Build a six-T field decoder and a complete 72-T four-address pilot

WORKSPACE.md

@@ -11,9 +11,9 @@ Manuscript drafting and the changed-decoder supplement are separate tasks.
 
 ## Current checkpoint
 
-Verified baseline: [PR #266](https://github.com/GoGoKo699/Hopf-Frame-Compilation/pull/266),
-`2217955f3c0149d80917b5edf88939f9f655098e`, 28 October 2026.
-Its reviewed tree passed 2,009 tests locally and in both CI versions, plus the full gate and presentation checks.
+Verified baseline: [PR #267](https://github.com/GoGoKo699/Hopf-Frame-Compilation/pull/267),
+`68f0ce342bf105e4b3aa3b1a7c5d76c3f579998b`, 28 October 2026.
+Its reviewed tree passed 2,020 tests locally and in both CI versions, plus the full gate and presentation checks.
 Fetch current main; this baseline is not a pin. Exact-tree receipts belong in the PR record.
 
 Read the [resource frontier](docs/OPEN_PROBLEM.md) for general theorems,
@@ -123,9 +123,9 @@ requires strong cross-commutators on every independent address pair;
 monomial query-eigenbasis seeds have a half-error floor. The full-Hadamard
 field seed passes the mixing test but still has a half-error spectral floor.
 
-The [four-address pilot](research/depth/FIELD_COPY_CLIFFORD_PILOT.md) pays
-the seed, copy and inverses: 288 T gates, at most 96 bands and seven dirty
-wires. Its cheaper query gives the direct decoder 300 T, 160 bands and three dirty wires.
+The [flat-seed four-address pilot](research/depth/FLAT_FIELD_BANK_PILOT.md) has a six-T,
+three-band decoder and a complete 72-T, 28-band indicator with three dirty wires.
+Its full intertwiner has a Clifford completion; no uniform decoder is yet paid.
 The [Clifford-seed bounds](research/depth/CLIFFORD_SEED_COPY_STRUCTURE.md) give
 error at least 1/64 for p at least seven for projectively involutive seeds with commuting symplectic orbits,
 even for an arbitrary copy. For p at least nine, every Clifford seed has one-band
@@ -139,11 +139,11 @@ and full copy error at least $`1/\sqrt6`$ for every unitary seed. This family is
 The [Clifford-dual grading](research/depth/CLIFFORD_DUAL_COPY_GRADING.md) requires each
 output dual's symplectic orbit group to have binary abelianization rank at least
 $`m-p-1`$ exactly. An odd-order pair gives an explicit robust word obstruction.
-**Next question:** construct changed joint walls with this grading and a genuinely
-non-Clifford $`R=VPV^\dagger`$, then solve the full carrier equations. Two-band V
-and a one-band copy would give at most 42 indicator bands; this remains conditional.
-Price every inverse and common budget. Empty kernels reject exact instances;
-approximation needs quantitative certificates. The [whole-table alternative](research/depth/TABLE_SPECIFIC_INDICATOR_INTERFACES.md#5-what-this-changes-and-what-it-leaves-unpaid) remains unpaid.
+The [seed fingerprint](research/depth/NATIVE_COPY_INTERACTION.md#seed-coefficient-rigidity)
+excludes product THT-dagger seeds below 3/32 for every one-band field copy at p at least two.
+**Next question:** extend the flat decoder uniformly, allowing the complete probe's
+commutant freedom, with sublogarithmic depth and polynomial discovery budgets.
+The joint seed/copy alternative remains open; two-band V and one-band C would give 42 bands conditionally.
 
 The [four-band Clifford front end](research/depth/CLIFFORD_INVOLUTION_SOURCE.md),
 [weighted outer-phase route](research/depth/JOINT_ADDRESS_HELPER_SOURCE.md),

