66834,
+    "paid_rank_mass": 240152,
+    "productive_inputs": 13444,
+    "separate_thin_owned_label": 13473,
+    "target_rank_mass": 240912
+  },
+  "schema": "selected-native-tensor-closure-result/1",
+  "source_sha256": {
+    "research/p104-caller-independent-audit/capsule/outputs/p104-caller-route-audit-v2-20261025/sources/DATA-PHASES.json": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+    "research/p104-helper470-twin-rung-removal/RESULT.json": "84fcbe2a0b64307b5703c363408042bdd91555b0419509aa50daa43b3b1924ad",
+    "research/p104-helper470-twin-rung-removal/WORD.json.gz": "7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f",
+    "research/p104-native-channel/vendor/clifford_exact.py": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+    "research/p104-productive-controller/CONTROLLER.json": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9",
+    "research/p104-productive-controller/caller.py": "300d40043953e4f6aecfde8f2d1004b8ad6dc5a79ec3c7666c9908c2466c91c8",
+    "research/p104-productive-controller/local.py": "878734dfd64c4b9d362d0ae476cc5c2a19fe93e48b7378763b23f3e0dd0a5a71"
+  },
+  "status": "PASS_SELECTED_NATIVE_TENSOR_CLOSURE",
+  "whole_job_algebra": {
+    "complete_idle_endpoints": 3120,
+    "complete_main_endpoints": 3328,
+    "complete_raw_contacts": 7488,
+    "composed_native_paths": 5616,
+    "full_coordinate_bijections": 4888,
+    "idle_boundary_sha256": "3b493ec6281025ba31b1a66b682d33292b58f99c982461f68018f340c8bbb677",
+    "idle_profile": {
+      "16": 416,
+      "24": 416,
+      "32": 416,
+      "40": 416,
+      "48": 416,
+      "56": 416,
+      "64": 416,
+      "72": 208
+    },
+    "limits": "algebraic whole-field bijections; physical scheduling and charged realization remain separate",
+    "main_boundary_sha256": "57d64f0cbbd7430e780837e64f8bd7336df73c88de52adbe91f8204b62770afd",
+    "main_profile": {
+      "17": 416,
+      "25": 416,
+      "33": 416,
+      "41": 416,
+      "49": 416,
+      "57": 416,
+      "65": 416,
+      "73": 208,
+      "145": 208
+    },
+    "omitted_identity_spectator_rows": 120,
+    "retained_source_rows": 145,
+    "source_dimension": 265,
+    "source_route_payload_sha256": "7f4f0ca768ffbc4f93dcff575592c4c67411f46b4289e45b73f715a2ad47b142",
+    "thin_query_boundaries": 104
+  }
+}
diff --git a/research/selected-native-tensor-closure/SOURCES.json b/research/selected-native-tensor-closure/SOURCES.json
new file mode 100644
index 0000000000..57558383d3
--- /dev/null
+++ b/research/selected-native-tensor-closure/SOURCES.json
@@ -0,0 +1,60 @@
+{
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "path": "docs/research/agent-task-catalog.json",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "path": "certificates/selected-result.json",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa",
+      "usage": "provenance"
+    },
+    {
+      "git_blob": "22f9ab02d1b73ea0aa4b07ce445a14cc2602c760",
+      "path": "research/p104-helper470-twin-rung-removal/WORD.json.gz",
+      "sha256": "7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f",
+      "usage": "input"
+    },
+    {
+      "git_blob": "e4a03118d585fb80c9ac932281b528ac394ccc5f",
+      "path": "research/p104-helper470-twin-rung-removal/RESULT.json",
+      "sha256": "84fcbe2a0b64307b5703c363408042bdd91555b0419509aa50daa43b3b1924ad",
+      "usage": "input"
+    },
+    {
+      "git_blob": "cc539db709cf7a6c42872c4380902736ad453af3",
+      "path": "research/p104-productive-controller/CONTROLLER.json",
+      "sha256": "a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9",
+      "usage": "input"
+    },
+    {
+      "git_blob": "d9a4f84211182ca200c081ea061d724c8a2cfb70",
+      "path": "research/p104-caller-independent-audit/capsule/outputs/p104-caller-route-audit-v2-20261025/sources/DATA-PHASES.json",
+      "sha256": "c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa",
+      "usage": "input"
+    },
+    {
+      "git_blob": "166841a037f069249f6f7f6dedb89248812f59e2",
+      "path": "research/p104-native-channel/vendor/clifford_exact.py",
+      "sha256": "2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962",
+      "usage": "input"
+    },
+    {
+      "git_blob": "dbb3d56a4f5b733bb0c32a91249d1f6a7e1ab5d2",
+      "path": "research/p104-productive-controller/local.py",
+      "sha256": "878734dfd64c4b9d362d0ae476cc5c2a19fe93e48b7378763b23f3e0dd0a5a71",
+      "usage": "input"
+    },
+    {
+      "git_blob": "6ac5e735691555ecafe2afbcff71edd923178113",
+      "path": "research/p104-productive-controller/caller.py",
+      "sha256": "300d40043953e4f6aecfde8f2d1004b8ad6dc5a79ec3c7666c9908c2466c91c8",
+      "usage": "input"
+    }
+  ],
+  "schema_version": 1
+}
diff --git a/research/selected-native-tensor-closure/TASK.json b/research/selected-native-tensor-closure/TASK.json
new file mode 100644
index 0000000000..0251f7fb6a
--- /dev/null
+++ b/research/selected-native-tensor-closure/TASK.json
@@ -0,0 +1,88 @@
+{
+  "actual_scope": "Prove and independently certify the selected #6030 algebraic native-to-18-call-caller implication for every positive column width, arbitrary spectators, signed phases and complete whole-job adapters.",
+  "ai_assistance": "Substantial OpenAI Codex assistance in derivation, implementation and adversarial review; no claim of independent human or proof-kernel acceptance.",
+  "author": "@CubeJerry",
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "bottleneck_link": "native_all_columns at the actual selected caller, with owned reset, physical selected-rate dispatch, compiler, precision_tape and analytic_recovery premises explicitly retained.",
+  "evidence": "PROOF.md and RESULT.json; full source-bound deterministic normal/optimized replay, exact Gauss-sum validation, complete recipe regeneration and rejecting phase/order/typing/endpoint/source controls.",
+  "open_questions": [
+    {
+      "blocked_by": "Selected-rate physical native and adapter realization, complete call/return accounting and current precision/storage contracts.",
+      "first_step": "Bind one full compiler occurrence to the actual call schedule, preserving exterior scalar coefficients and all complete coordinate maps; charge every introduced operation.",
+      "question": "Can the exact typed selected algebraic word be bound to a physical dispatcher at the selected accelerated tariff?",
+      "why_it_matters": "This package establishes native/caller equality but does not implement or price source access, staging, monomials and recursive returns."
+    },
+    {
+      "blocked_by": "The retained owned-erasure and precision_tape framework obligations; this package deliberately assumes ideal owned reset semantics.",
+      "first_step": "Construct one complete call/return with all 29 local owned labels and the separate thin scratch field, including retained information, initialization, rounding and returned ownership.",
+      "question": "Can the actual owned-stream and numerical contracts be composed with every owned lifetime of this selected caller?",
+      "why_it_matters": "Ideal INIT/CLEAR algebra does not prove physically valid erasure or control finite-precision errors."
+    }
+  ],
+  "original_card": {
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "existing_checks": [
+      {
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "status": "author-checked"
+      },
+      {
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "status": "reviewed"
+      }
+    ],
+    "first_experiment": "Choose one assumption ID (native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery), one quantified implication and exact dependency closure. Check current handoffs, including #5625 row reentry and #5707 root-prefix lowering, without treating their partial scope as the whole theorem. Prove or falsify that implication with a caller-bound argument; stop at the chosen obligation. Finite tests or root digits do not discharge an all-size statement.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "id": "uniform-transfer-audit",
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "overlap_terms": [
+      "native_all_columns",
+      "owned_erasure",
+      "compiler",
+      "precision_tape",
+      "analytic_recovery"
+    ],
+    "question": "Can one of the selected record's five inherited assumptions be proved at its actual caller?",
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ]
+  },
+  "outcome": "completed",
+  "schema_version": 1,
+  "scope_change_reason": "The card asks for one quantified inherited implication. Earlier dense censuses and the scalar common-chart lemma were already proved; this scope isolates the complete selected assembled-caller join and checks native operators by a distinct exact closed-trace method.",
+  "scope_status": "refined",
+  "stop_rule": "Complete the exact pinned native, local, raw-contact, free-Laurent and whole-job inventories; stop if their all-width interpretation cannot be proved without tensoring exterior scalar sums. No open-ended parameter search.",
+  "task_id": "uniform-transfer-audit",
+  "time_budget_minutes": 180
+}
diff --git a/research/selected-native-tensor-closure/VERIFY.json b/research/selected-native-tensor-closure/VERIFY.json
new file mode 100644
index 0000000000..4d06e71495
--- /dev/null
+++ b/research/selected-native-tensor-closure/VERIFY.json
@@ -0,0 +1,69 @@
+{
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "checks": "Complete exact selected-source native/caller/whole-job certificate and all 20 rejecting controls",
+      "timeout_seconds": 120
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "checks": "Complete exact selected-source native/caller/whole-job certificate and all 20 rejecting controls under optimized Python",
+      "timeout_seconds": 120
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/test_gauss_trace.py"
+      ],
+      "checks": "Independent dense rational oracle, exhaustive small Gauss sums, affine substitution and malformed-gate/operator controls",
+      "timeout_seconds": 60
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "-O",
+        "{package}/test_gauss_trace.py"
+      ],
+      "checks": "Independent dense rational oracle, exhaustive small Gauss sums, affine substitution and malformed-gate/operator controls under optimized Python",
+      "timeout_seconds": 60
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-B",
+        "{package}/emit_recipes.py"
+      ],
+      "checks": "Regenerate every complete native recipe and intrinsic adapter with the pinned credited compiler; require exact selected payload hash and full decompressed candidate bytes",
+      "timeout_seconds": 600
+    }
+  ],
+  "inputs": [
+    "research/p104-helper470-twin-rung-removal/WORD.json.gz",
+    "research/p104-helper470-twin-rung-removal/RESULT.json",
+    "research/p104-productive-controller/CONTROLLER.json",
+    "research/p104-caller-independent-audit/capsule/outputs/p104-caller-route-audit-v2-20261025/sources/DATA-PHASES.json",
+    "research/p104-native-channel/vendor/clifford_exact.py",
+    "research/p104-productive-controller/local.py",
+    "research/p104-productive-controller/caller.py"
+  ],
+  "python": "3.12",
+  "requirements": [],
+  "schema_version": 1,
+  "status": "ready"
+}
diff --git a/research/selected-native-tensor-closure/caller_check.py b/research/selected-native-tensor-closure/caller_check.py
new file mode 100644
index 0000000000..23ef0969bb
--- /dev/null
+++ b/research/selected-native-tensor-closure/caller_check.py
@@ -0,0 +1,212 @@
+#!/usr/bin/env python3
+"""Exact source contacts and whole-job positive endpoints, with no old imports."""
