Controlling terminal and closure grammar

SG6 is fully adjudicated, but the advertised full positive-stability claim is not positively closed. The unrestricted pure-curvature stationary branch has the exact shape-doublet eigenvalue -1/3; it is a saddle and is CLOSED-NEGATIVE. That is a complete negative result, not missing arithmetic.

An explicit construction adds V_stab=(kappa/2)(x-y)^2. At kappa=1/3 its declared zero-mode Hessian is positive definite with minimum eigenvalue 1/3, so that separate branch is PASS-CONDITIONAL. Target radii and modulus masses are MATCHING-PARAMETER, and the stabilizer’s independent physical origin is OPEN. The current SG1 constrained construction excludes the negative mode; it does not dynamically stabilize the unrestricted branch.

Authority order and cumulative SG5 lineage

The user-supplied building blocks are the governing source of truth. Authority is ordered as the complete source-of-truth archive, cumulative SG5 blocks, V4.1 tagged-slice execution protocol, SG2-SG8 challenge specification, frozen SG1 shape-mode record, SG5-to-SG6 dependency ledger, and generated witnesses.

The cumulative SG5 archive is frozen at 6123e8a8ceee83ca7725edfa4f1b0e40baa873c1332c34101672e4153b365483. Its manifest is replayed file by file and recursively preserves SG4, SG3, SG2, and the updated SG1 V3 lineage. SG5’s physical OPEN terminal and nineteen readiness debts are preserved rather than reset by the SG6 calculation.

The SG1 shape manifest explicitly records an unrestricted control in which the physical shape-doublet mass squared is -1/3, while its current construction is SECOND_CLASS_CONSTRAINED / CONSTITUTIVE under Actor GA-CA-1. SG6 uses both facts without contradiction: the unrestricted mode is unstable; the construction removes it from the admissible tangent space. Constraint exclusion and positive dynamical mass are different operations.

SG6 challenge, declared fields, and evidence boundary

The challenge asks whether the proposed internal shape is dynamically stable. The executable reconstruction tests four declared real zero-mode coordinates (x,y,u,v). The first two span the shape doublet, while the last two are positive spectator directions in the exact challenge Hessian. The point is stationary because all four gradient entries vanish exactly.

The evidence boundary is deliberately narrow. Exact rational diagonalization settles the declared finite Hessians. It does not create a full compactified action, normalize every KK mode, establish nonlinear coercivity, enumerate all competing extrema, calculate tunneling, or prove global stability. Those questions receive independent scope fields and cannot inherit the zero-mode terminal.

Eight blinded manifests implement five intended decoys, the pure-curvature incumbent, the explicit-stabilization incumbent, and one honest alternative saddle. A committed answer key, deterministic session order, manifest hashes, witness hashes, and first-hard-failure rules make the reconstruction replayable. It remains an internal reconstruction; independent randomized review is NOT-EVALUATED.

Exact Hessian method and negative-closure rule

For a stationary point, positive stability within a declared real finite sector requires the symmetric Hessian to be positive definite on the physical tangent space. SG6 computes its eigenvalues exactly as rational numbers and also records axis probes, leading principal minors, determinant, and named eigenvectors. Floating-point tolerances are unnecessary.

For a block [[a,b],[b,a]], the symmetric eigenvector (1,1)/sqrt(2) has eigenvalue a+b, and the antisymmetric vector (1,-1)/sqrt(2) has eigenvalue a-b. Positive diagonals alone test only coordinate axes. When b>a, the antisymmetric mixed direction is negative even though each axis is positive.

The negative-closure rule is strict: if a direction required by the claim has an exact negative eigenvalue, the positive-stability claim for that branch is CLOSED-NEGATIVE. A new action may define a repaired branch, but it cannot mutate the failed witness. An exact constraint may remove the mode, but it must be reported as a constrained construction rather than dynamical stabilization.

Branch certificate: pure-curvature unrestricted

Stationary: TRUE
Computed terminal: CLOSED-NEGATIVE
Classification: SADDLE
Exact eigenvalues: ['-1/3', '1', '2', '3']
Leading principal minors: ['1/3', '-1/3', '-2/3', '-2']
Determinant: -2
Shape antisymmetric eigenvector: (1,-1,0,0)/sqrt(2)
Shape antisymmetric eigenvalue: -1/3

The matrix is symmetric and evaluated at an exact stationary point. Its terminal applies only to the declared zero-mode Hessian. Full 13D/KK/global artifacts attached is FALSE. Stabilization attached is FALSE. These fields keep the algebraic classification separate from action ownership, physical scope, and nature-selection. A positive result in one constructed branch cannot erase a negative result in another.

