Dynamics V4.2 — Agent-Ready Gate-Closure Package

--- title: "BB-DYN-4.2 — Maximum-Rigor Full Gate-Closure Dynamics Object" buildingblockid: "BB-DYN-4.2" version: "4.2" date: "2026-08-02" status: "DEVELOPMENT AUTHORITY CANDIDATE — MAXIMUM-RIGOR PRESERVATION-AUDITED STANDALONE GATE-EXECUTION OBJECT — NOT FROZEN" supersedesfordevelopment: "BB-DYN-4.1" preserves: "All substantive BB-DYN-3.3.1 obligations, DYN-C01 through DYN-C08, boundary/orbifold authority, standalone interface, certificates, controls, status grammar, and anti-promotion rules" claim_ceiling: "Defines required Dynamics objects, algorithms, evidence, controls, preservation obligations, and terminals; does not assert that gate-specific calculations pass." ---

BB-DYN-4.2 — Maximum-Rigor Full Gate-Closure Dynamics Object

0. Executive verdict

Dynamics is the complete lawful-change object acting on the frozen Shape. It is not merely an action, an equation list, a Hessian, or a beta function. A gate receives Dynamics support only when the relevant chain is executed:

\[ \text{Shape inputs} \rightarrow\text{parent law} \rightarrow\text{constraint/measure reduction} \rightarrow\text{background} \rightarrow\text{physical operators} \rightarrow\text{interactions and transitions} \rightarrow\text{quantum/history/causal evolution} \rightarrow\text{refinement and regional balance} \rightarrow\text{finite Observer records}. \]

This file is standalone for the Dynamics object. It contains the owned physical questions, the complete seventeen-row constraint system, thought-experiment derivation, cross-block contracts, generation and saturation algorithms, packet schemas, all-gate profiles, destructive controls, residual vector, evidence tree, validator logic, and terminal rules. Definitions are not witnesses; gate-specific calculations must still be executed.

1. Why V4.0 was not complete

V4.0 correctly identified nine missing reusable obligations, but it remained an expanded specification. Several new rows were short prose requirements; packet schemas were incomplete or stored separately; the Stage–Rulebook–Actor compiler was not explicit; interaction and operator saturation were not algorithmic; all-gate profiles were outside the authority; and no standalone validator or residual theorem connected evidence to a terminal. V4.1 repaired those operational gaps without claiming new gate results. V4.2 additionally performs a full preservation audit against BB-DYN-3.3.1 and restores every legacy normative clause that V4.1 did not explicitly carry into its main execution text.

2. Dynamics identity, ownership, and non-ownership

2.1 Owned object

Dynamics owns the parent law and every lawful transformation from admitted configurations to physical evolution and response. Its owned object is

\[ \mathfrak D=(S{\mathrm{parent}},\mathcal F,\mathcal C,\mu{\mathrm{phys}},\bar z,\mathfrak O_{\mathrm{phys}},\mathfrak I,\mathfrak T,\mathfrak Q,\mathfrak H,\mathfrak R,\mathfrak J), \]

where the entries are respectively the parent law, equation vector, constraint system, physical measure, lawful background, physical operator shelf, interaction hypergraph, transition/mass system, quantum physicality and channels, nonequilibrium/causal histories, refinement system, and regional currents/fluxes.

2.2 Non-ownership firewall

• Shape owns supports, admissibility, and Actor identities; Dynamics cannot invent a support, rule, or Actor to make a gate pass.
• Granularity owns operational equivalence, candidate/mode exhaustion, and stopping; Dynamics supplies the calculated objects it audits.
• Scale owns rulers, numerical provenance, matching conventions, and uncertainty transport.
• Boundary A/B owns exact domains, fixed-set kernels, local/global anomaly and inflow closure, although Dynamics consumes and tests them.
• Rigidity owns deformation/stability terminals; Dynamics supplies the physical Hessian and evolution.
• Vacuum owns constant-shift protection; Dynamics supplies full equations, stress, and finite-time response.
• Observer owns record maps and calibration; Dynamics supplies lawful response and channels.
• Time Synchronization owns clock compatibility; Dynamics supplies characteristic support and evolution.
• Interdependence/governance owns branch hashes, stale propagation, evidence status, and terminal issuance.

3. Completeness theorem for the Dynamics object

For the frozen branch, a Dynamics responsibility is in scope if changing it while holding Shape and the other block inputs fixed can change at least one lawful physical trajectory, amplitude, spectrum, probability, history, or finite record. Such changes fall into exactly the following current families:

1. parent-law ownership;

1. constraint/gauge closure;

1. physical measure;

1. stress and source response;

1. background/reaction solvability;

1. quantum/domain/anomaly consistency;

1. ordinary causal response;

1. scope and tails;

1. physical operators and spectra;

1. interactions and sources;

1. masses/mixings/transitions;

1. quantum positivity and continuation;

1. closed/open probability evolution;

1. nonequilibrium and thermal history;

1. well-posed causal evolution;

1. RG/refinement stability;

1. regional/interface flux balance;

These families are represented by DYN-C01 through DYN-C17. Completeness is current-branch and grammar-relative, not metaphysical. The block reopens if a gate produces a lawful dynamics distinction not representable by these families.

Define the Dynamics closure operator Cl_DYN as repeated application of term generation, equation variation, constraint closure, physical reduction, interaction generation, history/refinement generation, and complement testing. The object is saturated at a gate only when a new pass produces no new in-grammar term, equation, constraint, physical operator, interaction edge, transition, channel, history variable, refinement direction, or regional current.

4. Frozen input contracts

4.1 Stage input

branch/support IDs; metric, topology, strata, incidence, volume, curvature, orientation, variational and deformation ledgers; cover/quotient normalization.

4.2 Rulebook input

representations, parities, domains, quotients, exact constraints, global form, positivity requirements, interaction permissions, regulator rules, claim ceilings.

4.3 Actor input

owner IDs, supports, representations, equations, split/merge criteria, modes versus identities, kill and reopen conditions.

4.4 Co-Actor input

all applicable parent, owner, domain, quotient, measure, spectrum, interaction, rigidity, regulator, and exhaustion attack classes.

4.5 Granularity input

candidate/mode grammar, cutoff, equivalence relation, tail obligations, omitted-rival and stopping rules.

4.6 Scale input

rulers, units, radii, gaps, renormalization/matching scheme, uncertainties, transport maps.

4.7 Boundary input

self-adjoint/elliptic domains, parity and fixed-set data, gluing, edge kernels, anomaly/inflow and relative-QME certificates.

4.8 Observer input

record space, projection, normalization, calibration, accessibility and no-signalling criteria.

4.9 Rigidity/Vacuum/TS input

deformation basis and terminal criteria; constant-shift tests and history scope; clock and causal-composition rules.

4.10 Governance input

authority hashes, gate contract, frozen assumptions, evidence registry, status grammar, reopen propagation.

Every input is hash-bound. A missing or stale input yields OPEN or RESTART-REQUIRED, never an inferred default.

5. Shape-to-Dynamics compilation contract

The compiler consumes the frozen three-layer Shape and returns physical Dynamics packets without reading the desired gate answer.

\[ \mathsf{Compile}_{\mathrm{DYN}}=\mathsf{Refine}\circ\mathsf{Evolve}\circ\mathsf{Quantize}\circ\mathsf{Transition}\circ\mathsf{Interact}\circ\mathsf{Reduce}\circ\mathsf{Vary}\circ\mathsf{Generate}. \]

5.1 Pass A — Generate

Generate every admissible parent term and primitive move from Stage supports, Rulebook permissions, Actor owners, tensor/representation grammar, derivative scope, strata, and global classes. No target-dependent term may be inserted.

5.2 Pass B — Vary

Vary every retained variable and boundary/global datum, or publish an explicit lawful equation disposition. Produce the complete bulk–boundary–global equation vector.

5.3 Pass C — Reduce

Run constraint/gauge/BV/BFV closure, construct the physical measure, solve or classify the background, and build the physical quotient.

5.4 Pass D — Interact and transition

Generate the complete interaction/source hypergraph, then derive masses, mixings, transitions, decays, and thresholds only from owned edges and solved backgrounds.

5.5 Pass E — Quantize and evolve

Construct positive physical sectors, closed/open channels, causal initial-boundary evolution, nonequilibrium histories, and regional flux balances.

5.6 Pass F — Refine and saturate

Run nested-basis, regulator, tower, history-resolution, and omitted-tail controls. Pass outputs to Granularity for equivalence and exhaustion.

6. Universal object and packet grammar

Every Dynamics object uses the following base record:

dynamics_object:
  object_id:
  branch_id:
  gate_id:
  owner_actor_ids: []
  stage_support_ids: []
  rulebook_rule_ids: []
  boundary_domain_ids: []
  scale_packet_id:
  granularity_scope_id:
  parent_object_ids: []
  assumptions: []
  equations_or_maps: []
  evidence_files: []
  evidence_hashes: []
  calculation_order:
  scope_ceiling:
  destructive_controls: []
  unresolved_residuals: []
  status: PASS|FAIL|OPEN|NOT-EVALUATED|NOT-APPLICABLE|RESTART-REQUIRED

No packet may use an Actor name, expected multiplicity, observed matrix, or desired terminal as its derivation. Such data may appear only as a later comparison target.

6A. Preserved full mathematical kernel from BB-DYN-3.3.1

The following kernel is retained normatively rather than summarized. It supplies the detailed parent-action, equation, constraint, measure, stress, reaction, background, radiative, stability, vacuum-response, probability, causality, conservation, and boundary/orbifold formulas that V4.0 did not fully integrate into its standalone file. V4.1 additions refine execution and evidence; they do not erase these equations.

1. Parent action

Every candidate must supply a typed parent action or an explicitly scoped non-action replacement.

A general action ledger is

\[ S{\mathrm{parent}}=S{\mathrm{grav}}+S{\mathrm{gauge}}+S{\mathrm{matter}}+S{\mathrm{scalar}} +S{\mathrm{boundary}}+S{\mathrm{constraint}}+S{\mathrm{global}}+S_{\mathrm{domain}}. \]

For every term publish:

• support and dimensionality;
• field owner;
• derivative order;
• exact and broken symmetries;
• coefficient provenance;
• boundary variation;
• whether microscopic, structural, global, or matching-only;
• stress-energy contribution;
• vacuum-energy contribution;
• cutoff and scheme scope.

An arbitrary dense kernel entry without a parent-law owner is prohibited.

2. Complete equation vector

Let the independent variables be

\[ z^I=(g{\mu\nu},\gamma{mn},g{\mu m},AM,\Psi,\phi,b,\lambda,\sigma,\ldots). \]

The complete background and fluctuation equations are

\[ \mathcal F_I[z]\equiv\frac{\delta\Gamma}{\delta z^I}=0, \]

supplemented by boundary, global, gauge, and constraint equations.

The candidate must publish an equation ledger:

equation:
  equation_id:
  varied_object:
  local_or_global:
  bulk_or_boundary:
  differential_or_algebraic:
  independent_after_identities:
  source_terms:
  constraint_couplings:
  observer_projection:
  solution_status:

2.1 Equation counting

Before claiming a background solution:

1. count independent unknown functions and global variables;
2. count independent equations after Noether and Bianchi identities;
3. classify gauge freedoms;
4. classify second-class constraints;
5. include boundary and junction equations;
6. verify rank on the claimed stratum;
7. prove or compute existence of a solution.

A visually plausible product ansatz is not a solution certificate.

3. Constraint classification and closure

Constraints are typed as:

• first-class gauge constraints;
• second-class physical constraints;
• constitutive restrictions;
• boundary/gluing constraints;
• global constraints;
• effective matching relations.

For Hamiltonian variables, the total Hamiltonian is

\[ HT=H0+u^A\Xi_A. \]

The Dirac–Bergmann algorithm must:

1. list primary constraints;
2. impose preservation \(\dot\Xi_A\approx0\);
3. generate every secondary and higher constraint;
4. terminate with constant rank on the claimed stratum;
5. classify first- and second-class sectors;
6. determine multipliers and residual gauge functions;
7. verify boundary compatibility;
8. compute the physical degree-of-freedom count.

For second-class constraints,

\[ C{AB}=\{\XiA,\Xi_B\} \]

must be invertible on the physical stratum and

\[ \{F,G\}D=\{F,G\}-\{F,\XiA\}(C^{-1})^{AB}\{\Xi_B,G\}. \]

Rank change is a branch change or singular stratum, not a small correction.

4. Physical measure

For a second-class system,

\[ D\mu{\mathrm{phys}}=D\mu0\prodA\delta(\XiA)\sqrt{\det C}. \]

The determinant must be included in effective actions, matching, and stress calculations.

If \(C\) depends on retained fields,

\[ \Delta\Gamma_C=-\frac{i}{2}\operatorname{Tr}\log C \]

or the appropriate Euclidean counterpart contributes physical terms.

A field-dependent determinant cannot be dismissed as normalization.

5. No-work and stress are separate

For ideal constraints,

\[ \lambda^A d\Phi_A=0 \]

along admissible displacements.

This proves no virtual work in constrained directions. It does not establish zero stress-energy.

For

\[ S{\rm constraint}=\int d^Dx\sqrt{-g}\,\lambda^A\PhiA, \]

\[ T^{\rm constraint}{MN} =-\frac{2}{\sqrt{-g}}\frac{\delta S{\rm constraint}}{\delta g^{MN}}. \]

The calculation must include:

• metric variation of the volume form;
• explicit metric dependence of \(\Phi_A\);
• metric dependence of multipliers when induced by solved equations;
• determinant and ghost terms;
• boundary contributions.

Zero constraint stress is a result with conditions, not a default property.

6. Constrained reaction equations

For retained variables \(q^a\), constrained variables \(\phi^r\), and constraints \(\Phi_\alpha(q,\phi)=0\), the renormalized equations are

\[ \frac{\delta\Gamma0}{\delta q^a} +\lambda^\alpha\frac{\delta\Phi\alpha}{\delta q^a}=0, \]

\[ \frac{\delta\Gamma0}{\delta\phi^r} +\lambda^\alpha\frac{\delta\Phi\alpha}{\delta\phi^r}=0, \]

\[ \Phi_\alpha=0. \]

6.1 Block-separable case

If

\[ \frac{\delta\Phi_\alpha}{\delta q^a}=0 \]

and the normal Jacobian

\[ N{\alpha r}=\frac{\delta\Phi\alpha}{\delta\phi^r} \]

has full rank, the normal equations determine reactions when

\[ -\left.\frac{\delta\Gamma0}{\delta\phi^r}\right|{\Phi=0} \in\operatorname{im}N^T. \]

This prevents normal-force overdetermination only under that span condition.

6.2 General case

If constraints depend on retained variables, multipliers modify retained equations. The full coupled system must be solved.

6.3 Global solvability

Pointwise algebra is insufficient. The multiplier system must satisfy:

• regularity;
• boundary and fixed-set parity;
• junction conditions;
• global integrability;
• Ward identities;
• anomaly cancellation;
• conserved-charge balance.

7. Background-equation completeness

A candidate background receives one of four Dynamics statuses:

7.1 Parent solution

Every varied parent equation and boundary equation is satisfied.

7.2 Constrained reaction solution

Displaced equations are replaced by a complete, solvable reaction system, including stress and boundary effects.

7.3 Effective external background

The Stage is externally supplied. The reduced theory may be useful, but it does not derive the background as a parent vacuum.

7.4 Inconsistent background

The complete equation system has no solution at the claimed object.

A closure statement must name which status is used.

8. Radiative solvability

Let \(\Gamma_k\) be the effective action at scale \(k\). Radiative consistency requires more than preservation of the classical constraint surface.

At each claimed order or Wilsonian step, test:

• closure of the corrected constraint algebra;
• rank stability;
• normal-force span;
• retained equation solvability;
• boundary compatibility;
• anomaly and Ward identities;
• generation of forbidden operators;
• determinant contributions;
• parameter and counterterm provenance.

A forbidden mode need not be regenerated for the theory to fail. Equations with no solution are a separate failure.

9. Physical stability interface

Dynamics supplies the effective Hessian to BB-RIG-1.1.

After gauge quotient and exact constraints,

\[ Q{\rm phys}=P{\rm phys}^{T}\,\delta^2\Gamma\,P_{\rm phys}. \]

The supplied packet must include:

• kinetic metric;
• gauge fixing and ghosts;
• mixed blocks;
• boundary domain;
• scale and scheme;
• zero modes;
• negative modes;
• intended protected scalars;
• tower scope;
• uncertainty.

Dynamics may not label an isolated critical point stable without the physical Hessian result.

10. Vacuum-offset response interface

A field-independent shift of a protected matter sector is

\[ \mathcal Lm\rightarrow\mathcal Lm-c. \]

Dynamics must expose how it enters the complete equations.

For ordinary Einstein coupling, the external equation shifts by a term proportional to

\[ c\,g_{\mu\nu}. \]

Constraint reactions in internal directions do not automatically remove this external response.

The vacuum-response packet sent to BB-VAC-1.1 contains:

vacuum_response_packet:
  protected_sector:
  constant_shift_parameter:
  external_metric_equation:
  internal_equations:
  constraint_reaction_equations:
  global_auxiliary_equations:
  boundary_equations:
  determinant_terms:
  matching_terms:
  graviton_loop_scope:
  ordinary_stress_response:
  solved_response_operator:

11. Ordinary-gravity control

Any vacuum-offset mechanism must preserve lawful response to nonconstant stress.

The negative control is:

• add a constant \(-c\) to the protected Lagrangian;
• verify local curvature observables do not change in the protected scope;
• add a localized excitation with nontrivial traceless or gradient stress;
• verify ordinary gravitational response remains.

A mechanism that removes all stress from the gravitational equations fails.

12. Constant shifts versus phase transitions

A constant threshold correction and a finite-time phase transition are distinct.

Dynamics must classify:

• static additive shift;
• changing condensate;
• latent heat;
• radiation production;
• nonadiabatic boundary or global response;
• historical averages;
• residual present value.

A constant-shift identity does not automatically solve cosmological transition history.

13. Probability, causality, and conservation

For closed quantum sectors, the generator must be self-adjoint on the physical domain or possess an equivalent unitary kernel.

For open sectors, declare:

• environment;
• channel;
• normalization;
• exchange currents;
• observer access;
• approximation bound.

The law must preserve, at the claimed scope:

• probability;
• gauge and BRST constraints;
• exact charges;
• boundary flux balance;
• causal support;
• operational no-signalling.

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 — stratified quantum gauge consistency

1. Complete gauge variation

For a \(d+1\)-dimensional parent with \(d\)-dimensional fixed sets \(F_a\), the renormalized gauge variation is decomposed as

\[ \delta\epsilon\Gamma{\rm tot} = \delta\epsilon\Gamma{\rm bulk} + \suma \delta\epsilon\Gamma{Fa} + \delta\epsilon S{\rm inflow} + \delta\epsilon S{\rm ct}. \]

For four-dimensional fixed sets, write the descent data

\[ I6=dI5^{(0)}, \qquad \delta\epsilon I5^{(0)}=dI_4^{(1)}(\epsilon,A). \]

A bulk inflow action of the form

\[ S{\rm inflow} = 2\pi i\,k\int{Y5}I5^{(0)} \]

has boundary variation

\[ \delta\epsilon S{\rm inflow} = 2\pi i\,k \suma sa \int{Fa}I_4^{(1)}, \]

where \(s_a\) is fixed by the outward-normal orientation. The coefficient \(k\) must satisfy the appropriate quantization and global-form conditions.

2. Fixed-set local equations

For each fixed set define

\[ \mathcal Aa^{(1)} = \mathcal A{a,\rm projected\ bulk}^{(1)} + \mathcal A{a,\rm localized}^{(1)} + \mathcal A{a,\rm measure}^{(1)} + \mathcal A{a,\rm counterterm}^{(1)} + \mathcal A{a,\rm inflow}^{(1)}. \]

The local pass condition is

\[ \boxed{\mathcal A_a^{(1)}=0\quad\text{for every }a} \]

as a cohomology class under all allowed gauge transformations.

The weaker condition

\[ \suma\mathcal Aa^{(1)}=0 \]

is only an integrated check. It does not prove local gauge invariance.

3. Quantum master equation with boundaries

The candidate must formulate the quantum gauge system using BV–BFV or an equivalent boundary-complete formalism.

Schematically,

\[ \frac12(S,S)-i\hbar\Delta S = \pi^*S_{\partial} \]

before boundary completion. Full closure requires a compatible boundary state/functional and counterterm/inflow system such that the total quantum master equation is satisfied.

A bulk statement \(Q_{\rm BRST}^2=0\) is insufficient when the measure has fixed-set support.

4. Consistent versus covariant anomalies

The ledger must state whether it uses:

• consistent anomalies;
• covariant anomalies;
• Bardeen–Zumino shifted currents.

All fixed-set and inflow entries must use one compatible convention. A Bardeen–Zumino polynomial may redistribute anomaly between currents or strata, but it cannot remove the total anomaly class.

5. Regulator and parity dependence

Localized anomalies can depend on the parity/domain assignment and on how bulk fermions are regulated. The final anomaly class must be regulator-independent after allowed local counterterms are included.

The certificate must report:

• regulator;
• consistent-current convention;
• parity signs;
• fixed-set weights;
• localized spectrum;
• allowed counterterms;
• invariant remainder.

6. Global relative anomaly equation

Perturbative cancellation is not enough. The exponentiated fermion determinant is a section of an anomaly/determinant line.

The complete parent-plus-boundary system must provide a gluing-compatible trivialization:

\[ Z{\rm boundary}\, Z{\rm inflow} \in\mathbb C \]

rather than an untrivialized line.

For every closed test obtained by gluing or forming a mapping torus, require

\[ \exp\!\left(2\pi i\,\frac{\eta+h}{2}\right) \, \exp(2\pi i S_{\rm top}) = 1 \]

in the declared convention and tangential structure.

7. Dynamics certificate additions

boundary_quantum_dynamics:
  fixed_set_ids:
  allowed_gauge_transformations:
  anomaly_convention: consistent|covariant
  regulator:
  bulk_anomaly_polynomial:
  projected_bulk_contributions_by_fixed_set:
  localized_field_contributions:
  measure_and_ghost_contributions:
  counterterms:
  inflow_action:
  inflow_coefficient_quantization:
  orientation_signs:
  local_residuals:
  bv_bfv_or_equivalent_completion:
  determinant_line:
  relative_eta_or_bordism_test:
  gluing_test:
  quantum_master_equation_result:
  evidence_hashes:
  verdict:

8. New Dynamics falsifiers

Dynamics fails if:

• only the integrated anomaly is zero;
• an inflow coefficient is selected after reading the residual;
• the coefficient is not globally quantized;
• a boundary condition is not compatible with the Dirac/gauge operator domain;
• the local anomaly convention changes between rows;
• the determinant line remains untrivialized;
• the BV quantum master equation is asserted from classical Jacobi closure alone.

6B. Normative preservation layer from BB-DYN-3.3.1

This section restores every substantive legacy obligation that was not explicit in the V4.1 execution text. It is additive and binding. V4.2 may strengthen a legacy requirement, but may not silently weaken or delete it. Where a V4.2 clause and a legacy clause overlap, the stricter evidentiary requirement controls unless a gate-scoped supersession record identifies the exact clause, reason, replacement witness, and claim-ceiling effect.

6B.1 Original physical question and four anti-substitution firewalls

Dynamics answers:

Which lawful changes occur on the complete Shape, which equations define its backgrounds, how exact constraints are preserved, how quantum corrections modify the equations, and how the system responds separately to vacuum offsets and ordinary excitations?

This block prohibits four substitutions:

\[ \text{kinematic permission}\neq\text{nonzero amplitude}, \]

\[ \text{constraint no-work}\neq\text{zero stress}, \]

\[ \text{no propagator}\neq\text{radiative solvability}, \]

\[ \text{internal reaction solvable}\neq\text{vacuum energy degravitated}. \]

6B.2 Legacy Dynamics certificate remains a mandatory compatibility view

The V4.2 integrated certificate must be able to emit every field of the original dynamicscertificatev3. A missing legacy field is not excused merely because a newer packet contains related information.

dynamics_certificate_v3:
  candidate_id:
  branch_id:
  parent_action:
  primitive_term_ledger:
  complete_equation_vector:
  equation_count_and_identities:
  constraint_types:
  dirac_bergmann_closure:
  rank_strata:
  physical_measure:
  determinant_contributions:
  constraint_stress:
  multiplier_boundary_problem:
  background_status:
  background_solution:
  radiative_solvability:
  ward_identities:
  anomaly_result:
  physical_hessian_packet:
  vacuum_response_packet:
  ordinary_gravity_control:
  phase_transition_scope:
  full_tower_scope:
  evidence_hashes:
  verdict:

6B.3 Legacy destructive controls

The following four controls remain mandatory whenever their types apply. They supplement, rather than replace, the V4.2 thought-experiment suite.

15.1 No propagator but no solution

Remove a deformation field from the propagator but choose a radiative normal force outside the multiplier span.

Expected result: FAIL / RADIATIVE OVERDETERMINATION.

15.2 Constraint no-work with nonzero stress

Use a metric-dependent constraint that does no virtual work along admissible motion.

Expected result: no-work passes, stress-zero claim fails.

15.3 Vacuum degravitation that also removes matter gravity

Project out every trace and traceless source indiscriminately.

Expected result: vacuum shift passes, ordinary-gravity control fails.

15.4 Frozen Stage with abandoned internal equations

Do not vary the internal metric and supply no reaction or background equation.

Expected result: EFFECTIVE EXTERNAL BACKGROUND, not parent solution.

6B.4 Legacy falsifiers and reopen triggers

This block fails or reopens if:

• the parent action is incomplete;
• a primitive move lacks an owner;
• equation counting omits a boundary or global equation;
• Dirac closure or rank is incomplete;
• determinant terms are omitted;
• no-work is used as a stress proof;
• no-propagator is used as a solvability proof;
• a background is called a solution without all equations;
• a constant vacuum shift changes local curvature in the claimed protected scope;
• ordinary nonconstant stress fails to gravitate;
• a phase-transition claim is inferred from constant-shift cancellation;
• a zero-mode result is promoted to the full tower.