+from __future__ import annotations
+
+from collections import Counter
+from functools import lru_cache
+import hashlib
+import json
+
+from gauss_trace import Quadratic, adjoint, equal, require
+from native_check import transpose
+
+
+DIMENSION = 265
+AXES = tuple(1 << i for i in range(DIMENSION))
+
+
+def terms(entries):
+    coefficients = Counter()
+    for axis, value in entries:
+        require(type(axis) is int and 0 < axis < 1 << DIMENSION and type(value) is int, 'spectral term domain')
+        coefficients[axis] += value
+    return tuple(sorted((axis, value % 4) for axis, value in coefficients.items() if value % 4))
+
+
+@lru_cache(maxsize=None)
+def phase(entries):
+    polynomial = Quadratic.zero(DIMENSION)
+    for axis, coefficient in entries:
+        polynomial.add_affine(coefficient, (axis, 0))
+    return tuple(polynomial.linear), frozenset(polynomial.pairs), polynomial.constant
+
+
+def apply(columns, value):
+    result = 0
+    while value:
+        low = value & -value
+        result ^= columns[low.bit_length()-1]
+        value ^= low
+    return result
+
+
+def positive(vectors):
+    return terms((axis, 1) for axis in vectors)
+
+
+def stage_terms(stage, q):
+    require(type(stage) is int and 0 <= stage <= 32, 'controller stage domain')
+    return terms([(axis, 1) for axis in AXES[:9]]+[(q, -1)]) if stage == 0 else positive(AXES[9+8*(stage-1):9+8*stage])
+
+
+def orthogonal_base(q, recipe):
+    basis = tuple(recipe['idle_children'][0]['orthogonal_axes'][:8])
+    complete = (q,)+basis
+    require(len(set(complete)) == 9 and all(type(v) is int and 0 < v < 512 for v in complete),
+            'complete nine-axis port basis')
+    require(all((a & b).bit_count() % 2 == int(i == j)
+                for i, a in enumerate(complete) for j, b in enumerate(complete)), 'orthonormal port basis')
+    return basis
+
+
+def route(stage, q, basis):
+    """Swap eight orthonormal pairs; fix their orthogonal complement."""
+    if stage == 0:
+        return AXES
+    fresh = AXES[9+8*(stage-1):9+8*stage]
+    differences = tuple(a ^ b for a, b in zip(basis, fresh, strict=True))
+    columns = tuple(axis ^ apply(differences, sum(((axis & v).bit_count() % 2) << j
+                                                for j, v in enumerate(differences))) for axis in AXES)
+    require(all(apply(columns, a) == b and apply(columns, b) == a
+                for a, b in zip(basis, fresh, strict=True)), 'actual eight swaps')
+    require(apply(columns, q) == q and all(apply(columns, columns[j]) == AXES[j] for j in range(DIMENSION)),
+            'complete involutive route and central port')
+    # The checked two orthonormal families are disjoint. Their eight swaps and
+    # fixed orthogonal complement prove orthogonality, without a 265^2 replay.
+    return columns
+
+
+def contact(before, after, columns):
+    moved = terms((apply(columns, axis), coefficient) for axis, coefficient in before)
+    left, right = phase(moved), phase(after)
+    require(left[1:] == right[1:], 'contact full mixed and constant phase')
+    difference = tuple((b-a) % 4 for a, b in zip(left[0], right[0], strict=True))
+    require(all(value in (0, 2) for value in difference), 'contact even linear residual')
+    translation = sum((value//2) << j for j, value in enumerate(difference))
+    correction = [(translation, 2)] if translation else []
+    require(phase(terms(list(moved)+correction)) == right, 'complete forward contact')
+    inverse = apply(columns, translation)
+    backward = [(apply(columns, axis), coefficient) for axis, coefficient in after]
+    if inverse:
+        backward.append((inverse, 2))
+    require(phase(terms(backward)) == phase(before), 'complete inverse contact')
+    return {'translation': translation, 'inverse_translation': inverse}
+
+
+def transformed_path(local, entry, exit_, columns):
+    """Exit * native * entry, including both actual address translations."""
+    translation = exit_['translation'] ^ apply(columns, entry['translation'])
+    entries = [(apply(columns, axis), coefficient) for axis, coefficient in local]
+    if translation:
+        entries.append((translation, 2))
+    return phase(terms(entries))
+
+
+def complete_positive_basis(base, active, retained):
+    """Return a complete coordinate map, including all spectator columns."""
+    require(len(base) == 9 and len(set(active)) == len(active)
+            and all(type(j) is int and 0 <= j < DIMENSION for j in active), 'positive coordinate selection')
+    require(all((a & b).bit_count() % 2 == int(i == j)
+                for i, a in enumerate(base) for j, b in enumerate(base)), 'complete orthogonal local basis')
+    require(set(active) <= set(retained), 'positive job uses retained source rows')
+    order = (tuple(active)+tuple(j for j in retained if j not in set(active))
+             +tuple(j for j in range(DIMENSION) if j not in set(retained)))
+    require(sorted(order) == list(range(DIMENSION)), 'all spectator coordinates retained')
+    columns = tuple(base)+AXES[9:]
+    return tuple(columns[j] for j in order)
+
+
+def check_caller(word, controller, phase_source, source_result):
+    recipes = {entry['q']: entry for entry in phase_source['recipes']}
+    require(len(recipes) == len(phase_source['recipes']) and set(word['ports']) <= set(recipes), 'original source port data')
+    bases = {q: orthogonal_base(q, recipes[q]) for q in word['ports']}
+    local_stages = sorted(event[1] for event in controller['physical_events'] if event[0] == 'LOCAL')
+    require(local_stages == [0]+list(range(1, 32, 2))+[32], 'all selected actual stages')
+    retained = tuple(range(9))+tuple(j for stage in local_stages if stage
+                                   for j in range(9+8*(stage-1), 9+8*stage))
+    require(len(retained) == len(set(retained)) == controller['ambient_rank'], 'actual 145-row retained support')
+    retained_mask = sum(1 << j for j in retained)
+    contacts = []
+    paths = 0
+    endpoint_records = []
+    main_profile, idle_profile = Counter(), Counter()
+    for _, stage, pair, backward in (event for event in controller['physical_events'] if event[0] == 'LOCAL'):
+        require(not backward, 'actual forward source supplier')
+        spectators = positive(AXES[9:9+8*stage])
+        for q in word['ports']:
+            columns = route(stage, q, bases[q])
+            raw_x = ((q, 1),) if stage == 0 else positive(AXES[:9+8*(stage-1)])
+            raw_y = () if stage == 0 else terms(list(raw_x)+[(q, -1)])
+            entry_x = terms(list(spectators)+[(q, 1)])
+            entry_y = spectators
+            exit_x = terms(list(spectators)+[(axis, 1) for axis in AXES[:9]])
+            exit_y = terms(list(exit_x)+[(q, -1)])
+            out_x = positive(AXES[:9+8*stage])
+            out_y = terms(list(out_x)+[(q, -1)])
+            maps = [contact(a, b, columns) for a, b in
+                    ((raw_x, entry_x), (raw_y, entry_y), (exit_x, out_x), (exit_y, out_y))]
+            require(all(columns[j] == AXES[j] for j in range(DIMENSION) if j not in retained)
+                    and all((item['translation'] | item['inverse_translation']) & ~retained_mask == 0 for item in maps),
+                    'all actual contacts fix omitted spectator rows')
+            expected = phase(stage_terms(stage, q))
+            d0 = stage_terms(0, q)
+            require(transformed_path(d0, maps[0], maps[2], columns) == expected, 'executed X diagonal D_s')
+            require(transformed_path(d0, maps[1], maps[3], columns) == expected, 'executed Y diagonal D_s')
+            require(transformed_path(terms(list(d0)+[(q, -1)]), maps[0], maps[3], columns)
+                    == phase(terms(list(stage_terms(stage, q))+[(q, -1)])), 'executed cross D_s Q inverse')
+            require(sum(columns[j] != AXES[j] for j in range(DIMENSION)) <= 17
+                    and all(item['translation'].bit_count() <= 17 and item['inverse_translation'].bit_count() <= 17
+                            for item in maps), 'source seventeen-axis contacts')
+            contacts.append([stage, q, maps])
+            paths += 3
+    digest = hashlib.sha256(json.dumps(contacts, separators=(',', ':')).encode()).hexdigest()
+    require(digest == source_result['physical_caller']['route_payload_sha256'], 'complete selected raw contact payload')
+
+    for q in word['ports']:
+        # A rank-one positive query plus its full ordinary normalizer gives Q^-1.