Pure-curvature Hessian entry H(x,x)

The exact entry in row x and column x is 1/3. Its transposed partner is 1/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(x,y)

The exact entry in row x and column y is 2/3. Its transposed partner is 2/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(x,u)

The exact entry in row x and column u is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(x,v)

The exact entry in row x and column v is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(y,x)

The exact entry in row y and column x is 2/3. Its transposed partner is 2/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(y,y)

The exact entry in row y and column y is 1/3. Its transposed partner is 1/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(y,u)

The exact entry in row y and column u is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(y,v)

The exact entry in row y and column v is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(u,x)

The exact entry in row u and column x is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(u,y)

The exact entry in row u and column y is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(u,u)

The exact entry in row u and column u is 2. Its transposed partner is 2, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(u,v)

The exact entry in row u and column v is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(v,x)

The exact entry in row v and column x is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(v,y)

The exact entry in row v and column y is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(v,u)

The exact entry in row v and column u is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature Hessian entry H(v,v)

The exact entry in row v and column v is 3. Its transposed partner is 3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Pure-curvature declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Pure-curvature eigenpair: shape antisymmetric

Exact eigenvalue: -1/3
Exact eigenvector: (1,-1,0,0)/sqrt(2)
Sign: NEGATIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Pure-curvature eigenpair: shape symmetric

Exact eigenvalue: 1
Exact eigenvector: (1,1,0,0)/sqrt(2)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Pure-curvature eigenpair: spectator u

Exact eigenvalue: 2
Exact eigenvector: (0,0,1,0)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Pure-curvature eigenpair: spectator v

Exact eigenvalue: 3
Exact eigenvector: (0,0,0,1)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Pure-curvature diagnostic: Coordinate-axis probes

The diagonal probes are ['1/3', '1/3', '2', '3']. They are necessary local samples but not sufficient when the x-y mixed entries are nonzero.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Pure-curvature diagnostic: Leading principal minors

The exact Sylvester sequence is ['1/3', '-1/3', '-2/3', '-2']. A nonpositive leading minor prevents a positive-definite certificate.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Pure-curvature diagnostic: Determinant

The determinant is -2. Its sign is consistent with the exact eigenvalue product and provides an independent arithmetic cross-check.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Pure-curvature diagnostic: Stationarity

The declared gradient is (0,0,0,0). Hessian classification is therefore evaluated at a stationary point rather than along an arbitrary field-space location.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Pure-curvature diagnostic: Scope

The certificate evaluates the declared four-dimensional zero-mode sector. Complete 13D, KK, nonlinear, tunneling, and global artifacts are not attached.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Branch certificate: explicit positive stabilization

Stationary: TRUE
Computed terminal: PASS-CONDITIONAL
Classification: POSITIVE-DEFINITE
Exact eigenvalues: ['1/3', '1', '2', '3']
Leading principal minors: ['2/3', '1/3', '2/3', '2']
Determinant: 2
Shape antisymmetric eigenvector: (1,-1,0,0)/sqrt(2)
Shape antisymmetric eigenvalue: 1/3

The matrix is symmetric and evaluated at an exact stationary point. Its terminal applies only to the declared zero-mode Hessian. Full 13D/KK/global artifacts attached is FALSE. Stabilization attached is TRUE. These fields keep the algebraic classification separate from action ownership, physical scope, and nature-selection. A positive result in one constructed branch cannot erase a negative result in another.

Stabilized Hessian entry H(x,x)

The exact entry in row x and column x is 2/3. Its transposed partner is 2/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(x,y)

The exact entry in row x and column y is 1/3. Its transposed partner is 1/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(x,u)

The exact entry in row x and column u is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(x,v)

The exact entry in row x and column v is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(y,x)

The exact entry in row y and column x is 1/3. Its transposed partner is 1/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(y,y)

The exact entry in row y and column y is 2/3. Its transposed partner is 2/3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(y,u)

The exact entry in row y and column u is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(y,v)

The exact entry in row y and column v is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(u,x)

The exact entry in row u and column x is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(u,y)

The exact entry in row u and column y is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(u,u)

The exact entry in row u and column u is 2. Its transposed partner is 2, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(u,v)

The exact entry in row u and column v is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(v,x)

The exact entry in row v and column x is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(v,y)

The exact entry in row v and column y is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(v,u)

The exact entry in row v and column u is 0. Its transposed partner is 0, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized Hessian entry H(v,v)

The exact entry in row v and column v is 3. Its transposed partner is 3, so the symmetry check is TRUE. Zero entries are explicit evidence, not unwritten assumptions.

This entry is evaluated together with all fifteen other entries. The gate never treats a diagonal coefficient as an eigenvalue when mixed terms are present. In particular, the x-y block is diagonalized in symmetric and antisymmetric combinations. That operation is why the pure-curvature matrix exposes a negative direction even though both shape-axis probes are 1/3.

The entry belongs to the Stabilized declared zero-mode branch. It has no automatic authority over omitted KK modes, nonlinear excursions, competing stationary points, or tunneling paths. A candidate claim must reproduce this entry exactly; a correct final label cannot compensate for a wrong matrix.

