Granularity V4.0

--- title: "BB-GRN-4.0 — Maximum-Rigor Operational Granularity and Exhaustion" date: "2026-08-02" status: "DEVELOPMENT AUTHORITY; PRESERVES BB-GRN-3.3; ALL-GATE INTEGRATED" ---

BB-GRN-4.0 — Maximum-Rigor Operational Granularity and Exhaustion

0. Executive verdict

BB-GRN-3.3 contains the correct one-layer Granularity concept, but the thirty-gate audit exposed four execution defects:

1. its tolerance-based indistinguishability relation was used as an equivalence relation without a transitivity proof;
2. candidate exhaustion was explicit, but lawful test-family exhaustion was not a separate hard obligation;
3. histories, regional records, and effective/composite sectors were described but not first-class machine constraints;
4. the authority referenced the old eight-row Dynamics object rather than BB-DYN-4.2.

BB-GRN-4.0 preserves the old authority and repairs these defects without splitting Granularity into multiple physical layers.

1. Granularity identity

Granularity remains one layer. It owns the complete physical-record construction, exact and operational distinction, finite completion, candidate/test exhaustion, resolution scope, and stopping rule. It does not create Stage supports, physical dynamics, measurements, scales, or anomalies; it consumes their certified outputs.

2. Exact equivalence, pairwise closeness, and operational cells

Exact physical equivalence is denoted \(\cong_{\rm exact}\) and must be a proved isomorphism preserving state space, quotient, Dynamics, amplitudes, boundaries, global sectors, Scale, histories, regional records, and Observer map.

Pairwise tolerance closeness,

\[ dt(Ri,Rj)\le \varepsilont, \]

is symmetric but need not be transitive. It therefore cannot by itself define a quotient.

For operational quotienting, freeze a deterministic canonical record-cell map for every test,

\[ qt:Rt\rightarrow C_t, \]

including uncertainty/covariance and distributional data. Define the operational signature

\[ Q(i)=\left(I{\rm exact}(i),\{qt(Ri(t))\}{t\in\mathcal T}\right). \]

Then

\[ i\equiv_{\mathcal T,\Delta}j\quad\Longleftrightarrow\quad Q(i)=Q(j). \]

Equality of the frozen signature is reflexive, symmetric, and transitive. Pairwise distances remain diagnostics and robustness controls, not the quotient definition.

3. Fourteen Granularity constraints

GRN-C01 — Typed finite-completion and physical-record ontology

Requirement. Every candidate publishes complete objects, states, primitive transitions, constraints, measures, supports, global sectors, backgrounds, and unresolved tails.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C02 — Exact-record firewall

Requirement. Topology, charge lattice, anomaly class, index, parity/domain, determinant phase, constraint rank, and other exact data are compared before tolerances.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C03 — Complete mode, state, deformation, and stratum inventory

Requirement. Every primary mode, effective/composite state, deformation, boundary/edge sector, and stratum below scope receives one disposition.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C04 — Dynamic-history and transition-record completeness

Requirement. Static equality cannot imply physical equivalence when trajectories, phase transitions, transport, or finite-time records differ.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C05 — Regional, subsystem, horizon, and interface record completeness

Requirement. Global totals cannot substitute for region-resolved currents, edge sectors, horizons, interfaces, or subsystem records.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C06 — Observer-distribution and record-map completeness

Requirement. The record map, kernel, image, inaccessible sectors, uncertainty model, joint outcome distributions, and calibration are explicit.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C07 — Mathematically valid operational equivalence

Requirement. Exact equivalence is an isomorphism; operational classes use equality of pre-frozen canonical record-cell vectors, not a nontransitive pairwise epsilon relation.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C08 — Admitted-test-family saturation and omitted-test control

Requirement. Every lawful distinguishing test generated by Actor/Co-Actor, Dynamics, Boundary, Scale, Vacuum, TS, and Observer is included or proved inaccessible.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C09 — Candidate grammar exhaustion and omitted-rival control

Requirement. The roster/generator/class theorem terminates, canonicalizes duplicates, lists early failures, and fails when a known rival is removed.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C10 — Scale, cutoff, tail, and resolution-refinement lattice

Requirement. Every record and equivalence class states Scale, resolution, cutoff, tail theorem, and how classes map or split under lawful refinement.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C11 — Effective, composite, multiparticle, and superselection-sector completion

Requirement. Bound states, solitons, composites, microstates, many-body sectors, and effective Actors are inventoried and matched to Shape/Dynamics ownership.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C12 — Background, vacuum, ordinary-response, and phase-history signature

Requirement. Candidates differing in background solvability, constant-shift response, ordinary stress response, or phase history remain distinct.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C13 — Unknown-state firewall and quantitative stopping

Requirement. OPEN/NOT-EVALUATED is never collapsed into zero or equivalence; stopping requires zero unclassified candidates and tests.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

GRN-C14 — Provenance, branch, reproducibility, and stale invalidation

Requirement. Candidate, test, tolerance, quantizer, record-map, and refinement hashes are frozen and any comparison-relevant change creates a new branch.

Status rule. Missing evidence is OPEN; an explicit contradiction is FAIL; NOT-APPLICABLE requires a type proof. No unknown, omitted, inaccessible, or below-resolution row defaults to zero.

4. Packet schemas

4.1 finitecompletionpacket

finite_completion_packet:
  candidate_id:
  branch_hash:
  object_inventory:
  actor_and_state_inventory:
  primitive_transitions:
  exact_constraints:
  physical_measure:
  strata_and_derived_supports:
  observer_record_map:
  mode_state_deformation_inventory:
  global_topological_sectors:
  background_vacuum_history_signature:
  regional_interface_signature:
  cutoff_resolution_tail:
  unresolved_rows:

4.2 testfamilysaturation_packet

test_family_saturation_packet:
  generator_hash:
  actor_coactor_tests:
  dynamics_response_tests:
  boundary_global_tests:
  scale_vacuum_ts_observer_tests:
  generated_count:
  canonical_count:
  inaccessible_with_type_proof:
  omitted_test_control:
  saturation_residual:

4.3 operationalequivalencepacket

operational_equivalence_packet:
  exact_invariant_vector:
  record_cell_map_hash:
  tolerance_and_covariance_hash:
  test_family_hash:
  candidate_signature_vectors:
  exact_equivalence_classes:
  operational_cell_classes:
  pairwise_distance_diagnostics:
  transitivity_validation:
  robustness_under_refinement:

4.4 statehistoryregion_packet

state_history_region_packet:
  primary_modes:
  effective_composite_and_multiparticle_sectors:
  superselection_sectors:
  initial_ensembles:
  trajectory_and_transition_records:
  regions_horizons_interfaces:
  local_and_regional_flux_records:
  observer_distributions:
  unresolved_tails:

4.5 exhaustionandstopping_packet

exhaustion_and_stopping_packet:
  N_generated:
  N_canonical:
  N_evaluated:
  N_failed:
  N_open:
  N_survivor_exact_classes:
  N_survivor_operational_cells:
  N_unclassified_candidates:
  N_unclassified_tests:
  N_unclassified_modes_states_deformations:
  candidate_omission_control:
  test_omission_control:
  branch_and_refinement_hashes:
  terminal:

5. Destructive thought experiments

1. Nontransitive chain: construct records A, B, C with A close to B and B close to C but A not close to C. Pairwise epsilon closeness must not be quotienting.
2. Omitted lawful test: two candidates agree on every loaded test but differ under one lawful Co-Actor response. Test saturation must fail.
3. Static/history split: candidates share static endpoints but one has an unacceptable transient. They remain distinct.
4. Global/regional split: candidates have equal global charge but different horizon/interface flux. They remain distinct.
5. Primary/composite split: candidates share elementary modes but have different bound-state spectra. They remain distinct.
6. Unknown-as-zero: remove an unevaluated sector and attempt closure. The stopping certificate must fail.
7. Resolution split: refine the record cells so one class splits. A new branch/refinement record is mandatory.
8. Duplicate presentation: add a coordinate alias. Exact canonicalization must merge it without changing physical classes.