6B.5 Legacy terminal and delivery scope

The following is retained as the historical minimum delivery claim of the Dynamics block. The current authority identifier and status are controlled by V4.2 frontmatter and Section 19.

BB-DYN-3.3.1

DELIVERS:
  Parent law, complete equations, constraint and measure closure,
  background solvability, stress accounting, radiative response,
  and interfaces to rigidity and vacuum protection.

STATUS:
  DEVELOPMENT AUTHORITY CANDIDATE — NOT FROZEN.

6B.6 Complete standalone ownership interface

The full prior standalone interface is retained below. References to package v3.3.2 remain provenance statements; current evidence generation must additionally satisfy the V4.2 registries and validators.

Owned questions

This block alone owns:

• the typed parent action;
• the complete equation vector;
• constraint classification and preservation;
• physical measure and determinant ownership;
• stress of multiplier, boundary, and constraint sectors;
• retained/reaction-equation solvability;
• Ward identities and BV/BFV quantum consistency;
• ordinary probability, causality, conservation, and nonconstant-stress response.

Required inputs

• complete Shape and boundary manifests;
• constraint and permission packets;
• Scale/scheme packet;
• complete field and Actor inventory;
• candidate-specific parent terms.

Required outputs

• parent-action manifest;
• complete Euler–Lagrange and constraint-equation ledger;
• Dirac/BV/BFV closure;
• physical measure;
• stress and reaction calculations;
• background classification;
• radiative-solvability result;
• Ward/anomaly interface;
• Dynamics certificate.

Non-ownership boundary

Dynamics does not by itself prove:

• candidate uniqueness;
• intrinsic rigidity;
• full physical stability without the Rigidity block;
• vacuum-offset protection without the Vacuum block;
• local/global fixed-set anomaly closure without the Boundary block.

Observer-response handoff

Dynamics supplies the physical evolution, conserved currents, response functions, and physical measure consumed by OBS. OBS cannot repair an inconsistent Dynamics by projection. Conversely, a lawful parent response cannot be compared with data until OBS has produced the correct record functional.

For time-dependent vacuum tests, Dynamics must evolve the protecting sector and supply the constraint-algebra history required by VAC-C11.

Gauge-owner and kinetic-term consistency

Dynamics consumes the SHP-C09 gauge-owner ledger.

For every surviving generator \(T_A\), the parent action must identify exactly one independent connection variable and one owning kinetic structure. Metric Kaluza–Klein notation and principal-bundle notation may describe one field, but the action must not count both as independent unless a second gauge sector is physically intended and appears in the spectrum.

Required checks:

1. map every gauge-owner ID to a parent-action field;
2. count independent kinetic operators;
3. count first-class gauge parameters;
4. count four-dimensional vector zero modes;
5. verify equality of these counts after constraints and quotienting;
6. verify that center/global-lift data is compatible with every charged Actor;
7. run the duplicate-owner control by treating aliases as independent and confirm that the spectrum gains an extra gauge sector.

A gauge owner with no parent-action term fails Dynamics. A parent-action connection with no Shape owner fails the one-parent audit.

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. | | DYN-C01 | Parent-action manifest + term provenance table | Every primitive term has an owner, support, coefficient provenance, derivative order, symmetry, boundary status, and microscopic/effective label; every gauge connection references exactly one SHP-C09 owner ID. | A matrix element, potential, counterterm, or gauge connection has no parent-action owner, or an owner alias is independently double-counted. | | DYN-C02 | Dirac–Bergmann or BV–BFV closure calculation | All primary/secondary constraints, ranks, brackets, boundary terms, and residual gauges are computed until closure; the rank is constant on the claimed stratum. | The algorithm stops early, the rank changes, or boundary constraints are omitted. | | DYN-C03 | Physical-measure derivation + determinant artifact | The reduced measure includes all Faddeev–Popov/Dirac/BV determinants, and field dependence is retained in loops and matching. | A determinant is dropped as a constant without proof. | | DYN-C04 | Metric variation of all constraint/boundary sectors | The stress tensor is obtained by varying the full action and measure; the result is evaluated on the physical branch. | No-work is used as a substitute for zero stress. | | DYN-C05 | Solvability/Jacobian certificate | The full retained-plus-reaction equation system has a solution and a full-rank Jacobian or a theorem establishes existence in the declared domain. | Radiative normal forces lie outside the multiplier span or the retained equations have no solution. | | DYN-C06 | Ward/BV identity + boundary multiplier ledger | Allowed transformations preserve the domain, all measure anomalies cancel, and multiplier boundary conditions admit a global solution. | A Ward identity, anomaly, or boundary multiplier equation remains unresolved. | | DYN-C07 | Response-function calculation | Localized and nonconstant stress sources produce the correct causal/conservation response while protected constants are handled according to VAC. | The mechanism removes ordinary matter gravity, violates conservation, or produces acausal record response. | | DYN-C08 | Scope theorem or bounded truncation report | The certificate states exactly whether it covers zero modes, a cutoff tower, or all modes, with a tail theorem where a global claim is made. | A local, zero-mode, finite-EFT, or invariant-sector result is promoted beyond scope. | | VAC-C01 | Complete background-equation vector + equation-count/Jacobian report | External, internal, matter, boundary, constraint, and global equations are listed and solved or lawfully replaced; unknown/equation counts and rank are published. | An internal Einstein or boundary equation disappears when a Stage variable is frozen. | | VAC-C04 | Finite constant-shift response experiment | For each protected sector, the full equations are re-solved after \(\mathcal Ls o\mathcal Ls-c_s\), and the curvature-record derivative or finite difference is published. | Only an infinitesimal or tree-level shift is tested while auxiliary/global response is held fixed. | | VAC-C06 | Localized-source negative control | Dust, radiation, pressure gradients, and localized stress still source the correct gravitational response. | The protection mechanism degravitates ordinary matter or radiation. | | BND-C03 | Operator-domain certificate | Dirac, gauge, ghost, scalar, and load-bearing graviton domains are self-adjoint/elliptic as required and preserved by gauge/BRST transformations. | A parity table is supplied without a lawful operator domain. | | BND-C07 | Boundary BV/BFV quantum-master-equation output | Ghost, measure, boundary state, and inflow contributions satisfy the relative QME and compose under gluing. | Classical BRST closure is used while the measure/domain anomaly remains. | | VAC-C11 | Time-dependent constraint-algebra and cosmology simulation/theorem | A time-dependent vacuum shift \(c(t)\) representing QCD/electroweak transitions is inserted; the constraint algebra remains consistent and the Friedmann/perturbation history stays within frozen bounds, or the regime is explicitly excluded with quantified lost predictions. | The mechanism passes static shifts but becomes singular, acausal, or observationally unacceptable during a finite transition. |

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.

6B.7 Preservation rule for corrected source defects

Two old formulas contained line-wrap corruption in the source Markdown. V4.2 preserves the old source file byte-for-byte in 05_SOURCES/, but the corrected mathematical forms are normative:

\[ S{\mathrm{parent}}=S{\mathrm{grav}}+S{\mathrm{gauge}}+S{\mathrm{matter}}+S{\mathrm{scalar}} +S{\mathrm{boundary}}+S{\mathrm{constraint}}+S{\mathrm{global}}+S_{\mathrm{domain}}, \]

and

\[ D\mu{\mathrm{phys}}=D\mu0\prodA\delta(\XiA)\sqrt{\det C}. \]

No other legacy text is treated as a source defect without a separately hashed correction record.

7. Complete Dynamics constraints

DYN-C01 — Typed parent law and primitive-term ownership

Owned question: What is the complete parent law, and does every primitive term have one lawful owner?

Required inputs: Stage supports; Rulebook permissions; Actor owner IDs; Boundary supports; Scale/provenance packet.

Execution algorithm:

1. Generate the complete admissible term grammar before reading the gate target.

1. For every term, record fields, support, tensor contraction, derivative order, symmetry, boundary variation, coefficient provenance, and microscopic/effective status.

1. Canonicalize aliases and reject duplicate kinetic or interaction owners.

1. Vary every retained term or provide an explicit non-action replacement with equivalent equation and measure data.

Required outputs: Parent-action manifest; primitive-term ledger; variational-owner ledger; alias/duplicate report.

Minimal discharge: Every primitive term and gauge connection has one lawful owner, support, coefficient provenance, symmetry, derivative order, boundary status, and microscopic/effective scope.

Fail trigger: Unowned terms, double-counted aliases, or a gauge owner without a parent-action term.

Mandatory destructive control: Add an unowned but symmetry-allowed term, or duplicate one connection under metric and bundle notation. The validator must fail.

Gate families invoking this row: Gap-01, Gap-10/BG-10, Gap-11, SG-1, SG-2, SG-3, SG-4, SG-5, SG-7, SG-8, SG-9, UQF-10, UQF-5A/5B, UQF-9, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C02 — Constraint, gauge, and BV/BFV closure

Owned question: Does the constrained system close on the claimed stratum with the correct physical degree count?

Required inputs: Parent equations; constraint candidates; gauge generators; boundary domains; global restrictions.

Execution algorithm:

1. Run Dirac–Bergmann or BV/BFV closure to termination.

1. Compute all primary, secondary, and higher constraints and their rank.

1. Classify first- and second-class sectors, residual gauges, and boundary constraints.

1. Recompute physical degree counts on every rank stratum used by the claim.

Required outputs: Constraint algebra; bracket matrix; rank-stratum table; residual-gauge ledger; physical degree count.

Minimal discharge: All constraints, ranks, brackets, residual gauges, boundary terms, and physical degree counts are computed to closure on the claimed stratum.

Fail trigger: Early termination, rank change, omitted boundary constraint, or unresolved residual gauge.

Mandatory destructive control: Perturb to a rank-changing stratum or remove a boundary constraint. The claimed branch must reopen.

Gate families invoking this row: SG-2, UQF-10.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C03 — Physical measure and determinant ownership

Owned question: What measure lives on the reduced physical space, and which determinants contribute to loops, matching, and stress?

Required inputs: Constraint algebra; gauge fixing; field domains; boundary/ghost sectors.

Execution algorithm:

1. Derive FP, Dirac, BV, and boundary determinants.

1. Test whether each determinant is field dependent.

1. Carry field-dependent terms into the effective action, stress, anomaly, and matching calculations.

1. Publish regulator, phase, zero-mode, and normalization treatment.

Required outputs: Physical-measure packet; determinant ledger; effective-action contribution table.

Minimal discharge: All FP/Dirac/BV determinants and field dependence are retained in loops, matching, and stress calculations.

Fail trigger: A determinant is dropped as a constant without proof.

Mandatory destructive control: Drop a field-dependent determinant as normalization. The audit must produce a different loop or anomaly result and fail.

Gate families invoking this row: Born-rule, Gap-01, Gap-05-stability, Gap-13, SG-4, SG-7, UQF-3, UQF-4, UQF-5A/5B, UQF-7, UQF-9.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C04 — Full stress and metric-response derivation

Owned question: What stress is produced by every parent, constraint, boundary, ghost, and reaction sector?

Required inputs: Complete action/measure; solved multipliers; metric and boundary variations.

Execution algorithm:

1. Vary the full action and physical measure with respect to the metric.

1. Include volume-form, explicit metric, solved-multiplier, determinant, and boundary contributions.

1. Separate no-work statements from stress statements.

1. Evaluate on the lawful branch and compare constant versus localized sources.

Required outputs: Stress tensor decomposition; reaction-stress ledger; constant/nonconstant response split.

Minimal discharge: Stress is derived from the complete action and measure, including constraints, boundaries, ghosts, and solved multiplier dependence.

Fail trigger: No-work or absence from a propagator is substituted for zero stress.

Mandatory destructive control: Use a metric-dependent ideal constraint with zero virtual work but nonzero stress. No-work must not pass as zero-stress evidence.

Gate families invoking this row: Gap-05-stability, Lambda-catastrophe.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C05 — Retained-plus-reaction solvability and background closure

Owned question: Do all retained, constrained, global, and boundary equations admit a lawful solution?

Required inputs: Complete equation vector; constraint reactions; global variables; boundary equations.

Execution algorithm:

1. Count independent unknowns and equations after identities.

1. Construct the coupled retained-plus-reaction system.

1. Compute the Jacobian/Fredholm map or invoke an existence theorem with declared hypotheses.

1. Classify the result as parent solution, constrained-reaction solution, effective external background, or inconsistent background.

Required outputs: Equation ledger; unknown/equation count; solvability/Jacobian certificate; background classification.

Minimal discharge: The complete coupled equations have a solution with full-rank Jacobian or an existence theorem on the declared domain.

Fail trigger: Normal forces lie outside multiplier span or retained equations are unsolved.

Mandatory destructive control: Choose a radiative normal force outside the multiplier span. A no-propagator claim must still fail solvability.

Gate families invoking this row: Gap-05-stability, Lambda-catastrophe, SG-1, SG-5, SG-6, UQF-10, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C06 — Ward, anomaly, domain, and global quantum consistency

Owned question: Do gauge/BRST/BV identities survive the measure, boundaries, fixed sets, and global gluing?

Required inputs: Physical measure; domains; anomaly polynomials; boundary and inflow data; global-form data.

Execution algorithm:

1. Compute allowed transformations preserving every domain.

1. Evaluate bulk, fixed-set, ghost, measure, counterterm, and inflow variations in one convention.

1. Check local cancellation on every stratum and relative/global determinant-line trivialization.

1. Verify the quantum master equation and gluing compatibility.

Required outputs: Ward/BV packet; local anomaly ledger; inflow/quantization packet; relative-QME result.

Minimal discharge: The domain is preserved, anomalies cancel, and multiplier/boundary equations admit a global solution.

Fail trigger: Unresolved Ward identity, anomaly, boundary equation, or QME residual.

Mandatory destructive control: Arrange integrated anomaly cancellation while one fixed-set residual remains. The gate must stay open.

Gate families invoking this row: Born-rule, Gap-01, SG-2, SG-3, SG-4, SG-9, UQF-3, UQF-4, UQF-5C, UQF-7, UQF-9, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C07 — Physical response, conservation, and ordinary-source control

Owned question: How does the lawful system respond to localized and nonconstant sources without losing conservation or causality?

Required inputs: Solved background; physical operators; currents; Observer/TS interfaces; Vacuum packet where relevant.

Execution algorithm:

1. Derive retarded/causal response functions on the physical quotient.

1. Track exact currents and exchange terms.

1. Apply localized dust, radiation, pressure-gradient, and wave-packet controls.

1. Separate protected constants from ordinary excitations and publish record-level response.

Required outputs: Response-function packet; current-conservation ledger; ordinary-source negative control.

Minimal discharge: Localized and nonconstant sources yield the correct conserved causal response while protected constants follow VAC.

Fail trigger: Ordinary matter is degravitated, conservation fails, or response is acausal.

Mandatory destructive control: Project out all trace and traceless stress. Constant-shift protection may pass, but ordinary gravity must fail.

Gate families invoking this row: Black-hole-singularity, Born-rule, Gap-02, Gap-05-stability, Gap-05-value, Gap-08, Gap-10/BG-10, Gap-11, Gap-13, Lambda-catastrophe, UQF-14, UQF-5A/5B.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C08 — Scope, tower, and anti-promotion theorem

Owned question: What exact modal, energy, time, and approximation scope is proven?

Required inputs: Operator spectra; cutoff; Granularity inventory; Scale packet; tail information.

Execution algorithm:

1. Declare zero-mode, finite-tower, EFT, history-window, or all-mode scope.

1. List unresolved modes, sectors, and time intervals.

1. Supply a tail theorem or bounded remainder for any global claim.

1. Block every promotion from finite evidence to a stronger terminal.

Required outputs: Scope theorem; truncation ledger; tail/remainder certificate; claim ceiling.

Minimal discharge: The result declares zero-mode, finite-tower, EFT, or all-mode scope, with a tail theorem for global claims.

Fail trigger: A local or finite result is promoted beyond scope.

Mandatory destructive control: Use a finite prefix with two lawful tails giving opposite global behavior. The global claim must fail.

Gate families invoking this row: Black-hole-singularity, Gap-01, Gap-02, Gap-05-stability, Gap-13, SG-1, SG-10, SG-2, SG-3, SG-6, SG-7, SG-8, SG-9, UQF-10, UQF-14, UQF-3, UQF-5A/5B, UQF-5C, UQF-9.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C09 — Physical operator, kernel, spectrum, and response construction

Owned question: What physical operators follow from the complete equations, and what kernels and spectra survive every quotient and domain?

Required inputs: Solved background; full equations; constraint/gauge complex; Boundary domains; Actor owners; Granularity scope.

Execution algorithm:

1. Generate every gate-relevant fluctuation sector from the Stage–Actor support grammar.

1. Linearize the full equations, including mixed blocks and reaction sectors.

1. Apply gauge fixing, constraints, boundary/global domains, and physical projectors.

1. Publish raw, gauge, constrained, and physical kernels; zero modes; generalized modes; massive spectrum; response functions; and unresolved tails.

1. Prove operator-shelf saturation: every generated carrier/operator has one disposition.

Required outputs: Physical-operator packets; mixed-block matrix; kernel/spectrum ledger; operator-saturation certificate.

Minimal discharge: Linearize the complete parent equations, build the gauge-fixed and constraint-reduced operators on the lawful domains, and publish kernels, zero modes, spectra, response functions, multiplicities, and unresolved tails.

Fail trigger: A gate uses a declared Actor or expected spectrum without deriving the corresponding physical operator/kernel, or omits a mixed/operator block.

Mandatory destructive control: Add one lawful boundary-localized zero mode while keeping the owner list unchanged. The no-extra claim must fail.

Gate families invoking this row: Black-hole-singularity, Gap-01, Gap-02, Gap-08, SG-1, SG-10, SG-2, SG-3, SG-5, SG-6, SG-8, UQF-10, UQF-14, UQF-3, UQF-5A/5B, UQF-7, UQF-9.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C10 — Complete interaction, source, and dangerous-operator hypergraph

Owned question: What are all lawful interactions and sources, including theorem-zero and dangerous channels?

Required inputs: Parent term grammar; Actor/Co-Actor registry; Rulebook representation and selection rules; Boundary/Scale data.

Execution algorithm:

1. Generate all arity-1 and higher hyperedges allowed by support, representation, tensor, parity, charge, and derivative grammar.

1. Map every edge to a parent owner or a theorem-zero exclusion.

1. Track radiative generation, dangerous operators, portals, CP sources, baryon/lepton/flavor charges, and Observer records.

1. Run edge saturation and omitted-edge controls before evaluating the target gate.

Required outputs: Interaction hypergraph; theorem-zero ledger; dangerous-operator roster; edge-saturation certificate.

Minimal discharge: Enumerate every parent-owned interaction vertex, source edge, representation contraction, selection rule, theorem-zero edge, dangerous operator, support, scale, and boundary domain.

Fail trigger: A coupling, decay, portal, Yukawa, CP source, or dangerous operator is inserted or omitted without a complete parent-law disposition.

Mandatory destructive control: Hold the free spectrum fixed and add one allowed cubic/Yukawa edge to one candidate. Interaction-dependent gates must distinguish them.

Gate families invoking this row: Gap-10/BG-10, Gap-11, SG-5, SG-7, SG-8, SG-9, UQF-10, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C11 — Mass, mixing, transition, decay, and threshold derivation

Owned question: Do physical masses and transitions follow from owned interactions on a solved background?

Required inputs: Interaction hypergraph; background/order parameters; kinetic normalization; Scale/matching packet.

Execution algorithm:

1. Evaluate interaction derivatives on the solved background.

1. Construct kinetic and mass/mixing blocks after constraints and boundary reductions.

1. Canonically normalize, diagonalize, and identify physical eigenstates.

1. Derive widths, transition amplitudes, thresholds, and effective vertices with uncertainty and scope.

1. Run zero-condensate, wrong-representation, duplicate-owner, and omitted-edge controls.

Required outputs: Mass/mixing packet; eigenstate map; transition/decay ledger; threshold packet.

Minimal discharge: Mass matrices, mixings, symmetry breaking, decay widths, threshold transitions, and effective vertices are derived from C10 plus the solved background and are diagonalized/matched with stated scope.

Fail trigger: A matrix or transition is supplied by hand, the background does not solve the parent equations, or physical eigenstates/widths are not derived.

Mandatory destructive control: Insert a phenomenologically correct matrix without a parent vertex or solved order parameter. The row must fail.

Gate families invoking this row: Gap-10/BG-10, Gap-11, SG-5, SG-8, SG-9, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C12 — Quantum physicality, positivity, poles, and analytic continuation

Owned question: Does the quantum theory possess a positive physical sector with lawful continuation and pole structure?

Required inputs: Physical measure; gauge/BRST quotient; Euclidean/Lorentzian operator; domains; regulator family.

Execution algorithm:

1. Construct the physical pairing after quotient.

1. Test reflection positivity or an equivalent reconstruction theorem in the declared formalism.

1. Continue with an explicit prescription and track domains and branch cuts.

1. Compute physical poles, residues, complex modes, and negative-norm dispositions.

1. Repeat over required regulator/truncation families.

Required outputs: Quantum-physicality packet; reconstruction/positivity certificate; pole-residue ledger.

Minimal discharge: After gauge/BRST quotient, establish a positive physical pairing or reflection-positivity/equivalent reconstruction, pole and residue signs, continuation prescription, and absence/disposition of negative-norm states.

Fail trigger: Formal self-adjointness, Euclidean convergence, or BRST nilpotency is used without a physical positivity/pole certificate.

Mandatory destructive control: Use a formally self-adjoint gauge-fixed operator with a negative-residue physical pole. Self-adjointness alone must fail.

Gate families invoking this row: Born-rule, Gap-02, Gap-13, SG-5, SG-6, SG-8, UQF-10, UQF-14, UQF-3, UQF-5A/5B, UQF-5C.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C13 — Closed/open quantum evolution and probability instruments

Owned question: Is probability evolution lawful for closed and open sectors, including entangled extensions and Observer records?

Required inputs: Physical Hilbert/record spaces; Hamiltonian or channel generator; environment/interface ledger; Observer map.

Execution algorithm:

1. For closed sectors, prove unitary normalized evolution on the physical domain.

1. For open sectors, define the system–environment split, exchange currents, and CPTP channel or justified instrument.

1. Check complete positivity on admitted ancillas and composition across time/interface cuts.

1. Publish approximation errors and record normalization.

Required outputs: Quantum-channel packet; instrument/POVM packet; environment and exchange-current ledger.

Minimal discharge: Closed sectors have unitary normalized evolution; open sectors publish a completely positive trace-preserving channel or justified instrument, environment, exchange currents, approximation bound, and Observer handoff.

Fail trigger: A trace-preserving but nonphysical map, unnormalized probability rule, undeclared environment, or hidden information loss is used.

Mandatory destructive control: Use a trace-preserving but non-completely-positive map. Normalization must not be accepted as physical evolution.

Gate families invoking this row: Born-rule, Gap-10/BG-10, Gap-11, Gap-13, UQF-14, UQF-3.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C14 — Nonequilibrium, thermal, and cosmological-history evolution

Owned question: Does the proposed mechanism work through finite-time thermal and cosmological history?

Required inputs: Initial state/ensemble; background expansion; interaction graph; transport/collision data; Vacuum/Observer packets.

Execution algorithm:

1. Specify the initial ensemble and history interval.

1. Choose and justify Boltzmann, Schwinger–Keldysh, transport, lattice, kinetic, or equivalent evolution.

1. Compute production, washout, freeze-in/out, entropy, charge, perturbation, and transition histories.

1. Propagate uncertainties and compare finite records.

1. Separate static identities from history claims.

Required outputs: Nonequilibrium-history packet; abundance/charge trajectories; transition and perturbation records.

Minimal discharge: Publish initial state, thermal/kinetic/transport equations, entropy and charge production, phase-transition dynamics, production/washout/freeze-out, perturbations, and uncertainty through the claimed history.

Fail trigger: A static potential or constant-shift result is promoted to a finite-time cosmological, baryogenesis, dark-matter, or transition claim.

Mandatory destructive control: Use a mechanism that cancels static constants but produces an unacceptable transient. Static success must not close the history gate.

Gate families invoking this row: Gap-05-stability, Gap-05-value, Gap-08, Gap-10/BG-10, Gap-11, Lambda-catastrophe, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C15 — Well-posed causal initial-boundary-value evolution

Owned question: Do the equations define unique, stable, constraint-preserving predictions on the declared causal support?

Required inputs: Complete equations; principal symbol; initial surfaces; Boundary domains; TS and Observer handoffs.

Execution algorithm:

1. Compute the principal symbol or equivalent propagation object.

1. Establish hyperbolicity/well-posedness or a formalism-appropriate substitute.

1. Specify admissible initial data and boundary/fixed-set data.

1. Prove constraint propagation, uniqueness, continuous dependence, and support/no-signalling.

1. Compose clocks and interfaces through TS.

Required outputs: Causal-IBVP packet; characteristic-cone ledger; constraint-propagation certificate.

Minimal discharge: Prove or compute hyperbolicity/evolution well-posedness, constraint propagation, boundary compatibility, causal support, stability under allowed data, and synchronization/Observer handoffs.

Fail trigger: Equations exist but evolution is nonunique, unstable, superluminal outside the declared cone, domain-incompatible, or not globally composable.

Mandatory destructive control: Supply complete formal equations with nonunique evolution or non-preserved constraints. Equation completeness alone must fail.

Gate families invoking this row: Black-hole-singularity, Gap-02, Gap-08, Gap-10/BG-10, Gap-13, SG-1, SG-6, UQF-14, UQF-3, UQF-5A/5B, UQF-5C.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C16 — RG, regulator, refinement, and truncation stability

Owned question: Does the result survive lawful enlargement of operator bases, resolutions, schemes, and regulators?

Required inputs: Nested theory spaces; generated operators; regulators/schemes; Scale/Granularity packets.

Execution algorithm:

1. Construct nested operator bases or refinements before target inspection.

1. Track generated operators and closure residuals at every level.

1. Compute observables/flows/fixed points and differences between levels.

1. Vary admissible regulator and scheme families.

1. Provide convergence, critical-data, matching, and omitted-tail evidence.

Required outputs: RG/refinement packet; convergence table; scheme/regulator stability certificate.