docs/reference/VERIFICATION_CATALOGUE.md

@@ -281,6 +281,8 @@ without an extra native fixture.
 | Non-Clifford copy-axis span | [Proof](../../research/depth/COPY_AXIS_SPAN.md), [exact fixtures](../../tests/test_copy_axis_span.py) | Exhaustive small affine-map quotient counts and full-matrix noncommuting rotations test the span argument, including interference and mixed X/Z axes. The all-size norm bound is analytic and does not lower-bound growing depth |
 | Monomial query seed | [Proof](../../research/depth/MONOMIAL_QUERY_SEED_OBSTRUCTION.md), [exact fixtures](../../tests/test_monomial_query_seed_obstruction.py) | Non-Hermitian phases, shifted planes, Fourier field queries and arbitrary helper mixing check the alternating word. A nonmonomial escape guards the hypothesis. Rational/native commutator fixtures and an exact order-three Hadamard pilot check the stronger mixing requirement and its insufficiency. All-size floors are analytic; escape alone supplies no source |
 | Finite Clifford field copy | [Proof](../../research/depth/FIELD_COPY_CLIFFORD_PILOT.md), [algebra fixtures](../../tests/test_field_copy_clifford_pilot.py), [native fixtures](../../tests/test_field_copy_clifford_native.py) | Exact seed and central-sector algebra; all sector-controlled native factors and actual inverses; all 16 output-X copy sectors; every principal/dirty query column and the finite ledgers. The copy indicator has 288 T gates, 96 bands and seven dirty wires; reusing the improved query in the direct decoder gives 300 T, 160 bands and three dirty wires. Both assembled native directions are checked; neither is an all-size compiler |
+| Flat field bank and six-T decoder | [Proof](../../research/depth/FLAT_FIELD_BANK_PILOT.md), [native fixtures](../../tests/test_flat_field_bank_native.py), [dual and completion fixtures](../../tests/test_flat_field_bank_duals.py) | Exact flat seed, full commuting-bank grading, three physical decoder bands, literal copy action, full intertwiner completion and actual inverses. All 512 direct-indicator columns retain arbitrary address, output and dirty-mask inputs and literal phases. The 72-T, 28-band, 350-gate finite word uses three returned dirty wires. Exact signed-support enumeration proves no Clifford singleton dual; its gap is scoped to this seed. No uniform compiler or minimum-cost claim |
+| Seed coefficient and fingerprint rigidity | [Proof](../../research/depth/NATIVE_COPY_INTERACTION.md#seed-coefficient-rigidity), [exact fixtures](../../tests/test_seed_fingerprint.py) | Exact Pauli correlation levels, helper subgroup preservation, paired-axis normalizers, literal one-cell Clifford overlaps and affine scalar words challenge the analytic rigidity proof. Unique maxima give a coefficient-gap floor; product THT-dagger seeds give 3/32 independently of width, under Clifford output duals and arbitrary fixed Clifford embedding. Flat seeds and unrestricted copies are outside that product theorem |
 | Clifford seed copy structure | [Proof](../../research/depth/CLIFFORD_SEED_COPY_STRUCTURE.md), [nested-cube fixtures](../../tests/test_clifford_seed_copy_structure.py), [one-band fixtures](../../tests/test_one_band_clifford_seed.py) | Degree inflation, literal scalar phases and a complete 128-address field word support the commuting-orbit theorem. Exact signed affine-axis and modulo-four phase checks support the separate one-band theorem for every Clifford seed: error at least $`\sin(\pi/8)`$ for p at least nine. The analytic rigidity and degree-eight proof permit arbitrary dirty width and joint Clifford walls; non-Clifford seeds and general copies remain outside that second theorem |
 | Clifford output-dual grading | [Proof](../../research/depth/CLIFFORD_DUAL_COPY_GRADING.md), [exact fixtures](../../tests/test_clifford_dual_copy_grading.py) | Regular-module annihilators, Pauli-kernel transport and literal relation-word checks support the necessary symplectic-group quotient. Each exact one-band output dual needs binary abelianization rank at least $`m-p-1`$, for arbitrary unitary seeds. Quantitative gaps require explicit bounded words; the group-rank bound itself is exact and does not exclude every one-band copy |
 | Reduced pair-register gauge | [Proof](../../research/depth/COHERENT_CENTRAL_TRANSPORT.md#reduced-carrier-gauge), [exact fixtures](../../tests/test_reduced_pair_gauge.py) | Arbitrary pair/carrier action, helper-free native pair factors and root completion, interpolation and a nonrepresentation gauge check the complete probe. The low-row degree constraint is analytic and gauge-specific; no shallow joint schedule follows |

