Derive complete bank-mixing screens and Bell acceptance transfer

WORKSPACE.md

@@ -8,14 +8,13 @@ complete Hopf frames. Manuscript drafting is a separate task.
 
 <a id="current-checkpoint-native-source-scope-and-complete-readout"></a>
 
-## Current checkpoint: relative involution paid, lookup transitions unpaid
-
-**Verified starting checkpoint:** [PR #242](https://github.com/GoGoKo699/Hopf-Frame-Compilation/pull/242),
-`7fc9d32fd7bc70764ce09292897e248f315fa387`, fetched 24 October 2026.
-A resynthesized middle pays an exact Hermitian relative block in four
-native bands with zero clean work. Complete lookup still fails for
-the tested diagonal, Hadamard and cubic-phase dressings; a paid broad
-four-path Bell source also has the wrong Gram. R96 is unchanged.
+## Current checkpoint: joint mixing screens and the remaining lookup phase
+
+**Verified starting checkpoint:** [PR #243](https://github.com/GoGoKo699/Hopf-Frame-Compilation/pull/243),
+`4598492c870b81666f8b149a4c8ccd8efcd8d0b2`, fetched 25 October 2026;
+its post-merge validation and presentation passed. Affine/Fourier dressings have an all-angle transition gap.
+Two Clifford mixers and Bell acceptance expose the remaining lookup phase.
+Their scoped limitations leave R96 unchanged.
 These are scoped failures. Exact-tree receipts belong in the PR record. Fetch main before resuming; this is not a pin.
 
 Read in this order:
@@ -117,33 +116,34 @@ $`\tau_*=\sin(\pi/8)/(2\sqrt2)`$, the retained three-layer rounding
 theorem makes complete Pauli transport exact; two layers are excluded for
 p at least three. More source layers may use the approximate contract.
 
-**Latest partial construction.** The [corrected middle](research/depth/COUPLED_FINAL_ENCODER.md#relative-probe-involution-repair)
+**Retained partial construction.** The [corrected middle](research/depth/COUPLED_FINAL_ENCODER.md#relative-probe-involution-repair)
 makes both co-moving probe fibers equal, so $`F_s=E_sZ_\pi E_s^\dagger`$
 is an exact Hermitian involution. Its base source has $`T=4m`$,
 $`D_T\le4`$, $`G=4pm+20m-12`$, one arbitrary probe and zero
 clean work. The new involution statement concerns its relative block; the full transported Pauli was always an involution. Lookup is still unpaid.
-At the fixed native angles, [every diagonal dressing](research/depth/COUPLED_FINAL_ENCODER.md#relative-repair-diagonal-modulus)
-has direct error above one third for p at least three. Every nonempty
-[Hadamard mask](research/depth/COUPLED_FINAL_ENCODER.md#relative-repair-hadamard-mask)
-has error at least square root two; the tested cubic-phase/Hadamard
-interlayer also fails analytically. These are complete transition failures.
-
-Three [probe-asymmetric repairs](research/depth/COUPLED_FINAL_ENCODER.md#probe-asymmetric-paid-trials)
-have paid controls but exact nonreal relative entries at p=3; these finite failures do not exclude all sizes or all asymmetric words.
