Rigidity V4.1 — Maximum-Rigor All-Gate Package

--- title: "BB-RIG-4.0 — Maximum-Rigor Constraint-Derived All-Gate Rigidity Authority" buildingblockid: "BB-RIG-4.0" version: "4.0" date: "2026-08-02" status: "DEVELOPMENT AUTHORITY — MAXIMUM-RIGOR ALL-GATE CANDIDATE — NOT FROZEN" supersedesfordevelopment: "BB-RIG-1.3 while preserving it exactly" legacysourcesha256: "196ca8170197a3e88b441092600e1bf095ebfddacf3bb79073a58d007b0e86a8" protocolfreeze: false canonicalratification: false ---

BB-RIG-4.0 — Maximum-Rigor Constraint-Derived All-Gate Rigidity Authority

0. Controlling rule

This object determines whether the complete physical candidate is isolated, obstructed, dynamically stable, metastable, constitutively restricted, or unstable at a declared scope. It does not select a candidate, manufacture missing Dynamics, or convert a construction into a derivation.

The exact BB-RIG-1.3 authority is preserved in Appendix A and 06SOURCES/BBRIG13EXACTARCHIVAL_SOURCE.md. Every legacy requirement remains binding unless this authority explicitly strengthens it.

1. Core firewalls

\[ \text{unique point in an ansatz}\neq\text{physical rigidity}, \] \[ H^1{\rm phys}=0\neq Q{\rm phys}>0, \] \[ Q_{\rm phys}>0\text{ locally}\neq\text{nonlinear/global stability}, \] \[ \text{tree-level stability}\neq\text{quantum/radiative stability}, \] \[ \text{perturbative stability}\neq\text{nonperturbative longevity}, \] \[ \text{one scale, one time, one domain, or one truncation}\neq\text{complete rigidity}. \]

2. Owned object

For frozen parent object \(\mathfrak P\), define the typed physical deformation problem

\[ \mathfrak R(\mathfrak P)= (\mathcal E{\rm def},d0,d_1,\kappa,K,H,\mathcal D,\Sigma,\mathcal B,\mathcal T,\mathcal U), \]

where the entries are respectively the complete deformation grammar, redundancy map, linearized constraint/equation map, nonlinear obstruction, kinetic structure, full mixed Hessian/evolution generator, operator domain, physical spectrum, branch/bifurcation data, time/history data, and uncertainty/refinement data.

3. Constraint-derived method

1. Freeze authority hashes, candidate grammar, deformation grammar, gate scope, Observer, cutoff, histories, and accepted terminals.
2. Generate every deformation sector and every lawful stabilization/rigidity route without viewing the desired answer.
3. Build the gauge/redundancy map and exact constraint Jacobian.
4. Compute the physical tangent space, singular strata, and obstruction/integrability map.
5. Build the kinetic-normalized full mixed operator on the actual Boundary domain and function space.
6. Classify discrete/essential spectrum, ghosts, gradients, tachyons, complex modes, and coercivity.
7. Continue over scale, time/history, parameters, boundary supports, regulator/refinement, and branch changes.
8. Test nonlinear basins, metastability, topology change, composites, regions/horizons, and open-system effects when typed applicable.
9. Prove grammar and test-family exhaustion or retain explicit OPEN residuals.
10. Issue route-sensitive certificates and the weakest honest terminal.

4. Atomic Rigidity constraints

RIG-C01 — Complete deformation inventory across metric, bundle, boundary, Actor, Rulebook, and Observer sectors

Question. Complete deformation inventory across metric, bundle, boundary, Actor, Rulebook, and Observer sectors?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. deformationbasispacket, grammarexhaustionpacket.

Minimal witness. Every in-scope deformation sector is enumerated before reduction.

Fail trigger. An omitted sector can carry a modulus or instability.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C02 — Redundancy removal before physical counting

Question. Redundancy removal before physical counting?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. gaugequotientpacket, variationalownershippacket.

Minimal witness. Gauge, diffeomorphism, BRST-exact, presentation, boundary-gauge, and record-null directions are quotiented without removing physical modes.

Fail trigger. A gauge direction is counted physical or a physical direction is gauge-fixed away.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C03 — Infinitesimal physical deformation/cohomology or constraint-Jacobian calculation

Question. Infinitesimal physical deformation/cohomology or constraint-Jacobian calculation?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. constraintjacobianpacket, rankcertificatepacket, physicaltangentpacket.

Minimal witness. The physical tangent space is computed reproducibly as a kernel modulo gauge.

Fail trigger. Rigidity is asserted without a matrix, complex, rank, or nullspace witness.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C04 — Nonlinear obstruction and integrability

Question. Nonlinear obstruction and integrability?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. obstructionintegrabilitypacket, solutiongermpacket.

Minimal witness. Every surviving tangent direction is integrated or assigned a reproducible nonlinear obstruction.

Fail trigger. A nonzero tangent is called obstructed without a Kuranishi/integrability calculation.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C05 — Isolation kept separate from physical Hessian stability

Question. Isolation kept separate from physical Hessian stability?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. solutiongermpacket, mixedhessianpacket, stabilityroutepacket.

Minimal witness. Solution-germ isolation and off-shell physical stability are independently certified.

Fail trigger. An isolated saddle is called stable.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C06 — Complete mixed physical Hessian

Question. Complete mixed physical Hessian?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. kineticmetricpacket, mixedhessianpacket, spectrumcoercivitypacket.

Minimal witness. All diagonal and off-diagonal physical blocks are computed or eliminated by theorem, with kinetic normalization.

Fail trigger. Positive diagonal blocks hide a negative mixed mode.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C07 — Protected scalar whitelist

Question. Protected scalar whitelist?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. protectedscalarpacket.

Minimal witness. Every intended light scalar has pre-frozen identity, protection, positivity, mixing, and stability ownership.

Fail trigger. An unwanted modulus is relabeled protected after discovery.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C08 — Named radius, Wilson-line, boundary-position, and localized deformations

Question. Named radius, Wilson-line, boundary-position, and localized deformations?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. namedmodulipacket, boundarysupportrigidity_packet.

Minimal witness. Volume, relative radii, interval length, Wilson lines, boundary positions, and localized modes each receive explicit status.

Fail trigger. A fixed background value is treated as absence of its fluctuation.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C09 — Full-tower, boundary, and nonperturbative scope

Question. Full-tower, boundary, and nonperturbative scope?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. towertailpacket, metastabilitydecaypacket.

Minimal witness. The rigidity/stability claim declares zero-mode, tower, boundary, tunnelling, thermal, and cutoff scope with tails or OPEN residuals.

Fail trigger. A finite homogeneous calculation is promoted to full stability.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C10 — Constraint provenance and anti-circularity

Question. Constraint provenance and anti-circularity?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. anticircularitypacket.

Minimal witness. Only independently justified pre-frozen constraints may support derived constraint-rigidity; late constitutive rows are graded as construction.

Fail trigger. A post-failure restriction is called an intrinsic derivation.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C11 — Function-space, domain, Fredholm, essential-spectrum, and coercivity control

Question. Function-space, domain, Fredholm, essential-spectrum, and coercivity control?

Formal requirement. Rigidity and stability must be formulated on the actual operator domain and function space. The certificate must classify discrete and essential spectrum, kernel/cokernel, Fredholm index where applicable, boundary/domain dependence, and a coercive or gap estimate on the non-whitelisted physical subspace.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. functionspacedomainpacket, spectrumcoercivity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Finite matrices or a few eigenvalues can be positive while an essential-spectrum instability, non-Fredholm direction, or domain-dependent zero mode survives.

Destructive control. Two operators have identical first N eigenvalues; one has essential spectrum crossing zero. The audit must reject equivalence.

RIG-C12 — Singular strata, rank jumps, bifurcations, marginality, and spectral flow

Question. Singular strata, rank jumps, bifurcations, marginality, and spectral flow?

Formal requirement. The analysis must locate singular points of the constraint/deformation map, rank-changing strata, eigenvalue crossings, bifurcation branches, marginal directions, exceptional points, and spectral flow over the claimed parameter region.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. bifurcationspectralflowpacket, solutiongerm_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A full-rank Jacobian and positive Hessian at one point can sit arbitrarily close to a bifurcation or rank-changing branch.

Destructive control. Two candidates agree at the reference point; only one develops a zero eigenvalue under an arbitrarily small lawful parameter change.

RIG-C13 — Nonlinear finite-amplitude, Lyapunov/orbital stability, and basin control

Question. Nonlinear finite-amplitude, Lyapunov/orbital stability, and basin control?

Formal requirement. When a dynamical stability claim exceeds linear perturbation theory, publish a Lyapunov, energy, invariant-set, bootstrap, or equivalent nonlinear estimate; identify the attraction/orbital-stability basin, conserved quantities, runaway channels, and finite-amplitude thresholds.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. nonlinearbasinpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A positive Hessian can coexist with finite-amplitude runaway, nonlinear resonance, collapse, or blow-up.

Destructive control. Construct a potential with a positive local Hessian but an arbitrarily nearby escape channel; linear stability must not be promoted.

RIG-C14 — Time-dependent, parametric, Floquet, adiabatic, and history stability

Question. Time-dependent, parametric, Floquet, adiabatic, and history stability?

Formal requirement. For time-dependent backgrounds, moving boundaries, phase histories, periodic driving, cosmological evolution, or changing scales, evolve the perturbation system and test Floquet/parametric growth, adiabaticity, nonnormal transient amplification, memory, and history-dependent excitation.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. timedependentstability_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A static vacuum can be stable while a lawful transition or periodic background excites an otherwise harmless mode.

Destructive control. Two candidates share the same instantaneous Hessian; one has a parametric resonance during the required history.

RIG-C15 — Quantum, radiative, regulator, scheme, and refinement stability

Question. Quantum, radiative, regulator, scheme, and refinement stability?

Formal requirement. The effective-action Hessian, constraint rank, protected directions, and obstruction data must remain controlled under the claimed loop order, RG transport, regulator family, truncation enlargement, matching thresholds, and Granularity refinement.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. quantumradiativestability_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Tree-level rigidity can disappear through Coleman-Weinberg directions, anomalous mixing, regulator artifacts, or truncation-dependent fixed points.

Destructive control. Use two lawful regulators that agree at tree level but give opposite loop-corrected mass signs; the claim remains OPEN without stability.

RIG-C16 — Metastability, tunnelling, bounce, topology change, and lifetime

Question. Metastability, tunnelling, bounce, topology change, and lifetime?

Formal requirement. Separate absolute stability from metastability. Enumerate lower-action branches and topology-changing channels; compute or bound bounce/instanton actions, prefactors, decay rates, recurrence assumptions, and lifetime relative to the gate timescale.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. metastabilitydecaypacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. No perturbative negative mode does not exclude nonperturbative decay to another branch.

Destructive control. Two vacua have identical perturbative spectra; one has a finite-action bounce to a lower state.

RIG-C17 — Topological, index, symmetry-protected, and defect-sector rigidity

Question. Topological, index, symmetry-protected, and defect-sector rigidity?

Formal requirement. Identify deformations forbidden or protected by topology, index, superselection, exact symmetry, charge, or defect data; prove invariance under the admitted deformation grammar and classify wall-crossing or topology-changing exceptions.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. topologicalprotectionpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A local Hessian can miss exact topological protection or an allowed wall-crossing that changes the sector.

Destructive control. Two locally identical configurations carry different index/defect classes; they must not share a rigidity certificate.

RIG-C18 — Regional, boundary, horizon, interface, edge, and state-dependent-support rigidity

Question. Regional, boundary, horizon, interface, edge, and state-dependent-support rigidity?

Formal requirement. Import the complete Boundary support/incidence grammar and test deformations of fixed sets, corners, horizons, entangling cuts, phase interfaces, asymptotic regions, edge modes, and state-dependent supports, including gluing and flux balance.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. boundarysupportrigidity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A bulk-stable solution can have an unstable edge, moving horizon, corner, or interface mode.

Destructive control. Keep the bulk Hessian fixed while adding a negative boundary-localized mode; full rigidity must fail.

RIG-C19 — Composite, condensate, soliton, bound-state, and effective-Actor rigidity

Question. Composite, condensate, soliton, bound-state, and effective-Actor rigidity?

Formal requirement. For every promoted effective/composite Actor, derive the collective-coordinate and internal-shape deformation complex, binding/decay channels, mixing with elementary sectors, scale of validity, and de-promotion/reopen conditions.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. compositeeffectiverigidity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Elementary fields can be stable while a composite, condensate, soliton, or bound-state sector is unstable or double-counted.

Destructive control. Two theories share the same elementary spectrum; one contains an unstable composite pole or soliton size modulus.

RIG-C20 — Conditioning, uncertainty, identifiability, and robustness margin

Question. Conditioning, uncertainty, identifiability, and robustness margin?

Formal requirement. Publish singular-value/eigenvalue intervals, condition numbers, covariance transport, measured-anchor dependence, numerical tolerance tests, structural identifiability, and a quantitative margin to every rank change or instability.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. robustnessuncertaintypacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. An exactly positive but ill-conditioned result can change sign within lawful uncertainty or numerical error.

Destructive control. A Hessian eigenvalue is positive at its central value but its certified interval crosses zero.

RIG-C21 — Candidate-neutral deformation, stabilization-route, and rival grammar exhaustion

Question. Candidate-neutral deformation, stabilization-route, and rival grammar exhaustion?

Formal requirement. Generate all admissible deformation sectors and all lawful rigidity/stabilization mechanisms before viewing outcomes; canonicalize duplicates, test omitted rivals, classify every candidate, and prove finite enumeration, bounded generation, or a class theorem.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. grammarexhaustionpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. One tested deformation basis or one successful stabilizer is not evidence that no omitted route or instability exists.

Destructive control. Hide a lawful Wilson-line or edge deformation from the initial roster; the omitted-rival control must detect it.

RIG-C22 — Cross-scale continuation, decoupling, RG spectral flow, and matching stability

Question. Cross-scale continuation, decoupling, RG spectral flow, and matching stability?

Formal requirement. Track physical modes, Hessian eigenvalues, constraints, and protected directions across Scale packets, thresholds, compactification levels, EFT matching, and RG flow; prove decoupling or record nondecoupling and eigenvalue crossings.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. crossscalecontinuation_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A mode stable at one scale can become tachyonic, ghostlike, or strongly mixed after threshold transport.

Destructive control. Two candidates match at one scale; only one eigenvalue crosses zero above a threshold.

RIG-C23 — Constrained Hamiltonian, symplectic, Krein, ghost/gradient/tachyon stability

Question. Constrained Hamiltonian, symplectic, Krein, ghost/gradient/tachyon stability?

Formal requirement. On constrained systems, construct the reduced symplectic/Dirac structure and distinguish negative norm, negative energy, tachyonic mass, gradient instability, complex frequency, Jordan block, and Krein collision; verify constraint-rank preservation.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. symplectickreinpacket, kineticmetricpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A positive Euclidean Hessian or real frequency list can hide a ghost, gradient instability, or Hamiltonian-Hopf/Krein collision.

Destructive control. Use two systems with identical squared frequencies but opposite kinetic signature; only one is physical.

RIG-C24 — Open-system, stochastic, noise, decoherence, and ensemble stability

Question. Open-system, stochastic, noise, decoherence, and ensemble stability?

Formal requirement. When the gate uses reduced states, stochastic sources, thermal ensembles, measurements, or environmental channels, test complete-positive evolution, stationary-state stability, noise-induced transitions, large deviations, decoherence-free sectors, and ensemble dependence.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. opensystemstability_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A closed-system stable mode can become unstable under a lawful environment or noise channel.

Destructive control. Two systems share the same Hamiltonian; one Lindblad/noise channel destabilizes the stationary state.

5. Universal execution pipeline

The agent must execute the selected gate rows in dependency order:

\[ \text{manifest}\to\text{deformation grammar}\to\text{domain/gauge quotient}\to T_{\rm phys}\to\kappa\to(K,H)\to\Sigma\to \text{history/scale/global controls}\to\text{terminal}. \]

A failed early row does not permit later rows to be assumed. Later calculations may be performed diagnostically but remain NOT-EVALUATED for closure unless their prerequisites pass.

