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Pay augmented field recoupling and prove fixed-sector batching limits #270

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82 changes: 41 additions & 41 deletions WORKSPACE.md
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Expand Up @@ -11,12 +11,12 @@ Manuscript drafting and the changed-decoder supplement are separate tasks.

## Current checkpoint

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.
Verified baseline: [PR #269](https://github.com/GoGoKo699/Hopf-Frame-Compilation/pull/269),
`5867adce7db176dc9983db851df451123ab9d272`, 29 October 2026.
Its reviewed tree passed 2,054 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.
Read the [resource frontier](docs/OPEN_PROBLEM.md) for general theorems and the
[route decisions](research/ROUTE_DECISIONS.md) for the chosen mechanism.
Use the [research catalogue](research/README.md) to find its proof dependencies.

<a id="current-depth-frontier"></a>
Expand Down Expand Up @@ -81,16 +81,17 @@ they do not construct the source.
### Latest substantive outcomes and resumption question

The [tree-probe algebra](research/depth/INTERLEAVED_TREE_PROBE_SOURCE.md),
[Clifford-orbit rank bound](research/depth/CLIFFORD_ORBIT_QUERY_RANK.md) and
[non-Clifford seed-spread bound](research/depth/NONCLIFFORD_SEED_SPREAD.md)
retain their exact query/seed scopes. Target coordinates and support do not pay decoding.

The [field shear](research/depth/FIELD_SHEAR_INTERTWINER.md) has a paid
eight-band query and an [exact recursive decoder](research/depth/RECURSIVE_FIELD_SHEAR_DECODER.md).
Dense tower queries and multiplication-XOR batches are paid; its nonlinear
in-place maps remain unpaid. Bell-zero-fiber preservation fails even the
weaker lookup probe. For p at least two, a seed diagonal in any query-invariant
stabilizer basis fails, including arbitrary helpers and a non-Clifford decoder.
[Clifford-orbit rank](research/depth/CLIFFORD_ORBIT_QUERY_RANK.md) and
[non-Clifford seed spread](research/depth/NONCLIFFORD_SEED_SPREAD.md) retain their exact query/seed scopes.
Target coordinates and support do not pay decoding.

The [field shear](research/depth/FIELD_SHEAR_INTERTWINER.md) has a paid eight-band query.
The [augmented recoupler](research/depth/AUGMENTED_FIELD_RECOUPLER.md) replaces the
original decoder's unpaid division maps with two multiplication-XOR batches per level.
A moving Pauli helper query appears on both sides; the complete indicator echo returns every helper.
The paid decoder has at most $`16(p-1)`$ bands, and the indicator has at most $`64(p-1)+16`$.
Its $`O(mp^3)`$ T count, $`O(mp^2)`$ dirty width and $`O(m^2p^2)`$ gates fit the relaxed fusion budgets;
its linear depth does not. Invariant-stabilizer-diagonal seeds and Bell-zero-fiber preservation still fail.

The [three-band source theorem](research/depth/THREE_BAND_LOOKUP_SOURCE_OBSTRUCTION.md)
excludes every zero-clean native source of depth at most three for p at least
Expand Down Expand Up @@ -122,47 +123,48 @@ 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 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.
The [projective recoupler](research/depth/PROJECTIVE_FIELD_RECOUPLER.md) retains two exact eight-address allocations:
172 T gates and at most 83 bands without extra helpers, or 148 T gates and at most 39 bands with six returned dirty helpers.
Common reflection selection and private quartic batching pay the second word. Exponential line enumeration remains a finite-size limit.

[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 [weak-probe domain](research/depth/WEAK_PROBE_PAULI_DOMAIN.md) also excludes independent four-wire cells:
complete-probe error is at least $`1/12`$ for p at least three, and $`1/4`$ for p at least five.
It permits arbitrary joint Clifford walls and moving decoded helpers; extensive axis rank alone is not a depth bound.
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 [sector-degree barrier](research/depth/FIELD_SECTOR_DEGREE_BARRIER.md) requires helper-tableau degree at least
$`p-1`$ for Clifford-sector field intertwiners. The augmented construction attains it.
Fixed-frame helper-affine stage words need at least $`p-1`$ stages; this does not bound unrestricted native T-depth.