-The [four-path Bell source](research/depth/CORANK_FOURIER_BELL_SOURCE.md#four-path-bell-relative-phase)
-has two clean signals, a paid depth-94 schedule and exact Gram error
-one half at every canonical size. Normalization does not select lookup charge.
-Earlier covariance, singlet, spectrum and thin-source failures retain
-their original hypotheses; none is an unrestricted native-depth bound.
-
-**Next concrete question: change joint bank transitions.**
-The relative-involution question is settled by the partial construction.
-Pause the tested single-dressing words and swap/weight-phase acceptance.
-Can two noncommuting bank-mixing stages, with a changed middle between
-them, give $`F_s\approx(-1)^{\pi(B)+B_{s+HB}}`$ on every bank column?
-Derive that complete action before a new fixture; alternatively construct
-a genuinely different accepted Gram satisfying the retained normalized contract.
-Do not infer lookup from Hermiticity, broad support or normalized acceptance.
+
+**Latest complete-action results.** [Affine/Fourier dressings](research/depth/COUPLED_FINAL_ENCODER.md#affine-fourier-complete-transition-gap)
+have error at least square root two for p at least two, even with arbitrary
+X-diagonal pulses and computational-diagonal middles in that word. No computational-monomial dressing completes the original native word
+exactly; this separate screen gives no uniform approximation gap.
+For [two bank Clifford mixers separated by diagonal stages](research/depth/COUPLED_FINAL_ENCODER.md#two-clifford-mixer-screen),
+error below one requires a rank-one final mixer aligned with full bank
+complement. This bank-only, fixed-address grammar has an explicit exact
+exception, but its [middle phase derivative](research/depth/COUPLED_FINAL_ENCODER.md#rank-one-mixer-lookup-derivative)
+still computes lookup. Two Hadamard masks in one common affine frame fail.
+The [Bell-envelope identity](research/depth/CORANK_FOURIER_BELL_SOURCE.md#bell-acceptance-charge-transfer)
+preserves the complete charge-readout error on arbitrary auxiliaries;
+address-independent acceptance has sharp Gram floor one quarter.
+
+**Next concrete question: pay a non-Clifford joint transition.**
+The earlier instruction to try two mixers is now qualified by these screens.
+Can a changed non-Clifford bank mixer or a nondiagonal middle supply
+$`F_s\approx(-1)^{\pi(B)+B_{s+HB}}`$ with a complete native ledger?
+Alternatively, construct an accepted source with genuinely interleaved
+data/auxiliary action beyond the Bell-envelope factorization. Reopen the
+rank-one exception only with a paid shallow lookup-dependent derivative.
+Derive the complete action before another fixture; phase declarations,
+charge preservation and broad support do not supply the missing operation.
 The source still needs $`o(\log(p+2))`$ depth and the simultaneous
 count, width, precision and arbitrary-work return ledger. Completed
 quadratic coordinate rounds retain a logarithmic round requirement;