6. Rigidity routes and terminal grammar

• DERIVED-INTRINSIC-RIGIDITY: complete physical deformation space is empty using pre-frozen independent constraints.
• OBSTRUCTION-RIGIDITY: nonzero infinitesimal directions are all nonlinearly obstructed; Hessian separately graded.
• DYNAMIC-STABILITY: physical modes exist but the complete typed physical operator is stable at scope.
• METASTABLE-LONG-LIVED: decay channels exist but lifetime exceeds the frozen requirement with controlled prefactor and history.
• CONSTITUTIVE-RIGIDITY: a declared constraint/Actor/projector removes the direction; construction, not derivation.
• CLOSED-NEGATIVE / INSTABILITY: a physical negative/ghost/gradient/runaway/decay failure is certified.
• OPEN: required deformation, domain, spectrum, history, uncertainty, exhaustion, or external evidence is incomplete.

No lower rung may be narrated as a higher rung.

7. Required evidence packets

Every gate bundle contains every packet below. Active packets are executed; inactive packets are NOT-APPLICABLE only with a gate-specific type proof.

parentmanifestpacket

Required fields: candidateid, branchid, authorityhashes, scope, cutoff, observer, freezehash.

deformationbasispacket

Required fields: basisid, sectors, basisvectors, units, normalizations, kineticinnerproducts, prereductionstatus, towerscope, ownerhashes, omittedsectorproofs.

functionspacedomain_packet

Required fields: operators, functionspaces, domains, boundaryconditions, selfadjointness, ellipticityorhyperbolicity, fredholmdata, essentialspectrum, domainvariation_tests.

gaugequotientpacket

Required fields: gaugebasis, gaugematrix, ghostoperator, residualsymmetries, boundarygaugedata, quotientbasis, normon_quotient.

constraintjacobianpacket

Required fields: constraintrows, provenancehashes, evaluationpoint, jacobian, arithmetictype, rank, pivots, nullspace, sensitivity.

rankcertificatepacket

Required fields: matrixshape, exactorintervalrank, determinantsorminors, singularvalues, conditionnumber, tolerance_tests, classification.

physicaltangentpacket

Required fields: kernelbasis, gaugeimagebasis, quotientbasis, dimension, sectorlabels, unresolveddirections.

obstructionintegrabilitypacket

Required fields: directions, nonlinearequations, ordersortheorem, obstructionmap, h2targets, integrablebranches, convergence_scope.

solutiongermpacket

Required fields: localbranches, singularstrata, branchintersections, localuniqueness, globaldisconnectedbranches, scope.

kineticmetricpacket

Required fields: fields, kineticmatrixoroperator, signature, zeromodes, normalization, ghosttests, constraintreduction.

mixedhessianpacket

Required fields: sectororder, diagonalblocks, offdiagonalblocks, vanishingtheorems, schurcomplements, eigenvalue_bounds, normalization.

spectrumcoercivitypacket

Required fields: discretespectrum, essentialspectrum, kernel, negativemodes, complexmodes, gaporcoercivitybound, tailbound, domain.

protectedscalarpacket

Required fields: scalarids, roles, protectingmechanisms, prefreezehashes, masses, mixing, radiativeowner, reopentriggers.

namedmodulipacket

Required fields: volume, relativeradii, radion, interval, wilsonlines, bundlemoduli, boundarypositions, localizedmodes, observermap_deformations, statuses.

boundarysupportrigidity_packet

Required fields: supports, incidence, supportdeformations, edgemodes, horizonmodes, interfacemodes, cornermodes, gluing, fluxbalance.

towertailpacket

Required fields: zeromodescope, kklevels, boundarytower, continuumoressentialspectrum, tailtheorem, cutoff, nonperturbative_channels.

anticircularitypacket

Required fields: constraintsused, freezetimes, rationales, candidateblindnesstests, laterows, constructionrows, restart_status.

variationalownershippacket

Required fields: variables, equationdispositions, constraints, auxiliaries, externalinputs, unowned_equations.

stabilityroutepacket

Required fields: route, intrinsicstatus, obstructionstatus, dynamicstatus, constitutivestatus, negativestatus, routejustification.

vacuumscaleinterface_packet

Required fields: deformationstatus, vacuumoffsetstatus, scalestatus, phasehistorystatus, external_dependencies.

bifurcationspectralflow_packet

Required fields: parameterregion, rankstrata, eigenvaluetracks, crossings, bifurcations, exceptionalpoints, margins.

nonlinearbasinpacket

Required fields: lyapunovorenergyfunction, basin, finiteamplitudethresholds, runawaychannels, resonances, globalexistencescope.

timedependentstability_packet

Required fields: backgroundhistory, evolutionoperator, floquetdata, adiabaticity, transientgrowth, memory, excitation_bounds.

quantumradiativestability_packet

Required fields: effectiveactionorders, loopcorrectedhessian, rgflow, regulators, schemes, truncations, wardidentities, stability_intervals.

metastabilitydecaypacket

Required fields: lowerbranches, bounceorinstanton, action, prefactor, decayrate, lifetime, topologychange, timescalecomparison.

topologicalprotectionpacket

Required fields: indices, charges, superselectionsectors, defects, symmetries, wallcrossing, invariance_proofs.

compositeeffectiverigidity_packet

Required fields: effectiveactorids, collectivecoordinates, internalmodes, binding, decaychannels, mixing, validityscale, promotion_status.

robustnessuncertaintypacket

Required fields: parametercovariance, eigenvalueintervals, singularvalueintervals, conditionnumbers, identifiability, numericaltolerances, robustness_margins.

grammarexhaustionpacket

Required fields: deformationgrammar, stabilizationgrammar, canonicalaliases, enumerationmethod, omittedrivaltests, unclassifiedcandidates, exhaustionproof.

crossscalecontinuation_packet

Required fields: scales, thresholds, modetracking, constraintranktracking, eigenvalueflow, decoupling_tests, matching, crossings.

symplectickreinpacket

Required fields: reducedsymplecticform, diracbrackets, kineticsignature, energysignature, gradienttests, kreinsignatures, jordanblocks, constraint_rank.

opensystemstability_packet

Required fields: systemenvironmentsplit, generatororchannel, completepositivity, stationarystates, spectralgap, noiseinducedtransitions, largedeviationrates, ensembledependence.

negativecontrolspacket

Required fields: controlsrun, expectedresults, actual_results, failures, artifacts.

rigiditycertificatev1_3

Required fields: candidateid, branchid, deformationcomplex, functionspaces, sectorsincluded, gaugeredundancyquotient, h1physical, obstructionmap, localmoduligerm, independentconstraintregister, constraintderivedtangentspace, exactconstraintclassification, kineticmetric, mixedphysicalhessian, schurcomplements, protectedscalarwhitelist, zeromodes, negativemodes, towerandboundaryscope, tunnellingscope, scaleinterface, vacuumoffsetstatus, evidencehashes, route_terminal, verdict.

integratedrigiditycertificatev40

Required fields: legacycertificate, constraintstates, packethashes, preservationresidual, rigidityresidual, scope, claimceiling, terminal, reopen_triggers.

rigidityresidualpacket

Required fields: residualvector, nonzerocomponents, owners, required_actions.

terminal_packet

Required fields: gateid, physicalendpoint, projectendpoint, scope, claimceiling, evidencehashes, openrows, reopen_triggers.

8. Completion residual

Define

\[ \mathcal R{\rm RIG}=(N{\rm missing\ sectors},N{\rm bad\ quotient},N{\rm tangent},N{\rm obstruction},N{\rm domain},N{\rm negative},N{\rm mixed},N{\rm essential},N{\rm bifurcation},N{\rm nonlinear},N{\rm history},N{\rm quantum},N{\rm decay},N{\rm boundary},N{\rm composite},N{\rm uncertain},N{\rm unexhausted}). \]

A positive maximum-scope Rigidity terminal requires every applicable component to vanish or be discharged by an exact theorem. NOT-APPLICABLE is not zero without a type proof.

9. All-gate profiles

The machine-readable controlling map is 02REGISTRIES/RIGIDITYGATECROSSWALKV4_0.json.

Black-hole-singularity

• Role: Test geometry, horizon, interior, boundary, nonlinear, topology-changing, and quantum stability.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Born-rule

• Role: Test probability/instrument structures against perturbations of state, algebra, environment, and regional factorization.
• Rows: RIG-C05, RIG-C09, RIG-C11, RIG-C13, RIG-C15, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: boundarysupportrigiditypacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-01

• Role: Test the a6 operator result against domain, mixed Hessian, regulator, tower, and scale continuation.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-02

• Role: Separate finite-volume/perturbative stability from a true nonperturbative Yang-Mills mass-gap result.
• Rows: RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C16, RIG-C17, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, topologicalprotectionpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-05-stability

• Role: Test vacuum-offset mechanism across moduli, histories, loops, boundaries, and nonperturbative decay.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-05-value

• Role: Keep value prediction separate from background/ensemble/metastability and uncertainty robustness.
• Rows: RIG-C05, RIG-C09, RIG-C11, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C20, RIG-C21, RIG-C22, RIG-C24
• Active packets: crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-08

• Role: Test inflationary background and perturbations for slow-roll, multifield, parametric, quantum, and stochastic stability.
• Rows: RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-10/BG-10

• Role: Test baryogenesis through phase-transition, transport, CP-source, washout, and stochastic-history stability.
• Rows: RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-11

• Role: Test dark-sector candidate, portal, relic history, composites, decays, and environmental stability.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-13

• Role: Test black-hole saddle, replica, horizon/edge, microstate, information, and nonperturbative stability.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, topologicalprotectionpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Lambda-catastrophe

• Role: Test vacuum response across scales, phases, histories, regions, and radiative/nonperturbative effects.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-1

• Role: Determine whether the selected Shape is intrinsically fixed, merely constitutively frozen, or dynamically stabilized.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C05, RIG-C06, RIG-C08, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-10

• Role: Enforce scope and anti-promotion across truncation, tower, refinement, and parameter continuation.
• Rows: RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-2

• Role: Check that gauge-sector ownership and vector kernels are stable against omitted/mixed/tower deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-3

• Role: Test chiral-family and mirror-sector robustness under domain, boundary, tower, and radiative deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C07, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C14, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, namedmodulipacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, protectedscalarpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket, variationalownershippacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-4

• Role: Test charge/anomaly/global-kernel results against bundle, boundary, topological, and spectral-flow deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-5

• Role: Test EWSB vacuum, Higgs/Wilson sector, mass mixing, hierarchy, and history stability.
• Rows: RIG-C01, RIG-C02, RIG-C05, RIG-C06, RIG-C07, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-6

• Role: Primary full moduli and vacuum-stability gate; execute the complete Rigidity object.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, obstructionintegrabilitypacket, opensystemstabilitypacket, parentmanifestpacket, physicaltangentpacket, protectedscalarpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket, variationalownershippacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-7

• Role: Test threshold-unification/proton-safety sensitivity to spectra, matching, and deformation routes.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-8

• Role: Test flavor fits and hierarchy against ruler, mixing, radiative, composite, and history instabilities.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C14, RIG-C15, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-9

• Role: Test dangerous-operator suppression and proton lifetime under deformations and nonperturbative channels.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C13, RIG-C15, RIG-C16, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-10

• Role: Primary compactification-consistency stress test across all geometric, bundle, boundary, tower, and quantum sectors.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, protectedscalarpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-14

• Role: Test above-cutoff causality/unitarity under spectral flow, nonlinear, boundary/horizon, and open-system effects.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-3

• Role: Test positivity reconstruction against domain, mixed-sector, essential-spectrum, regional, and open-system instabilities.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C13, RIG-C15, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-4

• Role: Test anomaly/global consistency under deformations, wall crossing, determinant-line, and boundary support changes.
• Rows: RIG-C03, RIG-C04, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-5A/5B

• Role: Test graviton-sector physical stability, norm, mixing, nonlinear basin, and quantum corrections.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-5C

• Role: Prevent truncation-local UV claims from being promoted without global/refinement/metastability robustness.
• Rows: RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C16, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-7

• Role: Test chiral-domain/anomaly-descent stability under boundary, topological, tower, and global deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, namedmodulipacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-9

• Role: Test heat-kernel/UV outputs under operator-domain, regulator, refinement, and spectral-flow changes.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

theta-bar-QCD

• Role: Test topological-angle mechanism under bundle, vacuum-branch, RG, threshold, and tunnelling deformations.
• Rows: RIG-C03, RIG-C04, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C16, RIG-C17, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, constraintjacobianpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

10. Validation and preservation theorem

The authority is preservation-complete relative to BB-RIG-1.3 only when:

\[ \mathcal R{\rm pres}=(N{\rm unmapped},N{\rm weakened},N{\rm lost\ controls},N{\rm lost\ fields},N{\rm hash\ mismatches})=0. \]

The exact legacy source is normative and archived. The validator must check its SHA-256, every constraint ID, every gate profile, every packet schema, every thought experiment, and all internal references.

Appendix A — Exact BB-RIG-1.3 authority

The following source is preserved verbatim and remains binding.

--- BEGIN EXACT LEGACY SOURCE ---

--- title: "BB-RIG-1.3 — Intrinsic Rigidity, Constraint Jacobian, Obstruction, and Physical Stability" buildingblockid: "BB-RIG-1.3" version: "1.3" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — EVIDENCE-CRITERIA EXTENDED — NOT FROZEN" scope: "Complete owned specification under package v3.3.2" protocolfreeze: false canonicalratification: false packageauthorityversion: "3.3.2" ---

BB-RIG-1.3 — Intrinsic Rigidity, Constraint Jacobian, Obstruction, and Physical Stability

0. Purpose

This block answers the high-information question:

Can the independently justified physical constraints force a candidate whose complete physical deformation space is empty or fully obstructed, or must remaining deformation fields be stabilized by Dynamics or a declared construction Actor?

It exists because the following statements are different:

\[ \text{one metric was selected}, \]

\[ \text{there are no neighboring exact solutions}, \]

\[ \text{there are no physical scalar deformations}, \]

\[ \text{the physical quadratic form is positive}. \]

A candidate may satisfy one and fail another.

1. Core distinctions

1.1 Mathematical extension

A candidate may be embedded in a broad mathematical configuration space \(\widetilde{\mathcal C}\).

Directions in \(T\widetilde{\mathcal C}\) are not automatically physical.

1.2 Physical configuration space

After gauge quotient, exact structural rules, boundary conditions, and domain restrictions, the physical configuration space is \(\mathcal C_{\rm phys}\).

1.3 Solution space

The physical vacuum solution set is

\[ \mathcal M{\rm vac}=\{z\in\mathcal C{\rm phys}:\mathcal F_I[z]=0\}/\text{gauge}. \]

1.4 Physical fluctuation space

Off-shell physical perturbations lie in the tangent space of \(\mathcal C{\rm phys}\), not merely the tangent space of \(\mathcal M{\rm vac}\).

A direction can fail to integrate into another exact solution and still be a physical unstable fluctuation.

Therefore:

\[ T{[z]}\mathcal M{\rm vac}=0 \not\Rightarrow Q_{\rm phys}>0. \]

2. Complete deformation complex

For each complete parent object \(\mathfrak P\), construct a deformation complex or an explicitly equivalent operator sequence:

\[ 0\longrightarrow \mathfrak{aut}(\mathfrak P) \xrightarrow{d0} T{\mathfrak P}\mathcal C \xrightarrow{d1} \mathcal E{\mathfrak P} \xrightarrow{d2} \mathcal O{\mathfrak P} \longrightarrow0. \]

Here:

• \(\mathfrak{aut}(\mathfrak P)\) contains infinitesimal gauge, diffeomorphism, and exact presentation redundancies;
• \(T_{\mathfrak P}\mathcal C\) contains complete candidate deformations;
• \(\mathcal E_{\mathfrak P}\) contains linearized equations, constraints, boundary, and domain conditions;
• \(\mathcal O_{\mathfrak P}\) contains nonlinear obstruction data.

The physical infinitesimal deformation space is

\[ H^1{\rm phys}(\mathfrak P) = \frac{\ker d1}{\operatorname{im}d_0}. \]

When a cohomological complex is unavailable, an equivalent gauge-fixed kernel/cokernel construction is permitted, but it must cover the same physical content.