research/README.md

@@ -191,13 +191,14 @@ These are complete interfaces and conditional transfers, not constructions of th
 - [Table-specific probes and the remaining joint-word freedom](depth/TABLE_SPECIFIC_INDICATOR_INTERFACES.md)
 - [Encoded dirty copy without a synthesized decoder](depth/ENCODED_DIRTY_COPY_INTERFACE.md) — jointly paid carrier/copy interactions supply a physical indicator; an address-independent Clifford copy has a uniform error floor for one field-query orbit
 - [Exact dirty copying as a parity grading](depth/DIRTY_COPY_GRADING.md) — a complete criterion for arbitrary unitary families, robust word tests, field-pair rigidity and a noncommuting all-size calibration
-- [Native copy interactions and dual readout](depth/NATIVE_COPY_INTERACTION.md) — paid non-Clifford components, an extensive copy T-count floor, and exact support screens for specified product-magic and coded families
+- [Native copy interactions and dual readout](depth/NATIVE_COPY_INTERACTION.md) — paid components, extensive copy T count, support screens, and a width-independent 3/32 floor for product THT-dagger seeds with Clifford output duals
 - [Complete one-band copy feasibility](depth/ONE_BAND_COPY_FEASIBILITY.md) — signed Walsh support spaces and spectra, the exact carrier fixed-point criterion, and a native triple-axis joint-wall family with zero complete seed space and a $`1/\sqrt6`$ copy-error floor despite passing the tensor screen
 - [Clifford output-dual grading](depth/CLIFFORD_DUAL_COPY_GRADING.md) — every exact one-band copy, even with an arbitrary unitary seed, requires an extensive binary abelianization of each dual's symplectic orbit group; explicit relation words give robust certificates
 - [Affine Pauli-orbit copy screen](depth/PAULI_ORBIT_COPY_SCREEN.md) — a one-quarter error floor with Clifford dual probes, the native one-bit endpoint and the paid dirty-mask gauge
 - [Extensive non-Clifford copy-axis span](depth/COPY_AXIS_SPAN.md) — quotient-sector weight gives a robust output-axis and T-count requirement, with no growing depth conclusion
 - [Monomial query-seed copy obstruction](depth/MONOMIAL_QUERY_SEED_OBSTRUCTION.md) — commuting original and seed-conjugated query algebras force a four-word identity and copy error at least one half, even with an arbitrary physical copy
 - [Four-address Clifford seed and physical copy](depth/FIELD_COPY_CLIFFORD_PILOT.md) — a noncommuting central-sector grading, a paid native copy and a complete 288-T, 96-band finite indicator with seven arbitrary dirty wires
+- [Flat-seed field bank and six-T decoder](depth/FLAT_FIELD_BANK_PILOT.md) — a genuinely non-Clifford graded four-address bank, a three-band decoder with a full Clifford intertwiner completion, and a complete 72-T, 28-band indicator with three returned dirty wires; no all-size depth improvement
 - [Clifford seed copy structure](depth/CLIFFORD_SEED_COPY_STRUCTURE.md) — projectively involutive seeds with commuting symplectic orbits have a 1/64 field-copy floor at p at least seven for arbitrary copies; every Clifford seed has a $`\sin(\pi/8)`$ floor at p at least nine when the copy has one physical T band
 - [Joint bank routing and the complete dirty indicator](depth/JOINT_ROUTING_INDICATOR_INTERFACES.md)
 