Minimal discharge: Use nested operator bases/resolutions and admissible regulator/scheme families; track generated operators, fixed points/flows, critical data, convergence, matching, and omitted-tail bounds.

Fail trigger: A result exists only in one truncation, basis, regulator, or scheme and changes materially under lawful refinement.

Mandatory destructive control: Choose two lawful higher-order completions sharing the finite prefix but with opposite fixed-point behavior. A completion claim must fail.

Gate families invoking this row: Gap-01, Gap-02, Gap-05-stability, Gap-08, Gap-11, Gap-13, Lambda-catastrophe, SG-10, SG-6, SG-7, SG-8, SG-9, UQF-10, UQF-14, UQF-3, UQF-5A/5B, UQF-5C, UQF-9, theta-bar-QCD.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

DYN-C17 — Regional, horizon, interface, and information-flux balance

Owned question: How do conserved quantities and information move across boundaries, horizons, defects, or coarse-graining cuts?

Required inputs: Global currents; region/interface geometry; Boundary/TS/Observer data; open-system packet.

Execution algorithm:

1. Define the region, interface, normals, and edge sectors.

1. Derive energy, charge, probability, entropy, and information balance laws.

1. Include boundary, horizon, gluing, environment, and edge contributions.

1. Distinguish global conservation from regional change and apparent subsystem nonunitarity.

Required outputs: Regional-flux packet; interface/gluing ledger; information-channel balance.

Minimal discharge: For regions separated by boundaries, defects, horizons, or cuts, publish charge/energy/probability/information currents, gluing law, edge terms, exchange channel, and finite Observer records.

Fail trigger: Local conservation is claimed while flux through an interface is omitted, or subsystem nonunitarity is mistaken for global loss without a channel ledger.

Mandatory destructive control: Omit flux through a horizon while retaining local conservation. The regional closure must fail.

Gate families invoking this row: Black-hole-singularity, Gap-13, UQF-14.

Status rule: A definition, qualitative argument, or borrowed terminal is insufficient. Missing gate-scoped evidence is OPEN; a reproducible counterexample is FAIL; NOT-APPLICABLE requires a type proof.

8. Full packet schemas

8.1 parentactionmanifest

parent_action_manifest:

  manifest_id:

  branch_id:

  fields:

  supports:

  terms:

  coefficient_provenance:

  symmetries:

  boundary_variations:

  variational_dispositions:

  aliases:

  generated_term_grammar:

  omitted_term_theorem:

  status:

8.2 constraintclosurepacket

constraint_closure_packet:

  constraint_ids:

  primary_constraints:

  secondary_constraints:

  brackets:

  rank_strata:

  first_class:

  second_class:

  residual_gauges:

  boundary_constraints:

  physical_dof_count:

  termination_proof:

  status:

8.3 physicalmeasurepacket

physical_measure_packet:

  measure_id:

  gauge_fixing:

  ghosts:

  fp_determinant:

  dirac_determinant:

  bv_bfv_measure:

  zero_modes:

  phases:

  field_dependence:

  effective_action_terms:

  stress_terms:

  status:

8.4 backgroundsolvabilitypacket

background_solvability_packet:

  unknowns:

  equations:

  identities:

  gauge_freedoms:

  boundary_global_equations:

  jacobian_or_fredholm_map:

  existence_domain:

  solution:

  classification:

  residuals:

  status:

8.5 physicaloperatorpacket

physical_operator_packet:

  operator_id:

  parent_equations:

  background:

  carrier_and_owner:

  raw_operator:

  mixed_blocks:

  constraint_complex:

  gauge_fixing:

  boundary_global_domain:

  physical_projector:

  raw_kernel:

  gauge_kernel:

  physical_kernel:

  zero_modes:

  spectrum:

  response:

  tail_certificate:

  status:

8.6 interactionhypergraphpacket

interaction_hypergraph_packet:

  edge_generator:

  actor_vertices:

  composite_vertices:

  edges:

  representation_contractions:

  charges:

  parities:

  supports:

  selection_rules:

  radiative_generation:

  dangerous_operators:

  theorem_zero_edges:

  omitted_edge_theorem:

  status:

8.7 massmixingtransition_packet

mass_mixing_transition_packet:

  background_order_parameters:

  kinetic_matrix:

  mass_matrix:

  mixing_blocks:

  normalization:

  eigenstates:

  eigenvalues:

  transition_amplitudes:

  decay_widths:

  thresholds:

  uncertainties:

  controls:

  status:

8.8 quantumphysicalitypacket

quantum_physicality_packet:

  formalism:

  physical_quotient:

  pairing:

  reflection_positivity:

  reconstruction:

  continuation:

  poles:

  residues:

  branch_cuts:

  negative_norms:

  complex_modes:

  regulator_scope:

  status:

8.9 quantumchannelpacket

quantum_channel_packet:

  closed_or_open:

  system_space:

  environment_space:

  generator_or_unitary:

  channel_or_instrument:

  complete_positivity:

  trace_normalization:

  exchange_currents:

  composition:

  observer_records:

  approximation_bound:

  status:

8.10 nonequilibriumhistorypacket

nonequilibrium_history_packet:

  initial_state:

  ensemble:

  time_interval:

  temperature_history:

  expansion_history:

  evolution_equations:

  collision_terms:

  sources:

  entropy:

  charges:

  production:

  washout:

  freeze_in_out:

  perturbations:

  records:

  uncertainty:

  status:

8.11 causalinitialboundary_packet

causal_initial_boundary_packet:

  principal_symbol:

  characteristics:

  causal_structure:

  initial_surface:

  initial_data:

  boundary_data:

  constraint_propagation:

  existence:

  uniqueness:

  continuous_dependence:

  support:

  ts_handoff:

  observer_handoff:

  status:

8.12 rgrefinementpacket

rg_refinement_packet:

  nested_spaces:

  generated_operators:

  regulators:

  schemes:

  flows:

  fixed_points:

  critical_data:

  observables:

  level_differences:

  convergence:

  matching:

  tail_bounds:

  status:

8.13 regionalfluxpacket

regional_flux_packet:

  region:

  interfaces:

  normals:

  currents:

  edge_terms:

  gluing:

  environment_channels:

  energy_balance:

  charge_balance:

  probability_balance:

  entropy_balance:

  information_balance:

  observer_records:

  status:

8.14 integrateddynamicscertificatev42

This is the required top-level gate handoff. It preserves the complete V3 view and binds every V4.2 packet to one branch, gate, scope, and calculation order.

integrated_dynamics_certificate_v4_2:
  candidate_id:
  branch_id:
  gate_id:
  authority_hashes: {}
  frozen_assumptions: []
  legacy_v3_compatibility:
    parent_action:
    primitive_term_ledger:
    complete_equation_vector:
    equation_count_and_identities:
    constraint_types:
    dirac_bergmann_closure:
    rank_strata:
    physical_measure:
    determinant_contributions:
    constraint_stress:
    multiplier_boundary_problem:
    background_status:
    background_solution:
    radiative_solvability:
    ward_identities:
    anomaly_result:
    physical_hessian_packet:
    vacuum_response_packet:
    ordinary_gravity_control:
    phase_transition_scope:
    full_tower_scope:
    evidence_hashes: []
    verdict:
  packets:
    parent_action_manifest:
    constraint_closure_packet:
    physical_measure_packet:
    background_solvability_packet:
    physical_operator_packet:
    interaction_hypergraph_packet:
    mass_mixing_transition_packet:
    quantum_physicality_packet:
    quantum_channel_packet:
    nonequilibrium_history_packet:
    causal_initial_boundary_packet:
    rg_refinement_packet:
    regional_flux_packet:
    boundary_quantum_dynamics:
  legacy_negative_controls:
    no_propagator_but_no_solution:
    no_work_but_nonzero_stress:
    vacuum_protection_but_matter_degravitation:
    frozen_stage_with_abandoned_equations:
  v4_2_destructive_controls: []
  residual_vector: {}
  claim_ceiling:
  physical_verdict:
  dynamics_sufficiency_verdict:
  project_dependency_verdict:
  reopen_triggers: []
  stale_dependencies: []
  calculation_order:
  validator_result:
  evidence_hashes: []

A gate may publish specialized subcertificates, but this integrated certificate is mandatory for a final Dynamics adjudication.

9. Candidate generation and saturation algorithms

9.1 Parent-term saturation

INPUT: frozen Stage, Rulebook, Actor, Scale, Boundary scope
1. Enumerate supports, fields/owners, tensor types, representations, derivative orders, strata, and global classes.
2. Form every invariant local/global/boundary monomial allowed by the grammar.
3. Canonicalize integrations by parts, identities, field redefinitions, cover/quotient aliases, and exact equivalences.
4. Assign owner, coefficient provenance, and disposition: retained / forbidden / theorem-zero / matching-only / open.
5. Repeat after radiative generation and refinement.
PASS only when no undisposed in-grammar term remains.

9.2 Operator-shelf saturation

For every Stage-supported Actor carrier and every lawful mixed pair:
  derive the complete linearized block from the full equations;
  attach constraint, gauge, boundary, global, and physical-pairing data;
  compute or theorem-classify kernel and spectrum;
  assign every zero/generalized/massive mode and unresolved tail;
  compare against Granularity mode inventory.
PASS only when generated operators and modes have one terminal disposition.

9.3 Interaction-edge saturation

Generate every hyperedge from Actor tuples, representation singlets, support overlap, parity, exact charges, derivative grammar, and Rulebook permissions.
For each edge, prove parent ownership, theorem-zero exclusion, or OPEN status.
Run dangerous-operator and radiative-generation closure.
PASS only when a new generation pass adds no inequivalent edge.

9.4 History/refinement saturation

Refine modal cutoff, operator basis, regulator/scheme, time resolution, thermal species, regional decomposition, and interface data.
Track observable and terminal changes.
PASS for a global claim only with convergence/tail theorem; otherwise emit the strongest finite-scope terminal.

10. Thought-experiment derivation and destructive controls

TE-DYN-01 → DYN-C09

Same owner/action labels, but one candidate has an extra boundary zero mode. The old rows can pass while the physical result changes; therefore DYN-C09 is irreducible under the current grammar.

TE-DYN-02 → DYN-C10

Identical quadratic theories, but only one has an allowed cubic or Yukawa vertex. The old rows can pass while the physical result changes; therefore DYN-C10 is irreducible under the current grammar.

TE-DYN-03 → DYN-C11

A correct-looking mass matrix is inserted with no parent vertex or solved condensate. The old rows can pass while the physical result changes; therefore DYN-C11 is irreducible under the current grammar.

TE-DYN-04 → DYN-C12

A formally self-adjoint gauge-fixed operator has a negative physical residue. The old rows can pass while the physical result changes; therefore DYN-C12 is irreducible under the current grammar.

TE-DYN-05 → DYN-C13

A trace-preserving subsystem map fails complete positivity on an entangled extension. The old rows can pass while the physical result changes; therefore DYN-C13 is irreducible under the current grammar.

TE-DYN-06 → DYN-C14

A static vacuum cancellation produces an unacceptable finite-time transient. The old rows can pass while the physical result changes; therefore DYN-C14 is irreducible under the current grammar.

TE-DYN-07 → DYN-C15

A complete equation list has nonunique or constraint-violating evolution. The old rows can pass while the physical result changes; therefore DYN-C15 is irreducible under the current grammar.

TE-DYN-08 → DYN-C16

A fixed point disappears under a lawful higher-order completion. The old rows can pass while the physical result changes; therefore DYN-C16 is irreducible under the current grammar.

TE-DYN-09 → DYN-C17

Local conservation is claimed while horizon/interface flux is omitted. The old rows can pass while the physical result changes; therefore DYN-C17 is irreducible under the current grammar.

The original controls remain mandatory: unowned term, early constraint termination, dropped determinant, no-work/zero-stress substitution, unsolved reaction system, integrated-only anomaly cancellation, ordinary-source degravitation, and finite-to-global scope promotion.

11. Gate execution compiler

11.1 Frozen audit

1. Load gate contract, authorities, hashes, scope, expected witness types, and allowed terminals.
2. Select every revised Dynamics row from the gate profile; NOT-APPLICABLE requires a type proof.
3. Build the parent-term, equation, constraint, operator, interaction, and history slices without reading the desired answer.
4. Evaluate row by row and freeze PASS/FAIL/OPEN plus evidence hashes.
5. Run all mandatory destructive controls.
6. Compute the Dynamics residual vector.

11.2 Repair loop

Repairs occur only after the immutable audit. A repair is admissible only if it is candidate-neutral, reusable across every gate invoking the row, does not add a target-driven Actor/term/constraint/regulator, and propagates through dependency hashes. Re-execute from the earliest changed pass.

11.3 Challenge calculation

Every gate must include one calculation capable of falsifying the current branch or improving a reusable building block. A challenge calculation is not a prose review. It must produce machine-readable inputs, code/order, outputs, controls, and residuals.

11.4 Terminal separation

• Physical endpoint: what the calculation actually establishes.
• Dynamics sufficiency endpoint: whether this block had all required capabilities.
• Project-dependency endpoint: whether the scoped gate dependency is resolved under governance.
• Frontier/external endpoint: any unsolved theorem, measurement, or UV completion wall.

12. All-gate Dynamics profiles

12.1 Black-hole-singularity

Dynamics role: LOAD-BEARING. Source terminal under test: DISSOLVED-GIVEN-root / RESOLVED +0. Required rows: DYN-C07, DYN-C08, DYN-C09, DYN-C15, DYN-C17.

Audit finding: Dissolving a continuum singularity does not supply a finite causal evolution, horizon/interface flux balance, or record map.

Required execution:

1. construct physical gravitational operator and regularized/finite-support kernel at claimed scope.

1. establish causal evolution on the non-continuum interior model.

1. publish horizon/interface flux and Observer-record packet.

Claim ceiling / action: Add C15/C17 and integrate TS/OBS; no claim of a UV interior solution.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.2 Born-rule

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C03, DYN-C06, DYN-C07, DYN-C12, DYN-C13.

Audit finding: A stronger Born derivation requires a positive physical measure and lawful normalized event instrument; the current irreducibility terminal can remain narrower.

Required execution:

1. derive positive physical measure and quotient.

1. construct normalized event instrument or identify irreducible operational anchor.

1. run non-CP and normalization destructive controls.

Claim ceiling / action: Genuine improvement for constructive derivation; no forced change to irreducibility terminal.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.3 Gap-01

Dynamics role: LOAD-BEARING. Source terminal under test: DERIVED-GIVEN-anchor / RESOLVED +0. Required rows: DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16.

Audit finding: The a6 coefficient depends on the complete operator, domain, measure, boundary data, scheme, and finite-order scope.

Required execution:

1. build complete Laplace-type operator and boundary domain.

1. include ghosts/measure and anomaly conventions.

1. calculate finite heat-kernel coefficient at declared order and test scheme/refinement scope.

Claim ceiling / action: Add generic operator/kernel and refinement certificates; calculation remains gate-specific.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.4 Gap-02

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16.

Audit finding: A mass-gap claim requires a physical spectral operator, positive physical sector, causal theory, and nonperturbative/refinement control. The external-wall terminal needs only proof that these are not supplied.

Required execution:

1. construct physical Yang–Mills spectral operator.

1. establish positive physical sector and causal dynamics.

1. supply nonperturbative/refinement evidence or issue external-wall terminal.

Claim ceiling / action: Add C09/C12/C15/C16; no claim that the Clay problem is solved.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.5 Gap-05-stability

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C03, DYN-C04, DYN-C05, DYN-C07, DYN-C08, DYN-C14, DYN-C16.

Audit finding: Radiative stability requires full measure/stress, solvable reactions, ordinary-source control, time-dependent transitions, and scheme/refinement stability.

Required execution:

1. derive full vacuum/reaction stress and radiative equations.

1. simulate finite-time phase transitions.

1. test regulator/scheme/refinement stability.

Claim ceiling / action: Add C14/C16 execution; VAC retains vacuum-specific ownership.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.6 Gap-05-value

Dynamics role: NOT-LOAD-BEARING-FOR-MEASURED-TERMINAL. Source terminal under test: MEASURED-ANCHOR / RESOLVED +0. Required rows: DYN-C07, DYN-C14.

Audit finding: The measured value terminal does not require a mechanism. A predictive/history claim does require response and cosmological evolution.

Required execution:

1. for measured terminal, verify ordinary response and history compatibility only.

1. for predictive terminal, derive mechanism and cosmological evolution.

Claim ceiling / action: No change for measured anchor; use C14 only for stronger claim.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.7 Gap-08

Dynamics role: LOAD-BEARING. Source terminal under test: DISSOLVED-GIVEN-root / RESOLVED +0. Required rows: DYN-C07, DYN-C09, DYN-C14, DYN-C15, DYN-C16.

Audit finding: Replacing or dissolving inflation still requires a complete causal perturbation/history map from initial data to finite records.

Required execution:

1. specify initial state and causal perturbation evolution.

1. derive record map replacing inflationary history.

1. test refinement and uncertainty.

Claim ceiling / action: Genuine improvement: nonequilibrium history and causal evolution packets.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.8 Gap-10/BG-10

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C15.

Audit finding: Baryogenesis requires CP-violating interactions, out-of-equilibrium transport, charge/entropy evolution, washout, freeze-out, and causal initial data.

Required execution:

1. generate CP-violating interaction hypergraph.

1. derive kinetic/transport equations and charge sources.

1. compute production, washout, freeze-out, and final asymmetry.

Claim ceiling / action: Genuine Dynamics improvement: interaction plus nonequilibrium/thermal module.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.9 Gap-11

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C16.

Audit finding: A dark-matter portal needs complete production/decay interactions, open/thermal evolution, abundance history, and running/matching.

Required execution:

1. generate portal/production/decay edges.

1. derive thermal/open-system abundance evolution.

1. run matching and RG stability.

Claim ceiling / action: Genuine Dynamics improvement: interaction and thermal production module.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.10 Gap-13

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / PROJECT-DEPENDENCY / CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C03, DYN-C07, DYN-C08, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17.

Audit finding: Microstate/Page claims require positive quantum evolution, subsystem channels, causal/horizon flux, refinement scope, and finite records.

Required execution:

1. construct positive quantum channel for horizon-separated regions.

1. track Page/information and energy flux.

1. state refinement/UV claim ceiling.

Claim ceiling / action: Add quantum physicality, open-system, causal, and horizon-flux interfaces; constructive QG remains external.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.11 Lambda-catastrophe

Dynamics role: LOAD-BEARING. Source terminal under test: DISSOLVED-GIVEN-root / RESOLVED +0. Required rows: DYN-C04, DYN-C05, DYN-C07, DYN-C14, DYN-C16.

Audit finding: A degravitation mechanism must solve full equations, retain ordinary gravity, survive phase transitions, and remain radiatively stable.

Required execution:

1. solve complete reaction/background equations.

1. retain ordinary localized gravity.

1. simulate finite vacuum transitions and radiative stability.

Claim ceiling / action: Add C14/C16; VAC remains the mechanism owner.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.12 SG-1

Dynamics role: SUPPORTING. Source terminal under test: CLOSED / DERIVED-GIVEN-SHAPE / RESOLVED +0. Required rows: DYN-C01, DYN-C05, DYN-C08, DYN-C09, DYN-C15.

Audit finding: Existing Shape terminal does not require a derived parent vacuum, but any stronger claim that the selected Stage is dynamically realized requires background solvability, physical operators, and well-posed evolution.

Required execution:

1. for stronger dynamical-realization claim, solve parent background.

1. construct geometric fluctuation operators.

1. prove causal/well-posed evolution at declared scope.

Claim ceiling / action: Crosswalk expansion only; no new gate-specific physics term.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.13 SG-10

Dynamics role: LOAD-BEARING. Source terminal under test: DERIVED-GIVEN-anchor / RESOLVED +0. Required rows: DYN-C08, DYN-C09, DYN-C16.

Audit finding: Scope consistency requires the actual operator/tower content and a refinement/tail certificate, not a label saying finite or all modes.

Required execution:

1. construct actual tower operator and mode inventory.

1. publish tail theorem or finite scope.

1. test refinement stability.

Claim ceiling / action: Add C09/C16 to make the existing anti-promotion rule executable.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.14 SG-2

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / CERTIFIED-IRREDUCIBLE-CATEGORY-FLOOR / RESOLVED +0. Required rows: DYN-C01, DYN-C02, DYN-C06, DYN-C08, DYN-C09.

Audit finding: Gauge-owner identities must map to one kinetic owner, lawful domains, quotient, and exactly the physical vector kernel; the current crosswalk omits Dynamics entirely.

Required execution:

1. map each gauge owner to one kinetic owner.

1. construct BRST/domain-reduced vector operator.

1. derive physical kernel and prove 8+3+1 with zero extra modes.

Claim ceiling / action: Add DYN-C09 and refresh the crosswalk; existing SG-2 evidence can populate it.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.15 SG-3

Dynamics role: LOAD-BEARING. Source terminal under test: DERIVED-GIVEN-E / RESOLVED +0. Required rows: DYN-C01, DYN-C06, DYN-C08, DYN-C09.

Audit finding: The chiral index and no-mirror claim require the physical Dirac operator/domain/kernel and quantum gauge consistency, not only a topological count.

Required execution:

1. construct Dirac operator on exact parity/domain data.

1. derive chiral kernel and mirror disposition.

1. pass measure/anomaly handoff to AD/BND.

Claim ceiling / action: Add DYN-C09; integrate BND/AD evidence.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.16 SG-4

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / DERIVED-GIVEN-E / RESOLVED +0. Required rows: DYN-C01, DYN-C03, DYN-C06.

Audit finding: Charges and anomalies require one parent coupling, the fermion/ghost measure, Ward identities, and boundary/global anomaly closure.

Required execution:

1. derive charge couplings from parent terms.

1. include fermion/ghost measure.

1. evaluate Ward/anomaly consistency with BND-B.

Claim ceiling / action: Crosswalk expansion; BND-B remains decisive.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.17 SG-5

Dynamics role: LOAD-BEARING. Source terminal under test: DERIVED-GIVEN-anchor / RESOLVED +0. Required rows: DYN-C01, DYN-C05, DYN-C09, DYN-C10, DYN-C11, DYN-C12.

Audit finding: EWSB cannot close from Actor identity alone: the parent interaction, solved background, mass/mixing matrices, physical scalar/vector spectrum, and positivity are required.

Required execution:

1. generate complete electroweak interaction graph.

1. solve order-parameter background.

1. derive vector/scalar mass and mixing spectra and positivity.

Claim ceiling / action: Genuine Dynamics improvement: C10-C12 and executable packets.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.18 SG-6

Dynamics role: SUPPORTING. Source terminal under test: DERIVED-GIVEN-anchor / RESOLVED +0. Required rows: DYN-C05, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16.

Audit finding: Rigidity owns the terminal, but Dynamics must supply the full physical Hessian/operator, quantum physicality, tower scope, and refinement stability.

Required execution:

1. construct full mixed physical Hessian/operator.

1. supply positivity and causal evolution.

1. test tower/refinement stability before Rigidity terminal.

Claim ceiling / action: Strengthen Dynamics-to-Rigidity handoff; no replacement of RIG.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.19 SG-7

Dynamics role: LOAD-BEARING. Source terminal under test: DISSOLVED-GIVEN-root / RESOLVED +0. Required rows: DYN-C01, DYN-C03, DYN-C08, DYN-C10, DYN-C16.

Audit finding: Threshold unification and proton-scale matching depend on complete interaction content, determinants, tower scope, and scheme/refinement stability.

Required execution:

1. enumerate threshold-relevant interactions and determinants.

1. run same-ruler matching.

1. test tower/regulator/scheme stability.

Claim ceiling / action: Add C10/C16 packet; Scale still owns ruler/matching provenance.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.20 SG-8

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / RESCUED / DISSOLVED-WRONG-RULER + DERIVED-GIVEN-13D-ACTOR / RESOLVED +0. Required rows: DYN-C01, DYN-C08, DYN-C09, DYN-C10, DYN-C11, DYN-C12, DYN-C16.

Audit finding: Flavor masses and mixings require a complete Yukawa/interaction graph, generated matrices, physical eigenstates, and same-ruler/refinement controls.

Required execution:

1. generate complete Yukawa/flavor graph.

1. derive mass and mixing matrices from solved background.

1. test physical eigenstates, positivity, and refinement.

Claim ceiling / action: Genuine Dynamics improvement: interaction and mass-generation execution.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.21 SG-9

Dynamics role: LOAD-BEARING. Source terminal under test: DERIVED-GIVEN-anchor / RESOLVED +0. Required rows: DYN-C01, DYN-C06, DYN-C08, DYN-C10, DYN-C11, DYN-C16.

Audit finding: Proton safety is a universal statement over dangerous operators, selection rules, induced vertices, thresholds, and running.

Required execution:

1. generate all dangerous baryon-violating operators.

1. prove selection-rule/theorem-zero dispositions.

1. derive induced amplitudes, thresholds, and RG stability.

Claim ceiling / action: Genuine Dynamics improvement: complete dangerous-operator hypergraph and RG stability.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.22 UQF-10

Dynamics role: LOAD-BEARING. Source terminal under test: RESOLVED +0 / DERIVED-GIVEN-anchor + CLOSED-NEGATIVE leg. Required rows: DYN-C01, DYN-C02, DYN-C05, DYN-C08, DYN-C09, DYN-C10, DYN-C12, DYN-C16.

Audit finding: Compactification consistency requires complete mixed operators/interactions, constraint closure, solvable background, physical spectrum, and tower/refinement control.

Required execution:

1. construct all mixed compactification operators and interactions.

1. solve background/constraint system.

1. test physical spectrum, positivity, and tower stability.

Claim ceiling / action: Add C09/C10/C12/C16; Rigidity/Boundary remain co-owners.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.23 UQF-14

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17.

Audit finding: Above-cutoff causality requires a physical quantum channel, well-posed causal support, refinement/tail control, and regional/interface flux accounting.

Required execution:

1. construct physical high-energy channel.

1. establish causal initial-boundary evolution with TS.

1. track interface flux and refinement/tail scope.

Claim ceiling / action: Genuine improvement: TS-integrated causal and quantum-channel certificates.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.24 UQF-3

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16.

Audit finding: Reflection positivity is not discharged by formal self-adjointness; the full physical measure, quotient, continuation, poles, and regulator/tail stability are needed.