3. Complete sector inventory

The complex must include, where present:

• internal metric shape deformations;
• total volume and breathing modes;
• relative radii;
• interval and radion modes;
• mixed external–internal metric components;
• bundle deformations;
• connection and Wilson-line deformations;
• flux and top-form branch deformations;
• complex, almost-complex, torsion, and structure deformations;
• matter-background deformations;
• boundary and fixed-set positions;
• junction and edge-mode deformations;
• finite Actor continuous parameters;
• Rulebook/domain deformations;
• observer-map deformations that alter physical records.

A metric-only deformation analysis is incomplete unless a theorem proves all omitted sectors irrelevant or absent.

4. Physical redundancy removal

Before counting deformations, quotient:

• diffeomorphisms;
• internal gauge transformations;
• BRST-exact directions;
• exact bundle automorphisms;
• boundary gauge redundancies;
• presentation equivalences;
• observer relabelings with no record consequence.

Gauge fixing must not remove physical modes.

The certificate must report:

• gauge condition;
• ghost/Faddeev–Popov operator;
• zero modes of the gauge-fixing operator;
• residual transformations;
• boundary behavior;
• norm on the quotient.

5. Intrinsic infinitesimal rigidity

A candidate is infinitesimally physically rigid when

\[ H^1_{\rm phys}(\mathfrak P)=0. \]

This means there is no nonredundant infinitesimal deformation satisfying the linearized physical conditions.

The claim must state scope:

• invariant deformations only;
• all smooth deformations in a stated function space;
• all modes below a cutoff;
• full tower;
• bulk only;
• bulk plus boundaries and global sectors.

An invariant-sector result cannot be promoted to all deformations.

6. Nonlinear obstruction and local rigidity

If \(H^1_{\rm phys}\neq0\), construct the nonlinear obstruction or Kuranishi map

\[ \kappa:H^1{\rm phys}\to H^2{\rm phys}. \]

A candidate is locally obstruction-rigid when the local zero set satisfies

\[ \kappa^{-1}(0)=\{0\} \]

near the reference point.

This shows there is no nearby family of exact solutions.

It does not prove stability of off-shell physical fluctuations.

6.1 Obstruction certificate

The obstruction analysis must specify:

• nonlinear equations used;
• expansion order or exact theorem;
• function space;
• gauge quotient;
• boundary and global conditions;
• whether all formal obstructions converge to an actual local statement;
• excluded topology-changing directions.

7. Constraint-derived rigidity

The highest-information route uses constraints justified independently of the candidate result.

Let

\[ C_a(\mathfrak P)=0 \]

be the pre-existing physical constraints: carriers, global group, chirality, family count, anomalies, boundary consistency, or other independently sourced obligations.

Define

\[ \mathcal M{\rm admiss} = \{\mathfrak P':Ca(\mathfrak P')=0\ \forall a\}/\text{redundancy}. \]

The infinitesimal admissible deformation space is

\[ T{[\mathfrak P]}\mathcal M{\rm admiss} = \frac{\bigcapa\ker(dCa)}{\operatorname{im}d_{\rm gauge}}. \]

7.1 Strong result

If

\[ T{[\mathfrak P]}\mathcal M{\rm admiss}=0, \]

the already justified constraint register infinitesimally fixes the complete physical object at the stated scope.

7.2 Obstructed result

If the tangent is nonzero but every nonzero direction is nonlinearly obstructed, the candidate is locally isolated under the independent constraint register.

7.3 Surviving deformation result

If integrable nonzero directions survive, constraint-first selection does not intrinsically fix the candidate.

Dynamics must stabilize them, a declared construction must remove them, or the candidate must fail the intended rigidity claim.

7.4 Anti-circularity conditions

A constraint can participate in intrinsic rigidity only if:

• it predates the rigidity calculation;
• its rationale does not name the preferred candidate;
• it applies equally to rivals;
• its pass rule was frozen before outcomes;
• it corresponds to an observed record, exact consistency condition, or prior foundational axiom;
• it was not introduced solely to kill a discovered negative direction.

A post-failure constitutive constraint is a construction route, not a derived rigidity theorem.

8. Exact constraints and constitutive rigidity

A candidate may lawfully define a physical constraint surface

\[ \Phi_A=0. \]

If the constraints are second class, physical evolution and measure use the Dirac reduction.

This may remove deformation directions from the physical phase space.

The result is classified as:

• DERIVED-CONSTRAINT-RIGIDITY when \(\Phi_A\) follows from independently justified prior constraints;
• CONSTITUTIVE-RIGIDITY when \(\Phi_A\) is a declared law of the candidate;
• TARGET-LOADED-RESTRICTION when \(\Phi_A\) was introduced because a known direction failed and lacks independent provenance.

Only the first can convert “rigid by decree” into a theorem-like result.

9. Physical Hessian and kinetic normalization

Stability is evaluated on physical off-shell perturbations after gauge quotient and exact constraints.

Let \(K\) be the kinetic metric and \(H\) the second variation of the effective action.

The physical mass operator is schematically

\[ \mathcal M^2{\rm phys}=K{\rm phys}^{-1}H_{\rm phys}. \]

The quadratic form is

\[ Q{\rm phys}[\delta z] = \langle\delta z,H{\rm phys}\delta z\rangle. \]

A stable sector requires:

• positive physical norm;
• no negative eigenvalue;
• correct treatment of gauge zero modes;
• declared treatment of exact physical zero modes;
• boundary stability;
• no uncontrolled complex frequencies;
• no rank-changing constraint direction.

9.1 Coercivity

For full stability on a domain \(\mathcal D\), the preferred result is a bound

\[ Q_{\rm phys}[\delta z] \ge c\,\|\delta z\|^2, \qquad c>0, \]

for all admissible non-whitelisted perturbations.

A finite matrix positivity result is scoped to the truncated sector unless a tail theorem is supplied.

10. Complete mixed Hessian

The full Hessian must include off-diagonal mixing among all physical sectors.

If

\[ H= \begin{pmatrix} A&B\\B^T&D \end{pmatrix}, \]

positivity of \(A\) and \(D\) separately is insufficient.

When \(D>0\), the Schur complement condition is

\[ A-BD^{-1}B^T>0. \]

Every omitted mixed block requires a theorem of vanishing or a controlled decoupling estimate.

11. Stability routes

11.1 Route I — intrinsic physical rigidity

The complete physical deformation space is empty at the claimed scope.

No stabilizing potential or constraint multiplier is required.

11.2 Route II — obstruction rigidity

The local solution germ is isolated, but off-shell physical fluctuation stability remains separately tested.

11.3 Route III — dynamical stabilization

Physical deformation fields exist and a lawful parent effective action yields a positive full mixed Hessian.

11.4 Route IV — constitutive or Actor realization

A declared exact constraint, projector, Actor, Co-Actor, boundary mechanism, or stiffness term removes or stabilizes directions.

The route is legitimate but must be graded as a construction unless independently derived.

11.5 Route V — negative result

A physical negative mode survives or the constraint system is inconsistent.

This is a legitimate closed-negative finding when scoped and recorded honestly.

12. Rigidity ladder

| Level | Result | Meaning | Stability implication | |---|---|---|---| | R0 | Coordinate or parameter convention fixed | Presentation only | None | | R1 | Unique point inside a restricted ansatz | Ansatze-level isolation | None outside ansatz | | R2 | \(H^1_{\rm phys}=0\) at stated scope | Infinitesimal rigidity | Strong, but physical Hessian still checked | | R3 | Local solution germ isolated by obstruction | Nonlinear local rigidity | Does not exclude negative off-shell mode | | R4 | Complete physical scalar deformation space empty | Intrinsic physical rigidity | Closes scalar-modulus leg at stated scope | | R5 | Full physical Hessian coercive | Dynamical stability | Closes perturbative stability at stated scope | | R6 | Tower, boundary, mixed, and nonperturbative scope complete | Full scoped vacuum stability | Strongest scoped result |

No lower rung may be narrated as a higher rung.

13. Protected scalar whitelist

Intended physical scalars are not counted as unwanted moduli merely because they are geometric.

For every whitelisted scalar, require:

• exact identity;
• physical role;
• protecting symmetry;
• kinetic positivity;
• mass-generation mechanism;
• mixing analysis;
• observed or predicted coupling scope;
• vacuum and radiative stability owner.

A whitelist may not be created after discovering an unwanted zero mode without branch restart.

14. Scale and vacuum interfaces

Intrinsic rigidity does not by itself solve:

• the value of the compactification scale;
• radion stabilization when a scale fluctuation remains;
• vacuum-offset response;
• the residual cosmological constant;
• phase-transition history.

Every rigidity certificate must carry two independent labels:

DEFORMATION/PHYSICAL-STABILITY STATUS:
  intrinsic / obstructed / dynamic / construction / unstable / open.

VACUUM-OFFSET STATUS:
  parent-protected / scoped-construction / unprotected / open.

A valid terminal may be:

INTRINSICALLY-RIGID / LAMBDA-UNPROTECTED

which is not a complete protected vacuum.

15. Full-tower, boundary, and nonperturbative scope

A complete stability claim must state coverage of:

• zero modes;
• nonzero KK tower;
• boundary and fixed-set modes;
• mixed sectors;
• topology-changing tunnelling;
• finite-temperature or cosmological transitions;
• radiative order;
• cutoff and tail.

Possible honest scopes are:

• invariant zero-mode sector;
• complete zero-mode sector;
• all modes below cutoff;
• perturbative full tower;
• global/nonperturbative vacuum.

16. Rigidity certificate

rigidity_certificate:
  candidate_id:
  branch_id:
  deformation_complex:
  function_spaces:
  sectors_included:
  gauge_redundancy_quotient:
  h1_physical:
  obstruction_map:
  local_moduli_germ:
  independent_constraint_register:
  constraint_derived_tangent_space:
  exact_constraint_classification:
  kinetic_metric:
  mixed_physical_hessian:
  schur_complements:
  protected_scalar_whitelist:
  zero_modes:
  negative_modes:
  tower_and_boundary_scope:
  tunnelling_scope:
  scale_interface:
  vacuum_offset_status:
  evidence_hashes:
  route_terminal:
  verdict:

17. Mandatory negative controls

17.1 Isolated saddle

A solution is isolated among exact vacua but has one negative off-shell mode.

Expected classification:

OBSTRUCTED/ISOLATED: PASS
PHYSICAL STABILITY: FAIL

17.2 Frozen by decree

A negative deformation is removed by a newly declared constitutive constraint lacking prior provenance.

Expected classification:

CONSTITUTIVE-RIGIDITY / CONSTRUCTION-ANCHOR
NOT INTRINSICALLY DERIVED

17.3 Constraint-derived rigid candidate

Independent carrier and anomaly constraints have a full-rank Jacobian whose physical kernel is zero.

Expected classification:

DERIVED-CONSTRAINT-RIGIDITY

with the physical Hessian still separately recorded.

17.4 Positive diagonal, negative mixed mode

Metric and bundle diagonal Hessians are positive; their Schur complement is negative.

Expected result: stability fails.

17.5 Rigid but vacuum-unprotected

No physical moduli; constant matter shift changes curvature.

Expected result: rigidity passes, vacuum status fails.

18. Falsifiers and reopen triggers

This block fails or reopens if:

• a deformation sector is omitted;
• gauge fixing hides a physical mode;
• invariant rigidity is promoted to full rigidity;
• isolation is promoted to stability;
• an obstruction map is asserted but not constructed;
• a post-failure constraint is called independently derived;
• a mixed Hessian block is ignored;
• kinetic normalization is missing;
• an intended scalar is whitelisted post hoc;
• a zero-mode result is promoted to the tower;
• intrinsic rigidity is used as proof of vacuum-offset protection;
• a physical negative mode survives.

19. Terminal grammar

CLOSED / DERIVED-INTRINSIC-PHYSICAL-RIGIDITY / RESOLVED +0

CLOSED-SCOPED / OBSTRUCTED-LOCAL-RIGIDITY /
PHYSICAL-HESSIAN SEPARATELY GRADED

CLOSED / DERIVED-POSITIVE-FULL-PHYSICAL-HESSIAN / RESOLVED +0

CLOSED-SCOPED / REALIZED-GIVEN-STABILIZING-ACTOR /
CONSTRUCTION-ANCHOR

CLOSED-NEGATIVE / PHYSICAL INSTABILITY CERTIFIED

OPEN / DEFORMATION OR STABILITY CERTIFICATE INCOMPLETE

20. Current status

BB-RIG-1.3

STATUS:
  DEVELOPMENT AUTHORITY CANDIDATE — NOT FROZEN.

PURPOSE OF NEXT TEST:
  Determine whether independently justified constraints can eliminate the
  complete physical deformation space, rather than merely removing a known
  bad direction after failure.

Integrated boundary/orbifold authority

This block is part of the Candidate-Neutral Building-Block Authority v3.1 — Boundary and Inflow Development Candidate.

The following rule is binding across the set:

A bulk or quotient-space anomaly cancellation statement does not close a theory with orbifold fixed sets or physical boundaries. Each stratum requires a local anomaly ledger, and the complete quantum theory requires a relative/global trivialization compatible with gluing.

The dedicated owner is BB-BND-1 — Orbifold Fixed-Set, Relative-Anomaly, and Inflow Completion.

Every candidate containing a boundary, fixed set, defect, quotient singularity, interface, or parity projection must provide a boundaryanomalycompletion_certificate. Until that certificate passes:

• a chirality or no-mirror result is CONDITIONAL-ON-BND;
• quantum BRST/BV nilpotency is CONDITIONAL-ON-BND;
• a global-bordism result for the closed bulk target is not a complete boundary result;
• cancellation after integrating over the compact direction is not accepted as local gauge invariance;
• no gate may claim that the interval/orbifold construction is fully quantum consistent.

Boundary/orbifold completion — boundary data in the deformation complex

The complete deformation complex must include:

• fixed-set position and normal-bundle deformations;
• parity/domain deformations;
• localized bundle and connection deformations;
• inflow coefficient branches;
• boundary counterterm branches;
• determinant-line trivialization data.

Anomaly cancellation may legitimately obstruct a deformation only when the anomaly constraint:

1. is independently justified;
2. predates the candidate outcome;
3. is applied equally to all candidates;
4. is computed on the full stratified parent object.

A boundary condition chosen after discovering an unstable or anomalous mode is a construction repair, not intrinsic rigidity.

The physical Hessian and deformation complex are conditional on BND-C01 through BND-C10; an anomalous boundary domain is not a physical vacuum domain.

Block-Specific Standalone Interface — Intrinsic Rigidity and Stability

Owned questions

This block alone owns:

• complete physical deformation space;
• gauge/constraint redundancy quotient;
• infinitesimal deformation cohomology;
• nonlinear obstruction and integrability;
• distinction between isolation and stability;
• complete mixed physical Hessian;
• intrinsic, obstruction, dynamical, constitutive, and negative routes;
• protected-scalar whitelist.

Required outputs

• deformation-complex manifest;
• \(H^1_{\rm phys}\) or equivalent tangent-space result;
• obstruction map;
• mixed Hessian and kinetic normalization;
• full-tower/boundary/tunnelling scope;
• route-sensitive terminal;
• rigidity certificate.

Non-ownership boundary

Rigidity does not imply:

• vacuum-offset protection;
• background-equation completeness;
• anomaly-free boundary domains;
• candidate selection.

Algorithmic computation of constraint-derived rigidity

Definitions are not enough. Every concrete candidate must publish the finite or discretized linear map that decides rigidity.

1. Freeze the deformation basis

Choose an ordered basis

\[ \mathcal E{\rm def}=(e1,\ldots,e_N) \]

covering every in-scope deformation. For the current compact branch, the minimal homogeneous/low-mode basis should include at least:

\[ (\delta\log u1,\delta\log u2,\delta\log u3, \delta\log R{S^2},\delta\log RI, \delta WY,\delta WH, \delta b1,\delta x0,\delta x\pi, \delta\Pi_{\rm obs}), \]

with any additional bundle, boundary, or Actor modes required by the full manifest.

Each basis vector must state:

• sector;
• normalization and units;
• kinetic inner product;
• gauge or physical status before reduction;
• cutoff/tower scope;
• parent-manifest owner.