6. All-gate profiles

Black-hole-singularity

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C05, GRN-C12.
• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Dynamics handoff: DYN-C07, DYN-C08, DYN-C09, DYN-C15, DYN-C17.
• Gate-specific discipline: A dissolved continuum singularity claim does not discharge finite causal, record, or horizon obligations.

Born-rule

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11.
• Source terminal: CLOSED / CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C03, DYN-C06, DYN-C07, DYN-C12, DYN-C13.
• Gate-specific discipline: Operational probabilities require a lawful physical measure and observer record map; ontology alone is not a witness.

Gap-01

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16.
• Gate-specific discipline: Order-six boundary terms require the BND-A domain manifest.

Gap-02

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16.
• Gate-specific discipline:

Gap-05-stability

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C12.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C03, DYN-C04, DYN-C05, DYN-C07, DYN-C08, DYN-C14, DYN-C16.
• Gate-specific discipline:

Gap-05-value

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C12.
• Source terminal: MEASURED-ANCHOR / RESOLVED +0.
• Dynamics handoff: DYN-C07, DYN-C14.
• Gate-specific discipline: Measured value does not discharge stability.

Gap-08

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C12.
• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Dynamics handoff: DYN-C07, DYN-C09, DYN-C14, DYN-C15, DYN-C16.
• Gate-specific discipline:

Gap-10/BG-10

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11, GRN-C12.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C15.
• Gate-specific discipline:

Gap-11

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11, GRN-C12.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C16.
• Gate-specific discipline:

Gap-13

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C05, GRN-C11.
• Source terminal: CLOSED / PROJECT-DEPENDENCY / CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C03, DYN-C07, DYN-C08, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17.
• Gate-specific discipline:

Lambda-catastrophe

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C12.
• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Dynamics handoff: DYN-C04, DYN-C05, DYN-C07, DYN-C14, DYN-C16.
• Gate-specific discipline: Static offset protection, cosmological history, ordinary gravity, and residual ownership are separate rows.

SG-1

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: CLOSED / DERIVED-GIVEN-SHAPE / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C05, DYN-C08, DYN-C09, DYN-C15.
• Gate-specific discipline: A constitutive-rigidity construction terminal remains possible; derived grading requires RIG-C10 PASS.

SG-10

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Dynamics handoff: DYN-C08, DYN-C09, DYN-C16.
• Gate-specific discipline:

SG-2

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: CLOSED / CERTIFIED-IRREDUCIBLE-CATEGORY-FLOOR / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C02, DYN-C06, DYN-C08, DYN-C09.
• Gate-specific discipline: Closed under a layer-typed construction terminal; Stage-only derivation is explicitly negative.

SG-3

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: DERIVED-GIVEN-E / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C06, DYN-C08, DYN-C09.
• Gate-specific discipline: Unconditional closure requires both BND-A domain and BND-B anomaly/global phases.

SG-4

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: CLOSED / DERIVED-GIVEN-E / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C03, DYN-C06.
• Gate-specific discipline:

SG-5

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11.
• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C05, DYN-C09, DYN-C10, DYN-C11, DYN-C12.
• Gate-specific discipline:

SG-6

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11.
• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Dynamics handoff: DYN-C05, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16.
• Gate-specific discipline:

SG-7

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C03, DYN-C08, DYN-C10, DYN-C16.
• Gate-specific discipline:

SG-8

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11.
• Source terminal: CLOSED / RESCUED / DISSOLVED-WRONG-RULER + DERIVED-GIVEN-13D-ACTOR / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C08, DYN-C09, DYN-C10, DYN-C11, DYN-C12, DYN-C16.
• Gate-specific discipline: Same-ruler projection is load-bearing.

SG-9

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11.
• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C06, DYN-C08, DYN-C10, DYN-C11, DYN-C16.
• Gate-specific discipline:

UQF-10

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11.
• Source terminal: RESOLVED +0 / DERIVED-GIVEN-anchor + CLOSED-NEGATIVE leg.
• Dynamics handoff: DYN-C01, DYN-C02, DYN-C05, DYN-C08, DYN-C09, DYN-C10, DYN-C12, DYN-C16.
• Gate-specific discipline:

UQF-14

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C05.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17.
• Gate-specific discipline:

UQF-3

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16.
• Gate-specific discipline:

UQF-4

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C05.
• Source terminal: CLOSED / DERIVED-GIVEN-13D-SHAPE / GLOBAL-ANOMALY-CONSISTENT / RESOLVED +0.
• Dynamics handoff: DYN-C03, DYN-C06.
• Gate-specific discipline: The BND-B certificate is the decisive open object.

UQF-5A/5B

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C03, DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16.
• Gate-specific discipline:

UQF-5C

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04.
• Source terminal: CLOSED / CERTIFIED-IRREDUCIBLE-FLOOR / RESOLVED +0.
• Dynamics handoff: DYN-C06, DYN-C08, DYN-C12, DYN-C15, DYN-C16.
• Gate-specific discipline:

UQF-7

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C05.
• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Dynamics handoff: DYN-C03, DYN-C06, DYN-C09.
• Gate-specific discipline:

UQF-9

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14.
• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16.
• Gate-specific discipline:

theta-bar-QCD

• Required Granularity rows: GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11, GRN-C12.
• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Dynamics handoff: DYN-C01, DYN-C05, DYN-C06, DYN-C10, DYN-C11, DYN-C14, DYN-C16.
• Gate-specific discipline: Any top-form or relaxation construction must retain anomaly, global-group, and observer-map consistency.

7. Granularity residual and closure

\[ \mathcal R{\rm GRN4}=\left( N{\rm incomplete\ candidates}, N{\rm incomplete\ tests}, N{\rm invalid\ equivalence\ relations}, N{\rm unclassified\ modes}, N{\rm unclassified\ states}, N{\rm unclassified\ deformations}, N{\rm unclassified\ histories}, N{\rm unclassified\ regions}, N{\rm unresolved\ tails}, N_{\rm stale\ branches} \right). \]

A positive uniqueness/selection terminal requires \(\mathcal R_{\rm GRN4}=0\) and exactly one permitted survivor class under the frozen exact/cell logic. External-wall, measured-anchor, negative, and dissolution terminals use their constitutionally permitted scoped residuals but may not relabel missing work as zero.

8. Preservation and claim ceiling

The full BB-GRN-3.3 source is preserved verbatim in Appendix A with SHA-256:

2f7be11cbbeb3fedb0acce9605416ba81e4552a4fb41952547fb29e7954032ec

BB-GRN-4.0 is complete relative to the current thirty-gate corpus and the frozen source set used in this audit. It is not a proof that no future experiment, gate, Actor, support, or record type can reopen Granularity.

