Compile a 172-T eight-address field decoder and certify coupled-axis requirements

WORKSPACE.md

@@ -11,15 +11,13 @@ Manuscript drafting and the changed-decoder supplement are separate tasks.
 
 ## Current checkpoint
 
-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.
+Verified baseline: [PR #268](https://github.com/GoGoKo699/Hopf-Frame-Compilation/pull/268),
+`156f2cd4f5bc2e3c3d1f4afec17b31d0627764d3`, 28 October 2026.
+Its reviewed tree passed 2,035 tests locally and in both CI versions, plus the full gate and presentation checks.
 
 Read the [resource frontier](docs/OPEN_PROBLEM.md) for general theorems,
 then the [route decisions](research/ROUTE_DECISIONS.md) for the chosen mechanism.
 Use the [research catalogue](research/README.md) to find its proof dependencies.
-The source, decoder and copy screens are integrated; none improves R96.
 
 <a id="current-depth-frontier"></a>
 
@@ -123,27 +121,29 @@ 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 [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
-copy error at least $`\sin(\pi/8)`$, without an orbit-commutation assumption.
-
-[One-band copy feasibility](research/depth/ONE_BAND_COPY_FEASIBILITY.md)
-gives signed Walsh slices, their spectrum and the complete carrier fixed point.
-All output probes alone can miss wrong carrier action. The triple-axis joint walls
-pass the tensor screen but have zero complete seed space for p at least two,
-and full copy error at least $`1/\sqrt6`$ for every unitary seed. This family is stopped.
-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.
-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 [flat four-address pilot](research/depth/FLAT_FIELD_BANK_PILOT.md) retains its six-T decoder and complete 72-T indicator.
+The [projective recoupler](research/depth/PROJECTIVE_FIELD_RECOUPLER.md) now pays an exact eight-address full decoder:
+172 T gates, at most 83 bands, eight arbitrary wires and no extra helpers. Selected reflection phases cancel jointly.
+Projective-line enumeration grows exponentially in field degree; this is a finite calibration.
+
+[Pauli-domain rigidity](research/depth/FIELD_DECODER_PAULI_DOMAIN.md) requires absorbed-axis rank at least $`2m-2`$,
+including moving Clifford helpers. Independent finite cells fail even with joint outer Clifford walls.
+The [two-band screen](research/depth/TWO_BAND_FIELD_DECODER_OBSTRUCTION.md) excludes disjoint
+absorbed spans at fixed error; shared-span collisions and coupled three-band words remain open.
+The [scalar-field radical model](research/depth/SCALAR_FIELD_RADICAL_COPY_SCREEN.md)
+has a four-address identity for every address embedding; changing its address space does not repair it.
+These restrictions do not exclude the weaker complete-probe decoder contract.
+
+The [Clifford-seed bounds](research/depth/CLIFFORD_SEED_COPY_STRUCTURE.md) retain their
+commuting-orbit and one-band-copy scopes. The [seed fingerprint](research/depth/NATIVE_COPY_INTERACTION.md#seed-coefficient-rigidity)
+excludes product THT-dagger seeds below 3/32 for one-band field copies at p at least two.
+[Complete copy feasibility](research/depth/ONE_BAND_COPY_FEASIBILITY.md) still requires
+the carrier equation; triple-axis joint walls have zero full seed space.
+[Clifford-dual grading](research/depth/CLIFFORD_DUAL_COPY_GRADING.md) requires binary
+abelianization rank at least $`m-p-1`$; explicit relation words give robust floors.
+**Next question:** jointly resynthesize the projective corrections and recursive levels,
+using zero-sector and complete-probe freedom, within sublogarithmic depth and polynomial budgets.
+The joint seed/copy alternative remains open; no uniform decoder or improved R96 bound is supplied.
 
 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

@@ -282,6 +282,10 @@ without an extra native fixture.
 | 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 |
+| Projective field recoupler | [Proof](../../research/depth/PROJECTIVE_FIELD_RECOUPLER.md), [native fixtures](../../tests/test_projective_field_recoupler.py) | Exact field Fourier walls, all projective fibers, literal paired-reflection tables, borrowed-wire quartic phases, all 240 nonzero-U recoupler columns, every actual-inverse column and all eight full-query intertwinings retain scalar phases and arbitrary inputs. The 172-T, 83-band finite full decoder uses eight wires and no extra helpers; exponential line enumeration gives no uniform compiler |
+| Complete field decoder Pauli domain | [Proof](../../research/depth/FIELD_DECODER_PAULI_DOMAIN.md), [exact fixtures](../../tests/test_field_decoder_pauli_domain.py) | Literal field derivatives, second-difference masks, the moving-helper quotient, its essential boundary example, and the finite decoder domain check the robust all-size proof. The absorbed-axis and T-count floor is $`2m-2`$, including arbitrary Clifford target-helper motion; it is not a depth bound or a restriction on the weaker probe contract |
+| Disjoint-span two-band field decoder | [Proof](../../research/depth/TWO_BAND_FIELD_DECODER_OBSTRUCTION.md), [exact fixtures](../../tests/test_disjoint_two_band_decoder.py) | Exact collision-free Pauli supports, the sharp two-term fingerprint gap, partial-band normalization, moving residual-query deviations and actual shared-span collisions test the scope. The analytic fixed error floor permits arbitrary Clifford helper motion; overlapping bands remain open |
+| Scalar-field radical copy screen | [Proof](../../research/depth/SCALAR_FIELD_RADICAL_COPY_SCREEN.md), [exact fixtures](../../tests/test_scalar_field_radical_copy.py) | Complete small equivariant bilinear-product kernels, multiple banks, fixed helpers and changed address scalars support the square-zero image lemma. A one-address counterexample guards the threshold, and all-column regular affine words check the four-address relation. Direct-copy and output-dual floors concern the stated associative-algebra model only |
 | 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 |

research/README.md

@@ -199,6 +199,10 @@ These are complete interfaces and conditional transfers, not constructions of th
 - [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
+- [Projective field recoupler](depth/PROJECTIVE_FIELD_RECOUPLER.md) — projective homogeneity, an allowed emitted zero fiber, and joint selected-reflection phases give a complete eight-address native decoder with 172 T gates, at most 83 bands, eight arbitrary wires and no extra helpers; line enumeration is not a uniform polynomial-budget construction
+- [Complete field decoder Pauli domain](depth/FIELD_DECODER_PAULI_DOMAIN.md) — a robust $`2m-2`$ absorbed-axis and T-count requirement, including arbitrary Clifford motion of decoded helpers; independent finite cells fail even with joint outer Clifford walls
+- [Disjoint-span two-band field decoders](depth/TWO_BAND_FIELD_DECODER_OBSTRUCTION.md) — a uniform error floor for partial or full bands with zero span intersection, including arbitrary source identity helpers and Clifford target-helper motion; shared-span cancellation remains outside the theorem
+- [Scalar-field radical copy screen](depth/SCALAR_FIELD_RADICAL_COPY_SCREEN.md) — algebra-automorphism equivariance forces a square-zero shear image and a four-address relation for every address embedding, giving direct-copy error at least one half and output-dual copy error at least one quarter
 - [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)