Required execution:

1. derive full physical measure and operator quotient.

1. test reflection positivity/reconstruction and poles.

1. test causal/channel and regulator stability.

Claim ceiling / action: Genuine Dynamics improvement: executable positivity/continuation packet.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.25 UQF-4

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / DERIVED-GIVEN-13D-SHAPE / GLOBAL-ANOMALY-CONSISTENT / RESOLVED +0. Required rows: DYN-C03, DYN-C06.

Audit finding: Global anomalies are quantum-measure and Ward/QME obstructions; Boundary B owns local/global relative trivialization.

Required execution:

1. derive quantum measure/Ward residuals.

1. supply local/global anomaly data to Boundary B.

1. do not treat classical BRST closure as quantum closure.

Claim ceiling / action: Crosswalk expansion; no new Dynamics submodule.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.26 UQF-5A/5B

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C01, DYN-C03, DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16.

Audit finding: A graviton sector needs the full physical operator, gauge quotient, positive poles/residues, causal evolution, and stable tower/truncation scope.

Required execution:

1. construct gauge-reduced graviton operator.

1. derive positive poles/residues and causal propagation.

1. test tower/truncation stability.

Claim ceiling / action: Genuine improvement: positivity and causal/refinement packets.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.27 UQF-5C

Dynamics role: LOAD-BEARING. Source terminal under test: CLOSED / CERTIFIED-IRREDUCIBLE-FLOOR / RESOLVED +0. Required rows: DYN-C06, DYN-C08, DYN-C12, DYN-C15, DYN-C16.

Audit finding: A constructive UV completion requires nonperturbative/refinement evidence and quantum physicality. The existing irreducible-floor terminal only requires that finite evidence not be promoted.

Required execution:

1. for constructive claim, test nonperturbative/refinement stability and quantum physicality.

1. for irreducible-floor terminal, prove finite evidence cannot be promoted.

Claim ceiling / action: Add C16 for any constructive claim; current external-wall terminal remains honest.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.28 UQF-7

Dynamics role: LOAD-BEARING. Source terminal under test: DERIVED-GIVEN-anchor / RESOLVED +0. Required rows: DYN-C03, DYN-C06, DYN-C09.

Audit finding: Anomaly descent consumes the chiral operator/domain/kernel and quantum measure, then passes to AD/BND for descent and gluing.

Required execution:

1. construct chiral operator/domain/kernel.

1. derive measure contribution.

1. handoff descent and gluing to AD/BND.

Claim ceiling / action: Add C09 crosswalk; integrate AD-1.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.29 UQF-9

Dynamics role: LOAD-BEARING. Source terminal under test: CERTIFIED-IRREDUCIBLE / RESOLVED +0. Required rows: DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16.

Audit finding: Seeley–DeWitt/UV calculations require the complete operator, domain, measure, boundary terms, and strict finite-order/refinement scope.

Required execution:

1. construct complete heat-kernel operator/domain/measure.

1. calculate finite coefficients.

1. enforce strict finite-order and refinement scope.

Claim ceiling / action: Add C09/C16; no promotion to UV completion.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

12.30 theta-bar-QCD

Dynamics role: LOAD-BEARING. Source terminal under test: DISSOLVED-GIVEN-root / RESOLVED +0. Required rows: DYN-C01, DYN-C05, DYN-C06, DYN-C10, DYN-C11, DYN-C14, DYN-C16.

Audit finding: A constructive strong-CP mechanism needs topological interactions, anomaly consistency, relaxation/transition dynamics, and radiative stability. A dissolution terminal may use a narrower subset.

Required execution:

1. generate topological and anomalous interactions.

1. derive relaxation/transition dynamics for constructive route.

1. test radiative and refinement stability or preserve narrower dissolution route.

Claim ceiling / action: Add C10/C11/C14/C16 for constructive leg; preserve scoped dissolution route.

Minimum destructive controls: run the controls attached to every selected row, plus omitted-parent-term, omitted-operator/mode, omitted-interaction-edge, and scope-promotion controls where type-applicable.

13. Cross-block handoff matrix

| Interface | Dynamics consumes | Dynamics emits | Firewall | |---|---|---|---|

| Shape → Dynamics | supports, rules, Actor owners | parent terms, equations, physical kernels | Dynamics may not create new Shape content |

| Dynamics → Granularity | candidate packets, spectra, histories | equivalence/exhaustion verdict | Dynamics does not declare candidate exhaustion |

| Dynamics ↔ Boundary | operator variation and currents | domains, kernels, anomaly/inflow, gluing | local/global boundary closure remains Boundary-owned |

| Dynamics → Rigidity | physical Hessian, mixed operators, evolution | stability/rigidity terminal | Dynamics does not declare rigidity |

| Dynamics ↔ Vacuum | full equations, stress, history | constant-shift tests and mechanism terminal | static cancellation cannot replace history |

| Dynamics ↔ Scale | running, thresholds, matching objects | rulers, provenance, uncertainty | no unowned numerical convention |

| Dynamics → Observer | response, channels, currents | finite records/calibration | projection cannot repair inconsistent Dynamics |

| Dynamics ↔ TS | characteristics, causal support | clock compatibility and synchronization | causal evolution requires both |

| Dynamics → Interdependence | dependency graph and evidence hashes | stale propagation and branch coherence | Dynamics cannot self-ratify |

14. Dynamics residual vector and closure condition

Define \[\mathcal R{\mathrm{DYN}}=(r1,\ldots,r_{19}).\]

• r_1: count or quantified norm of unowned parent terms.

• r_2: count or quantified norm of undisposed variables/equations.

• r_3: count or quantified norm of unclosed constraints/rank strata.

• r_4: count or quantified norm of omitted determinant/measure terms.

• r_5: count or quantified norm of unsolved background/reaction equations.

• r_6: count or quantified norm of Ward/anomaly/domain residuals.

• r_7: count or quantified norm of incorrect ordinary-source response.

• r_8: count or quantified norm of scope/tail residuals.

• r_9: count or quantified norm of unclassified operators/modes.

• r_10: count or quantified norm of undisposed interaction edges.

• r_11: count or quantified norm of unowned masses/mixings/transitions.

• r_12: count or quantified norm of negative/complex physical-mode residuals.

• r_13: count or quantified norm of invalid quantum channels/instruments.

• r_14: count or quantified norm of unresolved history variables/intervals.

• r_15: count or quantified norm of ill-posed causal data/domains.

• r_16: count or quantified norm of unstable regulator/refinement directions.

• r_17: count or quantified norm of unbalanced regional/interface currents.

• r_18: count or quantified norm of failed destructive controls.

• r_19: count or quantified norm of stale or missing authority/evidence hashes.

A Dynamics-supported positive terminal requires every hard applicable residual to vanish and every claimed global remainder to satisfy its bound. A negative terminal requires a reproducible failed hard constraint. An external-wall terminal requires proof that the gate demands evidence outside the current executable/known class, together with exact reopen conditions. Missing evidence is OPEN, not zero.

15. Machine validator logic

def adjudicate_dynamics(gate, selected_rows, evidence, residuals, controls):
    assert gate.authority_hashes_are_frozen
    for row in selected_rows:
        if row not in evidence:
            return "OPEN", f"missing {row}"
        if evidence[row].status == "NOT-APPLICABLE" and not evidence[row].type_proof:
            return "FAIL", f"unproved N/A for {row}"
        if evidence[row].status == "FAIL":
            return "FAIL", row
        if evidence[row].status != "PASS":
            return "OPEN", row
    if any(c.status != "PASS" for c in controls):
        return "FAIL", "destructive control"
    if residuals.any_hard_nonzero():
        return "OPEN", "nonzero residual"
    if gate.claim_is_global and not evidence["DYN-C08"].tail_theorem:
        return "OPEN", "scope promotion"
    return "DYNAMICS-SUFFICIENT", gate.scope_ceiling

The validator does not declare the full gate terminal. It emits a Dynamics adjudication consumed by the cross-block status engine.

16. Required evidence tree

<GATE>/DYNAMICS/
  00_CONTRACT/
    GATE_CONTRACT.yaml
    AUTHORITY_HASHES.json
    DYNAMICS_ROW_SELECTION.json
  01_PARENT_LAW/
    PARENT_ACTION_MANIFEST.yaml
    PRIMITIVE_TERM_LEDGER.csv
    VARIATIONAL_DISPOSITION.csv
  02_CONSTRAINT_MEASURE_BACKGROUND/
    EQUATION_LEDGER.csv
    CONSTRAINT_CLOSURE.json
    PHYSICAL_MEASURE.json
    BACKGROUND_SOLVABILITY.json
  03_OPERATORS/
    PHYSICAL_OPERATOR_PACKETS/
    MIXED_BLOCK_LEDGER.csv
    OPERATOR_SATURATION.json
  04_INTERACTIONS_TRANSITIONS/
    INTERACTION_HYPERGRAPH.json
    DANGEROUS_OPERATOR_LEDGER.csv
    MASS_MIXING_TRANSITION.json
  05_QUANTUM_HISTORY_CAUSAL/
    QUANTUM_PHYSICALITY.json
    QUANTUM_CHANNEL.json
    NONEQUILIBRIUM_HISTORY.json
    CAUSAL_IBVP.json
  06_REFINEMENT_FLUX/
    RG_REFINEMENT.json
    REGIONAL_FLUX.json
  07_CONTROLS/
    DESTRUCTIVE_CONTROLS.json
  08_ADJUDICATION/
    DYNAMICS_ROW_STATES.csv
    DYNAMICS_RESIDUAL_VECTOR.json
    DYNAMICS_ADJUDICATION.md
    SHA256SUMS.txt

17. Status grammar, terminals, and anti-promotion rules

PASS, FAIL, OPEN, NOT-EVALUATED, NOT-APPLICABLE, CONSTRUCTION-ANCHOR, CONDITIONAL-ON-BND-A, CONDITIONAL-ON-BND-B, and RESTART-REQUIRED retain their package meanings.

• An action is not a solved background.

• An equation list is not a well-posed evolution.

• A declared Actor is not a derived physical kernel.

• A free spectrum is not an interaction theory.

• A fitted matrix is not a parent-derived mass matrix.

• Formal self-adjointness is not physical positivity.

• Trace preservation is not a physical open-system channel.

• A static identity is not a cosmological history.

• Local conservation is not regional closure.

• One truncation or regulator is not a continuum/UV theorem.

• A zero-mode result is not a full-tower result.

• An integrated anomaly cancellation is not local or global boundary closure.

18. Reopen conditions

• a new Stage support, Actor, Rulebook rule, boundary stratum, or variational owner appears.

• a new invariant parent term or interaction edge survives the grammar.

• a constraint rank, gauge complex, or physical measure changes.

• a new physical pole, zero mode, mirror, edge mode, or mixed block appears.

• a claimed theorem-zero interaction is generated radiatively.

• a mass/mixing/transition changes under lawful normalization or matching.

• positivity, channel, causal, history, refinement, or flux controls fail.

• a gate produces a Dynamics distinction not representable by DYN-C01–DYN-C17.

• authority hashes or upstream branch objects change.

18A. Legacy-preservation theorem and revision discipline

V4.2 is valid as a successor only if all of the following hold:

1. Source retention: the exact BB-DYN-3.3.1 source is present and hash-verified.
2. Section mapping: every substantive old section maps to an explicit V4.2 section or preserved clause.
3. Witness monotonicity: no old required witness is replaced by a weaker witness.
4. Failure monotonicity: every old fail trigger remains capable of producing FAIL or OPEN.
5. Control monotonicity: every old destructive control remains executable.
6. Status compatibility: all old status states retain their meanings.
7. Claim-ceiling monotonicity: no local, zero-mode, finite-tower, boundary-incomplete, or construction-anchor result is promoted by the revision.
8. Correction transparency: source defects are corrected only through an explicit correction ledger.
9. Gate regression: SG-2, SG-5, UQF-3, BG-10, and UQF-14 must be rerun target-blind before canonical ratification.
10. Hostile review: an independent review must attempt to find omitted terms, equations, determinants, interactions, domains, histories, regulators, and fluxes.

Define the preservation residual

\[ \mathcal R{\mathrm{pres}}=(N{\mathrm{unmapped\ legacy\ clauses}},N{\mathrm{weakened\ witnesses}},N{\mathrm{lost\ fail\ triggers}},N{\mathrm{lost\ controls}},N{\mathrm{uncorrected\ source\ defects}},N_{\mathrm{hash\ mismatches}}). \]

A V4.2 package-level preservation pass requires

\[ \boxed{\mathcal R_{\mathrm{pres}}=0.} \]

This package satisfies the documentation-level preservation test. Gate-level physical sufficiency remains subject to execution.

19. Authority and claim ceiling

This is a development authority candidate. It is more complete than V4.0 as a standalone execution object, but it does not ratify itself and does not convert prior gate evidence into PASS. Ratification requires hostile review, schema/validator testing, target-blind calibration on SG-2, interaction calibration on SG-5, positivity calibration on UQF-3, nonequilibrium calibration on BG-10, and causal/interface calibration on UQF-14. Any failed calibration repairs the reusable row and propagates to all mapped gates.

Appendix A — Constraint registry

| ID | Witness | Minimal discharge | Fail trigger | |---|---|---|---|

| DYN-C01 | Parent-action manifest and term provenance | Every primitive term and gauge connection has one lawful owner, support, coefficient provenance, symmetry, derivative order, boundary status, and microscopic/effective scope. | Unowned terms, double-counted aliases, or a gauge owner without a parent-action term. |

| DYN-C02 | Dirac–Bergmann or BV–BFV closure | All constraints, ranks, brackets, residual gauges, boundary terms, and physical degree counts are computed to closure on the claimed stratum. | Early termination, rank change, omitted boundary constraint, or unresolved residual gauge. |

| DYN-C03 | Physical measure and determinant derivation | All FP/Dirac/BV determinants and field dependence are retained in loops, matching, and stress calculations. | A determinant is dropped as a constant without proof. |

| DYN-C04 | Full metric variation and stress | Stress is derived from the complete action and measure, including constraints, boundaries, ghosts, and solved multiplier dependence. | No-work or absence from a propagator is substituted for zero stress. |

| DYN-C05 | Retained-plus-reaction solvability | The complete coupled equations have a solution with full-rank Jacobian or an existence theorem on the declared domain. | Normal forces lie outside multiplier span or retained equations are unsolved. |

| DYN-C06 | Ward/BV identity and global boundary multiplier closure | The domain is preserved, anomalies cancel, and multiplier/boundary equations admit a global solution. | Unresolved Ward identity, anomaly, boundary equation, or QME residual. |

| DYN-C07 | Physical response, conservation, and ordinary-source control | Localized and nonconstant sources yield the correct conserved causal response while protected constants follow VAC. | Ordinary matter is degravitated, conservation fails, or response is acausal. |

| DYN-C08 | Scope theorem or bounded truncation report | The result declares zero-mode, finite-tower, EFT, or all-mode scope, with a tail theorem for global claims. | A local or finite result is promoted beyond scope. |

| DYN-C09 | Operator construction and physical kernel/spectrum certificate | Linearize the complete parent equations, build the gauge-fixed and constraint-reduced operators on the lawful domains, and publish kernels, zero modes, spectra, response functions, multiplicities, and unresolved tails. | A gate uses a declared Actor or expected spectrum without deriving the corresponding physical operator/kernel, or omits a mixed/operator block. |

| DYN-C10 | Complete interaction and source hypergraph | Enumerate every parent-owned interaction vertex, source edge, representation contraction, selection rule, theorem-zero edge, dangerous operator, support, scale, and boundary domain. | A coupling, decay, portal, Yukawa, CP source, or dangerous operator is inserted or omitted without a complete parent-law disposition. |

| DYN-C11 | Mass, mixing, transition, and decay derivation | Mass matrices, mixings, symmetry breaking, decay widths, threshold transitions, and effective vertices are derived from C10 plus the solved background and are diagonalized/matched with stated scope. | A matrix or transition is supplied by hand, the background does not solve the parent equations, or physical eigenstates/widths are not derived. |

| DYN-C12 | Quantum physicality, positivity, and analytic-continuation certificate | After gauge/BRST quotient, establish a positive physical pairing or reflection-positivity/equivalent reconstruction, pole and residue signs, continuation prescription, and absence/disposition of negative-norm states. | Formal self-adjointness, Euclidean convergence, or BRST nilpotency is used without a physical positivity/pole certificate. |

| DYN-C13 | Closed/open quantum evolution and probability instrument | Closed sectors have unitary normalized evolution; open sectors publish a completely positive trace-preserving channel or justified instrument, environment, exchange currents, approximation bound, and Observer handoff. | A trace-preserving but nonphysical map, unnormalized probability rule, undeclared environment, or hidden information loss is used. |

| DYN-C14 | Nonequilibrium, thermal, and cosmological-history evolution | Publish initial state, thermal/kinetic/transport equations, entropy and charge production, phase-transition dynamics, production/washout/freeze-out, perturbations, and uncertainty through the claimed history. | A static potential or constant-shift result is promoted to a finite-time cosmological, baryogenesis, dark-matter, or transition claim. |

| DYN-C15 | Well-posed causal initial-boundary-value problem | Prove or compute hyperbolicity/evolution well-posedness, constraint propagation, boundary compatibility, causal support, stability under allowed data, and synchronization/Observer handoffs. | Equations exist but evolution is nonunique, unstable, superluminal outside the declared cone, domain-incompatible, or not globally composable. |

| DYN-C16 | RG, refinement, regulator, and truncation stability | Use nested operator bases/resolutions and admissible regulator/scheme families; track generated operators, fixed points/flows, critical data, convergence, matching, and omitted-tail bounds. | A result exists only in one truncation, basis, regulator, or scheme and changes materially under lawful refinement. |

| DYN-C17 | Regional, horizon, interface, and information-flux balance | For regions separated by boundaries, defects, horizons, or cuts, publish charge/energy/probability/information currents, gluing law, edge terms, exchange channel, and finite Observer records. | Local conservation is claimed while flux through an interface is omitted, or subsystem nonunitarity is mistaken for global loss without a channel ledger. |

Appendix B — Gate-to-row crosswalk

| Gate | Role | Dynamics rows | Source terminal under test | |---|---|---|---|

| Black-hole-singularity | LOAD-BEARING | DYN-C07, DYN-C08, DYN-C09, DYN-C15, DYN-C17 | DISSOLVED-GIVEN-root / RESOLVED +0 |

| Born-rule | LOAD-BEARING | DYN-C03, DYN-C06, DYN-C07, DYN-C12, DYN-C13 | CLOSED / CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| Gap-01 | LOAD-BEARING | DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16 | DERIVED-GIVEN-anchor / RESOLVED +0 |

| Gap-02 | LOAD-BEARING | DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| Gap-05-stability | LOAD-BEARING | DYN-C03, DYN-C04, DYN-C05, DYN-C07, DYN-C08, DYN-C14, DYN-C16 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| Gap-05-value | NOT-LOAD-BEARING-FOR-MEASURED-TERMINAL | DYN-C07, DYN-C14 | MEASURED-ANCHOR / RESOLVED +0 |

| Gap-08 | LOAD-BEARING | DYN-C07, DYN-C09, DYN-C14, DYN-C15, DYN-C16 | DISSOLVED-GIVEN-root / RESOLVED +0 |

| Gap-10/BG-10 | LOAD-BEARING | DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C15 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| Gap-11 | LOAD-BEARING | DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C16 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| Gap-13 | LOAD-BEARING | DYN-C03, DYN-C07, DYN-C08, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17 | CLOSED / PROJECT-DEPENDENCY / CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| Lambda-catastrophe | LOAD-BEARING | DYN-C04, DYN-C05, DYN-C07, DYN-C14, DYN-C16 | DISSOLVED-GIVEN-root / RESOLVED +0 |

| SG-1 | SUPPORTING | DYN-C01, DYN-C05, DYN-C08, DYN-C09, DYN-C15 | CLOSED / DERIVED-GIVEN-SHAPE / RESOLVED +0 |

| SG-10 | LOAD-BEARING | DYN-C08, DYN-C09, DYN-C16 | DERIVED-GIVEN-anchor / RESOLVED +0 |

| SG-2 | LOAD-BEARING | DYN-C01, DYN-C02, DYN-C06, DYN-C08, DYN-C09 | CLOSED / CERTIFIED-IRREDUCIBLE-CATEGORY-FLOOR / RESOLVED +0 |

| SG-3 | LOAD-BEARING | DYN-C01, DYN-C06, DYN-C08, DYN-C09 | DERIVED-GIVEN-E / RESOLVED +0 |

| SG-4 | LOAD-BEARING | DYN-C01, DYN-C03, DYN-C06 | CLOSED / DERIVED-GIVEN-E / RESOLVED +0 |

| SG-5 | LOAD-BEARING | DYN-C01, DYN-C05, DYN-C09, DYN-C10, DYN-C11, DYN-C12 | DERIVED-GIVEN-anchor / RESOLVED +0 |

| SG-6 | SUPPORTING | DYN-C05, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | DERIVED-GIVEN-anchor / RESOLVED +0 |

| SG-7 | LOAD-BEARING | DYN-C01, DYN-C03, DYN-C08, DYN-C10, DYN-C16 | DISSOLVED-GIVEN-root / RESOLVED +0 |

| SG-8 | LOAD-BEARING | DYN-C01, DYN-C08, DYN-C09, DYN-C10, DYN-C11, DYN-C12, DYN-C16 | CLOSED / RESCUED / DISSOLVED-WRONG-RULER + DERIVED-GIVEN-13D-ACTOR / RESOLVED +0 |

| SG-9 | LOAD-BEARING | DYN-C01, DYN-C06, DYN-C08, DYN-C10, DYN-C11, DYN-C16 | DERIVED-GIVEN-anchor / RESOLVED +0 |

| UQF-10 | LOAD-BEARING | DYN-C01, DYN-C02, DYN-C05, DYN-C08, DYN-C09, DYN-C10, DYN-C12, DYN-C16 | RESOLVED +0 / DERIVED-GIVEN-anchor + CLOSED-NEGATIVE leg |

| UQF-14 | LOAD-BEARING | DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| UQF-3 | LOAD-BEARING | DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| UQF-4 | LOAD-BEARING | DYN-C03, DYN-C06 | CLOSED / DERIVED-GIVEN-13D-SHAPE / GLOBAL-ANOMALY-CONSISTENT / RESOLVED +0 |

| UQF-5A/5B | LOAD-BEARING | DYN-C01, DYN-C03, DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| UQF-5C | LOAD-BEARING | DYN-C06, DYN-C08, DYN-C12, DYN-C15, DYN-C16 | CLOSED / CERTIFIED-IRREDUCIBLE-FLOOR / RESOLVED +0 |

| UQF-7 | LOAD-BEARING | DYN-C03, DYN-C06, DYN-C09 | DERIVED-GIVEN-anchor / RESOLVED +0 |

| UQF-9 | LOAD-BEARING | DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16 | CERTIFIED-IRREDUCIBLE / RESOLVED +0 |

| theta-bar-QCD | LOAD-BEARING | DYN-C01, DYN-C05, DYN-C06, DYN-C10, DYN-C11, DYN-C14, DYN-C16 | DISSOLVED-GIVEN-root / RESOLVED +0 |

Appendix C — Standalone execution prompt

Read this authority and the frozen gate inputs. Execute only the Dynamics rows selected for the gate. Do not load the expected numerical answer until after calculation. Generate the complete parent-term, equation, constraint, physical-operator, interaction, transition, quantum/history/causal, refinement, and regional slices required by type. Freeze all failures before repair. A repair must be reusable across every gate invoking the affected row. Run every destructive control, compute the Dynamics residual vector, and report physical result, Dynamics sufficiency, project dependency, claim ceiling, evidence hashes, and exact reopen triggers separately. Never convert missing evidence into zero, a finite result into a global result, or a declared Actor/interaction into a derived one.

Appendix D — Exact archival BB-DYN-3.3.1 source

The following archival annex is byte-identical to the source file whose SHA-256 is f3ad4b47e7726a05128333a469c43d5e571877ce7aab5ecd5b1fadf96756847e. It is included for mechanical preservation review. The operative V4.2 clauses are the main-body sections above; the annex prevents silent historical loss.

<!-- BEGINEXACTBBDYN331 -->

---
title: "BB-DYN-3.3.1 — Parent Dynamics, Constraints, Gauge Ownership, Background, Boundary Anomaly, and Response"
building_block_id: "BB-DYN-3.3.1"
version: "3.3.1"
date: "2026-07-24"
status: "DEVELOPMENT AUTHORITY — SHP-C09 GAUGE-OWNER INTERFACE MERGED — NOT FROZEN"
scope: "Complete owned specification under package v3.3.2"
protocol_freeze: false
canonical_ratification: false
package_authority_version: "3.3.2"
---

# BB-DYN-3.3.1 — Parent Dynamics, Constraints, Gauge Ownership, Background, Boundary Anomaly, and Response

## 0. Purpose

Dynamics answers:

> **Which lawful changes occur on the complete Shape, which equations define its backgrounds, how exact constraints are preserved, how quantum corrections modify the equations, and how the system responds separately to vacuum offsets and ordinary excitations?**

This block prohibits four substitutions:

\[
\text{kinematic permission}\neq\text{nonzero amplitude},
\]

\[
\text{constraint no-work}\neq\text{zero stress},
\]

\[
\text{no propagator}\neq\text{radiative solvability},
\]

\[
\text{internal reaction solvable}\neq\text{vacuum energy degravitated}.
\]

## 1. Parent action

Every candidate must supply a typed parent action or an explicitly scoped non-action replacement.

A general action ledger is

\[
S_{
m parent}=S_{
m grav}+S_{
m gauge}+S_{
m matter}+S_{
m scalar}
+S_{
m boundary}+S_{
m constraint}+S_{
m global}+S_{
m domain}.
\]

For every term publish:

- support and dimensionality;
- field owner;
- derivative order;
- exact and broken symmetries;
- coefficient provenance;
- boundary variation;
- whether microscopic, structural, global, or matching-only;
- stress-energy contribution;
- vacuum-energy contribution;
- cutoff and scheme scope.

An arbitrary dense kernel entry without a parent-law owner is prohibited.