research/ROUTE_DECISIONS.md

@@ -25,6 +25,8 @@ Start with a complete interface; consult component families only when the propos
 | [Localized non-Clifford copy repair](depth/COPY_AXIS_SPAN.md) | A one-shear orbit requires at least $`m-p-1`$ independent physical-X directions in the copy rotation-axis span exactly, and an extensive span at constant error below square root two. | This span bound alone does not exclude a broad one-band copy; pay it jointly with the seed. This is a count/span screen, not a growing depth obstruction. |
 | [Monomial seed in query eigen-coordinates](depth/MONOMIAL_QUERY_SEED_OBSTRUCTION.md) | Copy error is at least one half for p at least two, even for arbitrary non-Clifford copies. The broader screen is commutation of original and seed-conjugated query algebras. | Break that algebra condition with genuine eigenspace mixing and certify the complete copy. Nonlinear reversible permutations or phases in the same query eigenbasis do not suffice. The all-Hadamard field seed passes the quantitative mixing test but still has a half-error floor from an order-three orbit-pair product. |
 | [Finite Clifford field copy](depth/FIELD_COPY_CLIFFORD_PILOT.md) | A noncommuting four-address Clifford orbit has a central-sector grading and a paid physical copy. Its indicator uses 288 T gates, at most 96 bands and seven arbitrary dirty wires. The same cheaper query improves the direct decoder to 300 T, 160 bands and three dirty wires; the copy's depth gain costs more workspace and total gates. | Use its explicit sector-control construction as a component, not an all-size compiler. Its Pauli-valued commutators fall under the eventual obstruction below. |
+| [Flat field bank and shallow finite decoder](depth/FLAT_FIELD_BANK_PILOT.md) | A genuinely non-Clifford four-address seed has full commuting-bank grading. A six-T three-band decoder yields a 12-T six-band complete copy and a cheaper direct indicator with 72 T gates, at most 28 bands, 350 total gates and three returned dirty wires. A separate Clifford completion makes the decoder a full field intertwiner. The same seed has one-band copy error at least $`1/(2\sqrt2)`$. | Extend the decoder uniformly within the polynomial discovery budgets and sublogarithmic depth, allowing the complete probe's commutant freedom. A full intertwiner below error one needs axis pairing rank at least $`m-2`$ on the query-fixed Pauli space; disjoint cells retaining all their core-fixed controls fail. This rank condition does not constrain the weaker one-probe decoder. The finite words establish no asymptotic bound or minimum cost. |
+| [Seed coefficient and fingerprint rigidity](depth/NATIVE_COPY_INTERACTION.md#seed-coefficient-rigidity) | A unique largest Pauli magnitude with gap g forces field-copy error at least g/2 for p at least two when a selected output dual is Clifford. Product THT-dagger seeds have the width-independent floor 3/32, even after an arbitrary Clifford embedding and with arbitrary identity helpers. | For a one-band copy, change the product seed and remove its distinguished Pauli witness; degeneracy alone is insufficient. Deeper copies with non-Clifford duals remain eligible, as the paid flat-bank pilot illustrates. This is a seed/readout restriction, not an all-two-band-seed or unrestricted-copy theorem. |
 | [Clifford seed copy structure](depth/CLIFFORD_SEED_COPY_STRUCTURE.md) | A projectively involutive Clifford seed with commuting symplectic orbit has field-copy error at least 1/64 for p at least seven, for an arbitrary copy. With one physical copy band, every Clifford seed has error at least $`\sin(\pi/8)`$ for p at least nine; no orbit commutation or seed-order assumption is needed. | A one-band field copy needs a genuinely non-Clifford seed at sufficiently small error. In $`R=VPV^\dagger`$, V must therefore use at least two T bands. A Clifford-seed route needs more copy bands and must still leave the older commuting-orbit class. Neither condition supplies a construction. |
 | [Clifford output-dual grading](depth/CLIFFORD_DUAL_COPY_GRADING.md) | Every exact one-band copy induces a copyable Clifford orbit at each physical output dual, even for an arbitrary unitary seed. Its symplectic orbit group needs elementary binary abelianization rank at least $`m-p-1`$ for the field query. Explicit symplectic relation words yield quantitative scalar-word tests; the rank bound alone has no uniform approximate gap. | Design a dual group that retains the required address grades, then satisfy the complete carrier equations and synthesize the seed. Noncommuting symplectic generators alone are insufficient. An odd-order orbit-pair product already gives a robust obstruction for p at least three, with its certified order in the denominator. |
 | [Projector-Gram readout](depth/NILPOTENT_PENCIL_OBSERVABLE_SOURCE.md#paired-projector-readout) | Four-call recovery and serial promotion are paid: complete error $`64\sqrt\delta`$ for one clean signal, $`128\sqrt\delta`$ for two jointly accepted signals. The six-call block-error alternative remains. | A normalized native Gram source, allowing arbitrary accepted private fibers; actual source synthesis remains essential. |

research/depth/ENCODED_DIRTY_COPY_INTERFACE.md

@@ -273,6 +273,11 @@ at p at least seven, a projectively involutive Clifford seed with commuting
 symplectic orbit actions has copy error at least 1/64, even with arbitrary C.
 
 The all-size task remains a jointly paid pair satisfying (2) within (4).
+The later [flat four-address bank](FLAT_FIELD_BANK_PILOT.md) supplies a
+genuinely non-Clifford seed, a 12-T six-band physical copy, and a cheaper
+direct indicator through its six-T three-band decoder. Extending that
+complete probe decoder uniformly is another concrete route; its finite
+word does not settle either all-size interface.
 For a Clifford seed, noncommuting symplectic dynamics are therefore necessary
 at arbitrarily small fixed error; non-Clifford seeds are another possibility.
 The complete dual readout, native inverse and every arbitrary-input sector

research/depth/FIELD_COPY_CLIFFORD_PILOT.md

@@ -15,6 +15,10 @@ not exclude every Clifford seed at this address size.
 
 This is a finite complete-operator pilot. Its four addresses do not give
 an all-size source or improve the general frame-depth frontier.
+The later [flat-seed pilot](FLAT_FIELD_BANK_PILOT.md) gives a different
+commuting bank and a cheaper 72-T, 28-band direct indicator with three
+dirty wires. The noncommuting central-sector construction here remains
+a distinct algebraic and native calibration.
 
 ## 1. The physical seed and query
 

research/depth/FIELD_SHEAR_INTERTWINER.md

@@ -303,6 +303,11 @@ does not compile that change into native gates.
 
 ## 6. A paid four-wire decoder for two address bits
 
+The later [flat-bank decoder](FLAT_FIELD_BANK_PILOT.md) supplies a different
+six-T, three-band full intertwiner at this size. The construction below
+retains its punctured-coordinate mechanism and exact native fixtures;
+neither finite word establishes an all-size shallow decoder.
+
 There is also a native p=2 decoder, independent of the canonical
 eigenbasis choice in (17). It uses only the four bank wires. The
 following construction explains its full action, including the zero