+        query_basis = transpose((q,)+bases[q])
+        query = (('P', tuple(query_basis)),)
+        equal(9, adjoint(query)+(('C', 1, 1),)+query+(('X', q),), (('C', q, -1),), 'complete positive thin query')
+        for output, diagonal in enumerate(controller['diagonal']):
+            qp, selected = diagonal[0], diagonal[1:]
+            require(qp in (-1, 1) and len(selected) == 33 and set(selected) <= {0, 1}, 'complete raw endpoint exponent grammar')
+            raw = terms([(q, qp)]+[entry for stage, yes in enumerate(selected) if yes for entry in stage_terms(stage, q)])
+            active = list(range(9)) if selected[0] else [0]
+            active += [j for stage in range(1, 33) if selected[stage] for j in range(9+8*(stage-1), 9+8*stage)]
+            base = AXES[:9] if selected[0] else (q,)+bases[q]
+            canonicalizer = complete_positive_basis(base, active, retained)
+            wanted = canonicalizer[:len(active)]
+            translation = q if qp == -1 else 0
+            corrected = terms(list(raw)+([(translation, 2)] if translation else []))
+            require(phase(corrected) == phase(positive(wanted)), 'entire main endpoint with final translation')
+            require(len(wanted) == 1+8*sum(selected), 'complete productive main rank')
+            main_profile[len(wanted)] += 1
+            endpoint_records.append([q, output, len(wanted), translation, list(canonicalizer)])
+    idle_endpoints = []
+    for event_index, event in enumerate(controller['physical_events']):
+        if event[0] != 'IDLE':
+            continue
+        stages = event[2]
+        require(stages and len(stages) == len(set(stages)) and set(stages) <= set(range(33)), 'entire idle window')
+        for q in word['ports']:
+            raw = terms(entry for stage in stages for entry in stage_terms(stage, q))
+            active = list(range(1, 9)) if 0 in stages else []
+            active += [j for stage in stages if stage for j in range(9+8*(stage-1), 9+8*stage)]
+            canonicalizer = complete_positive_basis((q,)+bases[q], active, retained)
+            wanted = positive(canonicalizer[:len(active)])
+            adapter = contact(wanted, raw, AXES)
+            require(len(active) == 8*len(stages), 'positive whole-idle rank')
+            idle_profile[len(active)] += 2
+            idle_endpoints.append([event_index, q, len(active), adapter, list(canonicalizer)])
+    registered = source_result['physical_caller']
+    require(dict(main_profile) == {int(k): v for k, v in registered['main_targets'].items()}, 'selected main output census')
+    require(dict(idle_profile) == {int(k): v for k, v in registered['whole_idle_jobs'].items()}, 'selected whole-idle census')
+    return {'source_dimension': DIMENSION, 'retained_source_rows': len(retained), 'omitted_identity_spectator_rows': DIMENSION-len(retained),
+            'complete_raw_contacts': 4*len(contacts), 'composed_native_paths': paths,
+            'source_route_payload_sha256': digest, 'thin_query_boundaries': len(word['ports']),
+            'complete_main_endpoints': sum(main_profile.values()), 'main_profile': dict(sorted(main_profile.items())),
+            'complete_idle_endpoints': sum(idle_profile.values()), 'idle_profile': dict(sorted(idle_profile.items())),
+            'full_coordinate_bijections': len(endpoint_records)+len(idle_endpoints),
+            'main_boundary_sha256': hashlib.sha256(json.dumps(endpoint_records, separators=(',', ':')).encode()).hexdigest(),
+            'idle_boundary_sha256': hashlib.sha256(json.dumps(idle_endpoints, separators=(',', ':')).encode()).hexdigest(),
+            'limits': 'algebraic whole-field bijections; physical scheduling and charged realization remain separate'}
diff --git a/research/selected-native-tensor-closure/controls.py b/research/selected-native-tensor-closure/controls.py
new file mode 100644
index 0000000000..b20e5a434a
--- /dev/null
+++ b/research/selected-native-tensor-closure/controls.py
@@ -0,0 +1,133 @@
+#!/usr/bin/env python3
+"""Mathematical faults rejected independently of provenance-hash failures."""
+from __future__ import annotations
+
+from copy import deepcopy
+
+from gauss_trace import adjoint, equal, require
+from native_check import check_native, frame, gaussian, lowered, monomial
+from source_algebra import free_laurent, local_gauge, ownership_and_invoice
+from caller_check import AXES, contact, orthogonal_base, phase, positive, route, stage_terms, terms, transformed_path
+
+
+def lift(n, f, circuit, repeat_constant=True):
+    """Small checked test witness, not a replacement for the universal proof."""
+    dimension = n*f
+    out = []
+    for column in range(f):
+        offset = n*column
+        for gate in circuit:
+            kind = gate[0]
+            if kind == 'P':
+                rows = [1 << j for j in range(dimension)]
+                rows[offset:offset+n] = [row << offset for row in gate[1]]
+                out.append(('P', tuple(rows)))
+            elif kind == 'D':
+                rows, linear = [0]*dimension, [0]*dimension
+                rows[offset:offset+n] = [row << offset for row in gate[1]]
+                linear[offset:offset+n] = gate[2]
+                out.append(('D', tuple(rows), tuple(linear)))
+            elif kind in ('X', 'C'):
+                out.append((kind, gate[1] << offset)+gate[2:])
+            else:
+                require(kind == 'phase', 'witness native gate')
+                if repeat_constant or column == 0:
+                    out.append(gate)
+    return tuple(out)
+
+
+def run(word, result, candidates, controller, phase_source):
+    rejected = {}
+
+    def reject(name, action):
+        try:
+            action()
+        except ValueError as error:
+            rejected[name] = str(error)
+            return
+        raise ValueError('negative control accepted: '+name)
+
+    records = {(a, b): recipe for a, b, recipe in candidates['pairs']}
+    # These are actual source pairs, chosen once after locating nondegenerate
+    # faults. They are fixed regression witnesses, not a continuing search.
+    a, b = 0, 249
+    recipe = records[a, b]
+    expected = adjoint(frame(tuple(word['frames'][a])))+frame(tuple(word['frames'][b]))
+    actual = lowered(recipe)
+    equal(9, actual, expected, 'native control baseline')
+    reject('native_global_i_preserving_conjugation_action',
+           lambda: equal(9, actual+(('phase', 1),), expected, 'lost absolute phase'))
+    reject('native_left_right_order',
+           lambda: equal(9, monomial(recipe['left'])+gaussian(recipe['rank'])+monomial(recipe['right']),
+                         expected, 'reversed native factors'))
+    reject('native_missing_positive_child',
+           lambda: equal(9, monomial(recipe['right'])+monomial(recipe['left']), expected, 'omitted native Gaussian'))
+    equal(18, lift(9, 2, actual), lift(9, 2, expected), 'actual two-column native baseline')
+    reject('native_constant_i_not_raised_to_column_width',
+           lambda: equal(18, lift(9, 2, actual, repeat_constant=False), lift(9, 2, expected), 'wrong i^(p*f)'))
+    require(any(event[0] == 'SIGNED_SWAP' for event in controller['physical_events']), 'actual exterior minus source')
+    reject('exterior_minus_incorrectly_tensor_powered',
+           lambda: equal(18, lift(9, 2, (('phase', 2),)), (('phase', 2),), 'exterior sign squared'))
+    reject('tensor_of_sum_replaced_by_sum_of_tensors',
+           lambda: equal(2, (('C', 3, 1),), (('C', 1, 1), ('C', 2, 1)), 'tensoring a Gaussian expansion termwise'))
+    reject('zero_column_agrees_but_quadratic_operator_differs',
+           lambda: equal(2, (), (('D', (2, 1), (0, 0)),), 'missing controlled sign'))
+
+    a, b = 3, 117
+    helper = records[a, b]
+    helper_path = adjoint(frame(tuple(word['frames'][a])))+frame(tuple(word['frames'][b]))
+    equal(9, adjoint(monomial(helper['right']))+helper_path+adjoint(monomial(helper['left'])),
+          gaussian(helper['rank']), 'whole-helper control baseline')
+    reject('whole_helper_pre_normalizer_not_inverted',
+           lambda: equal(9, monomial(helper['right'])+helper_path+adjoint(monomial(helper['left'])),
+                         gaussian(helper['rank']), 'wrong whole-helper entry'))
+
+    changed = deepcopy(candidates)
+    changed['pairs'][0][2]['left']['phase'] = (changed['pairs'][0][2]['left']['phase']+1) % 4
+    reject('registered_recipe_payload_drift', lambda: check_native(word, result, changed))
+    changed_word = deepcopy(word)
+    scalar = next(event for event in changed_word['program'] if event[0] == 'ADD')
+    scalar[-1] = (scalar[-1]+1) % len(word['frames'])
+    reject('scalar_chart_mismatch', lambda: local_gauge(changed_word, result))
+    changed_word = deepcopy(word)
+    move = next(event for event in changed_word['program'] if event[0] == 'MOVE')
+    move[2], move[3] = move[3], move[2]
+    reject('MOVE_reversed_source_and_target', lambda: local_gauge(changed_word, result))
+    changed_word = deepcopy(word)
+    changed_word['live'][-1] = changed_word['live'][0]
+    reject('aliased_local_public_role', lambda: ownership_and_invoice(changed_word, controller, result))
+
+    changed_controller = deepcopy(controller)
+    changed_controller['physical_events'].remove(next(event for event in changed_controller['physical_events'] if event[0] == 'Q_POWER'))
+    reject('caller_missing_central_translation', lambda: free_laurent(changed_controller))
+    changed_controller = deepcopy(controller)
+    next(event for event in changed_controller['physical_events'] if event[0] == 'SIGNED_SWAP')[0] = 'SWAP'
+    reject('caller_exterior_swap_sign_lost', lambda: free_laurent(changed_controller))
+    changed_controller = deepcopy(controller)
+    changed_controller['diagonal'][0][0] *= -1
+    reject('caller_wrong_raw_diagonal_Q_exponent', lambda: free_laurent(changed_controller))
+
+    q = word['ports'][0]
+    reject('canonical_rank_one_chart_mistaken_for_C_q',
+           lambda: equal(9, frame(tuple(word['frames'][word['initial'][word['live'][0]]])),
+                         (('C', q, 1),), 'omitted nontrivial intrinsic main adapter'))
+    port_recipe = next(item for item in phase_source['recipes'] if item['q'] == q)
+    basis = orthogonal_base(q, port_recipe)
+    stage = 1
+    columns = route(stage, q, basis)
+    spectators = positive(AXES[9:17])
+    entry = contact(positive(AXES[:9]), terms(list(spectators)+[(q, 1)]), columns)
+    exit_ = contact(terms(list(spectators)+[(axis, 1) for axis in AXES[:9]]), positive(AXES[:17]), columns)
+    desired = phase(stage_terms(stage, q))
+    require(transformed_path(stage_terms(0, q), entry, exit_, columns) == desired, 'actual contact control baseline')
+    changed_exit = dict(exit_)
+    changed_exit['translation'] ^= AXES[264]
+    reject('contact_corrupts_arbitrary_spectator_row',
+           lambda: require(transformed_path(stage_terms(0, q), entry, changed_exit, columns) == desired,
+                           'nonidentity spectator action'))
+    diagonal = next(term for term in controller['diagonal'] if term[0] == -1)
+    raw = terms([(q, -1)]+[item for s, yes in enumerate(diagonal[1:]) if yes for item in stage_terms(s, q)])
+    corrected = phase(terms(list(raw)+[(q, 2)]))
+    reject('main_whole_job_final_translation_omitted',
+           lambda: require(phase(raw) == corrected, 'omitted X_q final raw endpoint correction'))
+    return rejected
diff --git a/research/selected-native-tensor-closure/emit_recipes.py b/research/selected-native-tensor-closure/emit_recipes.py
new file mode 100644
index 0000000000..4cd1b96d90
--- /dev/null
+++ b/research/selected-native-tensor-closure/emit_recipes.py
@@ -0,0 +1,90 @@
+#!/usr/bin/env python3
+"""Rebuild the selected candidate recipes with the credited source compiler.
+
+This is a candidate producer, not the independent acceptance checker. The
+complete payload must match the already registered selected recipe digest.