## 2. Complete equation vector

Let the independent variables be

\[
z^I=(g_{\mu\nu},\gamma_{mn},g_{\mu m},A_M,\Psi,\phi,b,\lambda,\sigma,\ldots).
\]

The complete background and fluctuation equations are

\[
\mathcal F_I[z]\equiv\frac{\delta\Gamma}{\delta z^I}=0,
\]

supplemented by boundary, global, gauge, and constraint equations.

The candidate must publish an equation ledger:

equation: equationid: variedobject: localorglobal: bulkorboundary: differentialoralgebraic: independentafteridentities: sourceterms: constraintcouplings: observerprojection: solutionstatus:

### 2.1 Equation counting

Before claiming a background solution:

1. count independent unknown functions and global variables;
2. count independent equations after Noether and Bianchi identities;
3. classify gauge freedoms;
4. classify second-class constraints;
5. include boundary and junction equations;
6. verify rank on the claimed stratum;
7. prove or compute existence of a solution.

A visually plausible product ansatz is not a solution certificate.

## 3. Constraint classification and closure

Constraints are typed as:

- first-class gauge constraints;
- second-class physical constraints;
- constitutive restrictions;
- boundary/gluing constraints;
- global constraints;
- effective matching relations.

For Hamiltonian variables, the total Hamiltonian is

\[
H_T=H_0+u^A\Xi_A.
\]

The Dirac–Bergmann algorithm must:

1. list primary constraints;
2. impose preservation \(\dot\Xi_A\approx0\);
3. generate every secondary and higher constraint;
4. terminate with constant rank on the claimed stratum;
5. classify first- and second-class sectors;
6. determine multipliers and residual gauge functions;
7. verify boundary compatibility;
8. compute the physical degree-of-freedom count.

For second-class constraints,

\[
C_{AB}=\{\Xi_A,\Xi_B\}
\]

must be invertible on the physical stratum and

\[
\{F,G\}_D=\{F,G\}-\{F,\Xi_A\}(C^{-1})^{AB}\{\Xi_B,G\}.
\]

Rank change is a branch change or singular stratum, not a small correction.

## 4. Physical measure

For a second-class system,

\[
D\mu_{
m phys}=D\mu_0\prod_A\delta(\Xi_A)\sqrt{\det C}.
\]

The determinant must be included in effective actions, matching, and stress calculations.

If \(C\) depends on retained fields,

\[
\Delta\Gamma_C=-\frac{i}{2}\operatorname{Tr}\log C
\]

or the appropriate Euclidean counterpart contributes physical terms.

A field-dependent determinant cannot be dismissed as normalization.

## 5. No-work and stress are separate

For ideal constraints,

\[
\lambda^A d\Phi_A=0
\]

along admissible displacements.

This proves no virtual work in constrained directions. It does not establish zero stress-energy.

For

\[
S_{\rm constraint}=\int d^Dx\sqrt{-g}\,\lambda^A\Phi_A,
\]

\[
T^{\rm constraint}_{MN}
=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\rm constraint}}{\delta g^{MN}}.
\]

The calculation must include:

- metric variation of the volume form;
- explicit metric dependence of \(\Phi_A\);
- metric dependence of multipliers when induced by solved equations;
- determinant and ghost terms;
- boundary contributions.

Zero constraint stress is a result with conditions, not a default property.

## 6. Constrained reaction equations

For retained variables \(q^a\), constrained variables \(\phi^r\), and constraints \(\Phi_\alpha(q,\phi)=0\), the renormalized equations are

\[
\frac{\delta\Gamma_0}{\delta q^a}
+\lambda^\alpha\frac{\delta\Phi_\alpha}{\delta q^a}=0,
\]

\[
\frac{\delta\Gamma_0}{\delta\phi^r}
+\lambda^\alpha\frac{\delta\Phi_\alpha}{\delta\phi^r}=0,
\]

\[
\Phi_\alpha=0.
\]

### 6.1 Block-separable case

If

\[
\frac{\delta\Phi_\alpha}{\delta q^a}=0
\]

and the normal Jacobian

\[
N_{\alpha r}=\frac{\delta\Phi_\alpha}{\delta\phi^r}
\]

has full rank, the normal equations determine reactions when

\[
-\left.\frac{\delta\Gamma_0}{\delta\phi^r}\right|_{\Phi=0}
\in\operatorname{im}N^T.
\]

This prevents normal-force overdetermination only under that span condition.

### 6.2 General case

If constraints depend on retained variables, multipliers modify retained equations. The full coupled system must be solved.

### 6.3 Global solvability

Pointwise algebra is insufficient. The multiplier system must satisfy:

- regularity;
- boundary and fixed-set parity;
- junction conditions;
- global integrability;
- Ward identities;
- anomaly cancellation;
- conserved-charge balance.

## 7. Background-equation completeness

A candidate background receives one of four Dynamics statuses:

### 7.1 Parent solution

Every varied parent equation and boundary equation is satisfied.

### 7.2 Constrained reaction solution

Displaced equations are replaced by a complete, solvable reaction system, including stress and boundary effects.

### 7.3 Effective external background

The Stage is externally supplied. The reduced theory may be useful, but it does not derive the background as a parent vacuum.

### 7.4 Inconsistent background

The complete equation system has no solution at the claimed object.

A closure statement must name which status is used.

## 8. Radiative solvability

Let \(\Gamma_k\) be the effective action at scale \(k\). Radiative consistency requires more than preservation of the classical constraint surface.

At each claimed order or Wilsonian step, test:

- closure of the corrected constraint algebra;
- rank stability;
- normal-force span;
- retained equation solvability;
- boundary compatibility;
- anomaly and Ward identities;
- generation of forbidden operators;
- determinant contributions;
- parameter and counterterm provenance.

A forbidden mode need not be regenerated for the theory to fail. Equations with no solution are a separate failure.

## 9. Physical stability interface

Dynamics supplies the effective Hessian to `BB-RIG-1.1`.

After gauge quotient and exact constraints,

\[
Q_{\rm phys}=P_{\rm phys}^{T}\,\delta^2\Gamma\,P_{\rm phys}.
\]

The supplied packet must include:

- kinetic metric;
- gauge fixing and ghosts;
- mixed blocks;
- boundary domain;
- scale and scheme;
- zero modes;
- negative modes;
- intended protected scalars;
- tower scope;
- uncertainty.

Dynamics may not label an isolated critical point stable without the physical Hessian result.

## 10. Vacuum-offset response interface

A field-independent shift of a protected matter sector is

\[
\mathcal L_m\rightarrow\mathcal L_m-c.
\]

Dynamics must expose how it enters the complete equations.

For ordinary Einstein coupling, the external equation shifts by a term proportional to

\[
c\,g_{\mu\nu}.
\]

Constraint reactions in internal directions do not automatically remove this external response.

The vacuum-response packet sent to `BB-VAC-1.1` contains:

vacuumresponsepacket: protectedsector: constantshiftparameter: externalmetricequation: internalequations: constraintreactionequations: globalauxiliaryequations: boundaryequations: determinantterms: matchingterms: gravitonloopscope: ordinarystressresponse: solvedresponse_operator:

## 11. Ordinary-gravity control

Any vacuum-offset mechanism must preserve lawful response to nonconstant stress.

The negative control is:

- add a constant \(-c\) to the protected Lagrangian;
- verify local curvature observables do not change in the protected scope;
- add a localized excitation with nontrivial traceless or gradient stress;
- verify ordinary gravitational response remains.

A mechanism that removes all stress from the gravitational equations fails.

## 12. Constant shifts versus phase transitions

A constant threshold correction and a finite-time phase transition are distinct.

Dynamics must classify:

- static additive shift;
- changing condensate;
- latent heat;
- radiation production;
- nonadiabatic boundary or global response;
- historical averages;
- residual present value.

A constant-shift identity does not automatically solve cosmological transition history.

## 13. Probability, causality, and conservation

For closed quantum sectors, the generator must be self-adjoint on the physical domain or possess an equivalent unitary kernel.

For open sectors, declare:

- environment;
- channel;
- normalization;
- exchange currents;
- observer access;
- approximation bound.

The law must preserve, at the claimed scope:

- probability;
- gauge and BRST constraints;
- exact charges;
- boundary flux balance;
- causal support;
- operational no-signalling.

## 14. Dynamics certificate

dynamicscertificatev3: candidateid: branchid: parentaction: primitivetermledger: completeequationvector: equationcountandidentities: constrainttypes: diracbergmannclosure: rankstrata: physicalmeasure: determinantcontributions: constraintstress: multiplierboundaryproblem: backgroundstatus: backgroundsolution: radiativesolvability: wardidentities: anomalyresult: physicalhessianpacket: vacuumresponsepacket: ordinarygravitycontrol: phasetransitionscope: fulltowerscope: evidence_hashes: verdict:

## 15. Negative controls

### 15.1 No propagator but no solution

Remove a deformation field from the propagator but choose a radiative normal force outside the multiplier span.

Expected result: `FAIL / RADIATIVE OVERDETERMINATION`.

### 15.2 Constraint no-work with nonzero stress

Use a metric-dependent constraint that does no virtual work along admissible motion.

Expected result: no-work passes, stress-zero claim fails.

### 15.3 Vacuum degravitation that also removes matter gravity

Project out every trace and traceless source indiscriminately.

Expected result: vacuum shift passes, ordinary-gravity control fails.

### 15.4 Frozen Stage with abandoned internal equations

Do not vary the internal metric and supply no reaction or background equation.

Expected result: `EFFECTIVE EXTERNAL BACKGROUND`, not parent solution.

## 16. Falsifiers and reopen triggers

This block fails or reopens if:

- the parent action is incomplete;
- a primitive move lacks an owner;
- equation counting omits a boundary or global equation;
- Dirac closure or rank is incomplete;
- determinant terms are omitted;
- no-work is used as a stress proof;
- no-propagator is used as a solvability proof;
- a background is called a solution without all equations;
- a constant vacuum shift changes local curvature in the claimed protected scope;
- ordinary nonconstant stress fails to gravitate;
- a phase-transition claim is inferred from constant-shift cancellation;
- a zero-mode result is promoted to the full tower.

## 17. Terminal

BB-DYN-3.3.1

DELIVERS: Parent law, complete equations, constraint and measure closure, background solvability, stress accounting, radiative response, and interfaces to rigidity and vacuum protection.

STATUS: DEVELOPMENT AUTHORITY CANDIDATE — NOT FROZEN.

## 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 `boundary_anomaly_completion_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 — stratified quantum gauge consistency

### 1. Complete gauge variation

For a \(d+1\)-dimensional parent with \(d\)-dimensional fixed sets \(F_a\), the renormalized gauge variation is decomposed as

\[
\delta_\epsilon\Gamma_{\rm tot}
=
\delta_\epsilon\Gamma_{\rm bulk}
+
\sum_a
\delta_\epsilon\Gamma_{F_a}
+
\delta_\epsilon S_{\rm inflow}
+
\delta_\epsilon S_{\rm ct}.
\]

For four-dimensional fixed sets, write the descent data

\[
I_6=dI_5^{(0)},
\qquad
\delta_\epsilon I_5^{(0)}=dI_4^{(1)}(\epsilon,A).
\]

A bulk inflow action of the form

\[
S_{\rm inflow}
=
2\pi i\,k\int_{Y_5}I_5^{(0)}
\]

has boundary variation

\[
\delta_\epsilon S_{\rm inflow}
=
2\pi i\,k
\sum_a s_a
\int_{F_a}I_4^{(1)},
\]

where \(s_a\) is fixed by the outward-normal orientation. The coefficient \(k\) must satisfy the appropriate quantization and global-form conditions.

### 2. Fixed-set local equations

For each fixed set define

\[
\mathcal A_a^{(1)}
=
\mathcal A_{a,\rm projected\ bulk}^{(1)}
+
\mathcal A_{a,\rm localized}^{(1)}
+
\mathcal A_{a,\rm measure}^{(1)}
+
\mathcal A_{a,\rm counterterm}^{(1)}
+
\mathcal A_{a,\rm inflow}^{(1)}.
\]

The local pass condition is

\[
\boxed{\mathcal A_a^{(1)}=0\quad\text{for every }a}
\]

as a cohomology class under all allowed gauge transformations.

The weaker condition

\[
\sum_a\mathcal A_a^{(1)}=0
\]

is only an integrated check. It does not prove local gauge invariance.

### 3. Quantum master equation with boundaries

The candidate must formulate the quantum gauge system using BV–BFV or an equivalent boundary-complete formalism.

Schematically,

\[
\frac12(S,S)-i\hbar\Delta S
=
\pi^*S_{\partial}
\]

before boundary completion. Full closure requires a compatible boundary state/functional and counterterm/inflow system such that the total quantum master equation is satisfied.

A bulk statement \(Q_{\rm BRST}^2=0\) is insufficient when the measure has fixed-set support.

### 4. Consistent versus covariant anomalies

The ledger must state whether it uses:

- consistent anomalies;
- covariant anomalies;
- Bardeen–Zumino shifted currents.

All fixed-set and inflow entries must use one compatible convention. A Bardeen–Zumino polynomial may redistribute anomaly between currents or strata, but it cannot remove the total anomaly class.

### 5. Regulator and parity dependence

Localized anomalies can depend on the parity/domain assignment and on how bulk fermions are regulated. The final anomaly class must be regulator-independent after allowed local counterterms are included.

The certificate must report:

- regulator;
- consistent-current convention;
- parity signs;
- fixed-set weights;
- localized spectrum;
- allowed counterterms;
- invariant remainder.

### 6. Global relative anomaly equation

Perturbative cancellation is not enough. The exponentiated fermion determinant is a section of an anomaly/determinant line.

The complete parent-plus-boundary system must provide a gluing-compatible trivialization:

\[
Z_{\rm boundary}\,
Z_{\rm inflow}
\in\mathbb C
\]

rather than an untrivialized line.

For every closed test obtained by gluing or forming a mapping torus, require

\[
\exp\!\left(2\pi i\,\frac{\eta+h}{2}\right)
\,
\exp(2\pi i S_{\rm top})
=
1
\]

in the declared convention and tangential structure.

### 7. Dynamics certificate additions

boundaryquantumdynamics: fixedsetids: allowedgaugetransformations: anomalyconvention: consistent|covariant regulator: bulkanomalypolynomial: projectedbulkcontributionsbyfixedset: localizedfieldcontributions: measureandghostcontributions: counterterms: inflowaction: inflowcoefficientquantization: orientationsigns: localresiduals: bvbfvorequivalentcompletion: determinantline: relativeetaorbordismtest: gluingtest: quantummasterequationresult: evidencehashes: verdict:

### 8. New Dynamics falsifiers

Dynamics fails if:

- only the integrated anomaly is zero;
- an inflow coefficient is selected after reading the residual;
- the coefficient is not globally quantized;
- a boundary condition is not compatible with the Dirac/gauge operator domain;
- the local anomaly convention changes between rows;
- the determinant line remains untrivialized;
- the BV quantum master equation is asserted from classical Jacobi closure alone.

# Block-Specific Standalone Interface — Dynamics

## Owned questions

This block alone owns:

- the typed parent action;
- the complete equation vector;
- constraint classification and preservation;
- physical measure and determinant ownership;
- stress of multiplier, boundary, and constraint sectors;
- retained/reaction-equation solvability;
- Ward identities and BV/BFV quantum consistency;
- ordinary probability, causality, conservation, and nonconstant-stress response.

## Required inputs

- complete Shape and boundary manifests;
- constraint and permission packets;
- Scale/scheme packet;
- complete field and Actor inventory;
- candidate-specific parent terms.

## Required outputs

- parent-action manifest;
- complete Euler–Lagrange and constraint-equation ledger;
- Dirac/BV/BFV closure;
- physical measure;
- stress and reaction calculations;
- background classification;
- radiative-solvability result;
- Ward/anomaly interface;
- Dynamics certificate.

## Non-ownership boundary

Dynamics does not by itself prove:

- candidate uniqueness;
- intrinsic rigidity;
- full physical stability without the Rigidity block;
- vacuum-offset protection without the Vacuum block;
- local/global fixed-set anomaly closure without the Boundary block.

## Observer-response handoff

Dynamics supplies the physical evolution, conserved currents, response functions, and physical measure consumed by OBS. OBS cannot repair an inconsistent Dynamics by projection. Conversely, a lawful parent response cannot be compared with data until OBS has produced the correct record functional.

For time-dependent vacuum tests, Dynamics must evolve the protecting sector and supply the constraint-algebra history required by `VAC-C11`.

## Gauge-owner and kinetic-term consistency

Dynamics consumes the `SHP-C09` gauge-owner ledger.

For every surviving generator \(T_A\), the parent action must identify exactly one independent connection variable and one owning kinetic structure. Metric Kaluza–Klein notation and principal-bundle notation may describe one field, but the action must not count both as independent unless a second gauge sector is physically intended and appears in the spectrum.

Required checks:

1. map every gauge-owner ID to a parent-action field;
2. count independent kinetic operators;
3. count first-class gauge parameters;
4. count four-dimensional vector zero modes;
5. verify equality of these counts after constraints and quotienting;
6. verify that center/global-lift data is compatible with every charged Actor;
7. run the duplicate-owner control by treating aliases as independent and confirm that the spectrum gains an extra gauge sector.

A gauge owner with no parent-action term fails Dynamics. A parent-action connection with no Shape owner fails the one-parent audit.

## Owned hard constraints and acceptance evidence

The full controlling registry is `EVIDENCE_DISCHARGE_REGISTRY_V3_3_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. |
| `DYN-C01` | Parent-action manifest + term provenance table | Every primitive term has an owner, support, coefficient provenance, derivative order, symmetry, boundary status, and microscopic/effective label; every gauge connection references exactly one `SHP-C09` owner ID. | A matrix element, potential, counterterm, or gauge connection has no parent-action owner, or an owner alias is independently double-counted. |
| `DYN-C02` | Dirac–Bergmann or BV–BFV closure calculation | All primary/secondary constraints, ranks, brackets, boundary terms, and residual gauges are computed until closure; the rank is constant on the claimed stratum. | The algorithm stops early, the rank changes, or boundary constraints are omitted. |
| `DYN-C03` | Physical-measure derivation + determinant artifact | The reduced measure includes all Faddeev–Popov/Dirac/BV determinants, and field dependence is retained in loops and matching. | A determinant is dropped as a constant without proof. |
| `DYN-C04` | Metric variation of all constraint/boundary sectors | The stress tensor is obtained by varying the full action and measure; the result is evaluated on the physical branch. | No-work is used as a substitute for zero stress. |
| `DYN-C05` | Solvability/Jacobian certificate | The full retained-plus-reaction equation system has a solution and a full-rank Jacobian or a theorem establishes existence in the declared domain. | Radiative normal forces lie outside the multiplier span or the retained equations have no solution. |
| `DYN-C06` | Ward/BV identity + boundary multiplier ledger | Allowed transformations preserve the domain, all measure anomalies cancel, and multiplier boundary conditions admit a global solution. | A Ward identity, anomaly, or boundary multiplier equation remains unresolved. |
| `DYN-C07` | Response-function calculation | Localized and nonconstant stress sources produce the correct causal/conservation response while protected constants are handled according to VAC. | The mechanism removes ordinary matter gravity, violates conservation, or produces acausal record response. |
| `DYN-C08` | Scope theorem or bounded truncation report | The certificate states exactly whether it covers zero modes, a cutoff tower, or all modes, with a tail theorem where a global claim is made. | A local, zero-mode, finite-EFT, or invariant-sector result is promoted beyond scope. |
| `VAC-C01` | Complete background-equation vector + equation-count/Jacobian report | External, internal, matter, boundary, constraint, and global equations are listed and solved or lawfully replaced; unknown/equation counts and rank are published. | An internal Einstein or boundary equation disappears when a Stage variable is frozen. |
| `VAC-C04` | Finite constant-shift response experiment | For each protected sector, the full equations are re-solved after \(\mathcal L_s	o\mathcal L_s-c_s\), and the curvature-record derivative or finite difference is published. | Only an infinitesimal or tree-level shift is tested while auxiliary/global response is held fixed. |
| `VAC-C06` | Localized-source negative control | Dust, radiation, pressure gradients, and localized stress still source the correct gravitational response. | The protection mechanism degravitates ordinary matter or radiation. |
| `BND-C03` | Operator-domain certificate | Dirac, gauge, ghost, scalar, and load-bearing graviton domains are self-adjoint/elliptic as required and preserved by gauge/BRST transformations. | A parity table is supplied without a lawful operator domain. |
| `BND-C07` | Boundary BV/BFV quantum-master-equation output | Ghost, measure, boundary state, and inflow contributions satisfy the relative QME and compose under gluing. | Classical BRST closure is used while the measure/domain anomaly remains. |
| `VAC-C11` | Time-dependent constraint-algebra and cosmology simulation/theorem | A time-dependent vacuum shift \(c(t)\) representing QCD/electroweak transitions is inserted; the constraint algebra remains consistent and the Friedmann/perturbation history stays within frozen bounds, or the regime is explicitly excluded with quantified lost predictions. | The mechanism passes static shifts but becomes singular, acausal, or observationally unacceptable during a finite transition. |

## 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:

- `EVIDENCE_DISCHARGE_REGISTRY_V3_3_2.md`;
- `QUANTITATIVE_STOPPING_AND_EXHAUSTION_RULE_V3_3_2.md`;
- `MANDATORY_CONTROL_SUITE_V3_3_2.md`;
- `MIGRATION_AND_SUPERSESSION_RECORD_V3_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.

<!-- ENDEXACTBBDYN331 -->

Dynamics V4.2 Gate-Execution Prompt

Read 01CORE/BBDYN42MAXRIGORFULLGATECLOSUREDYNAMICS.md in full. Also read the frozen gate contract, Shape, Granularity, Scale, Boundary, Observer, Interdependence, Rigidity, Vacuum, anomaly/time authorities, assumptions ledger, and governance protocol that control the target gate.

Execute the gate target-blind. Do not use the expected answer, prior terminal, or desired particle/parameter result as an input.

Required procedure

1. Freeze all authority filenames, versions, SHA-256 hashes, gate scope, branch, Scale packet, boundary domains, Observer map, candidate grammar, allowed terminals, and claim ceiling.
2. Load every Dynamics constraint mapped to the gate from 02REGISTRIES/DYNAMICSGATECROSSWALKV4_2.json. Add any physically reachable Dynamics constraint omitted by the crosswalk and record the correction.
3. Build the complete Dynamics input packets specified by 02REGISTRIES/DYNAMICSPACKETSCHEMASV4_2.json.
4. Run the Stage–Rulebook–Actor-to-Dynamics compiler and construct the complete parent-term, equation, constraint, operator, interaction, history, refinement, and regional-flux ledgers required by the selected rows.
5. Perform the immutable row-by-row audit. Do not repair during this pass. Record witness, citation, scope, computation order, state, residual, and fail trigger for every row.
6. Execute every mandatory destructive control, including the four BB-DYN-3.3.1 legacy controls and the seven-step gauge-owner/kinetic consistency test wherever applicable.
7. Compute the full Dynamics residual vector and the legacy-preservation residual. A missing witness is OPEN, not zero.
8. Freeze all failures. In a separate repair phase, derive only candidate-neutral reusable repairs. Propagate every accepted repair to all dependent blocks and gates and issue new authority hashes.
9. Rerun the gate from the frozen contract using the repaired authorities.
10. Issue the strongest honest physical terminal and a separate project/governance terminal. Enforce every scope and anti-promotion rule in BB-DYN-4.2.

Required outputs

Produce at least:

• GATE_CONTRACT.md
• AUTHORITY_HASHES.json
• DYNAMICSROWSELECTION.json
• DYNAMICSINPUTPACKETS.json
• PARENTTERMLEDGER.json
• EQUATIONANDCONSTRAINT_LEDGER.json
• PHYSICALOPERATORSPECTRUM_LEDGER.json
• INTERACTIONSOURCEHYPERGRAPH.json
• MASSMIXINGTRANSITION_LEDGER.json
• POSITIVITYUNITARITYLEDGER.json
• QUANTUMEVOLUTIONLEDGER.json
• HISTORYTHERMALLEDGER.json
• CAUSALIBVPLEDGER.json
• REFINEMENTREGULATORLEDGER.json
• REGIONALFLUXLEDGER.json
• DESTRUCTIVE_CONTROLS.md
• IMMUTABLE_AUDIT.md
• FAILURE_FREEZE.json
• REPAIR_PROPOSALS.md
• DYNAMICS_RESIDUAL.json
• LEGACYPRESERVATIONRESIDUAL.json
• TERMINAL_ADJUDICATION.md
• PROPAGATIONANDSTALE_EVIDENCE.json

Before issuing a terminal, verify legacyv3compatibility, all legacy controls, applicable gauge-owner checks, shared status grammar, all anti-promotion rules, and preservation residual zero.

Gate-Execution Prompt — Shape, Granularity, and Dynamics Triad

Read the complete triad package and the selected gate dossier. Freeze all authority hashes before reading the expected terminal.

1. Generate the complete Shape slice: Stage supports, derived supports, Rulebook rules, Actors, states, effective/composite sectors, and eighteen-class Co-Actor envelope.
2. Execute every applicable Dynamics V4.2 row and packet. Do not infer an operator, interaction, history, channel, causal result, or flux from Actor identity.
3. Build the Granularity V4.0 finite-completion packet, including modes, states, composites, deformations, histories, regions, exact records, Observer distributions, cutoff, and tails.
4. Generate and saturate both the candidate grammar and the lawful test family. Run omitted-rival and omitted-test controls.
5. Use exact isomorphism for exact equivalence and equality of frozen canonical record-cell vectors for operational classes. Do not quotient on pairwise epsilon closeness.
6. Run all gate-relevant destructive controls. Unknown or unevaluated rows remain OPEN.
7. Emit the triad residual vector and separate physical, project-dependency, and evidence terminals.
8. Improve a building block only when a reusable capability is absent. Missing gate-specific calculations are evidence gaps, not automatically building-block gaps.

Shape–Granularity–Dynamics Triad Integration Contract

1. Role separation

• Shape: generates and types supports, rules, Actors, Co-Actors, states, and derived/effective sectors.
• Dynamics: derives complete equations, operators, interactions, histories, channels, causal evolution, refinement, and regional flux.
• Granularity: converts the complete certified outputs into finite physical records, validates equivalence, proves candidate/test exhaustion, and applies the stopping rule.