Stabilized eigenpair: shape antisymmetric

Exact eigenvalue: 1/3
Exact eigenvector: (1,-1,0,0)/sqrt(2)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Stabilized eigenpair: shape symmetric

Exact eigenvalue: 1
Exact eigenvector: (1,1,0,0)/sqrt(2)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Stabilized eigenpair: spectator u

Exact eigenvalue: 2
Exact eigenvector: (0,0,1,0)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Stabilized eigenpair: spectator v

Exact eigenvalue: 3
Exact eigenvector: (0,0,0,1)
Sign: POSITIVE

The engine substitutes this direction into the symmetric Hessian and records the result without decimal approximation. The sign contributes directly to the branch classification. One negative required direction is sufficient to close a positive-definiteness claim negatively; several positive directions cannot vote it away.

This eigenpair is also claim-aware. A saddle reported honestly as a saddle is valid negative evidence. A saddle reported as a minimum fails the full-Hessian rule. A direction removed by an exact constraint is absent only from that constrained branch; its unrestricted eigenpair remains an immutable control. The declared zero-mode scope does not settle tower or global questions.

Stabilized diagnostic: Coordinate-axis probes

The diagonal probes are ['2/3', '2/3', '2', '3']. They are necessary local samples but not sufficient when the x-y mixed entries are nonzero.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Stabilized diagnostic: Leading principal minors

The exact Sylvester sequence is ['2/3', '1/3', '2/3', '2']. A nonpositive leading minor prevents a positive-definite certificate.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Stabilized diagnostic: Determinant

The determinant is 2. Its sign is consistent with the exact eigenvalue product and provides an independent arithmetic cross-check.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Stabilized diagnostic: Stationarity

The declared gradient is (0,0,0,0). Hessian classification is therefore evaluated at a stationary point rather than along an arbitrary field-space location.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Stabilized diagnostic: Scope

The certificate evaluates the declared four-dimensional zero-mode sector. Complete 13D, KK, nonlinear, tunneling, and global artifacts are not attached.

The diagnostic is stored in the generated branch certificate and repeated in the blinded-session witness for the relevant incumbent. It is checked by the independent verifier after two byte-identical engine runs. Exact fractions remove ambiguity about tolerances and make every sign auditable.

No single diagnostic is allowed to overrun the others. Axis probes do not replace diagonalization; a positive determinant alone does not establish positive definiteness; stationarity does not imply stability; and a positive finite Hessian does not prove the full physical vacuum theorem. The terminal is obtained from the complete typed tuple.

Stabilization control: Potential Actor

The declared construction adds V_stab=(kappa/2)(x-y)^2 with primitive parent DECLARED-STABILIZATION-ACTOR. This is an explicit action change, not a reinterpretation of pure curvature.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

Stabilization control: Strict inequality

The exact requirement is kappa>1/6. The incumbent chooses kappa=1/3, and the computed inequality result is TRUE.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

Stabilization control: Eigenvalue lift

The shape antisymmetric eigenvalue changes from -1/3 to 1/3. The other declared eigenvalues remain positive.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

Stabilization control: Target radii

The target-radii class is MATCHING-PARAMETER. The chosen potential can match a target but does not independently predict it.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

Stabilization control: Modulus masses

The modulus-masses class is MATCHING-PARAMETER. Positive fitted masses are not nature-selected predictions without a parameter-fixing mechanism.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

Stabilization control: No-borrowing boundary

A physical promotion must derive the stabilizer from one primitive action, prove its symmetry permissions and sign, and show that no contribution is duplicated across effective descriptions.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

Stabilization control: Tower boundary

The positive zero-mode Hessian does not control KK operators, nonlinear runaways, competing extrema, tunneling exponents, or global field-space topology.

This control exists because a successful repair can still be overclaimed. The lawful terminal is PASS-CONDITIONAL for the declared stabilized zero-mode branch. Primitive ownership, predictive parameter fixing, and broader stability scopes remain independently testable interfaces. The original pure-curvature saddle stays CLOSED-NEGATIVE and is never overwritten.

Future evidence must preserve the frozen incumbent and attach a new branch manifest with exact action provenance. The verifier then recomputes the Hessian, strict inequalities, matching labels, scope flags, and first hard failure from that immutable candidate.

SG1-to-SG6 join: Frozen negative control

The SG1 record states: shape-doublet physical mass^2 = -1/3 in the fixed-volume tree-curvature control. The literal exact value -1/3 is present and is replayed as the SG6 unrestricted control.

The join is frozen by SHA-256 and checked by the independent verifier. It prevents a later dossier from forgetting the negative control while also preventing the negative control from being misapplied to a branch where the direction is exactly inadmissible.

Physical closure still requires constraint propagation, admissible phase space, measure, and coupling to the complete operator domain. Those artifacts are not supplied, so the constrained branch remains a construction anchor rather than a proof of positive dynamical stabilization.