2. Build the gauge matrix

Choose a basis of infinitesimal redundancies and compute

\[ G{I\alpha}=(d{\rm gauge}\epsilon\alpha)I. \]

Publish gauge_matrix.csv. Rank must be stable under the declared exact or interval-certified arithmetic.

3. Build the constraint Jacobian

For every independently justified active constraint \(C_a\), calculate

\[ J{aI} = dCa(eI) = \left.\frac{\partial Ca}{\partial q^I}\right|{\mathfrak P*}. \]

Publish the complete matrix—not only its rank—as constraint_jacobian.csv, with:

• row ID matching the evidence registry;
• constraint freeze hash;
• evaluation point;
• analytic expression or certified numerical interval;
• zero-row justification;
• provenance.

The package includes RIGIDITYCONSTRAINTJACOBIANTEMPLATEV332.csv.

4. Reduce exactly

Compute:

\[ K=\ker J, \qquad T_{\rm admiss}=K/\operatorname{im}G. \]

The artifact must include:

• exact pivots or certified singular-value intervals;
• rank \(J\);
• rank \(G\);
• a basis for \(K\);
• a basis for the quotient;
• sensitivity to numerical tolerance;
• branch hash.

5. Classify the result

DERIVED-CONSTRAINT-RIGIDITY:
  dim(T_admiss)=0 using only independently justified pre-frozen rows.

CONSTITUTIVE-RIGIDITY:
  dim(T_admiss)=0 only after candidate-specific constitutive rows are added.

OBSTRUCTION-RIGIDITY:
  dim(T_admiss)>0, but every nonzero direction has a certified nonlinear obstruction.

MODULI-PRESENT:
  At least one nonzero direction is integrable.

INCOMPLETE:
  The matrix, gauge quotient, rank stability, or deformation basis is missing.

6. Nonlinear and stability continuation

A zero tangent space still does not prove positive physical stability. If a tangent direction survives, calculate its obstruction map. Independently calculate the kinetic-normalized full mixed Hessian on the physical domain.

7. Required published artifacts

• deformation_basis.json;
• gauge_matrix.csv;
• constraint_jacobian.csv;
• rank_certificate.json;
• physicaltangentbasis.csv;
• obstruction_map.*, when required;
• mixed_hessian.*;
• reconstruction script and environment lock.

Owned hard constraints and acceptance evidence

The full controlling registry is EVIDENCEDISCHARGEREGISTRYV33_2.md. The rows below are embedded because this block owns or directly co-owns them.

| ID | Required witness | Minimal discharge | Fail trigger | |---|---|---|---| | GEN-C03 | Variational-ownership ledger + equation-disposition table | Every variable is marked varied, gauge, constrained, auxiliary, measured, matching, or external, with the displaced Euler–Lagrange equation explicitly retained, replaced, proved redundant, or abandoned. | A non-varied variable has no lawful equation disposition. | | SHP-C06 | Cross-sector zero-mode inventory | A merged mode table covers gauge, fermion, scalar, metric, boundary, edge, and topological sectors below cutoff. | An undeclared massless or parametrically light mode remains. | | RIG-C01 | Published deformation-basis manifest | A finite or function-space basis enumerates metric, radius, Wilson-line, bundle, boundary, Actor, Rulebook, and observer-map deformations. | A deformation sector is omitted because its background value was fixed. | | RIG-C02 | Gauge/constraint quotient matrix | The gauge image and exact constraint directions are computed and removed before physical rank is counted. | Coordinate, gauge, or reaction directions are counted as physical or vice versa. | | RIG-C03 | Constraint Jacobian matrix + rank-reduction output | The published matrix \(J{aI}=dCa(eI)\) is row-reduced with exact/rational or certified numerical arithmetic; its nullspace modulo gauge gives \(T{ m admiss}\). | The rigidity classification is asserted without the matrix or rank artifact. | | RIG-C04 | Kuranishi/obstruction calculation or integrability theorem | Every surviving infinitesimal direction is integrated or assigned a nonzero obstruction with reproducible coefficients. | A nonzero tangent direction is called obstructed without computing the obstruction map. | | RIG-C05 | Separate solution-space and Hessian certificates | The local moduli-germ result and the kinetic-normalized physical Hessian are published independently. | Isolation is used as evidence of positive stability. | | RIG-C06 | Full mixed Hessian artifact | All diagonal and off-diagonal sector blocks are computed, or symmetry theorems prove specified blocks vanish; eigenvalue bounds are published. | Only a metric or homogeneous sub-block is tested while mixed directions remain. | | RIG-C07 | Protected-scalar whitelist + symmetry proof | Every intended light scalar has an owning symmetry/mechanism and every unwanted modulus is absent, obstructed, or positively massive. | An unwanted scalar is relabeled as protected after its discovery. | | RIG-C08 | Named radius/Wilson/boundary deformation rows | Volume, relative radii, interval length, Wilson lines, boundary positions, and localized deformations each have an explicit row and status. | A fixed radius is treated as proof that its radion does not exist. | | RIG-C09 | Tower/boundary/tunnelling scope certificate | The claim states the highest KK level, boundary sectors, and nonperturbative channels covered, with a bound or honest OPEN rows. | A finite homogeneous calculation is promoted to full stability. | | RIG-C10 | Constraint-provenance audit + pre-freeze hash | Every constraint used in the rigidity matrix predates the candidate result and is independently justified; late-added rows force restart. | A negative mode is removed by a constraint written after the mode was found and then called intrinsic. | | OBS-C07 | Observer-map deformation Jacobian | Derivatives of records with respect to projection, frame, normalization, and apparatus choices are included in the deformation/uncertainty analysis. | A small convention change can move a prediction across the pass band but is not counted. |

Package status and authority boundary

This file belongs to the Candidate-Neutral Building-Block Authority v3.3.2.

The package is documentation-complete for the current development architecture, but it is not protocol-frozen and not canonically ratified. Technical results generated under it remain development evidence until the control suite, enumeration test, and hostile review pass.

The controlling package authorities are:

• EVIDENCEDISCHARGEREGISTRYV33_2.md;
• QUANTITATIVESTOPPINGANDEXHAUSTIONRULEV33_2.md;
• MANDATORYCONTROLSUITEV33_2.md;
• MIGRATIONANDSUPERSESSIONRECORDV3_3.md.

A block is standalone for its owned object: it includes its complete definitions, interfaces, certificates, falsifiers, and owned evidence criteria. The full cross-block registry is intentionally stored once rather than repeated verbatim in every block.

Shared status grammar

PASS:
  The named witness satisfies the minimal discharge criterion.

FAIL:
  A reproducible counterexample or failed acceptance test exists.

OPEN:
  The required witness is missing, incomplete, or inconclusive.

NOT-EVALUATED:
  Evaluation stopped after an earlier hard failure.

NOT-APPLICABLE:
  A type proof establishes that the requirement cannot apply.

CONSTRUCTION-ANCHOR:
  A declared Actor, constraint, boundary term, or global sector realizes
  the result but does not derive it from the pre-existing object.

CONDITIONAL-ON-BND-A:
  The spectrum/domain result awaits fixed-set and operator-domain closure.

CONDITIONAL-ON-BND-B:
  The quantum result awaits local and global anomaly/inflow closure.

RESTART-REQUIRED:
  A frozen object relevant to the result changed.

Universal anti-promotion rules

• A bare candidate is not a complete parent object.
• A definition is not a witness.
• A nonempty equation list is not a solvability proof.
• An isolated vacuum is not automatically stable.
• A fixed background value is not automatically an absent fluctuation.
• Static vacuum-offset protection is not automatically cosmological-history compatibility.
• Integrated anomaly cancellation is not local fixed-set cancellation.
• Local anomaly cancellation is not global determinant-line trivialization.
• A raw higher-dimensional quantity is not automatically the measured four-dimensional observable.
• One tested survivor is not a selected survivor without explicit grammar enumeration or a finiteness theorem.

--- END EXACT LEGACY SOURCE ---

--- title: "BB-RIG-4.0 — Maximum-Rigor Constraint-Derived All-Gate Rigidity Authority" buildingblockid: "BB-RIG-4.0" version: "4.0" date: "2026-08-02" status: "DEVELOPMENT AUTHORITY — MAXIMUM-RIGOR ALL-GATE CANDIDATE — NOT FROZEN" supersedesfordevelopment: "BB-RIG-1.3 while preserving it exactly" legacysourcesha256: "196ca8170197a3e88b441092600e1bf095ebfddacf3bb79073a58d007b0e86a8" protocolfreeze: false canonicalratification: false ---

BB-RIG-4.1 — Maximum-Rigor Preservation-Complete All-Gate Rigidity Authority

0. Controlling rule

This object determines whether the complete physical candidate is isolated, obstructed, dynamically stable, metastable, constitutively restricted, or unstable at a declared scope. It does not select a candidate, manufacture missing Dynamics, or convert a construction into a derivation.

The exact BB-RIG-1.3 authority is preserved in Appendix A and 06SOURCES/BBRIG13EXACTARCHIVAL_SOURCE.md. Every legacy requirement remains binding unless this authority explicitly strengthens it.

1. Core firewalls

\[ \text{unique point in an ansatz}\neq\text{physical rigidity}, \] \[ H^1{\rm phys}=0\neq Q{\rm phys}>0, \] \[ Q_{\rm phys}>0\text{ locally}\neq\text{nonlinear/global stability}, \] \[ \text{tree-level stability}\neq\text{quantum/radiative stability}, \] \[ \text{perturbative stability}\neq\text{nonperturbative longevity}, \] \[ \text{one scale, one time, one domain, or one truncation}\neq\text{complete rigidity}. \]

2. Owned object

For frozen parent object \(\mathfrak P\), define the typed physical deformation problem

\[ \mathfrak R(\mathfrak P)= (\mathcal E{\rm def},d0,d_1,\kappa,K,H,\mathcal D,\Sigma,\mathcal B,\mathcal T,\mathcal U), \]

where the entries are respectively the complete deformation grammar, redundancy map, linearized constraint/equation map, nonlinear obstruction, kinetic structure, full mixed Hessian/evolution generator, operator domain, physical spectrum, branch/bifurcation data, time/history data, and uncertainty/refinement data.

3. Constraint-derived method

1. Freeze authority hashes, candidate grammar, deformation grammar, gate scope, Observer, cutoff, histories, and accepted terminals.
2. Generate every deformation sector and every lawful stabilization/rigidity route without viewing the desired answer.
3. Build the gauge/redundancy map and exact constraint Jacobian.
4. Compute the physical tangent space, singular strata, and obstruction/integrability map.
5. Build the kinetic-normalized full mixed operator on the actual Boundary domain and function space.
6. Classify discrete/essential spectrum, ghosts, gradients, tachyons, complex modes, and coercivity.
7. Continue over scale, time/history, parameters, boundary supports, regulator/refinement, and branch changes.
8. Test nonlinear basins, metastability, topology change, composites, regions/horizons, and open-system effects when typed applicable.
9. Prove grammar and test-family exhaustion or retain explicit OPEN residuals.
10. Issue route-sensitive certificates and the weakest honest terminal.

4. Atomic Rigidity constraints

RIG-C01 — Complete deformation inventory across metric, bundle, boundary, Actor, Rulebook, and Observer sectors

Question. Complete deformation inventory across metric, bundle, boundary, Actor, Rulebook, and Observer sectors?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. deformationbasispacket, grammarexhaustionpacket.

Minimal witness. Every in-scope deformation sector is enumerated before reduction.

Fail trigger. An omitted sector can carry a modulus or instability.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C02 — Redundancy removal before physical counting

Question. Redundancy removal before physical counting?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. gaugequotientpacket, variationalownershippacket.

Minimal witness. Gauge, diffeomorphism, BRST-exact, presentation, boundary-gauge, and record-null directions are quotiented without removing physical modes.

Fail trigger. A gauge direction is counted physical or a physical direction is gauge-fixed away.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C03 — Infinitesimal physical deformation/cohomology or constraint-Jacobian calculation

Question. Infinitesimal physical deformation/cohomology or constraint-Jacobian calculation?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. constraintjacobianpacket, rankcertificatepacket, physicaltangentpacket.

Minimal witness. The physical tangent space is computed reproducibly as a kernel modulo gauge.

Fail trigger. Rigidity is asserted without a matrix, complex, rank, or nullspace witness.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C04 — Nonlinear obstruction and integrability

Question. Nonlinear obstruction and integrability?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. obstructionintegrabilitypacket, solutiongermpacket.

Minimal witness. Every surviving tangent direction is integrated or assigned a reproducible nonlinear obstruction.

Fail trigger. A nonzero tangent is called obstructed without a Kuranishi/integrability calculation.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C05 — Isolation kept separate from physical Hessian stability

Question. Isolation kept separate from physical Hessian stability?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. solutiongermpacket, mixedhessianpacket, stabilityroutepacket.

Minimal witness. Solution-germ isolation and off-shell physical stability are independently certified.

Fail trigger. An isolated saddle is called stable.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C06 — Complete mixed physical Hessian

Question. Complete mixed physical Hessian?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. kineticmetricpacket, mixedhessianpacket, spectrumcoercivitypacket.

Minimal witness. All diagonal and off-diagonal physical blocks are computed or eliminated by theorem, with kinetic normalization.

Fail trigger. Positive diagonal blocks hide a negative mixed mode.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C07 — Protected scalar whitelist

Question. Protected scalar whitelist?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. protectedscalarpacket.

Minimal witness. Every intended light scalar has pre-frozen identity, protection, positivity, mixing, and stability ownership.

Fail trigger. An unwanted modulus is relabeled protected after discovery.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C08 — Named radius, Wilson-line, boundary-position, and localized deformations

Question. Named radius, Wilson-line, boundary-position, and localized deformations?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. namedmodulipacket, boundarysupportrigidity_packet.

Minimal witness. Volume, relative radii, interval length, Wilson lines, boundary positions, and localized modes each receive explicit status.

Fail trigger. A fixed background value is treated as absence of its fluctuation.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C09 — Full-tower, boundary, and nonperturbative scope

Question. Full-tower, boundary, and nonperturbative scope?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. towertailpacket, metastabilitydecaypacket.

Minimal witness. The rigidity/stability claim declares zero-mode, tower, boundary, tunnelling, thermal, and cutoff scope with tails or OPEN residuals.

Fail trigger. A finite homogeneous calculation is promoted to full stability.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C10 — Constraint provenance and anti-circularity

Question. Constraint provenance and anti-circularity?

Formal requirement. Binding legacy requirement preserved verbatim in Appendix A and the exact archival source.

Inputs. frozen parent manifest; relevant cross-block packets.

Execution.

1. execute the legacy BB-RIG-1.3 procedure.
2. publish the named witness.
3. run the legacy negative control.
4. apply the legacy fail trigger.

Required outputs. anticircularitypacket.

Minimal witness. Only independently justified pre-frozen constraints may support derived constraint-rigidity; late constitutive rows are graded as construction.

Fail trigger. A post-failure restriction is called an intrinsic derivation.

Destructive control. Use the corresponding legacy negative control or construct the minimal counterexample described by the fail trigger.

RIG-C11 — Function-space, domain, Fredholm, essential-spectrum, and coercivity control

Question. Function-space, domain, Fredholm, essential-spectrum, and coercivity control?

Formal requirement. Rigidity and stability must be formulated on the actual operator domain and function space. The certificate must classify discrete and essential spectrum, kernel/cokernel, Fredholm index where applicable, boundary/domain dependence, and a coercive or gap estimate on the non-whitelisted physical subspace.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. functionspacedomainpacket, spectrumcoercivity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Finite matrices or a few eigenvalues can be positive while an essential-spectrum instability, non-Fredholm direction, or domain-dependent zero mode survives.

Destructive control. Two operators have identical first N eigenvalues; one has essential spectrum crossing zero. The audit must reject equivalence.

RIG-C12 — Singular strata, rank jumps, bifurcations, marginality, and spectral flow

Question. Singular strata, rank jumps, bifurcations, marginality, and spectral flow?