The execution order is:

\[ \text{Shape generation} \rightarrow \text{Dynamics derivation} \rightarrow \text{Granularity completion/equivalence/exhaustion}. \]

Feedback is mandatory: a Dynamics pole, effective owner, horizon, phase interface, or new state sector reopens Shape; a Granularity-distinguishing record can reopen either Shape or Dynamics.

2. Mandatory gate packets

Every gate must produce:

1. TRIADAUTHORITYFREEZE.json;
2. SHAPESLICE.md and ACTORCOACTOR_ENVELOPE.md;
3. DYNAMICSEXECUTIONPACKET.json using applicable DYN-C01–DYN-C17 schemas;
4. GRANULARITYFINITECOMPLETION_PACKET.json;
5. TESTFAMILYSATURATION_PACKET.json;
6. OPERATIONALEQUIVALENCEPACKET.json;
7. CANDIDATEANDTESTEXHAUSTIONPACKET.json;
8. TRIADRESIDUALVECTOR.json;
9. destructive-control results;
10. a terminal with separate physical, project-dependency, and evidence statuses.

3. Universal no-smuggling rules

• Shape cannot name a desired Actor as evidence for its Dynamics kernel.
• Dynamics cannot add an interaction, field, boundary, or regulator after viewing the desired gate result.
• Granularity cannot erase a zero mode, negative mode, anomaly, finite response, inaccessible sector, or missing calculation.
• Approximate pairwise closeness cannot be used as an equivalence quotient.
• Candidate exhaustion and test-family exhaustion are separate obligations.
• Unknown, unevaluated, and out-of-scope are never zero.
• Any authority/hash change marks dependent evidence stale.

4. Triad closure residual

\[ \mathcal R{\rm TRIAD}=\left( \mathcal R{\rm SHP}, \mathcal R{\rm SHP+}, \mathcal R{\rm DYN}, \mathcal R{\rm GRN4}, N{\rm interface\ mismatches}, N{\rm stale\ hashes}, N{\rm failed\ destructive\ controls} \right). \]

A gate cannot claim triad-complete evidence unless every component required by its terminal is zero or has an explicitly permitted scoped disposition.

BB-DYN-4.2 Preservation Audit

Verdict

V4.1 was not fully preservation-complete. It retained the detailed mathematical and boundary/orbifold kernel, but omitted explicit main-body copies of several normative legacy components. V4.2 restores them and adds a mechanical preservation theorem.

V4.1 omissions found

1. Original purpose question and four anti-substitution firewalls.
2. Complete dynamicscertificatev3 compatibility schema.
3. Four legacy negative controls.
4. Complete legacy falsifier/reopen list.
5. Historical delivery terminal.
6. Standalone owned-question, input, output, and non-ownership interface.
7. Observer-response handoff.
8. Seven-step gauge-owner/kinetic-term consistency test.
9. Embedded owned hard-constraint acceptance table.
10. Package authority boundary, full status grammar, and universal anti-promotion rules.

Legitimate corrections, not omissions

The V4.1/V4.2 text repairs two source-format defects in the old Markdown: the parent-action sum and physical-measure formula had broken \mathrm tokens. The original file is retained byte-for-byte, and the corrections are recorded in LEGACYCORRECTIONLEDGERV42.json.

Preservation controls

• Exact old source SHA-256: f3ad4b47e7726a05128333a469c43d5e571877ce7aab5ecd5b1fadf96756847e.
• Exact source copied to 05SOURCES/BBDYN331EXACTARCHIVALSOURCE.md.
• Exact source also embedded in Appendix D of the standalone V4.2 object.
• Every substantive old section is mapped in LEGACYPRESERVATIONMATRIXV42.json.
• The integrated V4.2 certificate must emit the entire V3 compatibility view.
• Legacy controls, falsifiers, statuses, and anti-promotion rules remain binding.

Claim

Documentation-level preservation is complete. This does not establish that any gate passes; target-blind regression and hostile review are still required before canonical ratification.

Validation Report — dynamics.zip

Status: PASS

Canonical authority: 01CORE/BBDYN42MAXRIGORFULLGATECLOSUREDYNAMICS.md

Canonical SHA-256: cbc41bd5faf197208d7651496a912ac9fd7e0d4fb7c48705d9722116ca00ea6c

Files inventoried: 22

All required files are present.

Notes

• All BB-DYN-3.3.1 legacy obligations preserved through the V4.2 preservation layer.
• All seventeen Dynamics constraints and all thirty gate profiles present.
• SG-2 calibration passed; SG-3 and SG-4 identified external Boundary/anomaly evidence gaps rather than Dynamics gaps.

BB-DYN-4.2 Validation Report

Status: PASS

Standalone lines: 3959

Legacy source lines: 865

• oldsourcefileshamatches: PASS
• exactannexpresent: PASS
• purposefirewallspresent: PASS
• legacycertificatepresent: PASS
• integratedcertificatepresent: PASS
• legacycontrolspresent: PASS
• gaugeownersevencheckspresent: PASS
• statusgrammarpresent: PASS
• antipromotionpresent: PASS
• all17constraints_present: PASS
• all30gateprofilespresent: PASS
• boundarykernelpresent: PASS
• preservationresidualpresent: PASS
• allcheckspass: PASS

Dynamics Audit Across All 30 Gates

Reading rule

A row added here does not declare the gate open or closed. It states which Dynamics witness must be available before the gate can receive a Dynamics-supported terminal. A current external-wall or measured-anchor terminal may require fewer rows than a stronger constructive claim; those distinctions are stated in the finding.

| Gate | Dynamics role | Current crosswalk | Revised crosswalk | Audit finding | Dynamics action | |---|---|---|---|---|---| | Black-hole-singularity | LOAD-BEARING | DYN-C07, DYN-C08 | DYN-C07, DYN-C08, DYN-C09, DYN-C15, DYN-C17 | Dissolving a continuum singularity does not supply a finite causal evolution, horizon/interface flux balance, or record map. | Add C15/C17 and integrate TS/OBS; no claim of a UV interior solution. | | Born-rule | LOAD-BEARING | DYN-C03, DYN-C06, DYN-C07 | DYN-C03, DYN-C06, DYN-C07, DYN-C12, DYN-C13 | A stronger Born derivation requires a positive physical measure and lawful normalized event instrument; the current irreducibility terminal can remain narrower. | Genuine improvement for constructive derivation; no forced change to irreducibility terminal. | | Gap-01 | LOAD-BEARING | DYN-C01, DYN-C08 | DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16 | The a6 coefficient depends on the complete operator, domain, measure, boundary data, scheme, and finite-order scope. | Add generic operator/kernel and refinement certificates; calculation remains gate-specific. | | Gap-02 | LOAD-BEARING | DYN-C07, DYN-C08 | DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | A mass-gap claim requires a physical spectral operator, positive physical sector, causal theory, and nonperturbative/refinement control. The external-wall terminal needs only proof that these are not supplied. | Add C09/C12/C15/C16; no claim that the Clay problem is solved. | | Gap-05-stability | LOAD-BEARING | VAC-C11 | DYN-C03, DYN-C04, DYN-C05, DYN-C07, DYN-C08, DYN-C14, DYN-C16 | Radiative stability requires full measure/stress, solvable reactions, ordinary-source control, time-dependent transitions, and scheme/refinement stability. | Add C14/C16 execution; VAC retains vacuum-specific ownership. | | Gap-05-value | NOT-LOAD-BEARING-FOR-MEASURED-TERMINAL | — | DYN-C07, DYN-C14 | The measured value terminal does not require a mechanism. A predictive/history claim does require response and cosmological evolution. | No change for measured anchor; use C14 only for stronger claim. | | Gap-08 | LOAD-BEARING | DYN-C07, VAC-C11 | DYN-C07, DYN-C09, DYN-C14, DYN-C15, DYN-C16 | Replacing or dissolving inflation still requires a complete causal perturbation/history map from initial data to finite records. | Genuine improvement: nonequilibrium history and causal evolution packets. | | Gap-10/BG-10 | LOAD-BEARING | DYN-C07, VAC-C11 | DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C15 | Baryogenesis requires CP-violating interactions, out-of-equilibrium transport, charge/entropy evolution, washout, freeze-out, and causal initial data. | Genuine Dynamics improvement: interaction plus nonequilibrium/thermal module. | | Gap-11 | LOAD-BEARING | DYN-C01, DYN-C07 | DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C16 | A dark-matter portal needs complete production/decay interactions, open/thermal evolution, abundance history, and running/matching. | Genuine Dynamics improvement: interaction and thermal production module. | | Gap-13 | LOAD-BEARING | DYN-C03, DYN-C07, DYN-C08 | DYN-C03, DYN-C07, DYN-C08, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17 | Microstate/Page claims require positive quantum evolution, subsystem channels, causal/horizon flux, refinement scope, and finite records. | Add quantum physicality, open-system, causal, and horizon-flux interfaces; constructive QG remains external. | | Lambda-catastrophe | LOAD-BEARING | VAC-C11 | DYN-C04, DYN-C05, DYN-C07, DYN-C14, DYN-C16 | A degravitation mechanism must solve full equations, retain ordinary gravity, survive phase transitions, and remain radiatively stable. | Add C14/C16; VAC remains the mechanism owner. | | SG-1 | SUPPORTING | — | DYN-C01, DYN-C05, DYN-C08, DYN-C09, DYN-C15 | Existing Shape terminal does not require a derived parent vacuum, but any stronger claim that the selected Stage is dynamically realized requires background solvability, physical operators, and well-posed evolution. | Crosswalk expansion only; no new gate-specific physics term. | | SG-10 | LOAD-BEARING | DYN-C08 | DYN-C08, DYN-C09, DYN-C16 | Scope consistency requires the actual operator/tower content and a refinement/tail certificate, not a label saying finite or all modes. | Add C09/C16 to make the existing anti-promotion rule executable. | | SG-2 | LOAD-BEARING | — | DYN-C01, DYN-C02, DYN-C06, DYN-C08, DYN-C09 | Gauge-owner identities must map to one kinetic owner, lawful domains, quotient, and exactly the physical vector kernel; the current crosswalk omits Dynamics entirely. | Add DYN-C09 and refresh the crosswalk; existing SG-2 evidence can populate it. | | SG-3 | LOAD-BEARING | — | DYN-C01, DYN-C06, DYN-C08, DYN-C09 | The chiral index and no-mirror claim require the physical Dirac operator/domain/kernel and quantum gauge consistency, not only a topological count. | Add DYN-C09; integrate BND/AD evidence. | | SG-4 | LOAD-BEARING | — | DYN-C01, DYN-C03, DYN-C06 | Charges and anomalies require one parent coupling, the fermion/ghost measure, Ward identities, and boundary/global anomaly closure. | Crosswalk expansion; BND-B remains decisive. | | SG-5 | LOAD-BEARING | DYN-C01, DYN-C05 | DYN-C01, DYN-C05, DYN-C09, DYN-C10, DYN-C11, DYN-C12 | EWSB cannot close from Actor identity alone: the parent interaction, solved background, mass/mixing matrices, physical scalar/vector spectrum, and positivity are required. | Genuine Dynamics improvement: C10-C12 and executable packets. | | SG-6 | SUPPORTING | — | DYN-C05, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | Rigidity owns the terminal, but Dynamics must supply the full physical Hessian/operator, quantum physicality, tower scope, and refinement stability. | Strengthen Dynamics-to-Rigidity handoff; no replacement of RIG. | | SG-7 | LOAD-BEARING | DYN-C08 | DYN-C01, DYN-C03, DYN-C08, DYN-C10, DYN-C16 | Threshold unification and proton-scale matching depend on complete interaction content, determinants, tower scope, and scheme/refinement stability. | Add C10/C16 packet; Scale still owns ruler/matching provenance. | | SG-8 | LOAD-BEARING | — | DYN-C01, DYN-C08, DYN-C09, DYN-C10, DYN-C11, DYN-C12, DYN-C16 | Flavor masses and mixings require a complete Yukawa/interaction graph, generated matrices, physical eigenstates, and same-ruler/refinement controls. | Genuine Dynamics improvement: interaction and mass-generation execution. | | SG-9 | LOAD-BEARING | DYN-C01, DYN-C06, DYN-C08 | DYN-C01, DYN-C06, DYN-C08, DYN-C10, DYN-C11, DYN-C16 | Proton safety is a universal statement over dangerous operators, selection rules, induced vertices, thresholds, and running. | Genuine Dynamics improvement: complete dangerous-operator hypergraph and RG stability. | | UQF-10 | LOAD-BEARING | — | DYN-C01, DYN-C02, DYN-C05, DYN-C08, DYN-C09, DYN-C10, DYN-C12, DYN-C16 | Compactification consistency requires complete mixed operators/interactions, constraint closure, solvable background, physical spectrum, and tower/refinement control. | Add C09/C10/C12/C16; Rigidity/Boundary remain co-owners. | | UQF-14 | LOAD-BEARING | DYN-C07, DYN-C08 | DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17 | Above-cutoff causality requires a physical quantum channel, well-posed causal support, refinement/tail control, and regional/interface flux accounting. | Genuine improvement: TS-integrated causal and quantum-channel certificates. | | UQF-3 | LOAD-BEARING | DYN-C03, DYN-C06, DYN-C08 | DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16 | Reflection positivity is not discharged by formal self-adjointness; the full physical measure, quotient, continuation, poles, and regulator/tail stability are needed. | Genuine Dynamics improvement: executable positivity/continuation packet. | | UQF-4 | LOAD-BEARING | BND-C07 | DYN-C03, DYN-C06 | Global anomalies are quantum-measure and Ward/QME obstructions; Boundary B owns local/global relative trivialization. | Crosswalk expansion; no new Dynamics submodule. | | UQF-5A/5B | LOAD-BEARING | DYN-C01, DYN-C03, DYN-C07, DYN-C08 | DYN-C01, DYN-C03, DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | A graviton sector needs the full physical operator, gauge quotient, positive poles/residues, causal evolution, and stable tower/truncation scope. | Genuine improvement: positivity and causal/refinement packets. | | UQF-5C | LOAD-BEARING | DYN-C08 | DYN-C06, DYN-C08, DYN-C12, DYN-C15, DYN-C16 | A constructive UV completion requires nonperturbative/refinement evidence and quantum physicality. The existing irreducible-floor terminal only requires that finite evidence not be promoted. | Add C16 for any constructive claim; current external-wall terminal remains honest. | | UQF-7 | LOAD-BEARING | DYN-C06 | DYN-C03, DYN-C06, DYN-C09 | Anomaly descent consumes the chiral operator/domain/kernel and quantum measure, then passes to AD/BND for descent and gluing. | Add C09 crosswalk; integrate AD-1. | | UQF-9 | LOAD-BEARING | DYN-C08 | DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16 | Seeley–DeWitt/UV calculations require the complete operator, domain, measure, boundary terms, and strict finite-order/refinement scope. | Add C09/C16; no promotion to UV completion. | | theta-bar-QCD | LOAD-BEARING | DYN-C01, DYN-C05, DYN-C06 | DYN-C01, DYN-C05, DYN-C06, DYN-C10, DYN-C11, DYN-C14, DYN-C16 | A constructive strong-CP mechanism needs topological interactions, anomaly consistency, relaxation/transition dynamics, and radiative stability. A dissolution terminal may use a narrower subset. | Add C10/C11/C14/C16 for constructive leg; preserve scoped dissolution route. |

Cross-gate result

The current crosswalk materially under-invokes Dynamics. The clearest example is SG-2, where the current gate crosswalk lists no DYN-* row although the closure dossier uses a kinetic-owner map, BRST/domain certificate, and physical vector zero-mode kernel.

The revision therefore improves two different objects:

1. Dynamics authority: nine new executable hard rows (DYN-C09–DYN-C17).
2. Gate crosswalk: all 30 gates receive an explicit Dynamics role, including conditional roles for measured-anchor and external-wall terminals.

--- title: "BB-DYN-3.3.1 — Parent Dynamics, Constraints, Gauge Ownership, Background, Boundary Anomaly, and Response" buildingblockid: "BB-DYN-3.3.1" version: "3.3.1" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — SHP-C09 GAUGE-OWNER INTERFACE MERGED — NOT FROZEN" scope: "Complete owned specification under package v3.3.2" protocolfreeze: false canonicalratification: false packageauthorityversion: "3.3.2" ---

BB-DYN-3.3.1 — Parent Dynamics, Constraints, Gauge Ownership, Background, Boundary Anomaly, and Response

0. Purpose

Dynamics answers:

Which lawful changes occur on the complete Shape, which equations define its backgrounds, how exact constraints are preserved, how quantum corrections modify the equations, and how the system responds separately to vacuum offsets and ordinary excitations?

This block prohibits four substitutions:

\[ \text{kinematic permission}\neq\text{nonzero amplitude}, \]

\[ \text{constraint no-work}\neq\text{zero stress}, \]

\[ \text{no propagator}\neq\text{radiative solvability}, \]

\[ \text{internal reaction solvable}\neq\text{vacuum energy degravitated}. \]

1. Parent action

Every candidate must supply a typed parent action or an explicitly scoped non-action replacement.

A general action ledger is

\[ S{ m parent}=S{ m grav}+S{ m gauge}+S{ m matter}+S{ m scalar} +S{ m boundary}+S{ m constraint}+S{ m global}+S_{ m domain}. \]

For every term publish:

• support and dimensionality;
• field owner;
• derivative order;
• exact and broken symmetries;
• coefficient provenance;
• boundary variation;
• whether microscopic, structural, global, or matching-only;
• stress-energy contribution;
• vacuum-energy contribution;
• cutoff and scheme scope.

An arbitrary dense kernel entry without a parent-law owner is prohibited.

2. Complete equation vector

Let the independent variables be

\[ z^I=(g{\mu\nu},\gamma{mn},g{\mu m},AM,\Psi,\phi,b,\lambda,\sigma,\ldots). \]

The complete background and fluctuation equations are

\[ \mathcal F_I[z]\equiv\frac{\delta\Gamma}{\delta z^I}=0, \]

supplemented by boundary, global, gauge, and constraint equations.

The candidate must publish an equation ledger:

equation:
  equation_id:
  varied_object:
  local_or_global:
  bulk_or_boundary:
  differential_or_algebraic:
  independent_after_identities:
  source_terms:
  constraint_couplings:
  observer_projection:
  solution_status:

2.1 Equation counting

Before claiming a background solution:

1. count independent unknown functions and global variables;
2. count independent equations after Noether and Bianchi identities;
3. classify gauge freedoms;
4. classify second-class constraints;
5. include boundary and junction equations;
6. verify rank on the claimed stratum;
7. prove or compute existence of a solution.

A visually plausible product ansatz is not a solution certificate.

3. Constraint classification and closure

Constraints are typed as:

• first-class gauge constraints;
• second-class physical constraints;
• constitutive restrictions;
• boundary/gluing constraints;
• global constraints;
• effective matching relations.

For Hamiltonian variables, the total Hamiltonian is

\[ HT=H0+u^A\Xi_A. \]

The Dirac–Bergmann algorithm must:

1. list primary constraints;
2. impose preservation \(\dot\Xi_A\approx0\);
3. generate every secondary and higher constraint;
4. terminate with constant rank on the claimed stratum;
5. classify first- and second-class sectors;
6. determine multipliers and residual gauge functions;
7. verify boundary compatibility;
8. compute the physical degree-of-freedom count.

For second-class constraints,

\[ C{AB}=\{\XiA,\Xi_B\} \]

must be invertible on the physical stratum and

\[ \{F,G\}D=\{F,G\}-\{F,\XiA\}(C^{-1})^{AB}\{\Xi_B,G\}. \]

Rank change is a branch change or singular stratum, not a small correction.

4. Physical measure

For a second-class system,

\[ D\mu{ m phys}=D\mu0\prodA\delta(\XiA)\sqrt{\det C}. \]

The determinant must be included in effective actions, matching, and stress calculations.

If \(C\) depends on retained fields,

\[ \Delta\Gamma_C=-\frac{i}{2}\operatorname{Tr}\log C \]

or the appropriate Euclidean counterpart contributes physical terms.

A field-dependent determinant cannot be dismissed as normalization.

5. No-work and stress are separate

For ideal constraints,

\[ \lambda^A d\Phi_A=0 \]

along admissible displacements.

This proves no virtual work in constrained directions. It does not establish zero stress-energy.

For

\[ S{\rm constraint}=\int d^Dx\sqrt{-g}\,\lambda^A\PhiA, \]

\[ T^{\rm constraint}{MN} =-\frac{2}{\sqrt{-g}}\frac{\delta S{\rm constraint}}{\delta g^{MN}}. \]

The calculation must include:

• metric variation of the volume form;
• explicit metric dependence of \(\Phi_A\);
• metric dependence of multipliers when induced by solved equations;
• determinant and ghost terms;
• boundary contributions.

Zero constraint stress is a result with conditions, not a default property.

6. Constrained reaction equations

For retained variables \(q^a\), constrained variables \(\phi^r\), and constraints \(\Phi_\alpha(q,\phi)=0\), the renormalized equations are

\[ \frac{\delta\Gamma0}{\delta q^a} +\lambda^\alpha\frac{\delta\Phi\alpha}{\delta q^a}=0, \]

\[ \frac{\delta\Gamma0}{\delta\phi^r} +\lambda^\alpha\frac{\delta\Phi\alpha}{\delta\phi^r}=0, \]

\[ \Phi_\alpha=0. \]

6.1 Block-separable case

If

\[ \frac{\delta\Phi_\alpha}{\delta q^a}=0 \]

and the normal Jacobian

\[ N{\alpha r}=\frac{\delta\Phi\alpha}{\delta\phi^r} \]

has full rank, the normal equations determine reactions when

\[ -\left.\frac{\delta\Gamma0}{\delta\phi^r}\right|{\Phi=0} \in\operatorname{im}N^T. \]

This prevents normal-force overdetermination only under that span condition.

6.2 General case

If constraints depend on retained variables, multipliers modify retained equations. The full coupled system must be solved.

6.3 Global solvability

Pointwise algebra is insufficient. The multiplier system must satisfy:

• regularity;
• boundary and fixed-set parity;
• junction conditions;
• global integrability;
• Ward identities;
• anomaly cancellation;
• conserved-charge balance.

7. Background-equation completeness

A candidate background receives one of four Dynamics statuses:

7.1 Parent solution

Every varied parent equation and boundary equation is satisfied.

7.2 Constrained reaction solution

Displaced equations are replaced by a complete, solvable reaction system, including stress and boundary effects.

7.3 Effective external background

The Stage is externally supplied. The reduced theory may be useful, but it does not derive the background as a parent vacuum.

7.4 Inconsistent background

The complete equation system has no solution at the claimed object.

A closure statement must name which status is used.

8. Radiative solvability

Let \(\Gamma_k\) be the effective action at scale \(k\). Radiative consistency requires more than preservation of the classical constraint surface.

At each claimed order or Wilsonian step, test:

• closure of the corrected constraint algebra;
• rank stability;
• normal-force span;
• retained equation solvability;
• boundary compatibility;
• anomaly and Ward identities;
• generation of forbidden operators;
• determinant contributions;
• parameter and counterterm provenance.

A forbidden mode need not be regenerated for the theory to fail. Equations with no solution are a separate failure.

9. Physical stability interface

Dynamics supplies the effective Hessian to BB-RIG-1.1.

After gauge quotient and exact constraints,

\[ Q{\rm phys}=P{\rm phys}^{T}\,\delta^2\Gamma\,P_{\rm phys}. \]

The supplied packet must include:

• kinetic metric;
• gauge fixing and ghosts;
• mixed blocks;
• boundary domain;
• scale and scheme;
• zero modes;
• negative modes;
• intended protected scalars;
• tower scope;
• uncertainty.

Dynamics may not label an isolated critical point stable without the physical Hessian result.

10. Vacuum-offset response interface

A field-independent shift of a protected matter sector is

\[ \mathcal Lm\rightarrow\mathcal Lm-c. \]

Dynamics must expose how it enters the complete equations.

For ordinary Einstein coupling, the external equation shifts by a term proportional to

\[ c\,g_{\mu\nu}. \]

Constraint reactions in internal directions do not automatically remove this external response.

The vacuum-response packet sent to BB-VAC-1.1 contains:

vacuum_response_packet:
  protected_sector:
  constant_shift_parameter:
  external_metric_equation:
  internal_equations:
  constraint_reaction_equations:
  global_auxiliary_equations:
  boundary_equations:
  determinant_terms:
  matching_terms:
  graviton_loop_scope:
  ordinary_stress_response:
  solved_response_operator:

11. Ordinary-gravity control

Any vacuum-offset mechanism must preserve lawful response to nonconstant stress.

The negative control is:

• add a constant \(-c\) to the protected Lagrangian;
• verify local curvature observables do not change in the protected scope;
• add a localized excitation with nontrivial traceless or gradient stress;
• verify ordinary gravitational response remains.

A mechanism that removes all stress from the gravitational equations fails.

12. Constant shifts versus phase transitions

A constant threshold correction and a finite-time phase transition are distinct.

Dynamics must classify:

• static additive shift;
• changing condensate;
• latent heat;
• radiation production;
• nonadiabatic boundary or global response;
• historical averages;
• residual present value.

A constant-shift identity does not automatically solve cosmological transition history.

13. Probability, causality, and conservation

For closed quantum sectors, the generator must be self-adjoint on the physical domain or possess an equivalent unitary kernel.

For open sectors, declare:

• environment;
• channel;
• normalization;
• exchange currents;
• observer access;
• approximation bound.

The law must preserve, at the claimed scope:

• probability;
• gauge and BRST constraints;
• exact charges;
• boundary flux balance;
• causal support;
• operational no-signalling.

14. Dynamics certificate

dynamics_certificate_v3:
  candidate_id:
  branch_id:
  parent_action:
  primitive_term_ledger:
  complete_equation_vector:
  equation_count_and_identities:
  constraint_types:
  dirac_bergmann_closure:
  rank_strata:
  physical_measure:
  determinant_contributions:
  constraint_stress:
  multiplier_boundary_problem:
  background_status:
  background_solution:
  radiative_solvability:
  ward_identities:
  anomaly_result:
  physical_hessian_packet:
  vacuum_response_packet:
  ordinary_gravity_control:
  phase_transition_scope:
  full_tower_scope:
  evidence_hashes:
  verdict:

15. Negative controls

15.1 No propagator but no solution

Remove a deformation field from the propagator but choose a radiative normal force outside the multiplier span.