SG1-to-SG6 join: Constrained variational status

The current SG1 factor status is SECOND_CLASS_CONSTRAINED / CONSTITUTIVE. The unstable shape-doublet is not treated as an unrestricted physical variation in that branch.

The join is frozen by SHA-256 and checked by the independent verifier. It prevents a later dossier from forgetting the negative control while also preventing the negative control from being misapplied to a branch where the direction is exactly inadmissible.

Physical closure still requires constraint propagation, admissible phase space, measure, and coupling to the complete operator domain. Those artifacts are not supplied, so the constrained branch remains a construction anchor rather than a proof of positive dynamical stabilization.

SG1-to-SG6 join: Constraint Actor

The governing constraint Actor is GA-CA-1. Its role is exact admissibility control, not a positive quadratic potential.

The join is frozen by SHA-256 and checked by the independent verifier. It prevents a later dossier from forgetting the negative control while also preventing the negative control from being misapplied to a branch where the direction is exactly inadmissible.

Physical closure still requires constraint propagation, admissible phase space, measure, and coupling to the complete operator domain. Those artifacts are not supplied, so the constrained branch remains a construction anchor rather than a proof of positive dynamical stabilization.

SG1-to-SG6 join: Reconciliation theorem

Unrestricted negative mass and constrained exclusion are compatible statements about different tangent spaces. Neither statement may be silently promoted into the other.

The join is frozen by SHA-256 and checked by the independent verifier. It prevents a later dossier from forgetting the negative control while also preventing the negative control from being misapplied to a branch where the direction is exactly inadmissible.

Physical closure still requires constraint propagation, admissible phase space, measure, and coupling to the complete operator domain. Those artifacts are not supplied, so the constrained branch remains a construction anchor rather than a proof of positive dynamical stabilization.

Gauntlet rule SG6-GNT-01: Full mixed-direction Hessian

Recompute the complete symmetric Hessian and its exact spectrum. Positive coordinate-axis probes do not establish a minimum when mixed terms are present.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Gauntlet rule SG6-GNT-02: Stationarity before stability

Require the exact gradient to vanish at the declared point and compare the computed stationarity record with the candidate claim before classifying its Hessian.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Gauntlet rule SG6-GNT-03: Stabilization inequality and parameters

For an attached positive potential, recompute its strict positivity inequality, require a primitive Actor, and label target radii and modulus masses MATCHING-PARAMETER.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Gauntlet rule SG6-GNT-04: Scope firewall

A zero-mode Hessian cannot establish full 13D, KK, nonlinear, tunneling, or global stability without the corresponding operators, domains, and witnesses.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Gauntlet rule SG6-GNT-05: Immutable candidate evidence

A repair is a new candidate. Reject any repair artifact that embeds its failure record or mutates the immutable original instead of preserving provenance.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Gauntlet rule SG6-GNT-06: SG1 negative-control join

Replay the frozen SG1 statement that the unrestricted shape-doublet mass squared is exactly -1/3 and distinguish constrained exclusion from dynamical stabilization.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Gauntlet rule SG6-GNT-07: Branch-terminal consistency

Require every claimed terminal to match its computed branch, spectrum, and scope; the unrestricted negative branch and stabilized construction must not be merged.

Rules run in fixed order and the witness records the first hard failure. The answer key is canonically serialized and committed before roles are opened. Every blinded manifest and generated witness is hash-escrowed, and the opened comparison requires all eight verdict/failure pairs to match.

The rule is claim-aware. Honest negative evidence can pass, whereas a positive overclaim fails. The internal deterministic sessions test the reconstructed specification and do not substitute for an independent reviewer-randomized execution, which remains NOT-EVALUATED.

Blind session 1: manifest and sealed expectation

Candidate ID: session-6f758164c80ee3a8e24c
Opened label / role: repair-containing-failure-record / DECOY
Manifest branch: explicit-positive-stabilization
Manifest scope: declared-zero-mode-stabilized-branch
Manifest SHA-256: 74720195ce7bdb7ae3931254d9193f7b2eaf92fda065e8a9beb005aa8c4d24dd
Expected verdict / first failure: FAIL / SG6-GNT-05

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 1: computed witness

Witness SHA-256: 939d22e0a2d767df29d9e1070c6444d70139366e9c8f377979b3031d93ace42d
Actual verdict / first failure: FAIL / SG6-GNT-05
Computed classification: POSITIVE-DEFINITE
Computed eigenvalues: ['1/3', '1', '2', '3']
Computed branch terminal: PASS-CONDITIONAL
Stabilization attached: TRUE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 2: manifest and sealed expectation

Candidate ID: session-8d0b6eda46ba4d40280e
Opened label / role: honest-alternative-saddle / INNOCENT-CLOSED-NEGATIVE
Manifest branch: honest-alternative-saddle
Manifest scope: declared-zero-mode-symmetric-branch
Manifest SHA-256: 4b98914cfd506ad22666997043cac6318be2a0a6e5af2b2fa8f7ce9d50417a92
Expected verdict / first failure: PASS / None