Formal requirement. The analysis must locate singular points of the constraint/deformation map, rank-changing strata, eigenvalue crossings, bifurcation branches, marginal directions, exceptional points, and spectral flow over the claimed parameter region.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. bifurcationspectralflowpacket, solutiongerm_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A full-rank Jacobian and positive Hessian at one point can sit arbitrarily close to a bifurcation or rank-changing branch.

Destructive control. Two candidates agree at the reference point; only one develops a zero eigenvalue under an arbitrarily small lawful parameter change.

RIG-C13 — Nonlinear finite-amplitude, Lyapunov/orbital stability, and basin control

Question. Nonlinear finite-amplitude, Lyapunov/orbital stability, and basin control?

Formal requirement. When a dynamical stability claim exceeds linear perturbation theory, publish a Lyapunov, energy, invariant-set, bootstrap, or equivalent nonlinear estimate; identify the attraction/orbital-stability basin, conserved quantities, runaway channels, and finite-amplitude thresholds.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. nonlinearbasinpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A positive Hessian can coexist with finite-amplitude runaway, nonlinear resonance, collapse, or blow-up.

Destructive control. Construct a potential with a positive local Hessian but an arbitrarily nearby escape channel; linear stability must not be promoted.

RIG-C14 — Time-dependent, parametric, Floquet, adiabatic, and history stability

Question. Time-dependent, parametric, Floquet, adiabatic, and history stability?

Formal requirement. For time-dependent backgrounds, moving boundaries, phase histories, periodic driving, cosmological evolution, or changing scales, evolve the perturbation system and test Floquet/parametric growth, adiabaticity, nonnormal transient amplification, memory, and history-dependent excitation.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. timedependentstability_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A static vacuum can be stable while a lawful transition or periodic background excites an otherwise harmless mode.

Destructive control. Two candidates share the same instantaneous Hessian; one has a parametric resonance during the required history.

RIG-C15 — Quantum, radiative, regulator, scheme, and refinement stability

Question. Quantum, radiative, regulator, scheme, and refinement stability?

Formal requirement. The effective-action Hessian, constraint rank, protected directions, and obstruction data must remain controlled under the claimed loop order, RG transport, regulator family, truncation enlargement, matching thresholds, and Granularity refinement.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. quantumradiativestability_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Tree-level rigidity can disappear through Coleman-Weinberg directions, anomalous mixing, regulator artifacts, or truncation-dependent fixed points.

Destructive control. Use two lawful regulators that agree at tree level but give opposite loop-corrected mass signs; the claim remains OPEN without stability.

RIG-C16 — Metastability, tunnelling, bounce, topology change, and lifetime

Question. Metastability, tunnelling, bounce, topology change, and lifetime?

Formal requirement. Separate absolute stability from metastability. Enumerate lower-action branches and topology-changing channels; compute or bound bounce/instanton actions, prefactors, decay rates, recurrence assumptions, and lifetime relative to the gate timescale.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. metastabilitydecaypacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. No perturbative negative mode does not exclude nonperturbative decay to another branch.

Destructive control. Two vacua have identical perturbative spectra; one has a finite-action bounce to a lower state.

RIG-C17 — Topological, index, symmetry-protected, and defect-sector rigidity

Question. Topological, index, symmetry-protected, and defect-sector rigidity?

Formal requirement. Identify deformations forbidden or protected by topology, index, superselection, exact symmetry, charge, or defect data; prove invariance under the admitted deformation grammar and classify wall-crossing or topology-changing exceptions.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. topologicalprotectionpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A local Hessian can miss exact topological protection or an allowed wall-crossing that changes the sector.

Destructive control. Two locally identical configurations carry different index/defect classes; they must not share a rigidity certificate.

RIG-C18 — Regional, boundary, horizon, interface, edge, and state-dependent-support rigidity

Question. Regional, boundary, horizon, interface, edge, and state-dependent-support rigidity?

Formal requirement. Import the complete Boundary support/incidence grammar and test deformations of fixed sets, corners, horizons, entangling cuts, phase interfaces, asymptotic regions, edge modes, and state-dependent supports, including gluing and flux balance.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. boundarysupportrigidity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A bulk-stable solution can have an unstable edge, moving horizon, corner, or interface mode.

Destructive control. Keep the bulk Hessian fixed while adding a negative boundary-localized mode; full rigidity must fail.

RIG-C19 — Composite, condensate, soliton, bound-state, and effective-Actor rigidity

Question. Composite, condensate, soliton, bound-state, and effective-Actor rigidity?

Formal requirement. For every promoted effective/composite Actor, derive the collective-coordinate and internal-shape deformation complex, binding/decay channels, mixing with elementary sectors, scale of validity, and de-promotion/reopen conditions.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. compositeeffectiverigidity_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. Elementary fields can be stable while a composite, condensate, soliton, or bound-state sector is unstable or double-counted.

Destructive control. Two theories share the same elementary spectrum; one contains an unstable composite pole or soliton size modulus.

RIG-C20 — Conditioning, uncertainty, identifiability, and robustness margin

Question. Conditioning, uncertainty, identifiability, and robustness margin?

Formal requirement. Publish singular-value/eigenvalue intervals, condition numbers, covariance transport, measured-anchor dependence, numerical tolerance tests, structural identifiability, and a quantitative margin to every rank change or instability.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. robustnessuncertaintypacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. An exactly positive but ill-conditioned result can change sign within lawful uncertainty or numerical error.

Destructive control. A Hessian eigenvalue is positive at its central value but its certified interval crosses zero.

RIG-C21 — Candidate-neutral deformation, stabilization-route, and rival grammar exhaustion

Question. Candidate-neutral deformation, stabilization-route, and rival grammar exhaustion?

Formal requirement. Generate all admissible deformation sectors and all lawful rigidity/stabilization mechanisms before viewing outcomes; canonicalize duplicates, test omitted rivals, classify every candidate, and prove finite enumeration, bounded generation, or a class theorem.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. grammarexhaustionpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. One tested deformation basis or one successful stabilizer is not evidence that no omitted route or instability exists.

Destructive control. Hide a lawful Wilson-line or edge deformation from the initial roster; the omitted-rival control must detect it.

RIG-C22 — Cross-scale continuation, decoupling, RG spectral flow, and matching stability

Question. Cross-scale continuation, decoupling, RG spectral flow, and matching stability?

Formal requirement. Track physical modes, Hessian eigenvalues, constraints, and protected directions across Scale packets, thresholds, compactification levels, EFT matching, and RG flow; prove decoupling or record nondecoupling and eigenvalue crossings.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. crossscalecontinuation_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A mode stable at one scale can become tachyonic, ghostlike, or strongly mixed after threshold transport.

Destructive control. Two candidates match at one scale; only one eigenvalue crosses zero above a threshold.

RIG-C23 — Constrained Hamiltonian, symplectic, Krein, ghost/gradient/tachyon stability

Question. Constrained Hamiltonian, symplectic, Krein, ghost/gradient/tachyon stability?

Formal requirement. On constrained systems, construct the reduced symplectic/Dirac structure and distinguish negative norm, negative energy, tachyonic mass, gradient instability, complex frequency, Jordan block, and Krein collision; verify constraint-rank preservation.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. symplectickreinpacket, kineticmetricpacket.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A positive Euclidean Hessian or real frequency list can hide a ghost, gradient instability, or Hamiltonian-Hopf/Krein collision.

Destructive control. Use two systems with identical squared frequencies but opposite kinetic signature; only one is physical.

RIG-C24 — Open-system, stochastic, noise, decoherence, and ensemble stability

Question. Open-system, stochastic, noise, decoherence, and ensemble stability?

Formal requirement. When the gate uses reduced states, stochastic sources, thermal ensembles, measurements, or environmental channels, test complete-positive evolution, stationary-state stability, noise-induced transitions, large deviations, decoherence-free sectors, and ensemble dependence.

Inputs. frozen parent manifest; typed deformation/support/state roster; Dynamics V4.2 packets; Boundary V4.1 packets; Scale V4.1 packets; Granularity V4.0 packets as applicable.

Execution.

1. generate the complete typed object for this constraint.
2. compute exact/interval-certified invariants.
3. run the destructive control.
4. publish residuals, scope, and reopen triggers.

Required outputs. opensystemstability_packet.

Minimal witness. A gate-scoped machine-readable certificate satisfying the formal requirement with no unclassified applicable object.

Fail trigger. A closed-system stable mode can become unstable under a lawful environment or noise channel.

Destructive control. Two systems share the same Hamiltonian; one Lindblad/noise channel destabilizes the stationary state.

RIG-C25 — Non-normal operators, pseudospectrum, resolvent bounds, semigroup growth, and transient amplification

Question. Can a spectrally stable but non-normal physical evolution exhibit physically disqualifying transient growth?

Formal requirement. For every non-self-adjoint, constrained, dissipative, gauge-fixed, horizon, transport, or open-system operator in scope, compute or rigorously bound the pseudospectrum, resolvent norm, semigroup/propagator growth, condition of eigenvectors or spectral projectors, and transient amplification over the frozen time window. Eigenvalue location alone is insufficient.

Inputs. physical evolution operator; actual domain and norm; constraint/gauge quotient; time window and Observer tolerance.

Execution.

1. classify normality and generator type.
2. compute resolvent or certified bounds on a frozen complex-frequency region.
3. bound semigroup/propagator norm and transient gain.
4. track exceptional points and Jordan sensitivity.
5. compare amplification with physical tolerances and nonlinear thresholds.

Required outputs. pseudospectraltransientpacket.

Minimal witness. Certified resolvent/pseudospectral and propagator-growth bounds over the actual physical norm and domain.

Fail trigger. All eigenvalues can lie in the stable half-plane while non-normal transient growth drives the system outside the admissible basin.

Destructive control. Use two matrices with identical stable eigenvalues, one normal and one highly non-normal; the latter must remain OPEN or FAIL when transient gain exceeds the frozen threshold.

RIG-C26 — Resonance, secular drift, small divisors, long-time asymptotics, and near-integrable stability

Question. Does short-time or perturbative stability survive resonances and the full frozen physical time horizon?

Formal requirement. Enumerate resonant and near-resonant mode combinations, secular terms, small divisors, parametric bands, nonlinear frequency shifts, and cumulative drift. Where applicable provide normal-form, averaging, KAM/Nekhoroshev, multiple-scale, or rigorous long-time estimates and state the time horizon explicitly.

Inputs. mode spectrum and couplings; background history; interaction vertices; frozen physical time horizon.

Execution.

1. generate resonance lattice from the complete mode grammar.
2. compute detunings and coupling coefficients.
3. derive secular/normal-form evolution.
4. bound drift and resonance overlap.
5. compare stability time with gate requirement.

Required outputs. resonancelongtime_packet.

Minimal witness. Complete resonance roster with certified long-time drift or stability bounds to the gate horizon.

Fail trigger. A linearly stable and finite-time bounded system can undergo secular growth, Arnold diffusion, resonance overlap, or delayed instability.

Destructive control. Use two systems identical through quadratic order and over a short interval; add one resonant nonlinear coupling that produces secular growth on the required long time.

RIG-C27 — Numerical, discretization, spectral-pollution, and computer-assisted proof certification

Question. Are numerical rigidity and stability claims certified against discretization, roundoff, spectral pollution, solver bias, and nonreproducibility?

Formal requirement. Every numerical or symbolic-computational witness must specify arithmetic, discretization, basis, solver, residuals, conditioning, convergence sequence, independent cross-check, and certified error bounds. Spectral claims require pollution controls; sign/rank claims near zero require exact or interval arithmetic or a theorem-backed enclosure.

Inputs. all computational artifacts; operator/domain specification; tolerances and uncertainty budget.

Execution.

1. freeze code and environment hashes.
2. run refinement and basis-order sequences.
3. compute residual and backward-error bounds.
4. apply interval/exact arithmetic to decisive signs and ranks.
5. run independent algorithm or theorem cross-check.
6. test for spectral pollution and spurious modes.

Required outputs. numericalcertificationpacket.

Minimal witness. Reproducible computer-assisted certificate with convergence, residual, enclosure, and pollution controls.

Fail trigger. A finite discretization can display a positive gap, full rank, or stable spectrum that disappears under refinement or is purely spectral pollution.

Destructive control. Construct a Galerkin sequence with a spurious eigenvalue in a spectral gap; the validator must reject the apparent stable mode without a pollution-free enclosure.

RIG-C28 — Singular limits, noncommuting limits, continuum/infinite-volume/extremal limits, and uniformity

Question. Does the claimed rigidity survive every limit required by the gate, independent of the order in which limits are taken?

Formal requirement. List all required limits—cutoff removal, tower completion, continuum/refinement, infinite volume, zero noise, zero temperature, adiabatic, weak coupling, extremal/horizon, late time—and prove uniform estimates or explicitly record order dependence. A result at fixed regulator, finite volume, finite tower, or nonextremal parameter cannot be promoted.

Inputs. limit grammar; family of operators/solutions; uniform norms and domains; Scale and Granularity refinement data.

Execution.

1. enumerate required limits and admissible orders.
2. compute limit commutators or counterexamples.
3. derive uniform coercivity/gap/basin/error bounds.
4. track domains and constraints through limits.
5. publish strongest order-specific ceiling.

Required outputs. singularlimitpacket.

Minimal witness. Uniform or order-resolved limit certificate with no hidden promotion across noncommuting limits.

Fail trigger. Each finite member can be stable while the limiting object loses the gap, changes domain, gains a zero mode, or becomes singular.

Destructive control. Use a family whose finite-volume gap is positive but tends to zero as volume grows; finite-volume stability must not imply an infinite-volume gap.

RIG-C29 — Principal-symbol, characteristic-cone, hyperbolicity, constraint-propagation, and causal rigidity

Question. Do admissible deformations preserve a well-posed causal evolution and the constraint-propagation subsystem?

Formal requirement. For every dynamical deformation in scope, compute the principal symbol and characteristic variety/cones, classify strong/symmetric/weak hyperbolicity, verify boundary compatibility and constraint propagation, and bound changes in causal structure. Lower-order Hessian positivity does not certify well-posedness.

Inputs. full PDE system; gauge/constraint reduction; boundary conditions; deformation family.

Execution.

1. construct reduced principal symbol.
2. compute characteristics and multiplicities.
3. test diagonalizability/symmetrizer bounds.
4. derive constraint-propagation operator.
5. track cone and boundary changes under deformations.
6. certify well-posedness margin.

Required outputs. causalprincipalsymbol_packet.

Minimal witness. Gate-scoped hyperbolicity, characteristic, and constraint-propagation certificate across the admitted deformation neighborhood.

Fail trigger. A positive potential/Hessian can coexist with loss of hyperbolicity, superluminal/ill-defined characteristics, or growing constraint violations.

Destructive control. Keep all lower-order terms and static Hessian fixed while deforming the principal symbol from strongly hyperbolic to weakly hyperbolic.

RIG-C30 — Global gauge-orbit, stabilizer, reducible-connection, Gribov, and quotient-stratification rigidity

Question. Is the physical quotient globally valid, or do Gribov copies, stabilizer jumps, reducible configurations, or orbit-space singularities alter the rigidity result?

Formal requirement. The gauge/diffeomorphism quotient must be tested globally on the claimed sector: identify stabilizers and reducible strata, Gribov regions/horizons or an equivalent global slice theorem, ghost/Faddeev-Popov zero modes, orbit-space topology, and boundary gauge sectors. Local gauge fixing alone cannot support a global rigidity claim.

Inputs. gauge group and action; configuration space and topology; ghost/Faddeev-Popov operator; boundary/global sectors.

Execution.

1. classify orbit types and stabilizers.
2. locate reducible strata and FP zero modes.
3. test global slice or enumerate Gribov ambiguity.
4. compute quotient stratification and its deformation effect.
5. separate local BRST result from global quotient ceiling.

Required outputs. globalgaugeorbit_packet.

Minimal witness. Global-orbit/stratification certificate or an explicit scope ceiling excluding unresolved Gribov sectors.

Fail trigger. A locally complete gauge slice can double-count copies or miss physical zero modes when the orbit type changes.

Destructive control. Use a gauge family with identical local FP operator near the reference point but a second Gribov copy or stabilizer jump elsewhere in the admitted sector.