Expected result: FAIL / RADIATIVE OVERDETERMINATION.

15.2 Constraint no-work with nonzero stress

Use a metric-dependent constraint that does no virtual work along admissible motion.

Expected result: no-work passes, stress-zero claim fails.

15.3 Vacuum degravitation that also removes matter gravity

Project out every trace and traceless source indiscriminately.

Expected result: vacuum shift passes, ordinary-gravity control fails.

15.4 Frozen Stage with abandoned internal equations

Do not vary the internal metric and supply no reaction or background equation.

Expected result: EFFECTIVE EXTERNAL BACKGROUND, not parent solution.

16. Falsifiers and reopen triggers

This block fails or reopens if:

• the parent action is incomplete;
• a primitive move lacks an owner;
• equation counting omits a boundary or global equation;
• Dirac closure or rank is incomplete;
• determinant terms are omitted;
• no-work is used as a stress proof;
• no-propagator is used as a solvability proof;
• a background is called a solution without all equations;
• a constant vacuum shift changes local curvature in the claimed protected scope;
• ordinary nonconstant stress fails to gravitate;
• a phase-transition claim is inferred from constant-shift cancellation;
• a zero-mode result is promoted to the full tower.

17. Terminal

BB-DYN-3.3.1

DELIVERS:
  Parent law, complete equations, constraint and measure closure,
  background solvability, stress accounting, radiative response,
  and interfaces to rigidity and vacuum protection.

STATUS:
  DEVELOPMENT AUTHORITY CANDIDATE — NOT FROZEN.

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 — stratified quantum gauge consistency

1. Complete gauge variation

For a \(d+1\)-dimensional parent with \(d\)-dimensional fixed sets \(F_a\), the renormalized gauge variation is decomposed as

\[ \delta\epsilon\Gamma{\rm tot} = \delta\epsilon\Gamma{\rm bulk} + \suma \delta\epsilon\Gamma{Fa} + \delta\epsilon S{\rm inflow} + \delta\epsilon S{\rm ct}. \]

For four-dimensional fixed sets, write the descent data

\[ I6=dI5^{(0)}, \qquad \delta\epsilon I5^{(0)}=dI_4^{(1)}(\epsilon,A). \]

A bulk inflow action of the form

\[ S{\rm inflow} = 2\pi i\,k\int{Y5}I5^{(0)} \]

has boundary variation

\[ \delta\epsilon S{\rm inflow} = 2\pi i\,k \suma sa \int{Fa}I_4^{(1)}, \]

where \(s_a\) is fixed by the outward-normal orientation. The coefficient \(k\) must satisfy the appropriate quantization and global-form conditions.

2. Fixed-set local equations

For each fixed set define

\[ \mathcal Aa^{(1)} = \mathcal A{a,\rm projected\ bulk}^{(1)} + \mathcal A{a,\rm localized}^{(1)} + \mathcal A{a,\rm measure}^{(1)} + \mathcal A{a,\rm counterterm}^{(1)} + \mathcal A{a,\rm inflow}^{(1)}. \]

The local pass condition is

\[ \boxed{\mathcal A_a^{(1)}=0\quad\text{for every }a} \]

as a cohomology class under all allowed gauge transformations.

The weaker condition

\[ \suma\mathcal Aa^{(1)}=0 \]

is only an integrated check. It does not prove local gauge invariance.

3. Quantum master equation with boundaries

The candidate must formulate the quantum gauge system using BV–BFV or an equivalent boundary-complete formalism.

Schematically,

\[ \frac12(S,S)-i\hbar\Delta S = \pi^*S_{\partial} \]

before boundary completion. Full closure requires a compatible boundary state/functional and counterterm/inflow system such that the total quantum master equation is satisfied.

A bulk statement \(Q_{\rm BRST}^2=0\) is insufficient when the measure has fixed-set support.

4. Consistent versus covariant anomalies

The ledger must state whether it uses:

• consistent anomalies;
• covariant anomalies;
• Bardeen–Zumino shifted currents.

All fixed-set and inflow entries must use one compatible convention. A Bardeen–Zumino polynomial may redistribute anomaly between currents or strata, but it cannot remove the total anomaly class.

5. Regulator and parity dependence

Localized anomalies can depend on the parity/domain assignment and on how bulk fermions are regulated. The final anomaly class must be regulator-independent after allowed local counterterms are included.

The certificate must report:

• regulator;
• consistent-current convention;
• parity signs;
• fixed-set weights;
• localized spectrum;
• allowed counterterms;
• invariant remainder.

6. Global relative anomaly equation

Perturbative cancellation is not enough. The exponentiated fermion determinant is a section of an anomaly/determinant line.

The complete parent-plus-boundary system must provide a gluing-compatible trivialization:

\[ Z{\rm boundary}\, Z{\rm inflow} \in\mathbb C \]

rather than an untrivialized line.

For every closed test obtained by gluing or forming a mapping torus, require

\[ \exp\!\left(2\pi i\,\frac{\eta+h}{2}\right) \, \exp(2\pi i S_{\rm top}) = 1 \]

in the declared convention and tangential structure.

7. Dynamics certificate additions

boundary_quantum_dynamics:
  fixed_set_ids:
  allowed_gauge_transformations:
  anomaly_convention: consistent|covariant
  regulator:
  bulk_anomaly_polynomial:
  projected_bulk_contributions_by_fixed_set:
  localized_field_contributions:
  measure_and_ghost_contributions:
  counterterms:
  inflow_action:
  inflow_coefficient_quantization:
  orientation_signs:
  local_residuals:
  bv_bfv_or_equivalent_completion:
  determinant_line:
  relative_eta_or_bordism_test:
  gluing_test:
  quantum_master_equation_result:
  evidence_hashes:
  verdict:

8. New Dynamics falsifiers

Dynamics fails if:

• only the integrated anomaly is zero;
• an inflow coefficient is selected after reading the residual;
• the coefficient is not globally quantized;
• a boundary condition is not compatible with the Dirac/gauge operator domain;
• the local anomaly convention changes between rows;
• the determinant line remains untrivialized;
• the BV quantum master equation is asserted from classical Jacobi closure alone.

Block-Specific Standalone Interface — Dynamics

Owned questions

This block alone owns:

• the typed parent action;
• the complete equation vector;
• constraint classification and preservation;
• physical measure and determinant ownership;
• stress of multiplier, boundary, and constraint sectors;
• retained/reaction-equation solvability;
• Ward identities and BV/BFV quantum consistency;
• ordinary probability, causality, conservation, and nonconstant-stress response.

Required inputs

• complete Shape and boundary manifests;
• constraint and permission packets;
• Scale/scheme packet;
• complete field and Actor inventory;
• candidate-specific parent terms.

Required outputs

• parent-action manifest;
• complete Euler–Lagrange and constraint-equation ledger;
• Dirac/BV/BFV closure;
• physical measure;
• stress and reaction calculations;
• background classification;
• radiative-solvability result;
• Ward/anomaly interface;
• Dynamics certificate.

Non-ownership boundary

Dynamics does not by itself prove:

• candidate uniqueness;
• intrinsic rigidity;
• full physical stability without the Rigidity block;
• vacuum-offset protection without the Vacuum block;
• local/global fixed-set anomaly closure without the Boundary block.

Observer-response handoff

Dynamics supplies the physical evolution, conserved currents, response functions, and physical measure consumed by OBS. OBS cannot repair an inconsistent Dynamics by projection. Conversely, a lawful parent response cannot be compared with data until OBS has produced the correct record functional.

For time-dependent vacuum tests, Dynamics must evolve the protecting sector and supply the constraint-algebra history required by VAC-C11.

Gauge-owner and kinetic-term consistency

Dynamics consumes the SHP-C09 gauge-owner ledger.

For every surviving generator \(T_A\), the parent action must identify exactly one independent connection variable and one owning kinetic structure. Metric Kaluza–Klein notation and principal-bundle notation may describe one field, but the action must not count both as independent unless a second gauge sector is physically intended and appears in the spectrum.

Required checks:

1. map every gauge-owner ID to a parent-action field;
2. count independent kinetic operators;
3. count first-class gauge parameters;
4. count four-dimensional vector zero modes;
5. verify equality of these counts after constraints and quotienting;
6. verify that center/global-lift data is compatible with every charged Actor;
7. run the duplicate-owner control by treating aliases as independent and confirm that the spectrum gains an extra gauge sector.

A gauge owner with no parent-action term fails Dynamics. A parent-action connection with no Shape owner fails the one-parent audit.

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. | | DYN-C01 | Parent-action manifest + term provenance table | Every primitive term has an owner, support, coefficient provenance, derivative order, symmetry, boundary status, and microscopic/effective label; every gauge connection references exactly one SHP-C09 owner ID. | A matrix element, potential, counterterm, or gauge connection has no parent-action owner, or an owner alias is independently double-counted. | | DYN-C02 | Dirac–Bergmann or BV–BFV closure calculation | All primary/secondary constraints, ranks, brackets, boundary terms, and residual gauges are computed until closure; the rank is constant on the claimed stratum. | The algorithm stops early, the rank changes, or boundary constraints are omitted. | | DYN-C03 | Physical-measure derivation + determinant artifact | The reduced measure includes all Faddeev–Popov/Dirac/BV determinants, and field dependence is retained in loops and matching. | A determinant is dropped as a constant without proof. | | DYN-C04 | Metric variation of all constraint/boundary sectors | The stress tensor is obtained by varying the full action and measure; the result is evaluated on the physical branch. | No-work is used as a substitute for zero stress. | | DYN-C05 | Solvability/Jacobian certificate | The full retained-plus-reaction equation system has a solution and a full-rank Jacobian or a theorem establishes existence in the declared domain. | Radiative normal forces lie outside the multiplier span or the retained equations have no solution. | | DYN-C06 | Ward/BV identity + boundary multiplier ledger | Allowed transformations preserve the domain, all measure anomalies cancel, and multiplier boundary conditions admit a global solution. | A Ward identity, anomaly, or boundary multiplier equation remains unresolved. | | DYN-C07 | Response-function calculation | Localized and nonconstant stress sources produce the correct causal/conservation response while protected constants are handled according to VAC. | The mechanism removes ordinary matter gravity, violates conservation, or produces acausal record response. | | DYN-C08 | Scope theorem or bounded truncation report | The certificate states exactly whether it covers zero modes, a cutoff tower, or all modes, with a tail theorem where a global claim is made. | A local, zero-mode, finite-EFT, or invariant-sector result is promoted beyond scope. | | VAC-C01 | Complete background-equation vector + equation-count/Jacobian report | External, internal, matter, boundary, constraint, and global equations are listed and solved or lawfully replaced; unknown/equation counts and rank are published. | An internal Einstein or boundary equation disappears when a Stage variable is frozen. | | VAC-C04 | Finite constant-shift response experiment | For each protected sector, the full equations are re-solved after \(\mathcal Ls o\mathcal Ls-c_s\), and the curvature-record derivative or finite difference is published. | Only an infinitesimal or tree-level shift is tested while auxiliary/global response is held fixed. | | VAC-C06 | Localized-source negative control | Dust, radiation, pressure gradients, and localized stress still source the correct gravitational response. | The protection mechanism degravitates ordinary matter or radiation. | | BND-C03 | Operator-domain certificate | Dirac, gauge, ghost, scalar, and load-bearing graviton domains are self-adjoint/elliptic as required and preserved by gauge/BRST transformations. | A parity table is supplied without a lawful operator domain. | | BND-C07 | Boundary BV/BFV quantum-master-equation output | Ghost, measure, boundary state, and inflow contributions satisfy the relative QME and compose under gluing. | Classical BRST closure is used while the measure/domain anomaly remains. | | VAC-C11 | Time-dependent constraint-algebra and cosmology simulation/theorem | A time-dependent vacuum shift \(c(t)\) representing QCD/electroweak transitions is inserted; the constraint algebra remains consistent and the Friedmann/perturbation history stays within frozen bounds, or the regime is explicitly excluded with quantified lost predictions. | The mechanism passes static shifts but becomes singular, acausal, or observationally unacceptable during a finite transition. |

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.

--- title: "BB-DYN-3.3.1 — Parent Dynamics, Constraints, Gauge Ownership, Background, Boundary Anomaly, and Response" buildingblockid: "BB-DYN-3.3.1" version: "3.3.1" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — SHP-C09 GAUGE-OWNER INTERFACE MERGED — NOT FROZEN" scope: "Complete owned specification under package v3.3.2" protocolfreeze: false canonicalratification: false packageauthorityversion: "3.3.2" ---

BB-DYN-3.3.1 — Parent Dynamics, Constraints, Gauge Ownership, Background, Boundary Anomaly, and Response

0. Purpose

Dynamics answers:

Which lawful changes occur on the complete Shape, which equations define its backgrounds, how exact constraints are preserved, how quantum corrections modify the equations, and how the system responds separately to vacuum offsets and ordinary excitations?

This block prohibits four substitutions:

\[ \text{kinematic permission}\neq\text{nonzero amplitude}, \]

\[ \text{constraint no-work}\neq\text{zero stress}, \]

\[ \text{no propagator}\neq\text{radiative solvability}, \]

\[ \text{internal reaction solvable}\neq\text{vacuum energy degravitated}. \]

1. Parent action

Every candidate must supply a typed parent action or an explicitly scoped non-action replacement.

A general action ledger is

\[ S{ m parent}=S{ m grav}+S{ m gauge}+S{ m matter}+S{ m scalar} +S{ m boundary}+S{ m constraint}+S{ m global}+S_{ m domain}. \]

For every term publish:

• support and dimensionality;
• field owner;
• derivative order;
• exact and broken symmetries;
• coefficient provenance;
• boundary variation;
• whether microscopic, structural, global, or matching-only;
• stress-energy contribution;
• vacuum-energy contribution;
• cutoff and scheme scope.

An arbitrary dense kernel entry without a parent-law owner is prohibited.

2. Complete equation vector

Let the independent variables be

\[ z^I=(g{\mu\nu},\gamma{mn},g{\mu m},AM,\Psi,\phi,b,\lambda,\sigma,\ldots). \]

The complete background and fluctuation equations are

\[ \mathcal F_I[z]\equiv\frac{\delta\Gamma}{\delta z^I}=0, \]

supplemented by boundary, global, gauge, and constraint equations.

The candidate must publish an equation ledger:

equation:
  equation_id:
  varied_object:
  local_or_global:
  bulk_or_boundary:
  differential_or_algebraic:
  independent_after_identities:
  source_terms:
  constraint_couplings:
  observer_projection:
  solution_status:

2.1 Equation counting

Before claiming a background solution:

1. count independent unknown functions and global variables;
2. count independent equations after Noether and Bianchi identities;
3. classify gauge freedoms;
4. classify second-class constraints;
5. include boundary and junction equations;
6. verify rank on the claimed stratum;
7. prove or compute existence of a solution.

A visually plausible product ansatz is not a solution certificate.

3. Constraint classification and closure

Constraints are typed as:

• first-class gauge constraints;
• second-class physical constraints;
• constitutive restrictions;
• boundary/gluing constraints;
• global constraints;
• effective matching relations.

For Hamiltonian variables, the total Hamiltonian is

\[ HT=H0+u^A\Xi_A. \]

The Dirac–Bergmann algorithm must:

1. list primary constraints;
2. impose preservation \(\dot\Xi_A\approx0\);
3. generate every secondary and higher constraint;
4. terminate with constant rank on the claimed stratum;
5. classify first- and second-class sectors;
6. determine multipliers and residual gauge functions;
7. verify boundary compatibility;
8. compute the physical degree-of-freedom count.

For second-class constraints,

\[ C{AB}=\{\XiA,\Xi_B\} \]

must be invertible on the physical stratum and

\[ \{F,G\}D=\{F,G\}-\{F,\XiA\}(C^{-1})^{AB}\{\Xi_B,G\}. \]

Rank change is a branch change or singular stratum, not a small correction.

4. Physical measure

For a second-class system,

\[ D\mu{ m phys}=D\mu0\prodA\delta(\XiA)\sqrt{\det C}. \]

The determinant must be included in effective actions, matching, and stress calculations.

If \(C\) depends on retained fields,

\[ \Delta\Gamma_C=-\frac{i}{2}\operatorname{Tr}\log C \]

or the appropriate Euclidean counterpart contributes physical terms.

A field-dependent determinant cannot be dismissed as normalization.

5. No-work and stress are separate

For ideal constraints,

\[ \lambda^A d\Phi_A=0 \]

along admissible displacements.

This proves no virtual work in constrained directions. It does not establish zero stress-energy.

For

\[ S{\rm constraint}=\int d^Dx\sqrt{-g}\,\lambda^A\PhiA, \]

\[ T^{\rm constraint}{MN} =-\frac{2}{\sqrt{-g}}\frac{\delta S{\rm constraint}}{\delta g^{MN}}. \]

The calculation must include:

• metric variation of the volume form;
• explicit metric dependence of \(\Phi_A\);
• metric dependence of multipliers when induced by solved equations;
• determinant and ghost terms;
• boundary contributions.

Zero constraint stress is a result with conditions, not a default property.

6. Constrained reaction equations

For retained variables \(q^a\), constrained variables \(\phi^r\), and constraints \(\Phi_\alpha(q,\phi)=0\), the renormalized equations are

\[ \frac{\delta\Gamma0}{\delta q^a} +\lambda^\alpha\frac{\delta\Phi\alpha}{\delta q^a}=0, \]

\[ \frac{\delta\Gamma0}{\delta\phi^r} +\lambda^\alpha\frac{\delta\Phi\alpha}{\delta\phi^r}=0, \]

\[ \Phi_\alpha=0. \]

6.1 Block-separable case

If

\[ \frac{\delta\Phi_\alpha}{\delta q^a}=0 \]

and the normal Jacobian

\[ N{\alpha r}=\frac{\delta\Phi\alpha}{\delta\phi^r} \]

has full rank, the normal equations determine reactions when

\[ -\left.\frac{\delta\Gamma0}{\delta\phi^r}\right|{\Phi=0} \in\operatorname{im}N^T. \]

This prevents normal-force overdetermination only under that span condition.

6.2 General case

If constraints depend on retained variables, multipliers modify retained equations. The full coupled system must be solved.

6.3 Global solvability

Pointwise algebra is insufficient. The multiplier system must satisfy:

• regularity;
• boundary and fixed-set parity;
• junction conditions;
• global integrability;
• Ward identities;
• anomaly cancellation;
• conserved-charge balance.

7. Background-equation completeness

A candidate background receives one of four Dynamics statuses:

7.1 Parent solution

Every varied parent equation and boundary equation is satisfied.

7.2 Constrained reaction solution

Displaced equations are replaced by a complete, solvable reaction system, including stress and boundary effects.

7.3 Effective external background

The Stage is externally supplied. The reduced theory may be useful, but it does not derive the background as a parent vacuum.

7.4 Inconsistent background

The complete equation system has no solution at the claimed object.

A closure statement must name which status is used.

8. Radiative solvability

Let \(\Gamma_k\) be the effective action at scale \(k\). Radiative consistency requires more than preservation of the classical constraint surface.

At each claimed order or Wilsonian step, test:

• closure of the corrected constraint algebra;
• rank stability;
• normal-force span;
• retained equation solvability;
• boundary compatibility;
• anomaly and Ward identities;
• generation of forbidden operators;
• determinant contributions;
• parameter and counterterm provenance.

A forbidden mode need not be regenerated for the theory to fail. Equations with no solution are a separate failure.

9. Physical stability interface

Dynamics supplies the effective Hessian to BB-RIG-1.1.

After gauge quotient and exact constraints,

\[ Q{\rm phys}=P{\rm phys}^{T}\,\delta^2\Gamma\,P_{\rm phys}. \]

The supplied packet must include:

• kinetic metric;
• gauge fixing and ghosts;
• mixed blocks;
• boundary domain;
• scale and scheme;
• zero modes;
• negative modes;
• intended protected scalars;
• tower scope;
• uncertainty.

Dynamics may not label an isolated critical point stable without the physical Hessian result.

10. Vacuum-offset response interface

A field-independent shift of a protected matter sector is

\[ \mathcal Lm\rightarrow\mathcal Lm-c. \]

Dynamics must expose how it enters the complete equations.

For ordinary Einstein coupling, the external equation shifts by a term proportional to

\[ c\,g_{\mu\nu}. \]

Constraint reactions in internal directions do not automatically remove this external response.

The vacuum-response packet sent to BB-VAC-1.1 contains:

vacuum_response_packet:
  protected_sector:
  constant_shift_parameter:
  external_metric_equation:
  internal_equations:
  constraint_reaction_equations:
  global_auxiliary_equations:
  boundary_equations:
  determinant_terms:
  matching_terms:
  graviton_loop_scope:
  ordinary_stress_response:
  solved_response_operator:

11. Ordinary-gravity control

Any vacuum-offset mechanism must preserve lawful response to nonconstant stress.

The negative control is:

• add a constant \(-c\) to the protected Lagrangian;
• verify local curvature observables do not change in the protected scope;
• add a localized excitation with nontrivial traceless or gradient stress;
• verify ordinary gravitational response remains.

A mechanism that removes all stress from the gravitational equations fails.

12. Constant shifts versus phase transitions

A constant threshold correction and a finite-time phase transition are distinct.

Dynamics must classify:

• static additive shift;
• changing condensate;
• latent heat;
• radiation production;
• nonadiabatic boundary or global response;
• historical averages;
• residual present value.

A constant-shift identity does not automatically solve cosmological transition history.

13. Probability, causality, and conservation

For closed quantum sectors, the generator must be self-adjoint on the physical domain or possess an equivalent unitary kernel.

For open sectors, declare:

• environment;
• channel;
• normalization;
• exchange currents;
• observer access;
• approximation bound.

The law must preserve, at the claimed scope:

• probability;
• gauge and BRST constraints;
• exact charges;
• boundary flux balance;
• causal support;
• operational no-signalling.

14. Dynamics certificate

dynamics_certificate_v3:
  candidate_id:
  branch_id:
  parent_action:
  primitive_term_ledger:
  complete_equation_vector:
  equation_count_and_identities:
  constraint_types:
  dirac_bergmann_closure:
  rank_strata:
  physical_measure:
  determinant_contributions:
  constraint_stress:
  multiplier_boundary_problem:
  background_status:
  background_solution:
  radiative_solvability:
  ward_identities:
  anomaly_result:
  physical_hessian_packet:
  vacuum_response_packet:
  ordinary_gravity_control:
  phase_transition_scope:
  full_tower_scope:
  evidence_hashes:
  verdict:

15. Negative controls

15.1 No propagator but no solution

Remove a deformation field from the propagator but choose a radiative normal force outside the multiplier span.

Expected result: FAIL / RADIATIVE OVERDETERMINATION.

15.2 Constraint no-work with nonzero stress

Use a metric-dependent constraint that does no virtual work along admissible motion.

Expected result: no-work passes, stress-zero claim fails.

15.3 Vacuum degravitation that also removes matter gravity

Project out every trace and traceless source indiscriminately.

Expected result: vacuum shift passes, ordinary-gravity control fails.

15.4 Frozen Stage with abandoned internal equations

Do not vary the internal metric and supply no reaction or background equation.

Expected result: EFFECTIVE EXTERNAL BACKGROUND, not parent solution.

16. Falsifiers and reopen triggers

This block fails or reopens if:

• the parent action is incomplete;
• a primitive move lacks an owner;
• equation counting omits a boundary or global equation;
• Dirac closure or rank is incomplete;
• determinant terms are omitted;
• no-work is used as a stress proof;
• no-propagator is used as a solvability proof;
• a background is called a solution without all equations;
• a constant vacuum shift changes local curvature in the claimed protected scope;
• ordinary nonconstant stress fails to gravitate;
• a phase-transition claim is inferred from constant-shift cancellation;
• a zero-mode result is promoted to the full tower.

17. Terminal

BB-DYN-3.3.1

DELIVERS:
  Parent law, complete equations, constraint and measure closure,
  background solvability, stress accounting, radiative response,
  and interfaces to rigidity and vacuum protection.

STATUS:
  DEVELOPMENT AUTHORITY CANDIDATE — NOT FROZEN.

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 — stratified quantum gauge consistency

1. Complete gauge variation

For a \(d+1\)-dimensional parent with \(d\)-dimensional fixed sets \(F_a\), the renormalized gauge variation is decomposed as

\[ \delta\epsilon\Gamma{\rm tot} = \delta\epsilon\Gamma{\rm bulk} + \suma \delta\epsilon\Gamma{Fa} + \delta\epsilon S{\rm inflow} + \delta\epsilon S{\rm ct}. \]

For four-dimensional fixed sets, write the descent data

\[ I6=dI5^{(0)}, \qquad \delta\epsilon I5^{(0)}=dI_4^{(1)}(\epsilon,A). \]

A bulk inflow action of the form

\[ S{\rm inflow} = 2\pi i\,k\int{Y5}I5^{(0)} \]

has boundary variation

\[ \delta\epsilon S{\rm inflow} = 2\pi i\,k \suma sa \int{Fa}I_4^{(1)}, \]

where \(s_a\) is fixed by the outward-normal orientation. The coefficient \(k\) must satisfy the appropriate quantization and global-form conditions.

2. Fixed-set local equations

For each fixed set define

\[ \mathcal Aa^{(1)} = \mathcal A{a,\rm projected\ bulk}^{(1)} + \mathcal A{a,\rm localized}^{(1)} + \mathcal A{a,\rm measure}^{(1)} + \mathcal A{a,\rm counterterm}^{(1)} + \mathcal A{a,\rm inflow}^{(1)}. \]

The local pass condition is

\[ \boxed{\mathcal A_a^{(1)}=0\quad\text{for every }a} \]

as a cohomology class under all allowed gauge transformations.

The weaker condition

\[ \suma\mathcal Aa^{(1)}=0 \]

is only an integrated check. It does not prove local gauge invariance.

3. Quantum master equation with boundaries

The candidate must formulate the quantum gauge system using BV–BFV or an equivalent boundary-complete formalism.