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 2: computed witness

Witness SHA-256: 0b5dc02fab1b51ee8252a3cab2c3faac89f15f68324c886ea1e7678d0aff2893
Actual verdict / first failure: PASS / None
Computed classification: SADDLE
Computed eigenvalues: ['-1/2', '3/2', '2', '3']
Computed branch terminal: CLOSED-NEGATIVE
Stabilization attached: FALSE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 3: manifest and sealed expectation

Candidate ID: session-94ada10700c0c8114a67
Opened label / role: incumbent-explicit-stabilization / INCUMBENT-STABILIZED-BRANCH
Manifest branch: explicit-positive-stabilization
Manifest scope: declared-zero-mode-stabilized-branch
Manifest SHA-256: 7980bf38c6740b50a145d3fba6b1f3652e612d377f7c367bfe50140563fafbe0
Expected verdict / first failure: PASS / None

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 3: computed witness

Witness SHA-256: 94a45f60c4b8b7ba4d2b9479f36a4b4fcfe4565b698ec08491879a2b7bac54d9
Actual verdict / first failure: PASS / None
Computed classification: POSITIVE-DEFINITE
Computed eigenvalues: ['1/3', '1', '2', '3']
Computed branch terminal: PASS-CONDITIONAL
Stabilization attached: TRUE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 4: manifest and sealed expectation

Candidate ID: session-9655814e6b3590733d24
Opened label / role: zero-mode-to-full-scope-overclaim / DECOY
Manifest branch: explicit-positive-stabilization
Manifest scope: declared-zero-mode-stabilized-branch
Manifest SHA-256: 8041913d219f1c39fca303efb7ff8e4b6726ae0e9337e7d596bfccb2f23fed6c
Expected verdict / first failure: FAIL / SG6-GNT-04

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 4: computed witness

Witness SHA-256: 914a9dbae8ad6ea20dd26ae05539a162b5145cc18bc12cfc1611b99c982e5d3f
Actual verdict / first failure: FAIL / SG6-GNT-04
Computed classification: POSITIVE-DEFINITE
Computed eigenvalues: ['1/3', '1', '2', '3']
Computed branch terminal: PASS-CONDITIONAL
Stabilization attached: TRUE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 5: manifest and sealed expectation

Candidate ID: session-ad7dced5fdb17e509e20
Opened label / role: axis-probes-miss-mixed-saddle / DECOY
Manifest branch: pure-curvature-unrestricted
Manifest scope: declared-zero-mode-symmetric-branch
Manifest SHA-256: 547875e621d7823681bad09d600cc6f3e238f9dd87ba634a8ebcce2601c6d893
Expected verdict / first failure: FAIL / SG6-GNT-01

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 5: computed witness

Witness SHA-256: de23062f4f5bb8f2742d7aefeaead06b27acf9c3543f1098de510d35a9534b9a
Actual verdict / first failure: FAIL / SG6-GNT-01
Computed classification: SADDLE
Computed eigenvalues: ['-1/3', '1', '2', '3']
Computed branch terminal: CLOSED-NEGATIVE
Stabilization attached: FALSE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 6: manifest and sealed expectation

Candidate ID: session-c7030af02ea3b6f0a5cc
Opened label / role: failed-stabilization-inequality / DECOY
Manifest branch: explicit-positive-stabilization
Manifest scope: declared-zero-mode-stabilized-branch
Manifest SHA-256: 97ae2a42c76edd43d61c3f9d919f88f9dd2af76e6ecbf1555fa26e75e362be8a
Expected verdict / first failure: FAIL / SG6-GNT-03

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 6: computed witness

Witness SHA-256: 9f00795735eb45611badfb6f8f65da26b2dd54e254ead943d65eafbad7108209
Actual verdict / first failure: FAIL / SG6-GNT-03
Computed classification: SADDLE
Computed eigenvalues: ['-1/6', '1', '2', '3']
Computed branch terminal: CLOSED-NEGATIVE
Stabilization attached: TRUE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 7: manifest and sealed expectation

Candidate ID: session-d75f90e5a27bd52572a2
Opened label / role: claimed-minimum-is-saddle / DECOY
Manifest branch: honest-alternative-saddle
Manifest scope: declared-zero-mode-symmetric-branch
Manifest SHA-256: f3a20e3269a669a629ce4d69a5c1f8e76a5c5e71965a053ba41b767b9b4b142b
Expected verdict / first failure: FAIL / SG6-GNT-01

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 7: computed witness

Witness SHA-256: 918df35d568d48fa1677a494f2ccd84786c5f9422c4d42d114d998155253bae3
Actual verdict / first failure: FAIL / SG6-GNT-01
Computed classification: SADDLE
Computed eigenvalues: ['-1/2', '3/2', '2', '3']
Computed branch terminal: CLOSED-NEGATIVE
Stabilization attached: FALSE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