5. Universal execution pipeline

The agent must execute the selected gate rows in dependency order:

\[ \text{manifest}\to\text{deformation grammar}\to\text{domain/gauge quotient}\to T_{\rm phys}\to\kappa\to(K,H)\to\Sigma\to \text{history/scale/global controls}\to\text{pseudospectrum/resonance/limits/causal quotient}\to\text{terminal}. \]

A failed early row does not permit later rows to be assumed. Later calculations may be performed diagnostically but remain NOT-EVALUATED for closure unless their prerequisites pass.

6. Rigidity routes and terminal grammar

• DERIVED-INTRINSIC-RIGIDITY: complete physical deformation space is empty using pre-frozen independent constraints.
• OBSTRUCTION-RIGIDITY: nonzero infinitesimal directions are all nonlinearly obstructed; Hessian separately graded.
• DYNAMIC-STABILITY: physical modes exist but the complete typed physical operator is stable at scope.
• METASTABLE-LONG-LIVED: decay channels exist but lifetime exceeds the frozen requirement with controlled prefactor and history.
• CONSTITUTIVE-RIGIDITY: a declared constraint/Actor/projector removes the direction; construction, not derivation.
• CLOSED-NEGATIVE / INSTABILITY: a physical negative/ghost/gradient/runaway/decay failure is certified.
• OPEN: required deformation, domain, spectrum, history, uncertainty, exhaustion, or external evidence is incomplete.

No lower rung may be narrated as a higher rung.

7. Required evidence packets

Every gate bundle contains every packet below. Active packets are executed; inactive packets are NOT-APPLICABLE only with a gate-specific type proof.

parentmanifestpacket

Required fields: candidateid, branchid, authorityhashes, scope, cutoff, observer, freezehash.

deformationbasispacket

Required fields: basisid, sectors, basisvectors, units, normalizations, kineticinnerproducts, prereductionstatus, towerscope, ownerhashes, omittedsectorproofs.

functionspacedomain_packet

Required fields: operators, functionspaces, domains, boundaryconditions, selfadjointness, ellipticityorhyperbolicity, fredholmdata, essentialspectrum, domainvariation_tests.

gaugequotientpacket

Required fields: gaugebasis, gaugematrix, ghostoperator, residualsymmetries, boundarygaugedata, quotientbasis, normon_quotient.

constraintjacobianpacket

Required fields: constraintrows, provenancehashes, evaluationpoint, jacobian, arithmetictype, rank, pivots, nullspace, sensitivity.

rankcertificatepacket

Required fields: matrixshape, exactorintervalrank, determinantsorminors, singularvalues, conditionnumber, tolerance_tests, classification.

physicaltangentpacket

Required fields: kernelbasis, gaugeimagebasis, quotientbasis, dimension, sectorlabels, unresolveddirections.

obstructionintegrabilitypacket

Required fields: directions, nonlinearequations, ordersortheorem, obstructionmap, h2targets, integrablebranches, convergence_scope.

solutiongermpacket

Required fields: localbranches, singularstrata, branchintersections, localuniqueness, globaldisconnectedbranches, scope.

kineticmetricpacket

Required fields: fields, kineticmatrixoroperator, signature, zeromodes, normalization, ghosttests, constraintreduction.

mixedhessianpacket

Required fields: sectororder, diagonalblocks, offdiagonalblocks, vanishingtheorems, schurcomplements, eigenvalue_bounds, normalization.

spectrumcoercivitypacket

Required fields: discretespectrum, essentialspectrum, kernel, negativemodes, complexmodes, gaporcoercivitybound, tailbound, domain.

protectedscalarpacket

Required fields: scalarids, roles, protectingmechanisms, prefreezehashes, masses, mixing, radiativeowner, reopentriggers.

namedmodulipacket

Required fields: volume, relativeradii, radion, interval, wilsonlines, bundlemoduli, boundarypositions, localizedmodes, observermap_deformations, statuses.

boundarysupportrigidity_packet

Required fields: supports, incidence, supportdeformations, edgemodes, horizonmodes, interfacemodes, cornermodes, gluing, fluxbalance.

towertailpacket

Required fields: zeromodescope, kklevels, boundarytower, continuumoressentialspectrum, tailtheorem, cutoff, nonperturbative_channels.

anticircularitypacket

Required fields: constraintsused, freezetimes, rationales, candidateblindnesstests, laterows, constructionrows, restart_status.

variationalownershippacket

Required fields: variables, equationdispositions, constraints, auxiliaries, externalinputs, unowned_equations.

stabilityroutepacket

Required fields: route, intrinsicstatus, obstructionstatus, dynamicstatus, constitutivestatus, negativestatus, routejustification.

vacuumscaleinterface_packet

Required fields: deformationstatus, vacuumoffsetstatus, scalestatus, phasehistorystatus, external_dependencies.

bifurcationspectralflow_packet

Required fields: parameterregion, rankstrata, eigenvaluetracks, crossings, bifurcations, exceptionalpoints, margins.

nonlinearbasinpacket

Required fields: lyapunovorenergyfunction, basin, finiteamplitudethresholds, runawaychannels, resonances, globalexistencescope.

timedependentstability_packet

Required fields: backgroundhistory, evolutionoperator, floquetdata, adiabaticity, transientgrowth, memory, excitation_bounds.

quantumradiativestability_packet

Required fields: effectiveactionorders, loopcorrectedhessian, rgflow, regulators, schemes, truncations, wardidentities, stability_intervals.

metastabilitydecaypacket

Required fields: lowerbranches, bounceorinstanton, action, prefactor, decayrate, lifetime, topologychange, timescalecomparison.

topologicalprotectionpacket

Required fields: indices, charges, superselectionsectors, defects, symmetries, wallcrossing, invariance_proofs.

compositeeffectiverigidity_packet

Required fields: effectiveactorids, collectivecoordinates, internalmodes, binding, decaychannels, mixing, validityscale, promotion_status.

robustnessuncertaintypacket

Required fields: parametercovariance, eigenvalueintervals, singularvalueintervals, conditionnumbers, identifiability, numericaltolerances, robustness_margins.

grammarexhaustionpacket

Required fields: deformationgrammar, stabilizationgrammar, canonicalaliases, enumerationmethod, omittedrivaltests, unclassifiedcandidates, exhaustionproof.

crossscalecontinuation_packet

Required fields: scales, thresholds, modetracking, constraintranktracking, eigenvalueflow, decoupling_tests, matching, crossings.

symplectickreinpacket

Required fields: reducedsymplecticform, diracbrackets, kineticsignature, energysignature, gradienttests, kreinsignatures, jordanblocks, constraint_rank.

opensystemstability_packet

Required fields: systemenvironmentsplit, generatororchannel, completepositivity, stationarystates, spectralgap, noiseinducedtransitions, largedeviationrates, ensembledependence.

pseudospectraltransientpacket

Required fields: operatorids, physicalnorm, normalityclass, resolventregion, resolventbounds, pseudospectrum, spectralprojectorconditioning, semigrouporpropagatorbounds, maximumtransientgain, timewindow, thresholdcomparison.

resonancelongtime_packet

Required fields: moderoster, interactioncoefficients, resonancelattice, detunings, smalldivisors, normalformoraveraging, secularterms, resonanceoverlap, longtimebound, stabilityhorizon, failure_channels.

numericalcertificationpacket

Required fields: artifacthashes, softwareenvironment, arithmetic, discretizationorbasis, solver, residuals, backwarderrors, refinementsequence, convergenceorder, intervalorexactenclosures, spectralpollutiontests, independentcrosscheck, reproductioninstructions.

singularlimitpacket

Required fields: requiredlimits, parameterfamilies, limitorders, commutatorsoflimits, uniformdomains, uniformbounds, gaporcoercivityuniformity, constraintrankuniformity, limitobjects, orderdependence, claim_ceiling.

causalprincipalsymbol_packet

Required fields: pdesystem, reducedprincipalsymbol, characteristicvariety, causalcones, hyperbolicityclass, symmetrizerordiagonalizerbounds, constraintpropagationoperator, boundarycompatibility, deformationmargin, failuremodes.

globalgaugeorbit_packet

Required fields: gaugegroup, configurationsector, orbittypes, stabilizers, reduciblestrata, fporghostspectrum, gribovregionsorslicetheorem, globalcopies, orbitspacetopology, boundarygaugesectors, quotientscope, unresolvedglobal_sectors.

integratedrigiditycertificatev41

Required fields: legacycertificate, v40certificate, constraintstates, packethashes, legacypreservationresidual, v40preservationresidual, rigidityresidual, scope, claimceiling, terminal, reopen_triggers.

negativecontrolspacket

Required fields: controlsrun, expectedresults, actual_results, failures, artifacts.

rigiditycertificatev1_3

Required fields: candidateid, branchid, deformationcomplex, functionspaces, sectorsincluded, gaugeredundancyquotient, h1physical, obstructionmap, localmoduligerm, independentconstraintregister, constraintderivedtangentspace, exactconstraintclassification, kineticmetric, mixedphysicalhessian, schurcomplements, protectedscalarwhitelist, zeromodes, negativemodes, towerandboundaryscope, tunnellingscope, scaleinterface, vacuumoffsetstatus, evidencehashes, route_terminal, verdict.

integratedrigiditycertificatev40

Required fields: legacycertificate, constraintstates, packethashes, preservationresidual, rigidityresidual, scope, claimceiling, terminal, reopen_triggers.

rigidityresidualpacket

Required fields: residualvector, nonzerocomponents, owners, required_actions.

terminal_packet

Required fields: gateid, physicalendpoint, projectendpoint, scope, claimceiling, evidencehashes, openrows, reopen_triggers.

8. Completion residual

Define

\[ \mathcal R{m RIG}^{(4.1)}=(N{m missing\ sectors},N{m bad\ quotient},N{m tangent},N{m obstruction},N{m domain},N{m negative},N{m mixed},N{m essential},N{m bifurcation},N{m nonlinear},N{m history},N{m quantum},N{m decay},N{m boundary},N{m composite},N{m uncertain},N{m unexhausted},N{m transient},N{m resonance},N{m numerical},N{m singular\ limits},N{m causal},N{m global\ gauge}). \]

A positive maximum-scope Rigidity terminal requires every applicable component to vanish or be discharged by an exact theorem. NOT-APPLICABLE is not zero without a type proof.

9. All-gate profiles

The machine-readable controlling map is 02REGISTRIES/RIGIDITYGATECROSSWALKV4_1.json.

Black-hole-singularity

• Role: Test geometry, horizon, interior, boundary, nonlinear, topology-changing, and quantum stability.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Born-rule

• Role: Test probability/instrument structures against perturbations of state, algebra, environment, and regional factorization.
• Rows: RIG-C05, RIG-C09, RIG-C11, RIG-C13, RIG-C15, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: boundarysupportrigiditypacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-01

• Role: Test the a6 operator result against domain, mixed Hessian, regulator, tower, and scale continuation.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-02

• Role: Separate finite-volume/perturbative stability from a true nonperturbative Yang-Mills mass-gap result.
• Rows: RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C16, RIG-C17, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, topologicalprotectionpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-05-stability

• Role: Test vacuum-offset mechanism across moduli, histories, loops, boundaries, and nonperturbative decay.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-05-value

• Role: Keep value prediction separate from background/ensemble/metastability and uncertainty robustness.
• Rows: RIG-C05, RIG-C09, RIG-C11, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C20, RIG-C21, RIG-C22, RIG-C24
• Active packets: crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-08

• Role: Test inflationary background and perturbations for slow-roll, multifield, parametric, quantum, and stochastic stability.
• Rows: RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-10/BG-10

• Role: Test baryogenesis through phase-transition, transport, CP-source, washout, and stochastic-history stability.
• Rows: RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-11

• Role: Test dark-sector candidate, portal, relic history, composites, decays, and environmental stability.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Gap-13

• Role: Test black-hole saddle, replica, horizon/edge, microstate, information, and nonperturbative stability.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, topologicalprotectionpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

Lambda-catastrophe

• Role: Test vacuum response across scales, phases, histories, regions, and radiative/nonperturbative effects.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-1

• Role: Determine whether the selected Shape is intrinsically fixed, merely constitutively frozen, or dynamically stabilized.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C05, RIG-C06, RIG-C08, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-10

• Role: Enforce scope and anti-promotion across truncation, tower, refinement, and parameter continuation.
• Rows: RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-2

• Role: Check that gauge-sector ownership and vector kernels are stable against omitted/mixed/tower deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-3

• Role: Test chiral-family and mirror-sector robustness under domain, boundary, tower, and radiative deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C07, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C14, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, namedmodulipacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, protectedscalarpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket, variationalownershippacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-4

• Role: Test charge/anomaly/global-kernel results against bundle, boundary, topological, and spectral-flow deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-5

• Role: Test EWSB vacuum, Higgs/Wilson sector, mass mixing, hierarchy, and history stability.
• Rows: RIG-C01, RIG-C02, RIG-C05, RIG-C06, RIG-C07, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-6

• Role: Primary full moduli and vacuum-stability gate; execute the complete Rigidity object.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, obstructionintegrabilitypacket, opensystemstabilitypacket, parentmanifestpacket, physicaltangentpacket, protectedscalarpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket, variationalownershippacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-7

• Role: Test threshold-unification/proton-safety sensitivity to spectra, matching, and deformation routes.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-8

• Role: Test flavor fits and hierarchy against ruler, mixing, radiative, composite, and history instabilities.
• Rows: RIG-C05, RIG-C06, RIG-C07, RIG-C09, RIG-C11, RIG-C12, RIG-C14, RIG-C15, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, protectedscalarpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

SG-9

• Role: Test dangerous-operator suppression and proton lifetime under deformations and nonperturbative channels.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C13, RIG-C15, RIG-C16, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: compositeeffectiverigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-10

• Role: Primary compactification-consistency stress test across all geometric, bundle, boundary, tower, and quantum sectors.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C05, RIG-C06, RIG-C07, RIG-C08, RIG-C09, RIG-C10, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C17, RIG-C18, RIG-C19, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: anticircularitypacket, bifurcationspectralflowpacket, boundarysupportrigiditypacket, compositeeffectiverigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, namedmodulipacket, negativecontrolspacket, nonlinearbasinpacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, protectedscalarpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-14

• Role: Test above-cutoff causality/unitarity under spectral flow, nonlinear, boundary/horizon, and open-system effects.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C14, RIG-C15, RIG-C16, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, timedependentstabilitypacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-3

• Role: Test positivity reconstruction against domain, mixed-sector, essential-spectrum, regional, and open-system instabilities.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C13, RIG-C15, RIG-C18, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: boundarysupportrigiditypacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-4

• Role: Test anomaly/global consistency under deformations, wall crossing, determinant-line, and boundary support changes.
• Rows: RIG-C03, RIG-C04, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-5A/5B

• Role: Test graviton-sector physical stability, norm, mixing, nonlinear basin, and quantum corrections.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, nonlinearbasinpacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-5C

• Role: Prevent truncation-local UV claims from being promoted without global/refinement/metastability robustness.
• Rows: RIG-C09, RIG-C11, RIG-C12, RIG-C13, RIG-C15, RIG-C16, RIG-C20, RIG-C21, RIG-C22, RIG-C23, RIG-C24
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, negativecontrolspacket, nonlinearbasinpacket, opensystemstabilitypacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, symplectickreinpacket, terminalpacket, towertail_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-7

• Role: Test chiral-domain/anomaly-descent stability under boundary, topological, tower, and global deformations.
• Rows: RIG-C01, RIG-C02, RIG-C03, RIG-C04, RIG-C08, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C17, RIG-C18, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, boundarysupportrigiditypacket, constraintjacobianpacket, crossscalecontinuationpacket, deformationbasispacket, functionspacedomainpacket, gaugequotientpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, namedmodulipacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertailpacket, variationalownership_packet
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

UQF-9

• Role: Test heat-kernel/UV outputs under operator-domain, regulator, refinement, and spectral-flow changes.
• Rows: RIG-C05, RIG-C06, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C20, RIG-C21, RIG-C22, RIG-C23
• Active packets: bifurcationspectralflowpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, kineticmetricpacket, metastabilitydecaypacket, mixedhessianpacket, negativecontrolspacket, parentmanifestpacket, quantumradiativestabilitypacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, stabilityroutepacket, symplectickreinpacket, terminalpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

theta-bar-QCD

• Role: Test topological-angle mechanism under bundle, vacuum-branch, RG, threshold, and tunnelling deformations.
• Rows: RIG-C03, RIG-C04, RIG-C09, RIG-C11, RIG-C12, RIG-C15, RIG-C16, RIG-C17, RIG-C20, RIG-C21, RIG-C22
• Active packets: bifurcationspectralflowpacket, constraintjacobianpacket, crossscalecontinuationpacket, functionspacedomainpacket, grammarexhaustionpacket, integratedrigiditycertificatev40, metastabilitydecaypacket, negativecontrolspacket, obstructionintegrabilitypacket, parentmanifestpacket, physicaltangentpacket, quantumradiativestabilitypacket, rankcertificatepacket, rigiditycertificatev13, rigidityresidualpacket, robustnessuncertaintypacket, solutiongermpacket, spectrumcoercivitypacket, terminalpacket, topologicalprotectionpacket, towertailpacket
• Discipline: Execute selected rows; retain every inactive packet as NOT-APPLICABLE with a gate-specific type proof. Do not use an inherited terminal as evidence.
• Claim ceiling: No rigidity or stability claim may exceed the weakest unclosed selected row, function-space scope, tower/tail scope, uncertainty interval, history, or nonperturbative channel.