+"""
+from __future__ import annotations
+
+from functools import lru_cache
+import gzip
+import hashlib
+import importlib.util
+import json
+from pathlib import Path
+import sys
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+WORD = ROOT/'research/p104-helper470-twin-rung-removal/WORD.json.gz'
+RESULT = ROOT/'research/p104-helper470-twin-rung-removal/RESULT.json'
+COMPILER = ROOT/'research/p104-native-channel/vendor/clifford_exact.py'
+
+
+def need(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(',', ':')).encode()
+
+
+def produce():
+    need(hashlib.sha256(WORD.read_bytes()).hexdigest() == '7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f',
+         'selected word pin')
+    need(hashlib.sha256(COMPILER.read_bytes()).hexdigest() == '2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962',
+         'candidate compiler pin')
+    need(hashlib.sha256(RESULT.read_bytes()).hexdigest() == '84fcbe2a0b64307b5703c363408042bdd91555b0419509aa50daa43b3b1924ad',
+         'selected result pin')
+    word = json.loads(gzip.decompress(WORD.read_bytes()))
+    result = json.loads(RESULT.read_bytes())
+    spec = importlib.util.spec_from_file_location('credited_candidate_compiler', COMPILER)
+    module = importlib.util.module_from_spec(spec)
+    sys.modules[spec.name] = module
+    spec.loader.exec_module(module)
+    # The candidate producer is deliberately the source implementation. All
+    # acceptance, including absolute phases, uses gauss_trace independently.
+    @lru_cache(None)
+    def chart(index):
+        return module.canonical_frame(9, tuple(word['frames'][index]))
+    main = set(word['live']) | {r+112 for r in word['live']}
+    pairs = {(event[2], event[3]) for event in word['program'] if event[0] == 'MOVE'}
+    pairs |= {(word['initial'][r], word['final'][r]) for r in range(word['public']) if r not in main}
+    records = []
+    for i, (a, b) in enumerate(sorted(pairs)):
+        recipe, _ = module.lower_child(chart(a).adjoint().then(chart(b)))
+        records.append([a, b, recipe])
+        if i % 250 == 0:
+            print('candidate_pairs', i, len(pairs), flush=True)
+    adapters = []
+    for role, q in zip(word['live'], word['ports'], strict=True):
+        entry = module.cgate(9, q, 1).adjoint().then(chart(word['initial'][role]))
+        exit_ = chart(word['final'][role+112]).adjoint().then(module.cgate(9, q, -1).then(module.fourier(9)))
+        for kind, op in (('input', entry), ('output', exit_)):
+            recipe, _ = module.lower_child(op)
+            adapters.append({'kind': kind, 'role': role, 'port': q, 'recipe': recipe})
+    payload = records+[item['recipe'] for item in adapters]
+    digest = hashlib.sha256(canonical(payload)).hexdigest()
+    need(digest == result['native']['recipe_payload_sha256'], 'complete payload equals selected registered recipes')
+    return {'schema': 'selected-native-tensor-closure/1', 'pairs': records, 'adapters': adapters,
+            'source_word_sha256': hashlib.sha256(WORD.read_bytes()).hexdigest(), 'recipe_payload_sha256': digest}
+
+
+def main():
+    value = produce()
+    raw = canonical(value)+b'\n'
+    target = HERE/'NATIVE.json.gz'
+    if '--write' in sys.argv:
+        target.write_bytes(gzip.compress(raw, compresslevel=9, mtime=0))
+    else:
+        need(gzip.decompress(target.read_bytes()) == raw, 'complete regenerated payload bytes')
+    print('PASS_SELECTED_NATIVE_RECIPE_EMISSION', value['recipe_payload_sha256'])
+
+
+if __name__ == '__main__':
+    with tempfile.TemporaryDirectory(prefix='native-candidate-pycache-') as cache:
+        sys.pycache_prefix = cache
+        sys.dont_write_bytecode = True
+        main()
diff --git a/research/selected-native-tensor-closure/gauss_trace.py b/research/selected-native-tensor-closure/gauss_trace.py
new file mode 100644
index 0000000000..f232a11bf5
--- /dev/null
+++ b/research/selected-native-tensor-closure/gauss_trace.py
@@ -0,0 +1,332 @@
+#!/usr/bin/env python3
+"""Exact circuit traces by constrained quadratic Gauss sums.
+
+No Clifford tableau, distinguished input column, floating point, or dense
+operator matrix is used. See PROOF.md. Apache-2.0.
+"""
+from __future__ import annotations
+
+from dataclasses import dataclass
+from itertools import combinations
+
+
+def require(ok: bool, reason: str) -> None:
+    if not ok:
+        raise ValueError(reason)
+
+
+def bits(mask: int) -> list[int]:
+    out = []
+    while mask:
+        low = mask & -mask
+        out.append(low.bit_length() - 1)
+        mask ^= low
+    return out
+
+
+def gaussian_mul(a: tuple[int, int], b: tuple[int, int]) -> tuple[int, int]:
+    x, y = a
+    z, w = b
+    return x*z-y*w, x*w+y*z
+
+
+def unit(power: int) -> tuple[int, int]:
+    return ((1, 0), (0, 1), (-1, 0), (0, -1))[power % 4]
+
+
+@dataclass
+class Quadratic:
+    """c + sum a_i x_i + 2 sum_(i<j) b_ij x_i x_j modulo four."""
+    linear: list[int]
+    pairs: set[tuple[int, int]]
+    constant: int = 0
+
+    @classmethod
+    def zero(cls, n: int) -> Quadratic:
+        require(type(n) is int and n >= 0, 'quadratic variable count')
+        return cls([0]*n, set())
+
+    def copy(self) -> Quadratic:
+        return Quadratic(self.linear[:], self.pairs.copy(), self.constant)
+
+    def toggle(self, i: int, j: int) -> None:
+        require(0 <= i < len(self.linear) and 0 <= j < len(self.linear), 'quadratic index')
+        if i == j:
+            self.linear[i] = (self.linear[i]+2) % 4
+            return
+        pair = (min(i, j), max(i, j))
+        if pair in self.pairs:
+            self.pairs.remove(pair)
+        else:
+            self.pairs.add(pair)
+
+    def add_affine(self, coefficient: int, form: tuple[int, int]) -> None:
+        """Add coefficient times a Boolean affine parity, modulo four."""
+        mask, constant = form
+        require(mask >= 0 and mask >> len(self.linear) == 0 and constant in (0, 1), 'affine form domain')
+        coefficient %= 4
+        if coefficient == 0:
+            return
+        support = bits(mask)
+        self.constant = (self.constant+coefficient*constant) % 4
+        signed = coefficient*(1-2*constant)
+        for i in support:
+            self.linear[i] = (self.linear[i]+signed) % 4
+        if coefficient % 2:
+            for i, j in combinations(support, 2):
+                self.toggle(i, j)
+
+    def add_product(self, a: tuple[int, int], b: tuple[int, int]) -> None:
+        """Add twice the product of two Boolean affine forms, modulo four."""
+        x, c = a
+        y, d = b
+        require(x >= 0 and y >= 0 and (x | y) >> len(self.linear) == 0 and c in (0, 1) and d in (0, 1),
+                'quadratic product domain')
+        self.constant = (self.constant+2*c*d) % 4
+        for i in bits((y if c else 0) ^ (x if d else 0)):
+            self.linear[i] = (self.linear[i]+2) % 4
+        for i in bits(x):
+            for j in bits(y):
+                self.toggle(i, j)
+
+    def substitute(self, forms: list[tuple[int, int]], n: int) -> Quadratic:
+        require(len(forms) == len(self.linear), 'complete substitution')
+        out = Quadratic.zero(n)
+        out.constant = self.constant
+        for coefficient, form in zip(self.linear, forms, strict=True):
+            out.add_affine(coefficient, form)
+        for i, j in sorted(self.pairs):
+            out.add_product(forms[i], forms[j])
+        return out
+
+    def value(self, x: int) -> int:
+        require(type(x) is int and 0 <= x < 1 << len(self.linear), 'quadratic assignment')
+        return (self.constant+sum(a for i, a in enumerate(self.linear) if x >> i & 1)
+                +2*sum((x >> i & 1)*(x >> j & 1) for i, j in self.pairs)) % 4
+
+
+def gauss_sum(poly: Quadratic) -> tuple[int, int]:
+    """Return sum_x i^poly(x) exactly using one- and two-variable pivots."""
+    q = poly.copy()
+    active = set(range(len(q.linear)))
+    factor = (1, 0)
+    while active:
+        odd = next((i for i in sorted(active) if q.linear[i] % 2), None)
+        if odd is not None:
+            a = q.linear[odd]
+            neighbours = 0
+            retained = set()
+            for i, j in q.pairs:
+                if odd == i:
+                    neighbours ^= 1 << j
+                elif odd == j:
+                    neighbours ^= 1 << i
+                else:
+                    retained.add((i, j))
+            q.pairs = retained
+            q.linear[odd] = 0
+            active.remove(odd)
+            z = unit(a)
+            factor = gaussian_mul(factor, (1+z[0], z[1]))
+            q.add_affine(-a, (neighbours, 0))
+            continue
+        if q.pairs:
+            i, j = min(q.pairs)
+            require(i in active and j in active, 'elimination support invariant')
+            u = v = 0
+            retained = set()
+            for a, b in q.pairs:
+                if (a, b) == (i, j):
+                    continue
+                if i == a:
+                    u ^= 1 << b
+                elif i == b:
+                    u ^= 1 << a
+                elif j == a:
+                    v ^= 1 << b
+                elif j == b:
+                    v ^= 1 << a
+                else:
+                    retained.add((a, b))
+            require((u | v) & ((1 << i) | (1 << j)) == 0, 'two-pivot complement')
+            ci, cj = q.linear[i]//2, q.linear[j]//2
+            q.pairs = retained
+            q.linear[i] = q.linear[j] = 0
+            active.remove(i)
+            active.remove(j)
+            q.add_product((u, ci), (v, cj))
+            factor = (2*factor[0], 2*factor[1])
+            continue
+        if any(q.linear[i] == 2 for i in active):
+            return 0, 0
+        require(all(q.linear[i] == 0 for i in active), 'isolated even phases')
+        size = 1 << len(active)
+        factor = factor[0]*size, factor[1]*size
+        active.clear()
+    require(not q.pairs and not any(q.linear), 'complete quadratic elimination')
+    return gaussian_mul(factor, unit(q.constant))
+
+
+def affine_solution(n: int, equations: list[tuple[int, int]]) -> tuple[list[tuple[int, int]], int] | None:
+    """RREF parametrization of every solution, with each free assignment once."""