Blind session 8: manifest and sealed expectation

Candidate ID: session-ef1ff6baca2c8081a040
Opened label / role: incumbent-pure-curvature / INCUMBENT-NEGATIVE-BRANCH
Manifest branch: pure-curvature-unrestricted
Manifest scope: declared-zero-mode-symmetric-branch
Manifest SHA-256: 235dea1a28b8838d3cb5cf03614efdee62b49b7457337f0862b0c020f637d642
Expected verdict / first failure: PASS / None

The session presented role UNDISCLOSED to the solver. Its exact gradient, Hessian, stabilization declaration, scope claims, SG1 negative-control value, and artifact-immutability record were evaluated without access to the opened label. This separation prevents a familiar narrative from replacing the typed evidence.

The answer-key commitment was fixed before opening. The manifest hash binds the candidate seen by the engine, so a later repair must enter as a new candidate rather than altering this record.

Blind session 8: computed witness

Witness SHA-256: 47e1c9461230eb7637575f8ca46354d8da18e6167ffb93a6c9024db9b3fd4a65
Actual verdict / first failure: PASS / None
Computed classification: SADDLE
Computed eigenvalues: ['-1/3', '1', '2', '3']
Computed branch terminal: CLOSED-NEGATIVE
Stabilization attached: FALSE

The actual verdict and first failure match the sealed expectation. A PASS can mean an honest CLOSED-NEGATIVE report or a conditionally positive stabilized branch; it does not mean every candidate describes the target stable vacuum. A FAIL identifies the earliest rule violated by the candidate claim.

The witness is immutable and hash-escrowed. It records exact matrix arithmetic, the SG1 cross-check, scope flags, and stabilization inequality where relevant.

SG5-to-SG6 cumulative dependency reducer

The reducer receives nineteen inherited OPEN rows from the cumulative SG5 package and six SG6-local OPEN rows. It returns positive physical closure OPEN with 25 blockers. Separately, the controlling full positive-stability claim is CLOSED-NEGATIVE-AS-WRITTEN because the required unrestricted branch has an exact negative direction.

The reducer therefore keeps negative adjudication and future positive repair readiness in separate fields. The full claim is not left vague; it is closed negatively as written. A new physically derived stabilizer could open a new positive branch, but every inherited and local readiness row would still need evidence before physical PASS.

Physical readiness row 1: SG5-SG6-D01

Status: OPEN
Requirement: Physical gauge-group realization

The cumulative gauge carriers and global quotient remain physically OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 2: SG5-SG6-D02

Status: OPEN
Requirement: Faithful global gauge group

The conditional Z6 kernel has not been promoted to the physical compactification quotient.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 3: SG5-SG6-D03

Status: OPEN
Requirement: Chiral representation inventory

The one-copy matter kernels and their complete physical domain remain OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 4: SG5-SG6-D04

Status: OPEN
Requirement: Three-family module and mirror completeness

Family multiplicity passes conditionally while action ownership remains OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 5: SG5-SG6-D05

Status: OPEN
Requirement: Fermion and anomaly completion

The regulated determinant, heavy tower, inflow, and complete chiral domain remain OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 6: SG5-SG6-D06

Status: OPEN
Requirement: SG1 geometric realization lineage

The selected internal shape is a constrained construction; unrestricted physical realization and full operator closure remain OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 7: SG5-SG6-D07

Status: OPEN
Requirement: Gauge action, fixing, and ghosts

The BRST-complete higher-dimensional gauge reduction has not reached PASS.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 8: SG5-SG6-D08

Status: OPEN
Requirement: Family Actor action ownership

The multiplicity Actor lacks a closed primitive action parent.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 9: SG5-SG6-D09

Status: OPEN
Requirement: No-excess light-state census

A complete spectrum excluding extra light vectors, scalars, and fermions is not supplied.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 10: SG5-SG6-D10

Status: OPEN
Requirement: Interaction graph derived from the action

The charge-defining interaction graph remains a declared construction input.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 11: SG5-SG6-D11

Status: OPEN
Requirement: U(1) normalization and coupling transport

Primitive charge normalization does not close the physical kinetic normalization.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 12: SG5-SG6-D12

Status: OPEN
Requirement: Regulated anomaly descent

Zero-mode cancellation does not close the higher-dimensional anomaly theorem.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 13: SG5-SG6-D13

Status: OPEN
Requirement: Physical Z6 quotient

The character kernel is exact but the global quotient realization is OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 14: SG5-SG6-D14

Status: OPEN
Requirement: Independent SG4 reviewer execution

SG4 reviewer-randomized execution remains NOT-EVALUATED.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 15: SG5-SG6-D15