Appendix A — Exact archival BB-GRN-3.3 source

--- title: "BB-GRN-3.3 — Operational Equivalence, Finite Completion, and Enumerated Exhaustion" buildingblockid: "BB-GRN-3.3" version: "3.3" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — EVIDENCE-CRITERIA EXTENDED — NOT FROZEN" scope: "Complete owned specification under package v3.3.2" protocolfreeze: false canonicalratification: false packageauthorityversion: "3.3.2" ---

BB-GRN-3.3 — Operational Equivalence, Finite Completion, and Enumerated Exhaustion

0. Purpose

Granularity answers:

Which distinctions correspond to possible physical records, what must be completely specified at the finest admitted physical level, and when may mathematically different candidates be treated as one physical class?

Granularity is a firewall against unpaid continuum ontology. It is not a solvent for finite contradictions.

It may dissolve:

• an exact-point distinction with no possible finite record;
• a choice among continuum interpolations that produce identical finite physics;
• unobservable labels below a declared operational floor.

It may not dissolve:

• an extra zero mode;
• a radion;
• a negative physical eigenvalue;
• an unsolved boundary equation;
• a finite vacuum-energy shift;
• a missing reaction equation;
• a measured contradiction;
• an unclassified candidate.

1. Physical record quotient

Let \(\mathcal R\) be the candidate record space and \(\mathcal T\) the frozen admitted test family.

At operational floor \(\Delta_0\), define

\[ r\sim{\Delta0}r' \]

when every admitted test yields outcome distributions indistinguishable within frozen tolerances.

The physical record space is

\[ \mathcal Q{\Delta0}=\mathcal R/\!\sim{\Delta0}. \]

No candidate selection, prediction, or branch distinction may depend on a label absent from \(\mathcal Q{\Delta0}\).

2. Finite completion packet

At the declared physical floor and cutoff, every candidate must publish:

• complete object inventory;
• legal primitive transitions;
• exact constraints;
• physical measure;
• boundary and fixed-set completion;
• observer-equivalence relation;
• complete zero-mode inventory;
• every physical mode below the cutoff;
• every physical deformation below the cutoff;
• global and topological sectors;
• matching maps;
• background-response signature;
• vacuum-offset response signature;
• unresolved tail obligations.

A formal continuum action does not substitute for this packet.

3. Complete mode inventory

For candidate \(i\), define

\[ \mathcal Si(\Lambda)= \left\{ (\lambdan,\rhon,\chin,bn,\nun,\Pin) \mid 0\le\lambdan\le\Lambda \right\}, \]

where:

• \(\lambda_n\) is the physical spectral value;
• \(\rho_n\) is representation and charge;
• \(\chi_n\) is chirality or parity;
• \(b_n\) is bulk, boundary, or fixed-set support;
• \(\nu_n\) is norm and constraint class;
• \(\Pi_n\) is observer visibility.

Every entry is classified as:

• required physical mode;
• forbidden physical mode;
• exact projection;
• gauge redundancy;
• exact constrained direction;
• gapped stable mode;
• massless intended mode;
• unstable physical mode;
• boundary-localized mode;
• matching-only mode;
• not evaluated.

No unclassified entry is compatible with a complete spectrum claim.

4. Complete deformation inventory

The deformation ledger is separate from the eigenmode ledger because an infinitesimal deformation, an exact modulus, and an unstable fluctuation are different objects.

For every deformation \(\delta z_a\), record:

deformation_record:
  deformation_id:
  sector: metric|volume|radion|bundle|connection|wilson_line|boundary|actor|rulebook|observer
  mathematical_domain:
  gauge_redundancy:
  linearized_equation_status:
  nonlinear_integrability:
  obstruction_status:
  physical_norm:
  hessian_eigenvalue:
  constraint_status:
  cutoff_status:
  observer_signature:
  verdict:

4.1 Mandatory distinctions

Granularity enforces:

\[ \text{structurally absent} \neq \text{gauge redundant} \neq \text{obstructed} \neq \text{exactly constrained} \neq \text{massive} \neq \text{sub-resolution}. \]

No category may be used as a synonym for another.

5. Vacuum shifts are finite records

For a protected-sector constant shift

\[ \mathcal Lm\to\mathcal Lm-c, \]

\(c\) is a finite parameter in the renormalized effective equations.

If the shift changes a local curvature observable, the difference is physical regardless of any continuum interpretation.

Therefore:

\[ \boxed{\text{Granularity cannot dissolve a nonzero vacuum-response operator.}} \]

Granularity may remove dependence on a bare regulator convention only after the complete finite renormalized observable has been shown invariant.

6. Background and vacuum signatures

Each candidate’s operational signature includes:

• existence or nonexistence of a complete background solution;
• external curvature response;
• internal reaction fields where observable through conserved effects;
• ordinary matter response;
• vacuum constant-shift response;
• phase-transition response at the declared cosmological scope;
• residual cosmological value role;
• boundary and global-sector records.

Two candidates cannot be operationally equivalent if one has vacuum-offset protection and the other produces a finite curvature shift inside the admitted test family.

7. Exact and operational equivalence

7.1 Exact physical equivalence

Candidates \(i\) and \(j\) are exactly physically equivalent when an isomorphism preserves:

• physical state space;
• constraint and gauge quotient;
• parent Dynamics;
• observables and amplitudes;
• boundaries and global sectors;
• Scale packet;
• deformation classification;
• background and vacuum response;
• observer map.

7.2 Operational equivalence

Let \(R_i(t)\) be candidate \(i\)’s record for admitted test \(t\). Define

\[ d(i,j)=\sup{t\in\mathcal T}dt(Ri(t),Rj(t)). \]

Then

\[ Gi\sim{\rm phys}G_j \]

only if:

1. all exact discrete invariants agree;
2. all test distances lie within frozen tolerances;
3. every test available to one candidate is represented for the other;
4. background, rigidity, and vacuum-response classes agree at the claimed scope.

Tolerances must be fixed before outputs are loaded.

8. No false mathematical uniqueness

Different mathematical descriptions may represent one physical class when they differ only by:

• coordinates;
• gauge presentation;
• equivalent bundle data;
• unobservable subfloor labels;
• continuum interpolation;
• canonical representative choice.

A simplicity metric may select one representative for exposition. It does not create new physical evidence.

9. Full-tower and tail obligations

A finite enumeration supports a full-tower claim only with:

• an exact theorem;
• a certified asymptotic or positivity bound;
• or a deterministic enumeration through a physical cutoff plus a rigorous tail estimate.

Prohibited promotions include:

• zero mode to full KK tower;
• invariant deformation to all smooth deformations;
• local stability to global stability;
• finite EFT to microscopic completion;
• constant-shift response to complete cosmological history.

10. Candidate exhaustion

A grammar is exhausted through one of:

10.1 Explicit finite roster

Every candidate identifier is listed and completeness is proven under the grammar.

10.2 Bounded generator

A deterministic generator enumerates candidates under finite bounds and quotients duplicates by a frozen canonicalization map.

10.3 Class theorem

An infinite shelf is partitioned into finitely many obstruction or equivalence classes with checked representatives.

“Obvious alternatives” is not an exhaustion proof.

11. Stopping rule

Let

\[ \mathcal G{\rm surv}=\{Gi:\text{all hard constraints pass}\}. \]

Then:

• no survivors: negative closure of the frozen grammar;
• one physical equivalence class: positive scoped closure;
• multiple distinguishable classes: gate remains open;
• exhaustion unproved: gate remains open even if only one tested candidate survives.

12. Negative controls

12.1 Hidden modulus relabeled sub-resolution

Include an exact massless radion but declare its displacement smaller than the measurement floor.

Expected result: fail. A massless field is a physical degree of freedom even when a particular displacement is small.

12.2 Vacuum shift relabeled regulator artifact

Change the renormalized protected-sector action by a finite constant and observe finite curvature response.