+    rows = [[mask, rhs] for mask, rhs in equations]
+    require(all(mask >= 0 and mask >> n == 0 and rhs in (0, 1) for mask, rhs in rows), 'linear equation domain')
+    pivots = []
+    k = 0
+    for j in range(n):
+        at = next((i for i in range(k, len(rows)) if rows[i][0] >> j & 1), None)
+        if at is None:
+            continue
+        rows[k], rows[at] = rows[at], rows[k]
+        for i in range(len(rows)):
+            if i != k and rows[i][0] >> j & 1:
+                rows[i][0] ^= rows[k][0]
+                rows[i][1] ^= rows[k][1]
+        pivots.append(j)
+        k += 1
+    if any(mask == 0 and rhs for mask, rhs in rows):
+        return None
+    free = [j for j in range(n) if j not in set(pivots)]
+    positions = {j: i for i, j in enumerate(free)}
+    forms = [(0, 0)]*n
+    for j, i in positions.items():
+        forms[j] = (1 << i, 0)
+    for pivot, (mask, rhs) in zip(pivots, rows, strict=False):
+        require(mask & sum(1 << j for j in pivots) == 1 << pivot, 'RREF pivot isolation')
+        forms[pivot] = (sum(1 << positions[j] for j in free if mask >> j & 1), rhs)
+    return forms, len(free)
+
+
+def binary_inverse(rows: tuple[int, ...]) -> tuple[int, ...]:
+    n = len(rows)
+    require(n > 0 and all(type(row) is int and 0 <= row < 1 << n for row in rows), 'binary matrix domain')
+    augmented = [row | (1 << (n+i)) for i, row in enumerate(rows)]
+    for j in range(n):
+        k = next((i for i in range(j, n) if augmented[i] >> j & 1), None)
+        require(k is not None, 'singular address matrix')
+        augmented[j], augmented[k] = augmented[k], augmented[j]
+        for i in range(n):
+            if i != j and augmented[i] >> j & 1:
+                augmented[i] ^= augmented[j]
+    require([row & ((1 << n)-1) for row in augmented] == [1 << i for i in range(n)], 'binary inverse endpoint')
+    return tuple(row >> n for row in augmented)
+
+
+def adjoint(circuit: tuple[tuple, ...]) -> tuple[tuple, ...]:
+    out = []
+    for gate in reversed(circuit):
+        kind = gate[0]
+        if kind == 'P':
+            out.append(('P', binary_inverse(tuple(gate[1]))))
+        elif kind == 'D':
+            out.append(('D', tuple(gate[1]), tuple(-a % 4 for a in gate[2])))
+        elif kind == 'C':
+            out.append(('C', gate[1], -gate[2]))
+        elif kind == 'phase':
+            out.append(('phase', -gate[1] % 4))
+        else:
+            require(kind == 'X', 'adjoint gate vocabulary')
+            out.append(gate)
+    return tuple(out)
+
+
+def circuit_trace(n: int, circuit: tuple[tuple, ...]) -> tuple[int, int, int]:
+    """Trace numerator real/imaginary and power-of-two denominator.
+
+    All accepted gates are unitary by their checked syntax and invertible
+    address maps. The trace construction works at arbitrary n.
+    """
+    require(type(n) is int and n > 0, 'positive circuit dimension')
+    addresses = [(1 << i, 0) for i in range(n)]
+    q = Quadratic.zero(n)
+    factor = (1, 0)
+    denominator = 0
+    for gate in circuit:
+        kind = gate[0]
+        if kind == 'P':
+            require(len(gate) == 2 and len(gate[1]) == n, 'permutation arity')
+            rows = tuple(gate[1])
+            binary_inverse(rows)
+            following = []
+            for row in rows:
+                mask = constant = 0
+                for j in bits(row):
+                    mask ^= addresses[j][0]
+                    constant ^= addresses[j][1]
+                following.append((mask, constant))
+            addresses = following
+        elif kind == 'X':
+            require(len(gate) == 2 and type(gate[1]) is int and 0 <= gate[1] < 1 << n, 'translation domain')
+            addresses = [(mask, constant ^ (gate[1] >> i & 1)) for i, (mask, constant) in enumerate(addresses)]
+        elif kind == 'phase':
+            require(len(gate) == 2 and type(gate[1]) is int and 0 <= gate[1] < 4, 'phase domain')
+            q.constant = (q.constant+gate[1]) % 4
+        elif kind == 'D':
+            require(len(gate) == 3 and len(gate[1]) == len(gate[2]) == n, 'diagonal arity')
+            rows, linear = gate[1:]
+            require(all(type(row) is int and 0 <= row < 1 << n for row in rows), 'quadratic matrix domain')
+            require(all(type(a) is int and 0 <= a < 4 for a in linear), 'quadratic linear phase domain')
+            require(all((rows[i] >> j & 1) == (rows[j] >> i & 1) for i in range(n) for j in range(n)),
+                    'symmetric source quadratic')
+            require(all((linear[i] & 1) == (rows[i] >> i & 1) for i in range(n)), 'source diagonal parity')
+            for coefficient, form in zip(linear, addresses, strict=True):
+                q.add_affine(coefficient, form)
+            for i in range(n):
+                for j in range(i+1, n):
+                    if rows[i] >> j & 1:
+                        q.add_product(addresses[i], addresses[j])
+        else:
+            require(kind == 'C' and len(gate) == 3 and type(gate[1]) is int and 0 < gate[1] < 1 << n
+                    and type(gate[2]) is int and gate[2] in (-1, 1), 'Gaussian gate domain')
+            direction, sign = gate[1:]
+            variable = 1 << len(q.linear)
+            q.linear.append(-sign % 4)
+            addresses = [(mask ^ (variable if direction >> i & 1 else 0), constant)
+                         for i, (mask, constant) in enumerate(addresses)]
+            factor = gaussian_mul(factor, (1, sign))
+            denominator += 1
+    constraints = [(mask ^ (1 << i), constant) for i, (mask, constant) in enumerate(addresses)]
+    solved = affine_solution(len(q.linear), constraints)
+    if solved is None:
+        return 0, 0, 0
+    forms, free = solved
+    result = gaussian_mul(factor, gauss_sum(q.substitute(forms, free)))
+    real, imag = result
+    while denominator and real % 2 == 0 and imag % 2 == 0:
+        real //= 2
+        imag //= 2
+        denominator -= 1
+    return real, imag, denominator
+
+
+def validate(n: int, circuit: tuple[tuple, ...]) -> None:
+    """Validate original syntax before an adjoint can normalize any field."""
+    require(type(n) is int and n > 0, 'positive circuit dimension')
+    for gate in circuit:
+        require(type(gate) in (tuple, list) and len(gate) > 0, 'gate record')
+        kind = gate[0]
+        if kind == 'P':
+            require(len(gate) == 2 and len(gate[1]) == n, 'permutation arity')
+            binary_inverse(tuple(gate[1]))
+        elif kind == 'X':
+            require(len(gate) == 2 and type(gate[1]) is int and 0 <= gate[1] < 1 << n, 'translation domain')
+        elif kind == 'phase':
+            require(len(gate) == 2 and type(gate[1]) is int and 0 <= gate[1] < 4, 'phase domain')
+        elif kind == 'C':
+            require(len(gate) == 3 and type(gate[1]) is int and 0 < gate[1] < 1 << n
+                    and type(gate[2]) is int and gate[2] in (-1, 1), 'Gaussian gate domain')
+        else:
+            require(kind == 'D' and len(gate) == 3 and len(gate[1]) == len(gate[2]) == n, 'diagonal arity')
+            rows, linear = gate[1:]
+            require(all(type(row) is int and 0 <= row < 1 << n for row in rows), 'quadratic matrix domain')
+            require(all(type(a) is int and 0 <= a < 4 for a in linear), 'quadratic linear phase domain')
+            require(all((rows[i] >> j & 1) == (rows[j] >> i & 1) for i in range(n) for j in range(n)),
+                    'symmetric source quadratic')
+            require(all((linear[i] & 1) == (rows[i] >> i & 1) for i in range(n)), 'source diagonal parity')
+
+
+def equal(n: int, actual: tuple[tuple, ...], expected: tuple[tuple, ...], label: str) -> tuple[int, int, int]:
+    validate(n, expected)
+    trace = circuit_trace(n, actual+adjoint(expected))
+    require(trace == (1 << n, 0, 0), label+': exact unitary overlap trace '+str(trace))
+    return trace
diff --git a/research/selected-native-tensor-closure/native_check.py b/research/selected-native-tensor-closure/native_check.py
new file mode 100644
index 0000000000..a771f88464
--- /dev/null
+++ b/research/selected-native-tensor-closure/native_check.py
@@ -0,0 +1,168 @@
+#!/usr/bin/env python3
+"""Source-native acceptance using the independent exact trace mechanism."""
+from __future__ import annotations
+
+from collections import Counter
+from functools import lru_cache
+import hashlib
+import json
+
+from gauss_trace import adjoint, binary_inverse, equal, require
+
+
+def canonical(value) -> bytes:
+    return json.dumps(value, sort_keys=True, separators=(',', ':')).encode()
+
+
+def transpose(rows: tuple[int, ...]) -> tuple[int, ...]:
+    return tuple(sum(((row >> j) & 1) << i for i, row in enumerate(rows)) for j in range(len(rows)))
+
+
+def span(vectors: list[int] | tuple[int, ...], n: int = 9) -> frozenset[int]:
+    out = {0}
+    for value in vectors:
+        require(type(value) is int and 0 < value < 1 << n and value not in out, 'independent binary frame')
+        out |= {x ^ value for x in tuple(out)}
+    return frozenset(out)
+
+
+def extended_columns(vectors: tuple[int, ...], n: int) -> tuple[int, ...]:
+    """The source's ordered greedy extension, reimplemented without imports."""