Status: OPEN
Requirement: Higgs doublet action parent

The minimal Higgs doublet is a gate input rather than a completed geometric zero-mode theorem.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 16: SG5-SG6-D16

Status: OPEN
Requirement: Neutral VEV selection and scalar stability

The scalar potential, competing extrema, and full scalar Hessian remain OPEN and feed directly into SG6.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 17: SG5-SG6-D17

Status: OPEN
Requirement: Electroweak kinetic normalization

The zero-mode mass matrix is exact given its inputs, while coupling transport remains OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 18: SG5-SG6-D18

Status: OPEN
Requirement: Loops and rho0 confrontation

Radiative corrections, matching, and experimental rho0 confrontation are NOT-CLAIMED.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 19: SG5-SG6-D19

Status: OPEN
Requirement: Independent SG5 reviewer execution

SG5 reviewer-randomized execution remains NOT-EVALUATED.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 20: SG6-L01

Status: OPEN
Requirement: Full moduli Hessian derived from one action

The declared four-mode Hessians are exact challenge certificates; the complete physical Hessian, gauge quotient, domains, and action derivation are not supplied.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 21: SG6-L02

Status: OPEN
Requirement: Stabilization potential primitive ownership

The positive potential is an explicit construction Actor with tunable kappa; its independent origin and no-borrowing proof are OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 22: SG6-L03

Status: OPEN
Requirement: Matching-parameter and prediction separation

Target radii and modulus masses are matching parameters, not derived predictions; a physical parameter-fixing mechanism remains OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 23: SG6-L04

Status: OPEN
Requirement: KK, nonlinear, tunneling, and global stability

A positive declared zero-mode Hessian does not prove the full 13D/KK/global problem, which remains CLOSED-NEGATIVE AS WRITTEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 24: SG6-L05

Status: OPEN
Requirement: Constrained-branch propagation and phase-space proof

The current SG1 branch removes the shape-doublet by exact constraints rather than stabilizing it; complete constraint propagation and physical-measure closure remain OPEN.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Physical readiness row 25: SG6-L06

Status: OPEN
Requirement: Independent SG6 reviewer-randomized gauntlet

The internal eight-session reconstruction passes; no independent randomized manifests and opened key were supplied.

This row contributes one blocker to positive physical closure. Exact closure of another row does not discharge it. Promotion requires an immutable evidence artifact, action/domain provenance where applicable, explicit claim scope, and cumulative replay against the frozen SG5 package.

The row does not weaken the exact negative theorem. It describes what a future positive physical construction would have to supply. Until then, the lawful combined report is branch-separated: negative as written, conditional for the explicit repair, and OPEN for positive physical completion.

Building-block evolution: BB-HSC-1

Adds exact full-Hessian diagonalization, negative closure, stabilization overlay, and constraint-versus-mass separation.

This cumulative successor keeps the entire SG5 archive under UPSTREAM/ and adds new blocks and amendments as an overlay. No earlier terminal is silently deleted. The package manifest hashes every file, and the final ZIP can be tested independently.

The central improvement is status precision. An exact negative result is a valid closure; an explicit repair is a different conditional branch; physical completion remains blocked by named evidence interfaces. This structure is intended for SG7 and later gates so they inherit both the successful arithmetic and the unresolved obligations.

Building-block evolution: BB-VDR-1

Carries all inherited physical debts and adds six SG6 action, parameter, tower, constraint, and review readiness rows.

This cumulative successor keeps the entire SG5 archive under UPSTREAM/ and adds new blocks and amendments as an overlay. No earlier terminal is silently deleted. The package manifest hashes every file, and the final ZIP can be tested independently.

The central improvement is status precision. An exact negative result is a valid closure; an explicit repair is a different conditional branch; physical completion remains blocked by named evidence interfaces. This structure is intended for SG7 and later gates so they inherit both the successful arithmetic and the unresolved obligations.

Building-block evolution: BB-AHG-1 SG6 amendment

Requires primitive ownership, sign, normalization, symmetry permission, and no-borrowing evidence for every stabilizer.

This cumulative successor keeps the entire SG5 archive under UPSTREAM/ and adds new blocks and amendments as an overlay. No earlier terminal is silently deleted. The package manifest hashes every file, and the final ZIP can be tested independently.

The central improvement is status precision. An exact negative result is a valid closure; an explicit repair is a different conditional branch; physical completion remains blocked by named evidence interfaces. This structure is intended for SG7 and later gates so they inherit both the successful arithmetic and the unresolved obligations.

Building-block evolution: BB-MAP-1 SG6 amendment

Requires branch-and-scope status tuples and forbids promotion from zero mode to full 13D/KK/global stability.

This cumulative successor keeps the entire SG5 archive under UPSTREAM/ and adds new blocks and amendments as an overlay. No earlier terminal is silently deleted. The package manifest hashes every file, and the final ZIP can be tested independently.