docs/OPEN_PROBLEM.md

@@ -229,6 +229,8 @@ The relevant exclusions have different hypotheses:
 | Specified shifted-syndrome commutator | [Four-band native trial](../research/depth/COUPLED_FINAL_ENCODER.md#shifted-syndrome-wide-commutator): 4m T gates and no helpers, but complete Pauli-transport error at least square root two for p at least two, even at arbitrary rotation angles on the same axes; the native p=1 gap is at least one quarter. The missing row response excludes this commutator family without a thin-source assumption. A changed middle action, axes, full word or accepted Gram remains outside |
 | Common-probe pulses and diagonal bank repair | [Paid four-band family](../research/depth/COUPLED_FINAL_ENCODER.md#common-probe-wide-source): both frames change the probe through every bank row. Two row-local frames retain an address-shifting covariance, giving direct error at least square root two and Gram error at least one quarter. The graph-CZ third frame breaks that covariance but has a paired-state error at least square root two for p at least two. At fixed native angles, [every static bank-diagonal repair](../research/depth/COUPLED_FINAL_ENCODER.md#static-diagonal-common-probe-spectrum) has direct error above one half for p at least three: the relative probe spectrum is invariant. A changed relative-fiber action or accepted interface remains outside; the spectral result gives no new general Gram or native-depth bound |
 | Corrected relative probe and changed acceptance | [Relative involution repair](../research/depth/COUPLED_FINAL_ENCODER.md#relative-probe-involution-repair) has an exact Hermitian relative block in four native bands, zero clean work and linear T-count. Complete lookup remains wrong: diagonal dressings have direct error above one third for p at least three, every nonempty Hadamard mask has error at least square root two, and the tested cubic interlayer also fails. [Three asymmetric trials](../research/depth/COUPLED_FINAL_ENCODER.md#probe-asymmetric-paid-trials) have finite p=3 failures, not all-size exclusions. The separate [four-path Bell source](../research/depth/CORANK_FOURIER_BELL_SOURCE.md#four-path-bell-relative-phase) pays two-clean normalization and depth-94 controls but has exact Gram error one half. Changed joint bank transitions or another complete accepted interface remain open |
+| Joint bank mixing screens | [Affine/Fourier dressing](../research/depth/COUPLED_FINAL_ENCODER.md#affine-fourier-complete-transition-gap) has error at least square root two for either affine/Hadamard order, even for arbitrary X-diagonal pulses and diagonal middles in the same word. The original native word has no exact monomial dressing; no uniform approximate exclusion follows from that separate result. The [bank-only two-Clifford grammar](../research/depth/COUPLED_FINAL_ENCODER.md#two-clifford-mixer-screen) has error at least one except for an aligned rank-one final mixer whose explicit completion still needs a lookup-dependent diagonal derivative. Active extra work, general address motion and unrestricted native sources remain outside these hypotheses |
+| Bell-envelope acceptance | [Arbitrary auxiliary effects](../research/depth/CORANK_FOURIER_BELL_SOURCE.md#bell-acceptance-charge-transfer) inherit exactly the same full operator error for the transported charge. Address-independent effects have sharp Gram floor one quarter. Pure charge-preserving permutation expansion with held address and bank-independent interlayers also has a quarter floor, regardless of its spectral gap. Interleaved data/auxiliary dynamics, bank phases and other source grammars remain outside |
 | Wide Clifford filters | [Sharp lookup correlation](../research/depth/CLIFFORD_LOOKUP_FIXED_SPACE.md) is $`(p+1)/2^p`$ for $`p\ge2`$; normalized Clifford averages cannot separate the two full lookup eigenspaces by a constant gap. Coefficient and call bounds are not native-depth bounds |
 
 Completed address return does not establish native read-only sharing.