Expected result: fail. The response is not dissolved.

12.3 Duplicate-coordinate candidate

Include two coordinate descriptions of one physical branch.

Expected result: equivalence quotient merges them.

12.4 Omitted-rival test

Remove a known grammar-generated candidate.

Expected result: exhaustion certificate fails.

12.5 Isolated solution with negative mode

Use a candidate whose solution moduli space is locally a point but whose Hessian is negative.

Expected result: distinct records OBSTRUCTED/ISOLATED and UNSTABLE; no category collapse.

13. Certificates

13.1 Finite completion certificate

finite_completion_certificate:
  candidate_id:
  floor:
  cutoff:
  object_inventory:
  primitive_moves:
  constraints:
  physical_measure:
  boundaries:
  zero_modes:
  modes_below_cutoff:
  deformation_inventory:
  background_signature:
  vacuum_response_signature:
  tail_bound:
  unresolved_items:
  evidence_hashes:
  verdict:

13.2 Operational equivalence certificate

operational_equivalence_certificate:
  grammar_version:
  test_family_version:
  exact_invariants:
  numerical_distances:
  tolerances:
  rigidity_classes:
  vacuum_response_classes:
  survivor_ids:
  equivalence_classes:
  unresolved_pairs:
  evidence_hashes:

13.3 Exhaustion certificate

exhaustion_certificate:
  grammar_version:
  method:
  bounds_or_classes:
  canonicalization_map:
  generated_count:
  unique_count:
  duplicate_controls:
  omitted_rival_control:
  coverage_proof:
  roster_hash:
  verdict:

14. Falsifiers and reopen triggers

This block fails or reopens if:

• a spectrum or deformation sector is omitted;
• a finite vacuum response is dissolved by Granularity;
• two distinct categories are conflated;
• full-tower scope lacks a theorem or bound;
• tolerances are changed after outputs;
• an exact discrete difference is hidden by tolerance;
• duplicate candidates are counted separately;
• the grammar is not exhausted;
• a known rival is missing;
• one survivor is called unique without coverage proof;
• a future admitted record separates a previously equivalent class.

15. Terminal

BB-GRN-3.3

DELIVERS:
  Finite physical completion, complete mode and deformation ledgers,
  physical equivalence classes, candidate exhaustion, and stopping rules.

FIREWALL:
  Granularity never dissolves finite instability, missing equations,
  or vacuum-energy response.

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 — finite stratified anomaly completion

1. Fixed sets are finite physical records

For an orbifold with \(NF\) connected fixed components, the anomaly ledger has \(NF\) separately addressable records.

Granularity may not identify two fixed sets merely because their contributions cancel after integration.

Define the finite record vector

\[ \mathbf A{\rm fixed} = (\mathcal A1,\ldots,\mathcal A{NF}). \]

The pass record is the zero vector, not merely

\[ \suma\mathcal Aa=0. \]

2. Stratum inventory

Every candidate must enumerate:

• regular bulk strata;
• fixed sets;
• boundaries;
• corners/intersections;
• localized field sectors;
• edge modes;
• inflow channels;
• anomaly lines.

No unenumerated stratum is permitted below the declared cutoff.

3. Exact versus operational equivalence

Anomaly classes, global phases, charge lattices, and parity/domain assignments are exact discrete records. They cannot be hidden inside numerical tolerance.

Two candidates are not operationally equivalent if they differ by a nontrivial anomaly line even when their perturbative spectra agree.

4. Finite fixed-set certificate

stratified_finite_completion:
  candidate_id:
  strata:
    - stratum_id:
      type: bulk|fixed_set|boundary|corner
      isotropy:
      local_fields:
      physical_modes:
      gauge_group:
      anomaly_record:
      inflow_record:
      gluing_record:
  integrated_anomaly:
  local_anomaly_vector:
  exact_global_phase_records:
  omitted_strata_test:
  evidence_hashes:
  verdict:

5. Negative controls

The development suite must include:

1. two fixed-set anomalies equal and opposite, so the integrated anomaly vanishes while each local entry is nonzero;
2. a parity assignment with no zero mode but nonzero localized anomaly;
3. a coordinate/covering duplicate that must be merged;
4. an omitted fixed set that must fail exhaustion;
5. a globally nontrivial determinant line with zero local polynomial.

6. New Granularity firewall

\[ \text{integrated cancellation} \neq \text{local cancellation}, \]

and

\[ \text{zero local polynomial} \neq \text{zero global anomaly}. \]

Block-Specific Standalone Interface — Granularity and Exhaustion

Owned questions

This block alone owns:

• what finite physical records distinguish candidates;
• complete mode, deformation, stratum, and observer inventories;
• exact versus operational equivalence;
• cutoff and tail scope;
• candidate grammar exhaustion;
• stopping rules after survivor quotienting.

Required inputs

• complete candidate manifests;
• all block certificate outputs;
• frozen test family and tolerances;
• candidate roster/generator/class theorem;
• observer map and cutoff.

Required outputs

• finite-completion certificate;
• complete mode and stratum inventories;
• exact-invariant and operational-distance matrices;
• physical equivalence classes;
• exhaustion certificate;
• omitted-rival and duplicate controls.

Non-ownership boundary

Granularity cannot dissolve:

• a finite measured contradiction;
• a physical negative mode;
• a zero mode;
• a finite vacuum-energy response;
• a local or global anomaly;
• a missing finite calculation.

Complete Reconstruction Procedure — Granularity and Exhaustion

1. Inventory reconstruction order

A new agent reconstructing this block must proceed in this order:

1. load the frozen candidate grammar;
2. generate or load the complete candidate roster;
3. canonicalize exact duplicates before physics evaluation;
4. load every candidate’s complete parent manifest;
5. enumerate all regular strata, boundaries, fixed sets, corners, and defect sectors;
6. enumerate all physical fields, gauge redundancies, exact constraints, boundary modes, and observer records;
7. build the complete spectrum through the frozen cutoff;
8. attach the full-tail theorem, certified bound, or explicit scope limit;
9. record every exact discrete invariant;
10. compute operational signatures only after normalizations and observer maps are aligned;
11. form the equivalence relation;
12. prove candidate-grammar coverage;
13. apply the stopping rule.

Skipping canonicalization can falsely create multiple survivor classes. Skipping exact discrete invariants can falsely merge anomalously distinct candidates. Skipping the exhaustion proof can falsely promote “only tested survivor” to “selected survivor.”

2. Required finite tables

The reconstruction must emit at least these tables.

2.1 Candidate table

| Field | Meaning | |---|---| | candidate ID | stable unique identifier | | branch hash | complete parent identity | | grammar production | generator rule or roster row | | duplicate representative | canonical representative if equivalent by exact presentation | | completion level | deepest completed evaluation stage | | hard-failure stage | first hard failure, if any |

2.2 Mode table

Every mode receives:

• sector;
• stratum;
• eigenvalue or exact zero status;
• representation;
• global charge;
• chirality/parity;
• gauge/constraint status;
• observer accessibility;
• classification;
• evidence;
• cutoff/tail ownership.

No mode may be omitted because it appears inconvenient, heavy, boundary-localized, or mixed.

2.3 Exact-record table

Exact records include:

• topology;
• global gauge form;
• charge lattice;
• index;
• anomaly class;
• determinant-line phase;
• parity/domain class;
• number of fixed components;
• quantized flux or inflow coefficient;
• exact constraint rank.

Exact records are compared before numerical distances.