The central improvement is status precision. An exact negative result is a valid closure; an explicit repair is a different conditional branch; physical completion remains blocked by named evidence interfaces. This structure is intended for SG7 and later gates so they inherit both the successful arithmetic and the unresolved obligations.

Building-block evolution: SG6 findings

Makes the exact negative theorem, conditional repair, and prime directive explicit for later gates.

This cumulative successor keeps the entire SG5 archive under UPSTREAM/ and adds new blocks and amendments as an overlay. No earlier terminal is silently deleted. The package manifest hashes every file, and the final ZIP can be tested independently.

The central improvement is status precision. An exact negative result is a valid closure; an explicit repair is a different conditional branch; physical completion remains blocked by named evidence interfaces. This structure is intended for SG7 and later gates so they inherit both the successful arithmetic and the unresolved obligations.

Building-block evolution: SG6 scope reconciliation

Joins unrestricted, constrained, and stabilized branches without treating them as contradictory or interchangeable.

This cumulative successor keeps the entire SG5 archive under UPSTREAM/ and adds new blocks and amendments as an overlay. No earlier terminal is silently deleted. The package manifest hashes every file, and the final ZIP can be tested independently.

The central improvement is status precision. An exact negative result is a valid closure; an explicit repair is a different conditional branch; physical completion remains blocked by named evidence interfaces. This structure is intended for SG7 and later gates so they inherit both the successful arithmetic and the unresolved obligations.

Execution validation control 1

Control: Engine and verifier compile
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 2

Control: Two byte-identical generated runs
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 3

Control: All generated JSON parses
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 4

Control: Answer-key commitment and escrow
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 5

Control: Eight blinded sessions and complete manifests
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 6

Control: Manifest and witness hash escrow
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 7

Control: All eight verdicts match sealed key
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 8

Control: Five challenge decoys caught at intended first failures
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 9

Control: Honest alternative saddle accepted as closed-negative
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 10

Control: Exact pure-curvature Hessian, negative eigenpair, and determinant
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 11

Control: Exact positive stabilized zero-mode Hessian and matching-parameter labels
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 12

Control: Branch-separated terminals preserve the exact negative control
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 13

Control: SG1 exact negative control and constrained-branch cross-check
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 14

Control: Full-scope and immutable-artifact firewall
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 15

Control: SG6 branch, physical, and reviewer status grammar
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 16

Control: Frozen input hashes
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 17

Control: Frozen SG5 package integrity and extracted manifest replay
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Execution validation control 18

Control: SG5-to-SG6 cumulative dependency reducer
Result: PASS

This control is executed by the standalone verifier rather than inferred from the prose dossier. The verifier compiles the scripts, regenerates the exact artifacts, parses JSON, checks cryptographic commitments and hashes, and compares branch certificates and cumulative dependencies against frozen inputs.

All eighteen controls pass. That pass certifies reproducibility and internal consistency of the reconstructed SG6 adjudication. It does not promote the reviewer-randomized terminal, derive a physical stabilizer, or supply omitted KK/global artifacts. Those boundaries remain explicit in every output status.

Closure answer to the SG6 challenge

The challenge is solved by a branch-split theorem. Pure unrestricted curvature does not stabilize the proposed shape: the exact mixed shape-doublet eigenvalue is -1/3, so that branch is a saddle and is CLOSED-NEGATIVE. Coordinate-axis positivity was the decoy.

An explicit positive potential can construct a stable declared zero-mode branch. For V_stab=(kappa/2)(x-y)^2, positivity requires kappa>1/6; the incumbent kappa=1/3 gives eigenvalues (1/3,1,2,3) and therefore PASS-CONDITIONAL. Its radii and masses are matching parameters, and its primitive physical derivation is not supplied.

The current SG1 constrained construction removes the unstable mode rather than giving it positive mass. Full 13D/KK/nonlinear/tunneling/global positive stability is CLOSED-NEGATIVE-AS-WRITTEN. A future positive physical branch remains OPEN behind twenty-five named dependencies.

Final controlling statement

SG6 has complete evidentiary adjudication but not positive physical closure. The immutable unrestricted witness is CLOSED-NEGATIVE; the explicit stabilized zero-mode construction is PASS-CONDITIONAL; the SG1 constrained branch is CONSTRUCTION-ANCHOR-MODE-EXCLUDED; the full claim is CLOSED-NEGATIVE-AS-WRITTEN; and positive physical completion is OPEN with nineteen inherited plus six local blockers.

The reconstructed eight-session gauntlet passes its sealed key and all eighteen independent execution controls. The independent reviewer-randomized gauntlet remains NOT-EVALUATED, and nature-selection is NOT-CLAIMED.

No statement in this dossier should be read as predicting target radii, modulus masses, a unique physical stabilizer, or the complete spectrum. The lawful accomplishment is sharper: it identifies exactly why the pure branch fails, exactly how the declared repair changes the Hessian, and exactly what evidence would be needed to pursue a new positive physical closure.