+    result, reached = [], {0}
+    for value in vectors+tuple(1 << i for i in range(n)):
+        if value not in reached:
+            result.append(value)
+            reached |= {x ^ value for x in tuple(reached)}
+    require(len(result) == n and len(reached) == 1 << n, 'complete source frame extension')
+    return tuple(result)
+
+
+def gaussian(rank: int) -> tuple[tuple, ...]:
+    return tuple(('C', 1 << i, 1) for i in range(rank))
+
+
+@lru_cache(maxsize=None)
+def frame(vectors: tuple[int, ...], n: int = 9) -> tuple[tuple, ...]:
+    span(vectors, n)
+    rank = len(vectors)
+    if rank == 0:
+        return ()
+    if rank == n:
+        return gaussian(n)
+    matrix = transpose(extended_columns(vectors, n))
+    inverse_columns = transpose(binary_inverse(matrix))
+    quadratic = tuple((1 << i) ^ sum(((a & b).bit_count() % 2) << j
+                                   for j, b in enumerate(inverse_columns))
+                      for i, a in enumerate(inverse_columns))
+    linear = tuple((row >> i) & 1 for i, row in enumerate(quadratic))
+    conjugator = (('P', matrix), ('D', quadratic, linear))
+    return adjoint(conjugator)+gaussian(rank)+conjugator
+
+
+def monomial(record: dict) -> tuple[tuple, ...]:
+    require(type(record) is dict and set(record) == {'matrix_rows', 'offset', 'quadratic_rows', 'linear_Z4', 'phase'},
+            'complete monomial record')
+    matrix = tuple(record['matrix_rows'])
+    quadratic = tuple(record['quadratic_rows'])
+    linear = tuple(record['linear_Z4'])
+    require(len(matrix) == len(quadratic) == len(linear) == 9, 'source monomial dimension')
+    binary_inverse(matrix)
+    return (('D', quadratic, linear), ('P', matrix), ('X', record['offset']), ('phase', record['phase']))
+
+
+def lowered(record: dict) -> tuple[tuple, ...]:
+    require(type(record) is dict and set(record) == {'rank', 'left', 'right'}, 'complete lowered record')
+    rank = record['rank']
+    require(type(rank) is int and 0 <= rank <= 9, 'source child rank')
+    if rank == 0:
+        require(record['right'] is None, 'rank-zero right convention')
+        return monomial(record['left'])
+    return monomial(record['right'])+gaussian(rank)+monomial(record['left'])
+
+
+def inventory(word: dict) -> tuple[set[tuple[int, int]], list[int], list[tuple[str, int, int]]]:
+    main = set(word['live']) | {r+112 for r in word['live']}
+    helpers = [r for r in range(word['public']) if r not in main]
+    pairs = {(event[2], event[3]) for event in word['program'] if event[0] == 'MOVE'}
+    pairs |= {(word['initial'][r], word['final'][r]) for r in helpers}
+    adapters = [(kind, r, q) for r, q in zip(word['live'], word['ports'], strict=True)
+                for kind in ('input', 'output')]
+    return pairs, helpers, adapters
+
+
+def check_native(word: dict, source_result: dict, candidates: dict, progress=False) -> dict:
+    pairs, helpers, adapter_ids = inventory(word)
+    require(set(candidates) == {'schema', 'pairs', 'adapters', 'source_word_sha256', 'recipe_payload_sha256'},
+            'complete native candidate envelope')
+    require(candidates['schema'] == 'selected-native-tensor-closure/1', 'candidate schema')
+    require(candidates['source_word_sha256'] == '7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f',
+            'candidate source word binding')
+    records = {(a, b): record for a, b, record in candidates['pairs']}
+    require(len(records) == len(candidates['pairs']) and set(records) == pairs, 'exact MOVE/helper recipe coverage')
+    require(all(set(item) == {'kind', 'role', 'port', 'recipe'} for item in candidates['adapters']),
+            'complete main adapter records')
+    require([(item['kind'], item['role'], item['port']) for item in candidates['adapters']] == adapter_ids,
+            'complete ordered main adapter coverage')
+    payload = candidates['pairs']+[item['recipe'] for item in candidates['adapters']]
+    digest = hashlib.sha256(canonical(payload)).hexdigest()
+    require(digest == source_result['native']['recipe_payload_sha256'] == candidates['recipe_payload_sha256'],
+            'full selected recipe payload hash')
+    charts = [frame(tuple(vectors)) for vectors in word['frames']]
+    max_branches = 0
+    traces = []
+    for count, (a, b) in enumerate(sorted(pairs)):
+        expected = adjoint(charts[a])+charts[b]
+        actual = lowered(records[a, b])
+        require(records[a, b]['rank'] == abs(len(word['frames'][a])-len(word['frames'][b])), 'paid or helper source rank')
+        trace = equal(9, actual, expected, f'frame pair {a}:{b}')
+        max_branches = max(max_branches, sum(g[0] == 'C' for g in actual+expected))
+        traces.append([a, b, *trace])
+        if progress and count % 500 == 0:
+            print('trace_pairs', count, len(pairs), flush=True)
+    adapters = {}
+    for item in candidates['adapters']:
+        kind, r, q = item['kind'], item['role'], item['port']
+        if kind == 'input':
+            expected = (('C', q, -1),)+charts[word['initial'][r]]
+        else:
+            expected = adjoint(charts[word['final'][r+112]])+(('C', q, -1),)+gaussian(9)
+        require(item['recipe']['rank'] == 0, 'ordinary intrinsic adapter')
+        actual = lowered(item['recipe'])
+        equal(9, actual, expected, f'main {kind} {r}')
+        adapters[kind, r] = actual
+
+    # Join the logical endpoint X'=X, Y'=Y+X to all three canonical paths.
+    main_paths = 0
+    for r, q in zip(word['live'], word['ports'], strict=True):
+        ix, iy = word['initial'][r], word['initial'][r+112]
+        ox, oy = word['final'][r], word['final'][r+112]
+        require(word['frames'][ix] == [q] and word['frames'][iy] == [] and len(word['frames'][ox]) == 9,
+                'actual main entrance and full source exit')
+        entry, exit_ = adapters['input', r], adapters['output', r]
+        d0 = (('C', q, -1),)+gaussian(9)
+        paths = [
+            (entry+adjoint(charts[ix])+charts[ox], d0, 'source diagonal'),
+            (adjoint(charts[iy])+charts[oy]+exit_, d0, 'target diagonal'),
+            (entry+adjoint(charts[ix])+charts[oy]+exit_, (('C', q, -1),)+d0, 'positive cross'),
+        ]
+        for actual, expected, name in paths:
+            equal(9, actual, expected, f'local {name} {r}')
+            main_paths += 1
+        equal(9, (('C', q, 1), ('C', q, 1)), (('X', q),), 'central square absolute phase')
+
+    helper_pairs = {(word['initial'][r], word['final'][r]) for r in helpers}
+    helper_profile = Counter()
+    for a, b in sorted(helper_pairs):
+        rec = records[a, b]
+        require(rec['rank'] > 0, 'positive productive helper')
+        complete = adjoint(monomial(rec['right']))+adjoint(charts[a])+charts[b]+adjoint(monomial(rec['left']))
+        equal(9, complete, gaussian(rec['rank']), f'complete helper normalization {a}:{b}')
+    for r in helpers:
+        helper_profile[records[word['initial'][r], word['final'][r]]['rank']] += 1
+    require(dict(sorted(helper_profile.items())) == {int(k): v for k, v in source_result['native']['helpers'].items()},
+            'selected arbitrary helper census')
+    return {'distinct_native_pairs': len(pairs), 'distinct_move_pairs': len({(e[2], e[3]) for e in word['program'] if e[0] == 'MOVE'}),
+            'main_adapters': len(adapters), 'complete_main_paths': main_paths, 'central_squares': len(word['ports']),
+            'helper_normalized_pairs': len(helper_pairs), 'arbitrary_helpers': len(helpers), 'helper_profile': dict(sorted(helper_profile.items())),
+            'maximum_closed_trace_branches': max_branches, 'complete_trace_digest': hashlib.sha256(canonical(traces)).hexdigest(),
+            'selected_recipe_payload_sha256': digest, 'method': 'exact constrained quadratic Gauss sum; no tableau or distinguished column'}
diff --git a/research/selected-native-tensor-closure/source_algebra.py b/research/selected-native-tensor-closure/source_algebra.py
new file mode 100644
index 0000000000..597ccfbd2d
--- /dev/null
+++ b/research/selected-native-tensor-closure/source_algebra.py
@@ -0,0 +1,210 @@
+#!/usr/bin/env python3
+"""Complete current-word gauge, ownership and free-Laurent certificates."""
+from __future__ import annotations
+
+from collections import Counter
+from fractions import Fraction
+import hashlib
+
+from gauss_trace import require
+from native_check import canonical, span
+
+
+def add_scaled(dst, src, factor):
+    for key, value in tuple(src.items()):
+        updated = dst.get(key, 0)+factor*value
+        if updated:
+            dst[key] = updated
+        else:
+            dst.pop(key, None)
+
+
+def local_gauge(word: dict, source_result: dict) -> dict:
+    public, owned = word['public'], word['owned_fields']
+    require((public, owned) == (770, 29), 'selected public/owned counts')
+    total = public+owned
+    require(len(word['initial']) == len(word['final']) == public, 'complete local boundaries')
+    spaces = [span(vectors) for vectors in word['frames']]
+    rows = [{i: Fraction(1)} for i in range(total)]
+    current = word['initial'][:]+[None]*owned
+    live_owned = set()
+    paid = Counter()
+    counts = Counter()
+    peak = lifetimes = 0
+    for index, event in enumerate(word['program']):
+        kind, target = event[:2]
+        require(type(target) is int and 0 <= target < total, 'local target range')
+        counts[kind] += 1
+        if kind == 'MOVE':
+            require(len(event) == 5, 'MOVE grammar')
+            _, _, old, new, private = event
+            require(type(private) is bool and current[target] == old and (target in live_owned) == private,
+                    f'current native source and ownership at {index}')
+            require(spaces[new] < spaces[old] if private else spaces[old] < spaces[new], 'proper nested MOVE')
+            paid[abs(len(word['frames'][new])-len(word['frames'][old]))] += 1
+            current[target] = new
+        elif kind == 'INIT':
+            require(len(event) == 3 and target >= public and target not in live_owned, 'fresh owned INIT')
+            live_owned.add(target)
+            rows[target] = {}
+            current[target] = event[2]
+            peak = max(peak, len(live_owned))
+            lifetimes += 1
+        elif kind == 'CLEAR':
+            require(len(event) == 3 and target in live_owned and current[target] == event[2], 'owned CLEAR at current chart')
+            live_owned.remove(target)
+            rows[target] = {}
+            current[target] = None
+        else:
+            require(kind in ('ADD', 'SCALE', 'COPY') and len(event) == (4 if kind == 'SCALE' else 5), 'scalar grammar')
+            coefficient, chart = Fraction(event[-2]), event[-1]
+            require(coefficient != 0 and current[target] == chart, 'scalar current chart')
+            require(coefficient.denominator & (coefficient.denominator-1) == 0, 'dyadic exterior coefficient')
+            if kind == 'SCALE':
+                rows[target] = {origin: coefficient*value for origin, value in rows[target].items()}
+            else:
+                source = event[2]
+                require(type(source) is int and 0 <= source < total and source != target and current[source] == chart,
+                        'exact common-chart source')
+                if kind == 'COPY':
+                    require(target in live_owned and rows[target] == {}, 'COPY into reached owned zero')
+                    rows[target] = {origin: coefficient*value for origin, value in rows[source].items()}
+                else:
+                    add_scaled(rows[target], rows[source], coefficient)
+    require(not live_owned and current[:public] == word['final'] and current[public:] == [None]*owned,
+            'all current charts and ownership returned')
+    for target, row in enumerate(rows):
+        expected = {target: Fraction(1)} if target < public else {}
+        if target-112 in word['live']:
+            expected[target-112] = Fraction(1)
+        require(row == expected, f'complete symbolic gauge endpoint {target}')
+    expected = source_result['local']
+    require({int(k): v for k, v in expected['paid'].items()} == dict(paid), 'selected paid occurrences')
+    require((peak, lifetimes) == (expected['peak_owned_fields'], expected['owned_lifetimes']), 'actual owned chronology')
+    encoded = [[[origin, str(coefficient)] for origin, coefficient in sorted(row.items())] for row in rows]
+    return {'events': len(word['program']), 'opcode_counts': dict(sorted(counts.items())), 'rows': total,
+            'arbitrary_public_columns': public, 'owned_entrance_columns_erased': owned,
+            'full_endpoint_coefficients': total*total, 'owned_lifetimes': lifetimes, 'peak_owned': peak,
+            'paid_profile': dict(sorted(paid.items())), 'gauge_endpoint_sha256': hashlib.sha256(canonical(encoded)).hexdigest(),
+            'rule': 'chart coefficients are formal inverse/forward products; every scalar remains exterior'}
+
+
+ZERO = (0,)*34
+
+
+def shift(poly, exponent, factor=1):
+    require(len(exponent) == 34, 'all Laurent variables present')
+    return {tuple(a+b for a, b in zip(term, exponent, strict=True)): factor*value
+            for term, value in poly.items() if factor*value}
+
+
+def event_pullback(weights, event):
+    """Substitute the actual forward assignment into a final linear form."""