2.4 Operational-distance table

For each admitted test \(t\), record:

\[ dt(Ri,Rj), \qquad \varepsilont, \qquad \text{pass/fail}. \]

The overall distance uses the frozen aggregation rule. Tolerances are never chosen after seeing the candidate separation.

3. Exhaustion proof requirements

An explicit roster must explain why no in-scope item is missing.

A bounded generator must provide:

• finite bounds;
• generation algorithm;
• duplicate canonicalization;
• deterministic seed/version;
• total generated count;
• total unique count;
• omitted-rival negative control.

A class theorem must provide:

• hypotheses;
• partition into obstruction/equivalence classes;
• representative for every class;
• proof that no residual class remains.

The statement “the natural candidates were checked” is never an exhaustion proof.

4. Full-tail discipline

A finite mode list can support only one of:

• an exact all-mode theorem;
• a cutoff claim with rigorous tail bound;
• an explicitly finite operational claim.

The block must reject any promotion from a zero-mode calculation to the full Kaluza–Klein tower, from an invariant sector to all smooth perturbations, or from local records to global equivalence.

5. Final reconstruction test

A fresh agent must be able to remove one known candidate and watch the exhaustion certificate fail; add an unobservable label and watch equivalence remove it; duplicate a coordinate presentation and watch canonicalization merge it; and add a nontrivial anomaly-line phase and watch exact equivalence keep the candidates distinct.

Only after all four controls pass is the Granularity/Exhaustion record reconstructible.

Enumerated exhaustion implementation

Granularity owns the canonical candidate table, but CSE owns the freeze. The two blocks share the same enumeration hash.

The exhaustion certificate must list every canonical candidate explicitly, including early failures. A candidate removed by canonical equivalence remains visible as an alias rather than disappearing from the audit trail.

Required machine counters and identities are defined in QUANTITATIVESTOPPINGANDEXHAUSTIONRULEV33_2.md.

The minimum executable controls are:

1. delete one known canonical row—the certificate must fail;
2. duplicate one coordinate presentation—the canonical count must remain unchanged;
3. add one physically null label—the physical equivalence classes must remain unchanged;
4. add one exact discrete anomaly or charge difference—the candidates must remain distinct;
5. change the generator bound—the branch and enumeration hash must change.

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-C08 | Deterministic spectrum enumerator + mode ledger + tail certificate | A script enumerates every physical eigenmode with eigenvalue below the frozen cutoff, labels representation/stratum/status, and matches an analytic count or certified tail bound. | A below-cutoff sector is omitted, duplicated, or inferred from heat-kernel asymptotics without kernel enumeration. | | GEN-C10 | Enumeration artifact or finiteness theorem + exhaustion certificate | The grammar generator terminates and lists every canonical candidate, or a theorem partitions the grammar into finitely many checked classes; the omitted-rival control fails when a row is removed. | The shelf is described only as ‘natural’ or ‘representative,’ or a known in-grammar candidate is absent. | | SHP-C06 | Cross-sector zero-mode inventory | A merged mode table covers gauge, fermion, scalar, metric, boundary, edge, and topological sectors below cutoff. | An undeclared massless or parametrically light mode remains. | | RIG-C09 | Tower/boundary/tunnelling scope certificate | The claim states the highest KK level, boundary sectors, and nonperturbative channels covered, with a bound or honest OPEN rows. | A finite homogeneous calculation is promoted to full stability. | | BND-C12 | Machine reconstruction + negative-control suite | A script rebuilds the fixed-set ledger and correctly classifies integrated-zero/local-nonzero, wrong-sign inflow, unquantized inflow, global-phase, domain, and gluing controls. | Any control is misclassified or a missing row defaults to zero. | | CLS-C04 | Survivor equivalence-class report | After all hard tests, exactly one operational equivalence class survives under the frozen map and tolerance. | Multiple distinguishable classes survive or equivalence was not computed. | | OBS-C06 | Kernel/image enumeration | The map’s kernel, image, degeneracies, inaccessible sectors, and information loss are explicitly listed through the claimed cutoff. | A null direction is treated as absent physics or an inaccessible sector is silently discarded. | | CLS-C07 | Explicit enumeration + quantitative completion counters | The grammar emits Ngenerated, Ncanonical, Nevaluated, Nfailed, N_survivors, and every canonical candidate ID; completion requires generator termination or a finiteness theorem, zero unclassified candidates, and a passing omitted-rival control. | The process stops because no new rivals were recently proposed or because one tested candidate remains. |

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.

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.

All-30-Gate Shape–Granularity–Dynamics Audit

Interpretation. TRIAD-CAPABILITY-COVERED means the improved triad contains the reusable capability and interface needed for the gate. It does not mean the gate calculation, Boundary/Observer/Scale/Vacuum/Rigidity/TS evidence, or external theorem is complete.