9A. V4.1 second-pass gate deltas

The V4.1 machine-readable crosswalk is controlling. The following rows strengthen the V4.0 profiles; they never remove an old row.

Black-hole-singularity — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Born-rule — V4.1 delta

• Added rows: RIG-C25, RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-01 — V4.1 delta

• Added rows: RIG-C27, RIG-C28
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-02 — V4.1 delta

• Added rows: RIG-C25, RIG-C27, RIG-C28, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, singularlimit_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-05-stability — V4.1 delta

• Added rows: RIG-C26, RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, resonancelongtime_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-05-value — V4.1 delta

• Added rows: RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-08 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-10/BG-10 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-11 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtime_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Gap-13 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

Lambda-catastrophe — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-1 — V4.1 delta

• Added rows: RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-10 — V4.1 delta

• Added rows: RIG-C27, RIG-C28
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-2 — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-3 — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-4 — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-5 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtime_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-6 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-7 — V4.1 delta

• Added rows: RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-8 — V4.1 delta

• Added rows: RIG-C27
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

SG-9 — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-10 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-14 — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29, RIG-C30
• Added active packets: causalprincipalsymbolpacket, globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimit_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-3 — V4.1 delta

• Added rows: RIG-C25, RIG-C27, RIG-C28, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, singularlimit_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-4 — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-5A/5B — V4.1 delta

• Added rows: RIG-C25, RIG-C26, RIG-C27, RIG-C28, RIG-C29
• Added active packets: causalprincipalsymbolpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, resonancelongtimepacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-5C — V4.1 delta

• Added rows: RIG-C25, RIG-C27, RIG-C28, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, singularlimit_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-7 — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

UQF-9 — V4.1 delta

• Added rows: RIG-C25, RIG-C27, RIG-C28
• Added active packets: integratedrigiditycertificatev41, numericalcertificationpacket, pseudospectraltransientpacket, singularlimitpacket
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

theta-bar-QCD — V4.1 delta

• Added rows: RIG-C27, RIG-C30
• Added active packets: globalgaugeorbitpacket, integratedrigiditycertificatev41, numericalcertification_packet
• Second-pass discipline: Execute each selected second-pass row or provide a gate-specific NOT-APPLICABLE type proof. Numerical certification is universal for every computed witness.

9B. Preservation and sufficiency boundary

V4.1 is preservation-complete only when both residuals vanish:

\[ \mathcal R{m pres}^{1.3 o4.1}=0, \qquad \mathcal R{m pres}^{4.0 o4.1}=0. \]

Every BB-RIG-1.3 heading is mapped to specific active constraints and evidence packets in LEGACYPRESERVATIONMATRIXV41.json. The complete prior V4.0 package is archived under 06SOURCES/PREVIOUSBBRIG40PACKAGE/ and hash-checked file by file.

Sufficiency ceiling. This authority is sufficient for the Rigidity-owned obligations of the present 30-gate corpus. It cannot by itself close gates whose remaining rows belong to Shape, Dynamics, Boundary, Scale, Granularity, Vacuum, Observer, Interdependence, Time Synchronization, measurement, or an external theorem. A gate may close only when every applicable cross-block row and candidate-specific witness passes.

10. Validation and preservation theorem

The authority is preservation-complete relative to BB-RIG-1.3 and BB-RIG-4.0 only when:

\[ \mathcal R{\rm pres}=(N{\rm unmapped},N{\rm weakened},N{\rm lost\ controls},N{\rm lost\ fields},N{\rm hash\ mismatches},N_{\rm v4.0\ mismatches})=0. \]

The exact legacy source is normative and archived. The validator must check its SHA-256, every constraint ID, every gate profile, every packet schema, every thought experiment, and all internal references.

Appendix A — Exact BB-RIG-1.3 authority

The following source is preserved verbatim and remains binding.

--- BEGIN EXACT LEGACY SOURCE ---

--- title: "BB-RIG-1.3 — Intrinsic Rigidity, Constraint Jacobian, Obstruction, and Physical Stability" buildingblockid: "BB-RIG-1.3" version: "1.3" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — EVIDENCE-CRITERIA EXTENDED — NOT FROZEN" scope: "Complete owned specification under package v3.3.2" protocolfreeze: false canonicalratification: false packageauthorityversion: "3.3.2" ---

BB-RIG-1.3 — Intrinsic Rigidity, Constraint Jacobian, Obstruction, and Physical Stability

0. Purpose

This block answers the high-information question:

Can the independently justified physical constraints force a candidate whose complete physical deformation space is empty or fully obstructed, or must remaining deformation fields be stabilized by Dynamics or a declared construction Actor?

It exists because the following statements are different:

\[ \text{one metric was selected}, \]

\[ \text{there are no neighboring exact solutions}, \]

\[ \text{there are no physical scalar deformations}, \]

\[ \text{the physical quadratic form is positive}. \]

A candidate may satisfy one and fail another.

1. Core distinctions

1.1 Mathematical extension

A candidate may be embedded in a broad mathematical configuration space \(\widetilde{\mathcal C}\).

Directions in \(T\widetilde{\mathcal C}\) are not automatically physical.

1.2 Physical configuration space

After gauge quotient, exact structural rules, boundary conditions, and domain restrictions, the physical configuration space is \(\mathcal C_{\rm phys}\).

1.3 Solution space

The physical vacuum solution set is

\[ \mathcal M{\rm vac}=\{z\in\mathcal C{\rm phys}:\mathcal F_I[z]=0\}/\text{gauge}. \]

1.4 Physical fluctuation space

Off-shell physical perturbations lie in the tangent space of \(\mathcal C{\rm phys}\), not merely the tangent space of \(\mathcal M{\rm vac}\).

A direction can fail to integrate into another exact solution and still be a physical unstable fluctuation.

Therefore:

\[ T{[z]}\mathcal M{\rm vac}=0 \not\Rightarrow Q_{\rm phys}>0. \]

2. Complete deformation complex

For each complete parent object \(\mathfrak P\), construct a deformation complex or an explicitly equivalent operator sequence:

\[ 0\longrightarrow \mathfrak{aut}(\mathfrak P) \xrightarrow{d0} T{\mathfrak P}\mathcal C \xrightarrow{d1} \mathcal E{\mathfrak P} \xrightarrow{d2} \mathcal O{\mathfrak P} \longrightarrow0. \]

Here:

• \(\mathfrak{aut}(\mathfrak P)\) contains infinitesimal gauge, diffeomorphism, and exact presentation redundancies;
• \(T_{\mathfrak P}\mathcal C\) contains complete candidate deformations;
• \(\mathcal E_{\mathfrak P}\) contains linearized equations, constraints, boundary, and domain conditions;
• \(\mathcal O_{\mathfrak P}\) contains nonlinear obstruction data.

The physical infinitesimal deformation space is

\[ H^1{\rm phys}(\mathfrak P) = \frac{\ker d1}{\operatorname{im}d_0}. \]

When a cohomological complex is unavailable, an equivalent gauge-fixed kernel/cokernel construction is permitted, but it must cover the same physical content.

3. Complete sector inventory

The complex must include, where present:

• internal metric shape deformations;
• total volume and breathing modes;
• relative radii;
• interval and radion modes;
• mixed external–internal metric components;
• bundle deformations;
• connection and Wilson-line deformations;
• flux and top-form branch deformations;
• complex, almost-complex, torsion, and structure deformations;
• matter-background deformations;
• boundary and fixed-set positions;
• junction and edge-mode deformations;
• finite Actor continuous parameters;
• Rulebook/domain deformations;
• observer-map deformations that alter physical records.

A metric-only deformation analysis is incomplete unless a theorem proves all omitted sectors irrelevant or absent.

4. Physical redundancy removal

Before counting deformations, quotient:

• diffeomorphisms;
• internal gauge transformations;
• BRST-exact directions;
• exact bundle automorphisms;
• boundary gauge redundancies;
• presentation equivalences;
• observer relabelings with no record consequence.

Gauge fixing must not remove physical modes.

The certificate must report:

• gauge condition;
• ghost/Faddeev–Popov operator;
• zero modes of the gauge-fixing operator;
• residual transformations;
• boundary behavior;
• norm on the quotient.

5. Intrinsic infinitesimal rigidity

A candidate is infinitesimally physically rigid when

\[ H^1_{\rm phys}(\mathfrak P)=0. \]

This means there is no nonredundant infinitesimal deformation satisfying the linearized physical conditions.

The claim must state scope:

• invariant deformations only;
• all smooth deformations in a stated function space;
• all modes below a cutoff;
• full tower;
• bulk only;
• bulk plus boundaries and global sectors.

An invariant-sector result cannot be promoted to all deformations.

6. Nonlinear obstruction and local rigidity

If \(H^1_{\rm phys}\neq0\), construct the nonlinear obstruction or Kuranishi map

\[ \kappa:H^1{\rm phys}\to H^2{\rm phys}. \]

A candidate is locally obstruction-rigid when the local zero set satisfies

\[ \kappa^{-1}(0)=\{0\} \]

near the reference point.

This shows there is no nearby family of exact solutions.

It does not prove stability of off-shell physical fluctuations.

6.1 Obstruction certificate

The obstruction analysis must specify:

• nonlinear equations used;
• expansion order or exact theorem;
• function space;
• gauge quotient;
• boundary and global conditions;
• whether all formal obstructions converge to an actual local statement;
• excluded topology-changing directions.

7. Constraint-derived rigidity

The highest-information route uses constraints justified independently of the candidate result.

Let

\[ C_a(\mathfrak P)=0 \]

be the pre-existing physical constraints: carriers, global group, chirality, family count, anomalies, boundary consistency, or other independently sourced obligations.

Define

\[ \mathcal M{\rm admiss} = \{\mathfrak P':Ca(\mathfrak P')=0\ \forall a\}/\text{redundancy}. \]

The infinitesimal admissible deformation space is

\[ T{[\mathfrak P]}\mathcal M{\rm admiss} = \frac{\bigcapa\ker(dCa)}{\operatorname{im}d_{\rm gauge}}. \]

7.1 Strong result

If

\[ T{[\mathfrak P]}\mathcal M{\rm admiss}=0, \]

the already justified constraint register infinitesimally fixes the complete physical object at the stated scope.

7.2 Obstructed result

If the tangent is nonzero but every nonzero direction is nonlinearly obstructed, the candidate is locally isolated under the independent constraint register.

7.3 Surviving deformation result

If integrable nonzero directions survive, constraint-first selection does not intrinsically fix the candidate.

Dynamics must stabilize them, a declared construction must remove them, or the candidate must fail the intended rigidity claim.

7.4 Anti-circularity conditions

A constraint can participate in intrinsic rigidity only if:

• it predates the rigidity calculation;
• its rationale does not name the preferred candidate;
• it applies equally to rivals;
• its pass rule was frozen before outcomes;
• it corresponds to an observed record, exact consistency condition, or prior foundational axiom;
• it was not introduced solely to kill a discovered negative direction.

A post-failure constitutive constraint is a construction route, not a derived rigidity theorem.

8. Exact constraints and constitutive rigidity

A candidate may lawfully define a physical constraint surface

\[ \Phi_A=0. \]

If the constraints are second class, physical evolution and measure use the Dirac reduction.

This may remove deformation directions from the physical phase space.

The result is classified as:

• DERIVED-CONSTRAINT-RIGIDITY when \(\Phi_A\) follows from independently justified prior constraints;
• CONSTITUTIVE-RIGIDITY when \(\Phi_A\) is a declared law of the candidate;
• TARGET-LOADED-RESTRICTION when \(\Phi_A\) was introduced because a known direction failed and lacks independent provenance.

Only the first can convert “rigid by decree” into a theorem-like result.

9. Physical Hessian and kinetic normalization

Stability is evaluated on physical off-shell perturbations after gauge quotient and exact constraints.

Let \(K\) be the kinetic metric and \(H\) the second variation of the effective action.

The physical mass operator is schematically

\[ \mathcal M^2{\rm phys}=K{\rm phys}^{-1}H_{\rm phys}. \]

The quadratic form is

\[ Q{\rm phys}[\delta z] = \langle\delta z,H{\rm phys}\delta z\rangle. \]

A stable sector requires:

• positive physical norm;
• no negative eigenvalue;
• correct treatment of gauge zero modes;
• declared treatment of exact physical zero modes;
• boundary stability;
• no uncontrolled complex frequencies;
• no rank-changing constraint direction.

9.1 Coercivity

For full stability on a domain \(\mathcal D\), the preferred result is a bound

\[ Q_{\rm phys}[\delta z] \ge c\,\|\delta z\|^2, \qquad c>0, \]

for all admissible non-whitelisted perturbations.

A finite matrix positivity result is scoped to the truncated sector unless a tail theorem is supplied.

10. Complete mixed Hessian

The full Hessian must include off-diagonal mixing among all physical sectors.

If

\[ H= \begin{pmatrix} A&B\\B^T&D \end{pmatrix}, \]

positivity of \(A\) and \(D\) separately is insufficient.

When \(D>0\), the Schur complement condition is

\[ A-BD^{-1}B^T>0. \]

Every omitted mixed block requires a theorem of vanishing or a controlled decoupling estimate.

11. Stability routes

11.1 Route I — intrinsic physical rigidity

The complete physical deformation space is empty at the claimed scope.

No stabilizing potential or constraint multiplier is required.

11.2 Route II — obstruction rigidity

The local solution germ is isolated, but off-shell physical fluctuation stability remains separately tested.

11.3 Route III — dynamical stabilization

Physical deformation fields exist and a lawful parent effective action yields a positive full mixed Hessian.

11.4 Route IV — constitutive or Actor realization

A declared exact constraint, projector, Actor, Co-Actor, boundary mechanism, or stiffness term removes or stabilizes directions.

The route is legitimate but must be graded as a construction unless independently derived.

11.5 Route V — negative result

A physical negative mode survives or the constraint system is inconsistent.

This is a legitimate closed-negative finding when scoped and recorded honestly.

12. Rigidity ladder

| Level | Result | Meaning | Stability implication | |---|---|---|---| | R0 | Coordinate or parameter convention fixed | Presentation only | None | | R1 | Unique point inside a restricted ansatz | Ansatze-level isolation | None outside ansatz | | R2 | \(H^1_{\rm phys}=0\) at stated scope | Infinitesimal rigidity | Strong, but physical Hessian still checked | | R3 | Local solution germ isolated by obstruction | Nonlinear local rigidity | Does not exclude negative off-shell mode | | R4 | Complete physical scalar deformation space empty | Intrinsic physical rigidity | Closes scalar-modulus leg at stated scope | | R5 | Full physical Hessian coercive | Dynamical stability | Closes perturbative stability at stated scope | | R6 | Tower, boundary, mixed, and nonperturbative scope complete | Full scoped vacuum stability | Strongest scoped result |

No lower rung may be narrated as a higher rung.