+    kind = event[0]
+    if kind in ('LOCAL', 'QUERY_SHEAR'):
+        _, stage, pair, backward = event
+        require(type(backward) is bool, 'literal controller orientation')
+        x, y = 2*pair, 2*pair+1
+        if backward:
+            x, y = y, x
+        d = ((0,)+tuple(int(j == stage) for j in range(33))) if kind == 'LOCAL' else ZERO
+        cross = (1 if backward else -1,)+(0,)*33
+        wx, wy = shift(weights[x], d), shift(weights[y], d)
+        add_scaled(wx, shift(wy, cross), -1 if backward else 1)
+        weights[x], weights[y] = wx, wy
+    elif kind == 'IDLE':
+        require(len(set(event[2])) == len(event[2]), 'no repeated idle stage')
+        d = (0,)+tuple(int(j in event[2]) for j in range(33))
+        for row in (2*event[1], 2*event[1]+1):
+            weights[row] = shift(weights[row], d)
+    elif kind == 'Q_POWER':
+        require(event[2] in (-2, 2), 'ordinary even central power')
+        weights[event[1]] = shift(weights[event[1]], (event[2],)+(0,)*33)
+    elif kind == 'ADD':
+        add_scaled(weights[event[2]], weights[event[1]], Fraction(event[3]))
+    elif kind == 'SCALE':
+        weights[event[1]] = shift(weights[event[1]], ZERO, Fraction(event[2]))
+    elif kind == 'SWAP':
+        weights[event[1]], weights[event[2]] = weights[event[2]], weights[event[1]]
+    else:
+        require(kind == 'SIGNED_SWAP', 'controller gate vocabulary')
+        weights[event[1]], weights[event[2]] = shift(weights[event[2]], ZERO, -1), weights[event[1]]
+
+
+def free_laurent(controller: dict) -> dict:
+    require((controller['kind'], controller['pairs'], controller['ambient_rank']) == ('complex', 16, 145), 'selected controller type')
+    for side in ('left', 'right'):
+        for output in range(32):
+            weights = [{ZERO: Fraction(int(row == output))} if row == output else {} for row in range(32)]
+            for event in reversed(controller['boundaries'][side+'_program']):
+                event_pullback(weights, event)
+            expected = [{ZERO: Fraction(c)} if Fraction(c) else {} for c in controller['boundaries'][side][output]]
+            require(weights == expected, 'literal '+side+' boundary matrix')
+    program = controller['boundaries']['right_program']+controller['physical_events']+controller['boundaries']['left_program']
+    outputs = []
+    for output in range(32):
+        weights = [{ZERO: Fraction(1)} if row == output else {} for row in range(32)]
+        for event in reversed(program):
+            event_pullback(weights, event)
+        expected = [{tuple(controller['diagonal'][output]): Fraction(1)} if row == output else {} for row in range(32)]
+        require(weights == expected, f'full free-Laurent reverse output {output}')
+        outputs.append([output, controller['diagonal'][output], '1'])
+    sites = []
+    local_count = query_count = 0
+    for i, event in enumerate(controller['physical_events']):
+        if event[0] == 'QUERY_SHEAR':
+            query_count += 1
+        if event[0] != 'LOCAL':
+            continue
+        _, stage, pair, backward = event
+        require(not backward, 'all actual suppliers forward')
+        local_count += 1
+        if stage in controller['wrapper_sites']:
+            x, y = 2*pair, 2*pair+1
+            require(controller['physical_events'][i-2:i+3] == [
+                ['Q_POWER', x, -2], ['SIGNED_SWAP', x, y], event,
+                ['Q_POWER', y, 2], ['SIGNED_SWAP', y, x]], 'literal five-step reverse interface')
+            sites.append(stage)
+    require(sites == controller['wrapper_sites'] and len(sites) == 16 and local_count == 18 and query_count == 15,
+            'complete all-forward sites and thin calls')
+    return {'controller_events': len(controller['physical_events']), 'literal_boundary_events': len(program)-len(controller['physical_events']),
+            'all_1024_laurent_entries_checked': True, 'forward_local_calls': local_count, 'thin_queries_per_port': query_count,
+            'five_step_forward_wrappers': len(sites), 'endpoint_sha256': hashlib.sha256(canonical(outputs)).hexdigest(),
+            'method': 'reverse linear-functional substitution; no finite-order quotient of Q'}
+
+
+def ownership_and_invoice(word: dict, controller: dict, source_result: dict) -> dict:
+    main_roles = set(word['live']) | {r+112 for r in word['live']}
+    helpers = [r for r in range(word['public']) if r not in main_roles]
+    calls = [e for e in controller['physical_events'] if e[0] == 'LOCAL']
+    mains = 32*len(word['live'])
+    productives = mains+len(calls)*len(helpers)
+    maps = []
+    for call, event in enumerate(calls):
+        pair = event[2]
+        mapping = {r: (2*pair)*len(word['live'])+j for j, r in enumerate(word['live'])}
+        mapping.update({r+112: (2*pair+1)*len(word['live'])+j for j, r in enumerate(word['live'])})
+        mapping.update({r: mains+call*len(helpers)+j for j, r in enumerate(helpers)})
+        require(set(mapping) == set(range(word['public'])) and len(set(mapping.values())) == word['public'],
+                'actual distinct local public owners')
+        mapping.update({r: productives+r-word['public'] for r in range(word['public'], word['public']+word['owned_fields'])})
+        require(len(set(mapping.values())) == word['public']+word['owned_fields'], 'owned bank disjoint from all public inputs')
+        maps.append([mapping[r] for r in range(word['public']+word['owned_fields'])])
+    paid = Counter({int(k): len(calls)*v for k, v in source_result['local']['paid'].items()})
+    targets = Counter({int(k): len(calls)*v for k, v in source_result['local']['helpers'].items()})
+    for event in controller['physical_events']:
+        if event[0] == 'QUERY_SHEAR':
+            paid[1] += len(word['ports'])
+        elif event[0] == 'IDLE':
+            paid[8*len(event[2])] += 2*len(word['ports'])
+    for diagonal in controller['diagonal']:
+        targets[1+8*sum(diagonal[1:])] += len(word['ports'])
+    cost = source_result['complete_cost']
+    require(dict(paid) == {int(k): v for k, v in cost['paid'].items()}, 'complete paid profile unchanged')
+    require(dict(targets) == {int(k): v for k, v in cost['targets'].items()}, 'complete productive profile unchanged')
+    require(sum(targets.values()) == productives == 13444, 'all productive owners')
+    return {'main_arbitrary_inputs': mains, 'fresh_helper_inputs': len(calls)*len(helpers), 'productive_inputs': productives,
+            'local_owned_labels': word['owned_fields'], 'separate_thin_owned_label': productives+word['owned_fields'],
+            'paid_calls': sum(paid.values()), 'paid_rank_mass': sum(r*n for r, n in paid.items()),
+            'target_rank_mass': sum(r*n for r, n in targets.items()), 'field_maps_sha256': hashlib.sha256(canonical(maps)).hexdigest(),
+            'limits': 'ideal source ownership only; allocation, physical erasure and retained backing remain premises'}
diff --git a/research/selected-native-tensor-closure/test_gauss_trace.py b/research/selected-native-tensor-closure/test_gauss_trace.py
new file mode 100644
index 0000000000..11d945e72b
--- /dev/null
+++ b/research/selected-native-tensor-closure/test_gauss_trace.py
@@ -0,0 +1,176 @@
+#!/usr/bin/env python3
+"""Independent bounded oracles and rejecting controls for gauss_trace.py."""
+from __future__ import annotations
+
+from fractions import Fraction
+from itertools import combinations, product
+import json
+from pathlib import Path
+import random
+import sys
+
+sys.path.insert(0, str(Path(__file__).resolve().parent))
+
+from gauss_trace import Quadratic, adjoint, affine_solution, circuit_trace, equal, gauss_sum, require
+
+
+def exhaustive_quadratics() -> int:
+    count = 0
+    for n in range(5):
+        edges = list(combinations(range(n), 2))
+        for linear in product(range(4), repeat=n):
+            for mask in range(1 << len(edges)):
+                q = Quadratic(list(linear), {edge for j, edge in enumerate(edges) if mask >> j & 1})
+                real = imag = 0
+                for x in range(1 << n):
+                    v = (sum(linear[j]*((x >> j) & 1) for j in range(n))
+                         +2*sum(((x >> a) & 1)*((x >> b) & 1) for a, b in q.pairs)) % 4
+                    real += (1, 0, -1, 0)[v]
+                    imag += (0, 1, 0, -1)[v]
+                require(gauss_sum(q) == (real, imag), 'exhaustive quadratic sum')
+                count += 1
+    return count
+
+
+def affine_and_substitution() -> dict[str, int]:
+    rng = random.Random(7606030)
+    systems = substitutions = 0
+    for n in range(8):
+        for _ in range(60):
+            equations = [(rng.randrange(1 << n), rng.randrange(2)) for __ in range(rng.randrange(n+3))]
+            expected = {x for x in range(1 << n)
+                        if all((x & mask).bit_count() % 2 == rhs for mask, rhs in equations)}
+            solved = affine_solution(n, equations)
+            if solved is None:
+                actual = set()
+            else:
+                forms, free = solved
+                values = [sum((((mask & t).bit_count() % 2)^c) << j for j, (mask, c) in enumerate(forms))
+                          for t in range(1 << free)]
+                require(len(set(values)) == len(values), 'unique affine enumeration')
+                actual = set(values)
+            require(actual == expected, 'complete affine solution set')
+            systems += 1
+        for _ in range(60):
+            q = Quadratic([rng.randrange(4) for __ in range(n)],
+                          {pair for pair in combinations(range(n), 2) if rng.randrange(2)}, rng.randrange(4))
+            free = rng.randrange(7)
+            forms = [(rng.randrange(1 << free), rng.randrange(2)) for __ in range(n)]
+            changed = q.substitute(forms, free)
+            for t in range(1 << free):
+                x = sum((((mask & t).bit_count() % 2)^c) << j for j, (mask, c) in enumerate(forms))
+                require(changed.value(t) == q.value(x), 'complete affine quadratic substitution')
+            substitutions += 1
+    return {'affine_systems': systems, 'affine_substitutions': substitutions}
+
+
+def dense_trace(n: int, circuit: tuple[tuple, ...]) -> tuple[Fraction, Fraction]:
+    """Separate direct column propagation; only used at n<=4."""