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.
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.
**Next question:** compress the complete source while leaving the fixed-frame helper-affine stage grammar,
allowing helper-basis motion or non-Clifford sector intermediates, with a complete native return ledger.
The weaker complete-probe and joint seed/copy alternatives remain open. R96 is unchanged.

The [four-band Clifford front end](research/depth/CLIFFORD_INVOLUTION_SOURCE.md),
The [four-band Clifford front end](research/depth/CLIFFORD_INVOLUTION_SOURCE.md) and
[weighted outer-phase route](research/depth/JOINT_ADDRESS_HELPER_SOURCE.md),
[reduced central gauge](research/depth/COHERENT_CENTRAL_TRANSPORT.md#reduced-carrier-gauge)
and [minimal-tree probe](research/depth/ORDERED_MATCHING_PROBE_SOURCE.md#minimal-tree-source)
remain component alternatives under their separate screens. In the separate
[precision route](research/depth/JOINT_PRECISION_ROTATION.md), joint recovery
beyond the separately paid two-path pilots remains open.
[reduced central gauge](research/depth/COHERENT_CENTRAL_TRANSPORT.md#reduced-carrier-gauge) and
[minimal-tree probe](research/depth/ORDERED_MATCHING_PROBE_SOURCE.md#minimal-tree-source) remain component alternatives under their separate screens.
In the [precision route](research/depth/JOINT_PRECISION_ROTATION.md), joint recovery beyond the separately paid two-path pilots remains open.

A finite-size identity needs a uniform all-size rule; serial grade computations retain the old cost. Any gain
needs a fresh complete-circuit ledger, including transfer, precision words,
actual inverses, leakage and work return.

## Stopped routes, reopening conditions and superseded instructions

The [route decisions](research/ROUTE_DECISIONS.md) own each failed mechanism
and its reopening condition. Detailed proofs and unique counterexamples stay
in their linked homes. Old instructions to add another mixer, improve a scalar
filter, or pay the rank-one phase do not override those complete-action screens.
The [route decisions](research/ROUTE_DECISIONS.md) own each failed mechanism and its reopening condition.
Detailed proofs and unique counterexamples stay in their linked homes. Old instructions to add another mixer,
improve a scalar filter, or pay the rank-one phase do not override those complete-action screens.
A scoped failure does not retire the unrestricted destination.

<a id="research-decision-and-stopping-rules"></a>
Expand All @@ -171,10 +173,9 @@ A scoped failure does not retire the unrestricted destination.

## Working contract and reading map

Prefer analytic identities and a few exact fixtures. Preserve literal phases,
every logical column, coherent address and arbitrary dirty/reference inputs,
actual native adjoints, and all clean leakage and work return. No QRAM, free
loading or precision states, unpriced clean history, resets or postselection.
Prefer analytic identities and a few exact fixtures. Preserve literal phases, every logical column,
coherent address and arbitrary dirty/reference inputs, actual native adjoints, and all clean leakage and work return.
No QRAM, free loading or precision states, unpriced clean history, resets or postselection.
A suffix is clean only on its active sector; inactive action needs its own proof.
Separate general theorems, conditional interfaces, useful components and
representation limits. Internal audits and finite checks do not replace
Expand All @@ -190,8 +191,7 @@ analytic proof or external technical review.

<a id="refurnishing-policy"></a>

Each claim has one proof home. Preserve unique attempts, stable identifiers, inbound anchors, tests and receipt hashes.
Change mechanisms when complete synthesis restores the old cost.
Each claim has one proof home. Preserve unique attempts, stable identifiers, inbound anchors, tests and receipt hashes; change mechanisms when complete synthesis restores the old cost.

## Restore and verify

Expand Down
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