13. Protected scalar whitelist

Intended physical scalars are not counted as unwanted moduli merely because they are geometric.

For every whitelisted scalar, require:

• exact identity;
• physical role;
• protecting symmetry;
• kinetic positivity;
• mass-generation mechanism;
• mixing analysis;
• observed or predicted coupling scope;
• vacuum and radiative stability owner.

A whitelist may not be created after discovering an unwanted zero mode without branch restart.

14. Scale and vacuum interfaces

Intrinsic rigidity does not by itself solve:

• the value of the compactification scale;
• radion stabilization when a scale fluctuation remains;
• vacuum-offset response;
• the residual cosmological constant;
• phase-transition history.

Every rigidity certificate must carry two independent labels:

DEFORMATION/PHYSICAL-STABILITY STATUS:
  intrinsic / obstructed / dynamic / construction / unstable / open.

VACUUM-OFFSET STATUS:
  parent-protected / scoped-construction / unprotected / open.

A valid terminal may be:

INTRINSICALLY-RIGID / LAMBDA-UNPROTECTED

which is not a complete protected vacuum.

15. Full-tower, boundary, and nonperturbative scope

A complete stability claim must state coverage of:

• zero modes;
• nonzero KK tower;
• boundary and fixed-set modes;
• mixed sectors;
• topology-changing tunnelling;
• finite-temperature or cosmological transitions;
• radiative order;
• cutoff and tail.

Possible honest scopes are:

• invariant zero-mode sector;
• complete zero-mode sector;
• all modes below cutoff;
• perturbative full tower;
• global/nonperturbative vacuum.

16. Rigidity certificate

rigidity_certificate:
  candidate_id:
  branch_id:
  deformation_complex:
  function_spaces:
  sectors_included:
  gauge_redundancy_quotient:
  h1_physical:
  obstruction_map:
  local_moduli_germ:
  independent_constraint_register:
  constraint_derived_tangent_space:
  exact_constraint_classification:
  kinetic_metric:
  mixed_physical_hessian:
  schur_complements:
  protected_scalar_whitelist:
  zero_modes:
  negative_modes:
  tower_and_boundary_scope:
  tunnelling_scope:
  scale_interface:
  vacuum_offset_status:
  evidence_hashes:
  route_terminal:
  verdict:

17. Mandatory negative controls

17.1 Isolated saddle

A solution is isolated among exact vacua but has one negative off-shell mode.

Expected classification:

OBSTRUCTED/ISOLATED: PASS
PHYSICAL STABILITY: FAIL

17.2 Frozen by decree

A negative deformation is removed by a newly declared constitutive constraint lacking prior provenance.

Expected classification:

CONSTITUTIVE-RIGIDITY / CONSTRUCTION-ANCHOR
NOT INTRINSICALLY DERIVED

17.3 Constraint-derived rigid candidate

Independent carrier and anomaly constraints have a full-rank Jacobian whose physical kernel is zero.

Expected classification:

DERIVED-CONSTRAINT-RIGIDITY

with the physical Hessian still separately recorded.

17.4 Positive diagonal, negative mixed mode

Metric and bundle diagonal Hessians are positive; their Schur complement is negative.

Expected result: stability fails.

17.5 Rigid but vacuum-unprotected

No physical moduli; constant matter shift changes curvature.

Expected result: rigidity passes, vacuum status fails.

18. Falsifiers and reopen triggers

This block fails or reopens if:

• a deformation sector is omitted;
• gauge fixing hides a physical mode;
• invariant rigidity is promoted to full rigidity;
• isolation is promoted to stability;
• an obstruction map is asserted but not constructed;
• a post-failure constraint is called independently derived;
• a mixed Hessian block is ignored;
• kinetic normalization is missing;
• an intended scalar is whitelisted post hoc;
• a zero-mode result is promoted to the tower;
• intrinsic rigidity is used as proof of vacuum-offset protection;
• a physical negative mode survives.

19. Terminal grammar

CLOSED / DERIVED-INTRINSIC-PHYSICAL-RIGIDITY / RESOLVED +0

CLOSED-SCOPED / OBSTRUCTED-LOCAL-RIGIDITY /
PHYSICAL-HESSIAN SEPARATELY GRADED

CLOSED / DERIVED-POSITIVE-FULL-PHYSICAL-HESSIAN / RESOLVED +0

CLOSED-SCOPED / REALIZED-GIVEN-STABILIZING-ACTOR /
CONSTRUCTION-ANCHOR

CLOSED-NEGATIVE / PHYSICAL INSTABILITY CERTIFIED

OPEN / DEFORMATION OR STABILITY CERTIFICATE INCOMPLETE

20. Current status

BB-RIG-1.3

STATUS:
  DEVELOPMENT AUTHORITY CANDIDATE — NOT FROZEN.

PURPOSE OF NEXT TEST:
  Determine whether independently justified constraints can eliminate the
  complete physical deformation space, rather than merely removing a known
  bad direction after failure.

Integrated boundary/orbifold authority

This block is part of the Candidate-Neutral Building-Block Authority v3.1 — Boundary and Inflow Development Candidate.

The following rule is binding across the set:

A bulk or quotient-space anomaly cancellation statement does not close a theory with orbifold fixed sets or physical boundaries. Each stratum requires a local anomaly ledger, and the complete quantum theory requires a relative/global trivialization compatible with gluing.

The dedicated owner is BB-BND-1 — Orbifold Fixed-Set, Relative-Anomaly, and Inflow Completion.

Every candidate containing a boundary, fixed set, defect, quotient singularity, interface, or parity projection must provide a boundaryanomalycompletion_certificate. Until that certificate passes:

• a chirality or no-mirror result is CONDITIONAL-ON-BND;
• quantum BRST/BV nilpotency is CONDITIONAL-ON-BND;
• a global-bordism result for the closed bulk target is not a complete boundary result;
• cancellation after integrating over the compact direction is not accepted as local gauge invariance;
• no gate may claim that the interval/orbifold construction is fully quantum consistent.

Boundary/orbifold completion — boundary data in the deformation complex

The complete deformation complex must include:

• fixed-set position and normal-bundle deformations;
• parity/domain deformations;
• localized bundle and connection deformations;
• inflow coefficient branches;
• boundary counterterm branches;
• determinant-line trivialization data.

Anomaly cancellation may legitimately obstruct a deformation only when the anomaly constraint:

1. is independently justified;
2. predates the candidate outcome;
3. is applied equally to all candidates;
4. is computed on the full stratified parent object.

A boundary condition chosen after discovering an unstable or anomalous mode is a construction repair, not intrinsic rigidity.

The physical Hessian and deformation complex are conditional on BND-C01 through BND-C10; an anomalous boundary domain is not a physical vacuum domain.

Block-Specific Standalone Interface — Intrinsic Rigidity and Stability

Owned questions

This block alone owns:

• complete physical deformation space;
• gauge/constraint redundancy quotient;
• infinitesimal deformation cohomology;
• nonlinear obstruction and integrability;
• distinction between isolation and stability;
• complete mixed physical Hessian;
• intrinsic, obstruction, dynamical, constitutive, and negative routes;
• protected-scalar whitelist.

Required outputs

• deformation-complex manifest;
• \(H^1_{\rm phys}\) or equivalent tangent-space result;
• obstruction map;
• mixed Hessian and kinetic normalization;
• full-tower/boundary/tunnelling scope;
• route-sensitive terminal;
• rigidity certificate.

Non-ownership boundary

Rigidity does not imply:

• vacuum-offset protection;
• background-equation completeness;
• anomaly-free boundary domains;
• candidate selection.

Algorithmic computation of constraint-derived rigidity

Definitions are not enough. Every concrete candidate must publish the finite or discretized linear map that decides rigidity.

1. Freeze the deformation basis

Choose an ordered basis

\[ \mathcal E{\rm def}=(e1,\ldots,e_N) \]

covering every in-scope deformation. For the current compact branch, the minimal homogeneous/low-mode basis should include at least:

\[ (\delta\log u1,\delta\log u2,\delta\log u3, \delta\log R{S^2},\delta\log RI, \delta WY,\delta WH, \delta b1,\delta x0,\delta x\pi, \delta\Pi_{\rm obs}), \]

with any additional bundle, boundary, or Actor modes required by the full manifest.

Each basis vector must state:

• sector;
• normalization and units;
• kinetic inner product;
• gauge or physical status before reduction;
• cutoff/tower scope;
• parent-manifest owner.

2. Build the gauge matrix

Choose a basis of infinitesimal redundancies and compute

\[ G{I\alpha}=(d{\rm gauge}\epsilon\alpha)I. \]

Publish gauge_matrix.csv. Rank must be stable under the declared exact or interval-certified arithmetic.

3. Build the constraint Jacobian

For every independently justified active constraint \(C_a\), calculate

\[ J{aI} = dCa(eI) = \left.\frac{\partial Ca}{\partial q^I}\right|{\mathfrak P*}. \]

Publish the complete matrix—not only its rank—as constraint_jacobian.csv, with:

• row ID matching the evidence registry;
• constraint freeze hash;
• evaluation point;
• analytic expression or certified numerical interval;
• zero-row justification;
• provenance.

The package includes RIGIDITYCONSTRAINTJACOBIANTEMPLATEV332.csv.

4. Reduce exactly

Compute:

\[ K=\ker J, \qquad T_{\rm admiss}=K/\operatorname{im}G. \]

The artifact must include:

• exact pivots or certified singular-value intervals;
• rank \(J\);
• rank \(G\);
• a basis for \(K\);
• a basis for the quotient;
• sensitivity to numerical tolerance;
• branch hash.

5. Classify the result

DERIVED-CONSTRAINT-RIGIDITY:
  dim(T_admiss)=0 using only independently justified pre-frozen rows.

CONSTITUTIVE-RIGIDITY:
  dim(T_admiss)=0 only after candidate-specific constitutive rows are added.

OBSTRUCTION-RIGIDITY:
  dim(T_admiss)>0, but every nonzero direction has a certified nonlinear obstruction.

MODULI-PRESENT:
  At least one nonzero direction is integrable.

INCOMPLETE:
  The matrix, gauge quotient, rank stability, or deformation basis is missing.

6. Nonlinear and stability continuation

A zero tangent space still does not prove positive physical stability. If a tangent direction survives, calculate its obstruction map. Independently calculate the kinetic-normalized full mixed Hessian on the physical domain.

7. Required published artifacts

• deformation_basis.json;
• gauge_matrix.csv;
• constraint_jacobian.csv;
• rank_certificate.json;
• physicaltangentbasis.csv;
• obstruction_map.*, when required;
• mixed_hessian.*;
• reconstruction script and environment lock.

Owned hard constraints and acceptance evidence

The full controlling registry is EVIDENCEDISCHARGEREGISTRYV33_2.md. The rows below are embedded because this block owns or directly co-owns them.

| ID | Required witness | Minimal discharge | Fail trigger | |---|---|---|---| | GEN-C03 | Variational-ownership ledger + equation-disposition table | Every variable is marked varied, gauge, constrained, auxiliary, measured, matching, or external, with the displaced Euler–Lagrange equation explicitly retained, replaced, proved redundant, or abandoned. | A non-varied variable has no lawful equation disposition. | | SHP-C06 | Cross-sector zero-mode inventory | A merged mode table covers gauge, fermion, scalar, metric, boundary, edge, and topological sectors below cutoff. | An undeclared massless or parametrically light mode remains. | | RIG-C01 | Published deformation-basis manifest | A finite or function-space basis enumerates metric, radius, Wilson-line, bundle, boundary, Actor, Rulebook, and observer-map deformations. | A deformation sector is omitted because its background value was fixed. | | RIG-C02 | Gauge/constraint quotient matrix | The gauge image and exact constraint directions are computed and removed before physical rank is counted. | Coordinate, gauge, or reaction directions are counted as physical or vice versa. | | RIG-C03 | Constraint Jacobian matrix + rank-reduction output | The published matrix \(J{aI}=dCa(eI)\) is row-reduced with exact/rational or certified numerical arithmetic; its nullspace modulo gauge gives \(T{ m admiss}\). | The rigidity classification is asserted without the matrix or rank artifact. | | RIG-C04 | Kuranishi/obstruction calculation or integrability theorem | Every surviving infinitesimal direction is integrated or assigned a nonzero obstruction with reproducible coefficients. | A nonzero tangent direction is called obstructed without computing the obstruction map. | | RIG-C05 | Separate solution-space and Hessian certificates | The local moduli-germ result and the kinetic-normalized physical Hessian are published independently. | Isolation is used as evidence of positive stability. | | RIG-C06 | Full mixed Hessian artifact | All diagonal and off-diagonal sector blocks are computed, or symmetry theorems prove specified blocks vanish; eigenvalue bounds are published. | Only a metric or homogeneous sub-block is tested while mixed directions remain. | | RIG-C07 | Protected-scalar whitelist + symmetry proof | Every intended light scalar has an owning symmetry/mechanism and every unwanted modulus is absent, obstructed, or positively massive. | An unwanted scalar is relabeled as protected after its discovery. | | RIG-C08 | Named radius/Wilson/boundary deformation rows | Volume, relative radii, interval length, Wilson lines, boundary positions, and localized deformations each have an explicit row and status. | A fixed radius is treated as proof that its radion does not exist. | | RIG-C09 | Tower/boundary/tunnelling scope certificate | The claim states the highest KK level, boundary sectors, and nonperturbative channels covered, with a bound or honest OPEN rows. | A finite homogeneous calculation is promoted to full stability. | | RIG-C10 | Constraint-provenance audit + pre-freeze hash | Every constraint used in the rigidity matrix predates the candidate result and is independently justified; late-added rows force restart. | A negative mode is removed by a constraint written after the mode was found and then called intrinsic. | | OBS-C07 | Observer-map deformation Jacobian | Derivatives of records with respect to projection, frame, normalization, and apparatus choices are included in the deformation/uncertainty analysis. | A small convention change can move a prediction across the pass band but is not counted. |

Package status and authority boundary

This file belongs to the Candidate-Neutral Building-Block Authority v3.3.2.

The package is documentation-complete for the current development architecture, but it is not protocol-frozen and not canonically ratified. Technical results generated under it remain development evidence until the control suite, enumeration test, and hostile review pass.

The controlling package authorities are:

• EVIDENCEDISCHARGEREGISTRYV33_2.md;
• QUANTITATIVESTOPPINGANDEXHAUSTIONRULEV33_2.md;
• MANDATORYCONTROLSUITEV33_2.md;
• MIGRATIONANDSUPERSESSIONRECORDV3_3.md.

A block is standalone for its owned object: it includes its complete definitions, interfaces, certificates, falsifiers, and owned evidence criteria. The full cross-block registry is intentionally stored once rather than repeated verbatim in every block.

Shared status grammar

PASS:
  The named witness satisfies the minimal discharge criterion.

FAIL:
  A reproducible counterexample or failed acceptance test exists.

OPEN:
  The required witness is missing, incomplete, or inconclusive.

NOT-EVALUATED:
  Evaluation stopped after an earlier hard failure.

NOT-APPLICABLE:
  A type proof establishes that the requirement cannot apply.

CONSTRUCTION-ANCHOR:
  A declared Actor, constraint, boundary term, or global sector realizes
  the result but does not derive it from the pre-existing object.

CONDITIONAL-ON-BND-A:
  The spectrum/domain result awaits fixed-set and operator-domain closure.

CONDITIONAL-ON-BND-B:
  The quantum result awaits local and global anomaly/inflow closure.

RESTART-REQUIRED:
  A frozen object relevant to the result changed.

Universal anti-promotion rules

• A bare candidate is not a complete parent object.
• A definition is not a witness.
• A nonempty equation list is not a solvability proof.
• An isolated vacuum is not automatically stable.
• A fixed background value is not automatically an absent fluctuation.
• Static vacuum-offset protection is not automatically cosmological-history compatibility.
• Integrated anomaly cancellation is not local fixed-set cancellation.
• Local anomaly cancellation is not global determinant-line trivialization.
• A raw higher-dimensional quantity is not automatically the measured four-dimensional observable.
• One tested survivor is not a selected survivor without explicit grammar enumeration or a finiteness theorem.

--- END EXACT LEGACY SOURCE ---