| Gate | Shape additions | Granularity V4 rows | Dynamics V4.2 rows | External dependencies | Post-repair triad finding | |---|---|---|---|---|---| | Black-hole-singularity | SHP-X01, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C05, GRN-C12 | DYN-C07, DYN-C08, DYN-C09, DYN-C15, DYN-C17 | Boundary, Observer, Time Synchronization, Vacuum/Rigidity as applicable | TRIAD-CAPABILITY-COVERED | | Born-rule | SHP-X02, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11 | DYN-C03, DYN-C06, DYN-C07, DYN-C12, DYN-C13 | Observer, Scale, CSE/governance | TRIAD-CAPABILITY-COVERED | | Gap-01 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16 | Boundary, Scale, Observer | TRIAD-CAPABILITY-COVERED | | Gap-02 | SHP-X02, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11 | DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | Boundary, Observer, Scale, external nonperturbative theorem | TRIAD-CAPABILITY-COVERED | | Gap-05-stability | SHP-X01, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C12 | DYN-C03, DYN-C04, DYN-C05, DYN-C07, DYN-C08, DYN-C14, DYN-C16 | Vacuum, Scale, Observer | TRIAD-CAPABILITY-COVERED | | Gap-05-value | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C12 | DYN-C07, DYN-C14 | Vacuum, Observer, Scale | TRIAD-CAPABILITY-COVERED | | Gap-08 | SHP-X01, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C12 | DYN-C07, DYN-C09, DYN-C14, DYN-C15, DYN-C16 | Vacuum, Observer, Time Synchronization | TRIAD-CAPABILITY-COVERED | | Gap-10/BG-10 | SHP-X01, SHP-X02, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11, GRN-C12 | DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C15 | Vacuum, Observer, Scale | TRIAD-CAPABILITY-COVERED | | Gap-11 | SHP-X02, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11, GRN-C12 | DYN-C01, DYN-C07, DYN-C10, DYN-C11, DYN-C13, DYN-C14, DYN-C16 | Observer, Scale, Vacuum/history | TRIAD-CAPABILITY-COVERED | | Gap-13 | SHP-X01, SHP-X02, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C05, GRN-C11 | DYN-C03, DYN-C07, DYN-C08, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17 | Boundary, Observer, Time Synchronization, external quantum gravity | TRIAD-CAPABILITY-COVERED | | Lambda-catastrophe | SHP-X01, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C12 | DYN-C04, DYN-C05, DYN-C07, DYN-C14, DYN-C16 | Vacuum, Observer, Scale | TRIAD-CAPABILITY-COVERED | | SG-1 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C05, DYN-C08, DYN-C09, DYN-C15 | Rigidity, Scale, Observer, CSE | TRIAD-CAPABILITY-COVERED | | SG-10 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C08, DYN-C09, DYN-C16 | Scale, Observer | TRIAD-CAPABILITY-COVERED | | SG-2 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C02, DYN-C06, DYN-C08, DYN-C09 | Boundary, Observer, Interdependence | TRIAD-CAPABILITY-COVERED | | SG-3 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C06, DYN-C08, DYN-C09 | Boundary A/B, AD-1, Observer | TRIAD-CAPABILITY-COVERED | | SG-4 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C03, DYN-C06 | Boundary B, AD-1, Observer | TRIAD-CAPABILITY-COVERED | | SG-5 | SHP-X01, SHP-X02, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11 | DYN-C01, DYN-C05, DYN-C09, DYN-C10, DYN-C11, DYN-C12 | Rigidity, Boundary, Scale, Observer | TRIAD-CAPABILITY-COVERED | | SG-6 | SHP-X02, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11 | DYN-C05, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | Rigidity, Boundary, Scale | TRIAD-CAPABILITY-COVERED | | SG-7 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C03, DYN-C08, DYN-C10, DYN-C16 | Scale, Observer | TRIAD-CAPABILITY-COVERED | | SG-8 | SHP-X02, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11 | DYN-C01, DYN-C08, DYN-C09, DYN-C10, DYN-C11, DYN-C12, DYN-C16 | Scale, Observer | TRIAD-CAPABILITY-COVERED | | SG-9 | SHP-X02, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11 | DYN-C01, DYN-C06, DYN-C08, DYN-C10, DYN-C11, DYN-C16 | Scale, Observer, Boundary | TRIAD-CAPABILITY-COVERED | | UQF-10 | SHP-X02, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C11 | DYN-C01, DYN-C02, DYN-C05, DYN-C08, DYN-C09, DYN-C10, DYN-C12, DYN-C16 | Rigidity, Boundary, Scale, Observer | TRIAD-CAPABILITY-COVERED | | UQF-14 | SHP-X01, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C05 | DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16, DYN-C17 | Time Synchronization, Boundary, Observer, Scale | TRIAD-CAPABILITY-COVERED | | UQF-3 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04 | DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C12, DYN-C13, DYN-C15, DYN-C16 | Boundary, Observer, Scale | TRIAD-CAPABILITY-COVERED | | UQF-4 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C05 | DYN-C03, DYN-C06 | Boundary B, AD-1, Observer | TRIAD-CAPABILITY-COVERED | | UQF-5A/5B | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04 | DYN-C01, DYN-C03, DYN-C07, DYN-C08, DYN-C09, DYN-C12, DYN-C15, DYN-C16 | Boundary, Observer, Scale | TRIAD-CAPABILITY-COVERED | | UQF-5C | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04 | DYN-C06, DYN-C08, DYN-C12, DYN-C15, DYN-C16 | Scale, Observer, external nonperturbative completion | TRIAD-CAPABILITY-COVERED | | UQF-7 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C05 | DYN-C03, DYN-C06, DYN-C09 | Boundary A/B, AD-1 | TRIAD-CAPABILITY-COVERED | | UQF-9 | SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14 | DYN-C01, DYN-C03, DYN-C06, DYN-C08, DYN-C09, DYN-C16 | Boundary, Scale, Observer | TRIAD-CAPABILITY-COVERED | | theta-bar-QCD | SHP-X01, SHP-X03, SHP-X04, SHP-X05 | GRN-C01, GRN-C02, GRN-C03, GRN-C06, GRN-C07, GRN-C08, GRN-C09, GRN-C10, GRN-C13, GRN-C14, GRN-C04, GRN-C11, GRN-C12 | DYN-C01, DYN-C05, DYN-C06, DYN-C10, DYN-C11, DYN-C14, DYN-C16 | Boundary/anomaly, Vacuum, Scale, Observer | TRIAD-CAPABILITY-COVERED |

Gate-by-gate notes

Black-hole-singularity

• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, first-class regional/horizon/interface records.
• Dynamics finding: Dissolving a continuum singularity does not supply a finite causal evolution, horizon/interface flux balance, or record map.
• Execution profile: construct physical gravitational operator and regularized/finite-support kernel at claimed scope; establish causal evolution on the non-continuum interior model; publish horizon/interface flux and Observer-record packet.
• Remaining external dependencies: Boundary, Observer, Time Synchronization, Vacuum/Rigidity as applicable.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Born-rule

• Source terminal: CLOSED / CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, effective/composite/multiparticle state inventory.
• Dynamics finding: A stronger Born derivation requires a positive physical measure and lawful normalized event instrument; the current irreducibility terminal can remain narrower.
• Execution profile: derive positive physical measure and quotient; construct normalized event instrument or identify irreducible operational anchor; run non-CP and normalization destructive controls.
• Remaining external dependencies: Observer, Scale, CSE/governance.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-01

• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: The a6 coefficient depends on the complete operator, domain, measure, boundary data, scheme, and finite-order scope.
• Execution profile: build complete Laplace-type operator and boundary domain; include ghosts/measure and anomaly conventions; calculate finite heat-kernel coefficient at declared order and test scheme/refinement scope.
• Remaining external dependencies: Boundary, Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-02

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, effective/composite/multiparticle state inventory.
• Dynamics 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.
• Execution profile: construct physical Yang–Mills spectral operator; establish positive physical sector and causal dynamics; supply nonperturbative/refinement evidence or issue external-wall terminal.
• Remaining external dependencies: Boundary, Observer, Scale, external nonperturbative theorem.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-05-stability

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records.
• Dynamics finding: Radiative stability requires full measure/stress, solvable reactions, ordinary-source control, time-dependent transitions, and scheme/refinement stability.
• Execution profile: derive full vacuum/reaction stress and radiative equations; simulate finite-time phase transitions; test regulator/scheme/refinement stability.
• Remaining external dependencies: Vacuum, Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-05-value

• Source terminal: MEASURED-ANCHOR / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: The measured value terminal does not require a mechanism. A predictive/history claim does require response and cosmological evolution.
• Execution profile: for measured terminal, verify ordinary response and history compatibility only; for predictive terminal, derive mechanism and cosmological evolution.
• Remaining external dependencies: Vacuum, Observer, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-08

• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records.
• Dynamics finding: Replacing or dissolving inflation still requires a complete causal perturbation/history map from initial data to finite records.
• Execution profile: specify initial state and causal perturbation evolution; derive record map replacing inflationary history; test refinement and uncertainty.
• Remaining external dependencies: Vacuum, Observer, Time Synchronization.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-10/BG-10

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, effective/composite/multiparticle state inventory.
• Dynamics finding: Baryogenesis requires CP-violating interactions, out-of-equilibrium transport, charge/entropy evolution, washout, freeze-out, and causal initial data.
• Execution profile: generate CP-violating interaction hypergraph; derive kinetic/transport equations and charge sources; compute production, washout, freeze-out, and final asymmetry.
• Remaining external dependencies: Vacuum, Observer, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-11

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, effective/composite/multiparticle state inventory.
• Dynamics finding: A dark-matter portal needs complete production/decay interactions, open/thermal evolution, abundance history, and running/matching.
• Execution profile: generate portal/production/decay edges; derive thermal/open-system abundance evolution; run matching and RG stability.
• Remaining external dependencies: Observer, Scale, Vacuum/history.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Gap-13

• Source terminal: CLOSED / PROJECT-DEPENDENCY / CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, first-class regional/horizon/interface records, effective/composite/multiparticle state inventory.
• Dynamics finding: Microstate/Page claims require positive quantum evolution, subsystem channels, causal/horizon flux, refinement scope, and finite records.
• Execution profile: construct positive quantum channel for horizon-separated regions; track Page/information and energy flux; state refinement/UV claim ceiling.
• Remaining external dependencies: Boundary, Observer, Time Synchronization, external quantum gravity.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Lambda-catastrophe

• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records.
• Dynamics finding: A degravitation mechanism must solve full equations, retain ordinary gravity, survive phase transitions, and remain radiatively stable.
• Execution profile: solve complete reaction/background equations; retain ordinary localized gravity; simulate finite vacuum transitions and radiative stability.
• Remaining external dependencies: Vacuum, Observer, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-1

• Source terminal: CLOSED / DERIVED-GIVEN-SHAPE / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics 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.
• Execution profile: for stronger dynamical-realization claim, solve parent background; construct geometric fluctuation operators; prove causal/well-posed evolution at declared scope.
• Remaining external dependencies: Rigidity, Scale, Observer, CSE.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-10

• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: Scope consistency requires the actual operator/tower content and a refinement/tail certificate, not a label saying finite or all modes.
• Execution profile: construct actual tower operator and mode inventory; publish tail theorem or finite scope; test refinement stability.
• Remaining external dependencies: Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-2

• Source terminal: CLOSED / CERTIFIED-IRREDUCIBLE-CATEGORY-FLOOR / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics 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.
• Execution profile: map each gauge owner to one kinetic owner; construct BRST/domain-reduced vector operator; derive physical kernel and prove 8+3+1 with zero extra modes.
• Remaining external dependencies: Boundary, Observer, Interdependence.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-3

• Source terminal: DERIVED-GIVEN-E / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: The chiral index and no-mirror claim require the physical Dirac operator/domain/kernel and quantum gauge consistency, not only a topological count.
• Execution profile: construct Dirac operator on exact parity/domain data; derive chiral kernel and mirror disposition; pass measure/anomaly handoff to AD/BND.
• Remaining external dependencies: Boundary A/B, AD-1, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-4

• Source terminal: CLOSED / DERIVED-GIVEN-E / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: Charges and anomalies require one parent coupling, the fermion/ghost measure, Ward identities, and boundary/global anomaly closure.
• Execution profile: derive charge couplings from parent terms; include fermion/ghost measure; evaluate Ward/anomaly consistency with BND-B.
• Remaining external dependencies: Boundary B, AD-1, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-5

• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, effective/composite/multiparticle state inventory.
• Dynamics finding: EWSB cannot close from Actor identity alone: the parent interaction, solved background, mass/mixing matrices, physical scalar/vector spectrum, and positivity are required.
• Execution profile: generate complete electroweak interaction graph; solve order-parameter background; derive vector/scalar mass and mixing spectra and positivity.
• Remaining external dependencies: Rigidity, Boundary, Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-6

• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, effective/composite/multiparticle state inventory.
• Dynamics finding: Rigidity owns the terminal, but Dynamics must supply the full physical Hessian/operator, quantum physicality, tower scope, and refinement stability.
• Execution profile: construct full mixed physical Hessian/operator; supply positivity and causal evolution; test tower/refinement stability before Rigidity terminal.
• Remaining external dependencies: Rigidity, Boundary, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-7

• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: Threshold unification and proton-scale matching depend on complete interaction content, determinants, tower scope, and scheme/refinement stability.
• Execution profile: enumerate threshold-relevant interactions and determinants; run same-ruler matching; test tower/regulator/scheme stability.
• Remaining external dependencies: Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-8

• Source terminal: CLOSED / RESCUED / DISSOLVED-WRONG-RULER + DERIVED-GIVEN-13D-ACTOR / RESOLVED +0.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, effective/composite/multiparticle state inventory.
• Dynamics finding: Flavor masses and mixings require a complete Yukawa/interaction graph, generated matrices, physical eigenstates, and same-ruler/refinement controls.
• Execution profile: generate complete Yukawa/flavor graph; derive mass and mixing matrices from solved background; test physical eigenstates, positivity, and refinement.
• Remaining external dependencies: Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

SG-9

• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, effective/composite/multiparticle state inventory.
• Dynamics finding: Proton safety is a universal statement over dangerous operators, selection rules, induced vertices, thresholds, and running.
• Execution profile: generate all dangerous baryon-violating operators; prove selection-rule/theorem-zero dispositions; derive induced amplitudes, thresholds, and RG stability.
• Remaining external dependencies: Scale, Observer, Boundary.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-10

• Source terminal: RESOLVED +0 / DERIVED-GIVEN-anchor + CLOSED-NEGATIVE leg.
• Pre-repair triad gaps: effective/composite Actor promotion contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, effective/composite/multiparticle state inventory.
• Dynamics finding: Compactification consistency requires complete mixed operators/interactions, constraint closure, solvable background, physical spectrum, and tower/refinement control.
• Execution profile: construct all mixed compactification operators and interactions; solve background/constraint system; test physical spectrum, positivity, and tower stability.
• Remaining external dependencies: Rigidity, Boundary, Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-14

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, first-class regional/horizon/interface records.
• Dynamics finding: Above-cutoff causality requires a physical quantum channel, well-posed causal support, refinement/tail control, and regional/interface flux accounting.
• Execution profile: construct physical high-energy channel; establish causal initial-boundary evolution with TS; track interface flux and refinement/tail scope.
• Remaining external dependencies: Time Synchronization, Boundary, Observer, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-3

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records.
• Dynamics finding: Reflection positivity is not discharged by formal self-adjointness; the full physical measure, quotient, continuation, poles, and regulator/tail stability are needed.
• Execution profile: derive full physical measure and operator quotient; test reflection positivity/reconstruction and poles; test causal/channel and regulator stability.
• Remaining external dependencies: Boundary, Observer, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-4

• Source terminal: CLOSED / DERIVED-GIVEN-13D-SHAPE / GLOBAL-ANOMALY-CONSISTENT / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class regional/horizon/interface records.
• Dynamics finding: Global anomalies are quantum-measure and Ward/QME obstructions; Boundary B owns local/global relative trivialization.
• Execution profile: derive quantum measure/Ward residuals; supply local/global anomaly data to Boundary B; do not treat classical BRST closure as quantum closure.
• Remaining external dependencies: Boundary B, AD-1, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-5A/5B

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records.
• Dynamics finding: A graviton sector needs the full physical operator, gauge quotient, positive poles/residues, causal evolution, and stable tower/truncation scope.
• Execution profile: construct gauge-reduced graviton operator; derive positive poles/residues and causal propagation; test tower/truncation stability.
• Remaining external dependencies: Boundary, Observer, Scale.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-5C

• Source terminal: CLOSED / CERTIFIED-IRREDUCIBLE-FLOOR / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records.
• Dynamics 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.
• Execution profile: for constructive claim, test nonperturbative/refinement stability and quantum physicality; for irreducible-floor terminal, prove finite evidence cannot be promoted.
• Remaining external dependencies: Scale, Observer, external nonperturbative completion.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-7

• Source terminal: DERIVED-GIVEN-anchor / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class regional/horizon/interface records.
• Dynamics finding: Anomaly descent consumes the chiral operator/domain/kernel and quantum measure, then passes to AD/BND for descent and gluing.
• Execution profile: construct chiral operator/domain/kernel; derive measure contribution; handoff descent and gluing to AD/BND.
• Remaining external dependencies: Boundary A/B, AD-1.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

UQF-9

• Source terminal: CERTIFIED-IRREDUCIBLE / RESOLVED +0.
• Pre-repair triad gaps: Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control.
• Dynamics finding: Seeley–DeWitt/UV calculations require the complete operator, domain, measure, boundary terms, and strict finite-order/refinement scope.
• Execution profile: construct complete heat-kernel operator/domain/measure; calculate finite coefficients; enforce strict finite-order and refinement scope.
• Remaining external dependencies: Boundary, Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

theta-bar-QCD

• Source terminal: DISSOLVED-GIVEN-root / RESOLVED +0.
• Pre-repair triad gaps: derived/state-dependent support contract, Shape↔Dynamics V4.2 version integration, transitive operational-equivalence construction, test-family saturation/omitted-test control, first-class dynamic-history records, effective/composite/multiparticle state inventory.
• Dynamics 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.
• Execution profile: generate topological and anomalous interactions; derive relaxation/transition dynamics for constructive route; test radiative and refinement stability or preserve narrower dissolution route.
• Remaining external dependencies: Boundary/anomaly, Vacuum, Scale, Observer.
• Post-repair result: the triad has no identified unowned reusable capability for this gate; missing gate-scoped calculations remain OPEN until executed.