Schematically,

\[ \frac12(S,S)-i\hbar\Delta S = \pi^*S_{\partial} \]

before boundary completion. Full closure requires a compatible boundary state/functional and counterterm/inflow system such that the total quantum master equation is satisfied.

A bulk statement \(Q_{\rm BRST}^2=0\) is insufficient when the measure has fixed-set support.

4. Consistent versus covariant anomalies

The ledger must state whether it uses:

• consistent anomalies;
• covariant anomalies;
• Bardeen–Zumino shifted currents.

All fixed-set and inflow entries must use one compatible convention. A Bardeen–Zumino polynomial may redistribute anomaly between currents or strata, but it cannot remove the total anomaly class.

5. Regulator and parity dependence

Localized anomalies can depend on the parity/domain assignment and on how bulk fermions are regulated. The final anomaly class must be regulator-independent after allowed local counterterms are included.

The certificate must report:

• regulator;
• consistent-current convention;
• parity signs;
• fixed-set weights;
• localized spectrum;
• allowed counterterms;
• invariant remainder.

6. Global relative anomaly equation

Perturbative cancellation is not enough. The exponentiated fermion determinant is a section of an anomaly/determinant line.

The complete parent-plus-boundary system must provide a gluing-compatible trivialization:

\[ Z{\rm boundary}\, Z{\rm inflow} \in\mathbb C \]

rather than an untrivialized line.

For every closed test obtained by gluing or forming a mapping torus, require

\[ \exp\!\left(2\pi i\,\frac{\eta+h}{2}\right) \, \exp(2\pi i S_{\rm top}) = 1 \]

in the declared convention and tangential structure.

7. Dynamics certificate additions

boundary_quantum_dynamics:
  fixed_set_ids:
  allowed_gauge_transformations:
  anomaly_convention: consistent|covariant
  regulator:
  bulk_anomaly_polynomial:
  projected_bulk_contributions_by_fixed_set:
  localized_field_contributions:
  measure_and_ghost_contributions:
  counterterms:
  inflow_action:
  inflow_coefficient_quantization:
  orientation_signs:
  local_residuals:
  bv_bfv_or_equivalent_completion:
  determinant_line:
  relative_eta_or_bordism_test:
  gluing_test:
  quantum_master_equation_result:
  evidence_hashes:
  verdict:

8. New Dynamics falsifiers

Dynamics fails if:

• only the integrated anomaly is zero;
• an inflow coefficient is selected after reading the residual;
• the coefficient is not globally quantized;
• a boundary condition is not compatible with the Dirac/gauge operator domain;
• the local anomaly convention changes between rows;
• the determinant line remains untrivialized;
• the BV quantum master equation is asserted from classical Jacobi closure alone.

Block-Specific Standalone Interface — Dynamics

Owned questions

This block alone owns:

• the typed parent action;
• the complete equation vector;
• constraint classification and preservation;
• physical measure and determinant ownership;
• stress of multiplier, boundary, and constraint sectors;
• retained/reaction-equation solvability;
• Ward identities and BV/BFV quantum consistency;
• ordinary probability, causality, conservation, and nonconstant-stress response.

Required inputs

• complete Shape and boundary manifests;
• constraint and permission packets;
• Scale/scheme packet;
• complete field and Actor inventory;
• candidate-specific parent terms.

Required outputs

• parent-action manifest;
• complete Euler–Lagrange and constraint-equation ledger;
• Dirac/BV/BFV closure;
• physical measure;
• stress and reaction calculations;
• background classification;
• radiative-solvability result;
• Ward/anomaly interface;
• Dynamics certificate.

Non-ownership boundary

Dynamics does not by itself prove:

• candidate uniqueness;
• intrinsic rigidity;
• full physical stability without the Rigidity block;
• vacuum-offset protection without the Vacuum block;
• local/global fixed-set anomaly closure without the Boundary block.

Observer-response handoff

Dynamics supplies the physical evolution, conserved currents, response functions, and physical measure consumed by OBS. OBS cannot repair an inconsistent Dynamics by projection. Conversely, a lawful parent response cannot be compared with data until OBS has produced the correct record functional.

For time-dependent vacuum tests, Dynamics must evolve the protecting sector and supply the constraint-algebra history required by VAC-C11.

Gauge-owner and kinetic-term consistency

Dynamics consumes the SHP-C09 gauge-owner ledger.

For every surviving generator \(T_A\), the parent action must identify exactly one independent connection variable and one owning kinetic structure. Metric Kaluza–Klein notation and principal-bundle notation may describe one field, but the action must not count both as independent unless a second gauge sector is physically intended and appears in the spectrum.

Required checks:

1. map every gauge-owner ID to a parent-action field;
2. count independent kinetic operators;
3. count first-class gauge parameters;
4. count four-dimensional vector zero modes;
5. verify equality of these counts after constraints and quotienting;
6. verify that center/global-lift data is compatible with every charged Actor;
7. run the duplicate-owner control by treating aliases as independent and confirm that the spectrum gains an extra gauge sector.

A gauge owner with no parent-action term fails Dynamics. A parent-action connection with no Shape owner fails the one-parent audit.

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. | | DYN-C01 | Parent-action manifest + term provenance table | Every primitive term has an owner, support, coefficient provenance, derivative order, symmetry, boundary status, and microscopic/effective label; every gauge connection references exactly one SHP-C09 owner ID. | A matrix element, potential, counterterm, or gauge connection has no parent-action owner, or an owner alias is independently double-counted. | | DYN-C02 | Dirac–Bergmann or BV–BFV closure calculation | All primary/secondary constraints, ranks, brackets, boundary terms, and residual gauges are computed until closure; the rank is constant on the claimed stratum. | The algorithm stops early, the rank changes, or boundary constraints are omitted. | | DYN-C03 | Physical-measure derivation + determinant artifact | The reduced measure includes all Faddeev–Popov/Dirac/BV determinants, and field dependence is retained in loops and matching. | A determinant is dropped as a constant without proof. | | DYN-C04 | Metric variation of all constraint/boundary sectors | The stress tensor is obtained by varying the full action and measure; the result is evaluated on the physical branch. | No-work is used as a substitute for zero stress. | | DYN-C05 | Solvability/Jacobian certificate | The full retained-plus-reaction equation system has a solution and a full-rank Jacobian or a theorem establishes existence in the declared domain. | Radiative normal forces lie outside the multiplier span or the retained equations have no solution. | | DYN-C06 | Ward/BV identity + boundary multiplier ledger | Allowed transformations preserve the domain, all measure anomalies cancel, and multiplier boundary conditions admit a global solution. | A Ward identity, anomaly, or boundary multiplier equation remains unresolved. | | DYN-C07 | Response-function calculation | Localized and nonconstant stress sources produce the correct causal/conservation response while protected constants are handled according to VAC. | The mechanism removes ordinary matter gravity, violates conservation, or produces acausal record response. | | DYN-C08 | Scope theorem or bounded truncation report | The certificate states exactly whether it covers zero modes, a cutoff tower, or all modes, with a tail theorem where a global claim is made. | A local, zero-mode, finite-EFT, or invariant-sector result is promoted beyond scope. | | VAC-C01 | Complete background-equation vector + equation-count/Jacobian report | External, internal, matter, boundary, constraint, and global equations are listed and solved or lawfully replaced; unknown/equation counts and rank are published. | An internal Einstein or boundary equation disappears when a Stage variable is frozen. | | VAC-C04 | Finite constant-shift response experiment | For each protected sector, the full equations are re-solved after \(\mathcal Ls o\mathcal Ls-c_s\), and the curvature-record derivative or finite difference is published. | Only an infinitesimal or tree-level shift is tested while auxiliary/global response is held fixed. | | VAC-C06 | Localized-source negative control | Dust, radiation, pressure gradients, and localized stress still source the correct gravitational response. | The protection mechanism degravitates ordinary matter or radiation. | | BND-C03 | Operator-domain certificate | Dirac, gauge, ghost, scalar, and load-bearing graviton domains are self-adjoint/elliptic as required and preserved by gauge/BRST transformations. | A parity table is supplied without a lawful operator domain. | | BND-C07 | Boundary BV/BFV quantum-master-equation output | Ghost, measure, boundary state, and inflow contributions satisfy the relative QME and compose under gluing. | Classical BRST closure is used while the measure/domain anomaly remains. | | VAC-C11 | Time-dependent constraint-algebra and cosmology simulation/theorem | A time-dependent vacuum shift \(c(t)\) representing QCD/electroweak transitions is inserted; the constraint algebra remains consistent and the Friedmann/perturbation history stays within frozen bounds, or the regime is explicitly excluded with quantified lost predictions. | The mechanism passes static shifts but becomes singular, acausal, or observationally unacceptable during a finite transition. |

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.

Constraint and Thought-Experiment Derivation of the Dynamics Improvements

Method

The revision was not built by asking what would make a selected gate pass. It used the following target-neutral loop:

1. freeze the existing Dynamics object and its non-ownership boundaries;
2. identify pairs of candidate worlds that pass all current Dynamics rows but have different physical outcomes;
3. formulate the smallest reusable constraint that separates the pair;
4. assign one required witness and one destructive fail trigger;
5. apply the new constraint to every gate, not only the gate that exposed it;
6. reject any proposed constraint that merely restates a desired answer or belongs to Shape, Granularity, Scale, Boundary, Rigidity, Vacuum, Observer, or governance.

Thought experiments

| ID | Thought experiment | Failure construction | Derived row | Consequence | |---|---|---|---|---| | TE-DYN-01 | Parent action without physical kernel | Two candidates have the same declared gauge owners and action labels. Candidate A has a 12-dimensional physical vector kernel after quotient; Candidate B has an additional boundary-localized zero mode. A parent-action ledger alone cannot distinguish them. | DYN-C09 | Require operator, domain, quotient, kernel, multiplicity, and tail evidence before a spectrum or no-extra claim. | | TE-DYN-02 | Same free theory, different interactions | Take two theories with identical quadratic action and spectrum. Add a cubic or Yukawa vertex to only one. Their masses at the symmetric point are identical, but decays, EWSB, proton safety, flavor, baryogenesis, and portals differ. | DYN-C10 | The free/operator census cannot certify interaction-dependent gates; require a complete interaction/source hypergraph. | | TE-DYN-03 | Hand-inserted mass matrix | Supply a phenomenologically correct mass matrix while no parent interaction and no solved condensate produce it. The gate can be numerically fit while the theory has no lawful transition generating the matrix. | DYN-C11 | Masses, mixings, widths, and threshold transitions must be derived from parent vertices plus a solved background. | | TE-DYN-04 | Self-adjoint but physically indefinite | A gauge-fixed quadratic operator is formally self-adjoint, yet the propagator contains a pole with negative residue after quotient or continuation. Self-adjointness alone does not establish a positive physical Hilbert space. | DYN-C12 | Require physical pairing, pole/residue, reflection-positivity or equivalent reconstruction, and negative-norm disposition. | | TE-DYN-05 | Trace-preserving but not a physical subsystem channel | A map preserves trace on a subsystem but is not completely positive when the subsystem is entangled with an allowed ancilla. Probability normalization passes while physical evolution fails. | DYN-C13 | Open evolution requires a CPTP channel or justified instrument, declared environment, exchange currents, and approximation bound. | | TE-DYN-06 | Static protection, bad finite transition | A mechanism cancels every constant source but responds strongly to a finite-time change, producing oscillations, entropy, or curvature outside allowed bounds. Static shift tests pass; cosmological history fails. | DYN-C14 | Require a time-dependent thermal/kinetic history, not a static-potential inference. | | TE-DYN-07 | Equations without a well-posed evolution | A complete Euler–Lagrange system admits formal solutions but has nonunique evolution or a boundary condition that fails to preserve constraints. Equation counting passes; prediction fails. | DYN-C15 | Require a well-posed initial-boundary-value problem, constraint propagation, and causal support. | | TE-DYN-08 | Fixed point destroyed by lawful refinement | Two beta functions agree through the computed order; one has a nonzero fixed point and another lawful higher-order completion has none. A stable result in one truncation does not establish a completion. | DYN-C16 | Require nested-basis and regulator-family stability plus tail/control theorems. | | TE-DYN-09 | Local conservation with omitted horizon flux | A conserved current obeys a local continuity equation, but charge inside a region changes because flux crosses a horizon or interface. Treating the region as closed creates apparent loss. | DYN-C17 | Require regional flux, edge terms, gluing, and information-channel ledgers. |

Derived new hard rows

| ID | Required witness | Minimal discharge | Fail trigger | |---|---|---|---| | DYN-C09 | Operator construction and physical kernel/spectrum certificate | Linearize the complete parent equations, build the gauge-fixed and constraint-reduced operators on the lawful domains, and publish kernels, zero modes, spectra, response functions, multiplicities, and unresolved tails. | A gate uses a declared Actor or expected spectrum without deriving the corresponding physical operator/kernel, or omits a mixed/operator block. | | DYN-C10 | Complete interaction and source hypergraph | Enumerate every parent-owned interaction vertex, source edge, representation contraction, selection rule, theorem-zero edge, dangerous operator, support, scale, and boundary domain. | A coupling, decay, portal, Yukawa, CP source, or dangerous operator is inserted or omitted without a complete parent-law disposition. | | DYN-C11 | Mass, mixing, transition, and decay derivation | Mass matrices, mixings, symmetry breaking, decay widths, threshold transitions, and effective vertices are derived from C10 plus the solved background and are diagonalized/matched with stated scope. | A matrix or transition is supplied by hand, the background does not solve the parent equations, or physical eigenstates/widths are not derived. | | DYN-C12 | Quantum physicality, positivity, and analytic-continuation certificate | After gauge/BRST quotient, establish a positive physical pairing or reflection-positivity/equivalent reconstruction, pole and residue signs, continuation prescription, and absence/disposition of negative-norm states. | Formal self-adjointness, Euclidean convergence, or BRST nilpotency is used without a physical positivity/pole certificate. | | DYN-C13 | Closed/open quantum evolution and probability instrument | Closed sectors have unitary normalized evolution; open sectors publish a completely positive trace-preserving channel or justified instrument, environment, exchange currents, approximation bound, and Observer handoff. | A trace-preserving but nonphysical map, unnormalized probability rule, undeclared environment, or hidden information loss is used. | | DYN-C14 | Nonequilibrium, thermal, and cosmological-history evolution | Publish initial state, thermal/kinetic/transport equations, entropy and charge production, phase-transition dynamics, production/washout/freeze-out, perturbations, and uncertainty through the claimed history. | A static potential or constant-shift result is promoted to a finite-time cosmological, baryogenesis, dark-matter, or transition claim. | | DYN-C15 | Well-posed causal initial-boundary-value problem | Prove or compute hyperbolicity/evolution well-posedness, constraint propagation, boundary compatibility, causal support, stability under allowed data, and synchronization/Observer handoffs. | Equations exist but evolution is nonunique, unstable, superluminal outside the declared cone, domain-incompatible, or not globally composable. | | DYN-C16 | RG, refinement, regulator, and truncation stability | Use nested operator bases/resolutions and admissible regulator/scheme families; track generated operators, fixed points/flows, critical data, convergence, matching, and omitted-tail bounds. | A result exists only in one truncation, basis, regulator, or scheme and changes materially under lawful refinement. | | DYN-C17 | Regional, horizon, interface, and information-flux balance | For regions separated by boundaries, defects, horizons, or cuts, publish charge/energy/probability/information currents, gluing law, edge terms, exchange channel, and finite Observer records. | Local conservation is claimed while flux through an interface is omitted, or subsystem nonunitarity is mistaken for global loss without a channel ledger. |

Ownership firewall

These additions do not move the following objects into Dynamics:

• candidate uniqueness and exhaustion remain in Granularity/CSE;
• intrinsic rigidity and stability terminal remain in Rigidity;
• exact boundary domains and anomaly trivialization remain in Boundary A/B;
• vacuum-offset protection remains in Vacuum;
• measured records remain in Observer;
• rulers and matching provenance remain in Scale;
• causal clock compatibility and synchronization remain in TS-1, while Dynamics supplies the evolution and causal-support packet TS consumes.

Existing Dynamics Object — Exact Reconstruction Before Repair

Authority inspected

• BBDYN331PARENTDYNAMICSCONSTRAINTSGAUGEOWNERSHIPBACKGROUNDBOUNDARYANOMALYANDRESPONSE.md
• Version: 3.3.1
• Package: candidate-neutral authority 3.3.2
• Status in source: development authority, not frozen, not canonically ratified
• Length: 865 lines

Owned physical object

The existing block defines Dynamics as the lawful-change system on the complete Shape. Its owned object is not merely an operator. It includes:

1. typed parent action and coefficient provenance;
2. complete bulk, boundary, global, gauge, and constraint equations;
3. Dirac–Bergmann / BV–BFV closure and physical degree counting;
4. physical measure and determinant contributions;
5. stress of constraints, multipliers, ghosts, and boundaries;
6. retained-plus-reaction solvability and global multiplier compatibility;
7. background classification;
8. radiative solvability and generated-operator checks;
9. physical Hessian handoff to Rigidity;
10. vacuum-offset and ordinary-stress response;
11. static-shift versus finite-transition distinction;
12. probability, causality, conservation, and Observer handoff;
13. boundary/fixed-set anomaly and relative-QME integration.

Existing hard rows

| ID | Witness family | Minimal discharge | Fail trigger | |---|---|---|---| | DYN-C01 | Parent-action manifest and term provenance | Every primitive term and gauge connection has one lawful owner, support, coefficient provenance, symmetry, derivative order, boundary status, and microscopic/effective scope. | Unowned terms, double-counted aliases, or a gauge owner without a parent-action term. | | DYN-C02 | Dirac–Bergmann or BV–BFV closure | All constraints, ranks, brackets, residual gauges, boundary terms, and physical degree counts are computed to closure on the claimed stratum. | Early termination, rank change, omitted boundary constraint, or unresolved residual gauge. | | DYN-C03 | Physical measure and determinant derivation | All FP/Dirac/BV determinants and field dependence are retained in loops, matching, and stress calculations. | A determinant is dropped as a constant without proof. | | DYN-C04 | Full metric variation and stress | Stress is derived from the complete action and measure, including constraints, boundaries, ghosts, and solved multiplier dependence. | No-work or absence from a propagator is substituted for zero stress. | | DYN-C05 | Retained-plus-reaction solvability | The complete coupled equations have a solution with full-rank Jacobian or an existence theorem on the declared domain. | Normal forces lie outside multiplier span or retained equations are unsolved. | | DYN-C06 | Ward/BV identity and global boundary multiplier closure | The domain is preserved, anomalies cancel, and multiplier/boundary equations admit a global solution. | Unresolved Ward identity, anomaly, boundary equation, or QME residual. | | DYN-C07 | Physical response, conservation, and ordinary-source control | Localized and nonconstant sources yield the correct conserved causal response while protected constants follow VAC. | Ordinary matter is degravitated, conservation fails, or response is acausal. | | DYN-C08 | Scope theorem or bounded truncation report | The result declares zero-mode, finite-tower, EFT, or all-mode scope, with a tail theorem for global claims. | A local or finite result is promoted beyond scope. |

Important existing strengths

• It already prohibits no propagator => solvable, constraint no-work => zero stress, and finite result => all-mode result.
• It already distinguishes parent solution, constrained-reaction solution, external background, and inconsistent background.
• It already requires quantum measure, Ward/BV identities, boundary compatibility, ordinary-source response, and a full tower scope statement.
• It already contains the correct gauge-owner/kinetic-owner logic used by SG-2.

Exact limitation found

The object contains prose obligations for several later-gate functions, but those functions are not promoted to independent hard rows with reusable witness schemas. Consequently the gate crosswalk can omit them without producing a machine failure. The missing executable functions are:

• physical operator/kernel derivation;
• complete interaction/source enumeration;
• mass/mixing/transition derivation;
• quantum positivity and analytic continuation;
• closed/open quantum channels;
• nonequilibrium/thermal history;
• well-posed causal initial-boundary evolution;
• RG/refinement/regulator stability;
• horizon/interface flux and information balance.

This is an operational completeness gap, not evidence that the existing parent-law architecture is wrong.

SG-2 → SG-3 → SG-4 Calibration Synthesis

Calibration verdict

The repaired Shape–Granularity–Dynamics triad is operationally useful and no new reusable triad capability was missing in these three gates. It generated the correct candidate spaces, operators/kernels, exact record classes, omitted-rival tests, and finite exact calculations.

The sequence does expose two external evidence debts, not triad design defects:

1. SG-3 requires a candidate-specific, current BND-A parity/domain manifest and a consistent BND-B/AD handoff for unconditional quantum-global language.
2. SG-4's local/perturbative result is exact, but the global relative-anomaly leg is not presently reproducible from the available BND-B authority, whose status says execution is owed. The gate source also contains incompatible statements about whether the R4 class is resolved or still open.

Building-block decision

• Shape V8.1: no repair triggered by SG-2/3/4.
• Granularity V4.0: no repair triggered; the exact/canonical record-class and omitted-test machinery worked.
• Dynamics V4.2: no repair triggered; DYN-C01/C02/C03/C06/C08/C09 were sufficient.
• Boundary/Anomaly evidence: needs execution and source-of-truth reconciliation before SG-3/4 can carry unconditional global claims.

Gate terminals from this run

| Gate | Independent result | Strongest honest terminal | |---|---|---| | SG-2 | 12 vector zero modes, rank 4, no extra vector/scalar zero modes | PASS at declared/scoped algebra and kernel level | | SG-3 | family multiplicity 3, one-sided chiral kernel, no below-cutoff fundamental mirror | PASS given-E/topology; conditional on current BND/AD global certificates | | SG-4 | primitive hypercharge ray, Z6 kernel, exact perturbative anomaly cancellation | perturbative/local face PASS; full global face OPEN/CONDITIONAL |

SG-2 Target-Blind Regression

Result

The triad reconstructs

\[\mathfrak g_{4D}=\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak u(1),\qquad \dim=8+3+1=12,\quad \operatorname{rank}=4.\]

The BRST-compatible interval domain gives one constant even Amu zero mode per generator and no odd Ay zero mode, hence 12 vector modes and zero unwanted scalar zero modes. All declared alias, duplicate-owner, extra-U(1), wrong-parity, and even-A_y controls fail as required.

Terminal

PASS at the declared/scoped local-algebra and zero-mode level. The forces-as-isometries/connection-owner categorical floor remains an explicitly named premise; the regression does not derive that modeling category from nothing.

SG-3 Target-Blind Regression

Result

The frozen Stage gives

\[P{K6}(t)=1+2t^2+2t^4+t^6,\qquad b0+b2=1+2=3.\]

The exact Borel–Weil–Bott/Riemann–Roch polynomial

\[\chi(m1,m2)=\frac{(m1+1)(m2+1)(m1+m2+2)}2\]

was enumerated over |m1|,|m2|<=8; magnitudes 2 and 4 never occur, while magnitude 3 appears at the first nontrivial twist cost. On the folded interval only the even (+,+) sector has a constant mode. Tensoring it with the three-dimensional generation space gives three one-sided zero modes per matter species and no fundamental mirror zero mode. The first opposite-parity excitation lies at 1.2566370614359170e+17 GeV, 1.580135694410067 times the admitted cutoff.

Terminal

PASS for the given-E/topology family magnitude and the scoped no-mirror kernel. Unconditional quantum-global closure remains conditional on a current, hash-matched BND-A/B and anomaly-descent certificate.

SG-4 Target-Blind Regression

Result

The exact linear system consisting of Yukawa invariance, the SU(2)^2 U(1) condition, and hypercharge neutrality of nu^c has rank 6 and a one-dimensional nullspace. Its primitive integer generator is

\[(qQ,q{u^c},q{d^c},qL,q{e^c},q{\nu^c},q_H)=(1,-4,2,-3,6,0,3),\qquad q=6Y.\]

Enumeration of all 36 center elements finds exactly six that act trivially, generated by (1,1,1), so the finest faithful kernel is Z6. Exact rational arithmetic gives zero for the pure color, mixed color-hypercharge, mixed weak-hypercharge, gravitational-hypercharge, cubic-hypercharge, and Witten ledgers. The nonzero specificity control is

\[\sumf Yf^2=10/3.\]

Terminal

PASS for the charge ray, electric charges, Z6 kernel, and perturbative anomaly face. The full relative/global anomaly terminal is not reproduced: BND-B is explicitly execution owed, and the source dossier contains conflicting statements about the R4 bordism residual.

Validation Report

• Overall: PASS
• Verifier: PASS: SG-2, SG-3, SG-4 exact regression assertions
• Authority files frozen: 13
• Historical terminal labels excluded from calculation scripts: yes

Dynamics V4.2 — Agent-Ready Gate-Closure Package

This package contains the complete maximum-rigor Dynamics authority. It preserves BB-DYN-3.3.1, integrates all seventeen Dynamics constraints, and includes operator, interaction, quantum-physicality, history, causal-evolution, refinement, and regional-flux execution systems.

Use it together with shape.zip and granularity.zip. Dynamics derives lawful equations and responses from the frozen Shape and Rulebook. It may not invent a field, interaction, boundary, regulator, or desired endpoint after seeing a gate result.

The SG-2/3/4 calibration evidence is included under 06_REGRESSION/.

Start Here — Dynamics V4.2

Read in this order:

1. 01CORE/BBDYN42MAXRIGORFULLGATECLOSUREDYNAMICS.md
2. 02REGISTRIES/DYNAMICSCONSTRAINTREGISTRYV4_2.json
3. 02REGISTRIES/DYNAMICSGATECROSSWALKV4_2.json
4. 03EXECUTION/TRIADINTEGRATION_CONTRACT.md
5. 03EXECUTION/TRIADGATEEXECUTIONPROMPT.md
6. 03EXECUTION/GATEEXECUTION_PROMPT.md

For a gate, freeze the Shape/Granularity/Dynamics and external-block hashes, execute every applicable DYN-C01–DYN-C17 packet, run destructive controls, publish the Dynamics residual vector, and preserve the gate claim ceiling. Missing gate calculations remain OPEN; they do not justify silently changing Dynamics.

BB-DYN-4.2 is the controlling development authority in this package. The exact BB-DYN-3.3.1 source and preservation audit remain included.