+    size = 1 << n
+    total = [Fraction(0), Fraction(0)]
+    for column in range(size):
+        real = [Fraction(int(x == column)) for x in range(size)]
+        imag = [Fraction(0) for _ in range(size)]
+        for gate in circuit:
+            kind = gate[0]
+            ar, ai = [Fraction(0) for _ in range(size)], [Fraction(0) for _ in range(size)]
+            if kind == 'C':
+                q, sign = gate[1:]
+                for x in range(size):
+                    y = x ^ q
+                    ar[x] = (real[x]-sign*imag[x]+real[y]+sign*imag[y])/2
+                    ai[x] = (imag[x]+sign*real[x]+imag[y]-sign*real[y])/2
+            else:
+                for x in range(size):
+                    y, power = x, 0
+                    if kind == 'P':
+                        y = sum(((row & x).bit_count() % 2) << j for j, row in enumerate(gate[1]))
+                    elif kind == 'X':
+                        y ^= gate[1]
+                    elif kind == 'phase':
+                        power = gate[1]
+                    else:
+                        require(kind == 'D', 'dense oracle gate')
+                        power = sum(a for j, a in enumerate(gate[2]) if x >> j & 1)
+                        power += 2*sum(1 for a in range(n) for b in range(a+1, n)
+                                       if (x >> a & 1) and (x >> b & 1) and (gate[1][a] >> b & 1))
+                    a, b = real[x], imag[x]
+                    ar[y], ai[y] = ((a, b), (-b, a), (-a, -b), (b, -a))[power % 4]
+            real, imag = ar, ai
+        total[0] += real[column]
+        total[1] += imag[column]
+    return total[0], total[1]
+
+
+def dense_circuits() -> int:
+    rng = random.Random(7606546)
+    count = 0
+    for n in range(1, 5):
+        for _ in range(30):
+            gates = []
+            for __ in range(1+rng.randrange(18)):
+                kind = rng.randrange(5)
+                if kind == 0:
+                    gates.append(('C', rng.randrange(1, 1 << n), (-1, 1)[rng.randrange(2)]))
+                elif kind == 1:
+                    gates.append(('X', rng.randrange(1 << n)))
+                elif kind == 2:
+                    gates.append(('phase', rng.randrange(4)))
+                elif kind == 3:
+                    rows = [1 << j for j in range(n)]
+                    for ___ in range(2*n):
+                        a, b = rng.randrange(n), rng.randrange(n)
+                        if a != b:
+                            rows[a] ^= rows[b]
+                    gates.append(('P', tuple(rows)))
+                else:
+                    linear = tuple(rng.randrange(4) for __ in range(n))
+                    rows = [((a % 2) << j) for j, a in enumerate(linear)]
+                    for a, b in combinations(range(n), 2):
+                        if rng.randrange(2):
+                            rows[a] ^= 1 << b
+                            rows[b] ^= 1 << a
+                    gates.append(('D', tuple(rows), linear))
+            circuit = tuple(gates)
+            got = circuit_trace(n, circuit)
+            expected = dense_trace(n, circuit)
+            require((Fraction(got[0], 1 << got[2]), Fraction(got[1], 1 << got[2])) == expected,
+                    'Gauss trace equals separate dense-column oracle')
+            equal(n, circuit+adjoint(circuit), (), 'literal inverse')
+            count += 1
+    require(circuit_trace(300, (('C', 1, 1),)) == (1 << 299, 1 << 299, 0),
+            '300-axis symbolic Gaussian trace')
+    return count
+
+
+def rejecting_controls() -> int:
+    attempts = [
+        lambda: equal(1, (('phase', 1),), (), 'wrong global phase'),
+        lambda: equal(1, (('D', (0,), (2,)),), (), 'input phase invisible on zero'),
+        lambda: equal(1, (('C', 1, 1),), (('C', 1, -1),), 'wrong child direction'),
+        lambda: equal(1, (('X', 1), ('D', (1,), (1,))), (('D', (1,), (1,)), ('X', 1)), 'wrong order'),
+        lambda: circuit_trace(2, (('P', (1, 1)),)),
+        lambda: circuit_trace(1, (('C', 1, 0),)),
+        lambda: circuit_trace(2, (('D', (2, 0), (0, 0)),)),
+        lambda: circuit_trace(1, (('scale', 2),)),
+        lambda: equal(1, (), (('phase', 4),), 'expected phase before normalization'),
+        lambda: equal(1, (), (('phase', 0, 'discarded'),), 'expected arity before adjoint'),
+    ]
+    for run in attempts:
+        try:
+            run()
+        except ValueError:
+            continue
+        raise ValueError('trace negative control accepted')
+    return len(attempts)
+
+
+def main() -> None:
+    result = {'all_quadratics_through_four_variables': exhaustive_quadratics(),
+              **affine_and_substitution(), 'independent_dense_circuits': dense_circuits(),
+              'rejecting_controls': rejecting_controls(), 'seed': [7606030, 7606546]}
+    print('PASS_GAUSS_TRACE', json.dumps(result, sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/selected-native-tensor-closure/verify.py b/research/selected-native-tensor-closure/verify.py
new file mode 100644
index 0000000000..e70c3d3fbe
--- /dev/null
+++ b/research/selected-native-tensor-closure/verify.py
@@ -0,0 +1,110 @@
+#!/usr/bin/env python3
+"""Source-bound selected native tensor closure; see PROOF.md for all-size proof.
+
+Prepared for @CubeJerry with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+from __future__ import annotations
+
+import argparse
+import gzip
+import hashlib
+import json
+from pathlib import Path
+import sys
+import tempfile
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(HERE))
+# Direct replay also avoids reading stale preexisting package bytecode. The
+# repository's registered runner supplies its own private cache as well.
+_cache = tempfile.TemporaryDirectory(prefix='native-tensor-closure-pycache-')
+sys.pycache_prefix = _cache.name
+sys.dont_write_bytecode = True
+
+from gauss_trace import require
+from native_check import check_native
+from source_algebra import free_laurent, local_gauge, ownership_and_invoice
+from caller_check import check_caller
+from controls import run as negative_controls
+
+
+BASE = '7c49fe32c55de02c2c873655c804b16e4c3fef0a'
+PINS = {
+    'research/p104-helper470-twin-rung-removal/WORD.json.gz': '7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f',
+    'research/p104-helper470-twin-rung-removal/RESULT.json': '84fcbe2a0b64307b5703c363408042bdd91555b0419509aa50daa43b3b1924ad',
+    'research/p104-productive-controller/CONTROLLER.json': 'a84f6bbd547379597464e56c420cc665f96bc43a9a1351eed4b7cd0fce3cc4b9',
+    'research/p104-caller-independent-audit/capsule/outputs/p104-caller-route-audit-v2-20261025/sources/DATA-PHASES.json': 'c5916ff312678f12aba403847643fbbc29f643acec60bdf836fa56cd3e8251fa',
+    'research/p104-native-channel/vendor/clifford_exact.py': '2982ead07721e83358dc67c6f49c2b4c16a236a6bebdfd72f58221be1f5d0962',
+    'research/p104-productive-controller/local.py': '878734dfd64c4b9d362d0ae476cc5c2a19fe93e48b7378763b23f3e0dd0a5a71',
+    'research/p104-productive-controller/caller.py': '300d40043953e4f6aecfde8f2d1004b8ad6dc5a79ec3c7666c9908c2466c91c8',
+}
+
+
+def authenticate(path, data):
+    require(path in PINS and hashlib.sha256(data).hexdigest() == PINS[path], 'source byte pin: '+path)
+    return data
+
+
+def run():
+    raw = {path: authenticate(path, (ROOT/path).read_bytes()) for path in PINS}
+    word = json.loads(gzip.decompress(raw['research/p104-helper470-twin-rung-removal/WORD.json.gz']))
+    source_result = json.loads(raw['research/p104-helper470-twin-rung-removal/RESULT.json'])
+    controller = json.loads(raw['research/p104-productive-controller/CONTROLLER.json'])
+    phase_source = json.loads(raw['research/p104-caller-independent-audit/capsule/outputs/p104-caller-route-audit-v2-20261025/sources/DATA-PHASES.json'])
+    candidates = json.loads(gzip.decompress((HERE/'NATIVE.json.gz').read_bytes()))
+    local = local_gauge(word, source_result)
+    native = check_native(word, source_result, candidates)
+    laurent = free_laurent(controller)
+    caller = check_caller(word, controller, phase_source, source_result)
+    invoice = ownership_and_invoice(word, controller, source_result)
+    controls = negative_controls(word, source_result, candidates, controller, phase_source)
+    for path in ('research/p104-helper470-twin-rung-removal/WORD.json.gz', 'research/p104-productive-controller/caller.py'):
+        try:
+            authenticate(path, raw[path]+b' ')
+        except ValueError:
+            controls['source_drift:'+path] = 'Rejected before parsing or execution'
+        else:
+            raise ValueError('source mutation accepted: '+path)
+    return {
+        'schema': 'selected-native-tensor-closure-result/1',
+        'status': 'PASS_SELECTED_NATIVE_TENSOR_CLOSURE',
+        'base_commit': BASE,
+        'source_sha256': PINS,
+        'local_gauge': local,
+        'native_trace': native,
+        'complete_caller': laurent,
+        'whole_job_algebra': caller,
+        'ownership_and_invoice': invoice,
+        'negative_controls': dict(sorted(controls.items())),
+        'all_size_status': 'PROOF.md proves the typed algebraic implication for every positive f and arbitrary spectator factors; finite replay is the exact selected-source premise',
+        'limitations': ['ideal owned reset semantics remain assumed', 'physical selected-rate dispatcher not implemented',
+                        'compiler, precision_tape and analytic_recovery not discharged', 'No new kappa'],
+    }
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--write', action='store_true', help='write the deterministic new-package result receipt')
+    args = parser.parse_args()
+    result = run()
+    path = HERE/'RESULT.json'
+    encoded = (json.dumps(result, sort_keys=True, indent=2)+'\n').encode()
+    if args.write:
+        path.write_bytes(encoded)
+    else:
+        require(path.read_bytes() == encoded, 'complete deterministic result receipt')
+    print(result['status'], json.dumps({
+        'native_pairs': result['native_trace']['distinct_native_pairs'],
+        'raw_contacts': result['whole_job_algebra']['complete_raw_contacts'],
+        'productive_fields': result['ownership_and_invoice']['productive_inputs'],
+        'paid_calls': result['ownership_and_invoice']['paid_calls'],
+        'rejecting_controls': len(result['negative_controls']), 'kappa_changed': False,
+    }, sort_keys=True))
+
+
+if __name__ == '__main__':
+    try:
+        main()
+    finally:
+        _cache.cleanup()