docs/reference/VERIFICATION_CATALOGUE.md

@@ -255,6 +255,9 @@ without an extra native fixture.
 | Common-probe source and static diagonal repairs | [Proof home](../../research/depth/COUPLED_FINAL_ENCODER.md#common-probe-wide-source), [exact native fixtures](../../tests/test_common_probe_source.py) | All one-address native source columns, actual inverse and relative-fiber columns check phase and orientation. Two- and three-address singlet inputs check literal pulse phases, the complete echo and squared error two. All 2,048 three-address diagonal square entries and both single-wire spectral factors challenge the static-dressing proof; distinct T targets and ledgers are checked through six address bits. The covariance, paired-state and spectral statements are analytic with distinct hypotheses. These internal audits and finite checks supply no unrestricted depth bound or external review |
 | Relative-probe involution and changed lookup trials | [Proof home](../../research/depth/COUPLED_FINAL_ENCODER.md#relative-probe-involution-repair), [native repair fixtures](../../tests/test_relative_probe_repair.py), [asymmetric-fiber fixtures](../../tests/test_probe_asymmetric_trials.py) | Exact native phases, actual inverses, relative blocks, normalized witnesses and ledgers check the corrected source. Rational displacement tails challenge the diagonal-dressing theorem; the nonempty-Hadamard and specified cubic-phase witnesses test complete lookup. Three exact p=3 nonreal relative entries reject the stated asymmetric trials; small binary bases and degree restrictions challenge their analytic components. All-size theorems, finite failures and internal audits retain separate scopes; no external review or frontier improvement is asserted |
 | Broad four-path Bell acceptance | [Complete operator and ledger](../../research/depth/CORANK_FOURIER_BELL_SOURCE.md#four-path-bell-relative-phase), [exact multipath fixtures](../../tests/test_multipath_bell_source.py) | Literal controlled swaps and phases, actual inverses and signal acceptance check all one-cell logical columns at two arbitrary-helper assignments. Complete two-cell Grams and native Bell Pauli propagation challenge coherent normalization and orientation. The exact half Gram gap and shared-control native schedule are analytic; finite fixtures are internal evidence, not an asymptotic emitter or external review |
+| Affine/Fourier repair and exact monomial boundary | [Proof home](../../research/depth/COUPLED_FINAL_ENCODER.md#affine-fourier-complete-transition-gap), [exact fixtures](../../tests/test_affine_fourier_repair.py) | Both affine/Hadamard orders, arbitrary bank columns, a complete native coherent echo and its actual inverse, and distinct-target ledgers challenge the robust transition gap. A correlated X-basis cubic phase checks the broader pulse scope. Walsh/noise and rectangle identities test the exact-only monomial screen, including a nonlinear permutation that transports the target to parity but fails the remaining transition condition. The all-angle and exact-only statements have separate hypotheses; no unrestricted native-depth or fixed-accuracy monomial obstruction is asserted |
+| Two-Clifford mixer screen and rank-one exception | [Proof home](../../research/depth/COUPLED_FINAL_ENCODER.md#two-clifford-mixer-screen), [exact fixtures](../../tests/test_two_mixer_lookup_screen.py) | Every p=3 bank translation and Walsh coefficient, an actual Clifford-coset source, and common-affine parity slopes check the scoped diagonal-support theorem. All 256 bank columns and actual inverses check the genuine rank-one completion, whose middle is supplied only as a truth-table diagonal; degree checks expose its unpaid lookup phase. The bank-only, fixed-address theorem does not cover active extra work or arbitrary native mixing |
+| Complete Bell-envelope transfer | [Proof home](../../research/depth/CORANK_FOURIER_BELL_SOURCE.md#bell-acceptance-charge-transfer), [exact fixtures](../../tests/test_bell_acceptance_transfer.py) | Literal paired Bell matrices retain all arbitrary data and auxiliary columns. A non-diagonal coherent auxiliary effect tests the exact whole-space residual, and distinct-address comparisons test the sharp quarter floor. Uniform coset vectors check the separate phase-free permutation screen. These analytic reductions supply no new source or frontier bound |
 | Coupled final encoders and bank dependence | [Proof home](../../research/depth/COUPLED_FINAL_ENCODER.md), [exact native fixtures](../../tests/test_coupled_final_encoder.py) | Complete two-address coefficient-bank and three-address cubic-sign matrices check the hybrid-address kernel on every dirty-tag input and its actual inverse. Bounded full-bank fixtures check the syndrome rewrite, literal inverse, Fourier/Pauli expansion columns and native ledgers. The bank-fiber rank theorem and the separate rotated-bank centralizer obstruction are analytic, including arbitrary middle Clifford boundaries and identity helpers. These encoders are components, not qualifying sources or unrestricted depth lower bounds |
 | Boolean spectral reflection sources | [Proof home](../../research/depth/BOOLEAN_REFLECTION_GRAM_SOURCE.md), [six exact fixtures](../../tests/test_boolean_reflection_gram_source.py) | Native eight-point Fourier transform and actual reverse on every dirty column; spectral truth-table compute/uncompute; literal controlled quarter-phase echo; complete accepted normal form; high-bit and parity p=1 sources; and a commuting-reflection success fixture. Exact spectral evaluation, bounded-clean removal, simultaneous source costs, robust reflection-product criterion and the all-size flat-mixer gap one quarter are analytic. The changed source matches normalized trace but fails complete lookup sectors; finite checks and internal audits are not external mathematical review |
 | Scalar-kernel partial source | [Proof home](../../research/depth/SCALAR_KERNEL_FILTER_SOURCE.md), [four exact fixtures](../../tests/test_scalar_kernel_filter_source.py) | Native source/inverse, support counts and finite halfspace rigidity remain. The new rank-one determinant trial checks all 128 p=1 algebra columns, exact kernels and accepted ranks; it is not a native emitter. The binary determinant identity, depth-eight query, approximate source ledger and Gram gap one half in every lookup sector are analytic. Neither the original thin projector nor the changed determinant echo supplies complete lookup; the scalar expander itself is unchanged |