Constraint and Thought-Experiment Audit

Method

For each proposed completeness claim, construct two candidates that agree on everything the current blocks explicitly require but differ in one physically relevant way. If both can pass, a reusable constraint is missing.

Results

TE-TRIAD-01 — Nontransitive tolerance chain

Choose one scalar record with frozen tolerance \(\varepsilon=1\): \(A=0\), \(B=0.75\), \(C=1.5\). Pairwise closeness gives \(A\sim B\) and \(B\sim C\), but not \(A\sim C\). The old epsilon relation cannot support a quotient. Repair: GRN-C07 canonical record-cell equality plus pairwise-distance diagnostics.

TE-TRIAD-02 — Omitted lawful test

Two candidates agree on every loaded test but differ under one lawful Co-Actor source/response test. Candidate exhaustion alone falsely merges them. Repair: GRN-C08 test-family saturation and omitted-test control.

TE-TRIAD-03 — Same static endpoint, different history

Two mechanisms end with the same vacuum value or abundance, but one passes through a singular or observationally excluded transient. Static records falsely merge them. Repair: GRN-C04, consumed from DYN-C14.

TE-TRIAD-04 — Same global total, different regional flux

Two black-hole/interface candidates conserve the same global charge while one omits horizon flux. Global records falsely merge them. Repair: SHP-X01, DYN-C17, and GRN-C05.

TE-TRIAD-05 — Same elementary spectrum, different composite spectrum

Two candidates share primary Actors and one-particle poles but differ in bound states or microstate sectors. Mode-only completion is insufficient. Repair: SHP-X02 and GRN-C11, sourced by DYN-C09–DYN-C11.

TE-TRIAD-06 — Auxiliary-field overcount

A field redefinition introduces an auxiliary variable with no independent pole. Counting it as a new Actor changes the census without changing physics. Repair: SHP-X02 promotion/nonduplication packet.

TE-TRIAD-07 — State-dependent horizon omitted from Stage

The parent manifold is unchanged, but a solution develops a causal horizon with edge support and flux. Treating only fundamental Stage factors as supports loses gate-relevant records. Repair: SHP-X01 derived-support packet.

TE-TRIAD-08 — Version-skew false PASS

A Shape witness envelope passes DYN-C01–DYN-C08 while omitting positivity, history, causal, or regional rows introduced in V4.2. Repair: SHP-X04 and hash/stale rule SHP-X05.

TE-TRIAD-09 — Unknown treated as zero

An uncomputed mode or test is removed from the table and the residual becomes zero. Repair: GRN-C13; unknown/open is a positive residual, never absence.

TE-TRIAD-10 — Resolution refinement splits a class

Two candidates occupy one coarse record cell but split at a lawful finer resolution. Reusing the old class without a branch record hides the distinction. Repair: GRN-C10 and GRN-C14.

Outcome

The thought experiments identify interface and operational gaps, not a missing fourth Shape layer or a replacement Dynamics object. Dynamics V4.2 already supplies the required physical calculations. Shape requires two narrow execution contracts and V4.2 integration. Granularity requires a maximum-rigor operational upgrade.

Validation Report — granularity.zip

Status: PASS

Canonical authority: 01CORE/BBGRN40MAXRIGORALLGATE_GRANULARITY.md

Canonical SHA-256: 9546a8071ec4855e5ef2b23cf5b5b4b27ab7a090ae4c27a58f3c55d10597e20b

Files inventoried: 15

All required files are present.

Notes

• BB-GRN-3.3 preserved exactly as an archival source.
• Operational quotient uses transitive frozen record-cell signatures; pairwise epsilon closeness is diagnostic only.
• Candidate exhaustion and lawful test-family exhaustion are separate hard obligations.

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

--- title: "BB-GRN-3.3 — Operational Equivalence, Finite Completion, and Enumerated Exhaustion" buildingblockid: "BB-GRN-3.3" version: "3.3" date: "2026-07-24" status: "DEVELOPMENT AUTHORITY — EVIDENCE-CRITERIA EXTENDED — NOT FROZEN" scope: "Complete owned specification under package v3.3.2" protocolfreeze: false canonicalratification: false packageauthorityversion: "3.3.2" ---

BB-GRN-3.3 — Operational Equivalence, Finite Completion, and Enumerated Exhaustion

0. Purpose

Granularity answers:

Which distinctions correspond to possible physical records, what must be completely specified at the finest admitted physical level, and when may mathematically different candidates be treated as one physical class?

Granularity is a firewall against unpaid continuum ontology. It is not a solvent for finite contradictions.

It may dissolve:

• an exact-point distinction with no possible finite record;
• a choice among continuum interpolations that produce identical finite physics;
• unobservable labels below a declared operational floor.

It may not dissolve:

• an extra zero mode;
• a radion;
• a negative physical eigenvalue;
• an unsolved boundary equation;
• a finite vacuum-energy shift;
• a missing reaction equation;
• a measured contradiction;
• an unclassified candidate.

1. Physical record quotient

Let \(\mathcal R\) be the candidate record space and \(\mathcal T\) the frozen admitted test family.

At operational floor \(\Delta_0\), define

\[ r\sim{\Delta0}r' \]

when every admitted test yields outcome distributions indistinguishable within frozen tolerances.

The physical record space is

\[ \mathcal Q{\Delta0}=\mathcal R/\!\sim{\Delta0}. \]

No candidate selection, prediction, or branch distinction may depend on a label absent from \(\mathcal Q{\Delta0}\).

2. Finite completion packet

At the declared physical floor and cutoff, every candidate must publish:

• complete object inventory;
• legal primitive transitions;
• exact constraints;
• physical measure;
• boundary and fixed-set completion;
• observer-equivalence relation;
• complete zero-mode inventory;
• every physical mode below the cutoff;
• every physical deformation below the cutoff;
• global and topological sectors;
• matching maps;
• background-response signature;
• vacuum-offset response signature;
• unresolved tail obligations.

A formal continuum action does not substitute for this packet.

3. Complete mode inventory

For candidate \(i\), define

\[ \mathcal Si(\Lambda)= \left\{ (\lambdan,\rhon,\chin,bn,\nun,\Pin) \mid 0\le\lambdan\le\Lambda \right\}, \]

where:

• \(\lambda_n\) is the physical spectral value;
• \(\rho_n\) is representation and charge;
• \(\chi_n\) is chirality or parity;
• \(b_n\) is bulk, boundary, or fixed-set support;
• \(\nu_n\) is norm and constraint class;
• \(\Pi_n\) is observer visibility.

Every entry is classified as:

• required physical mode;
• forbidden physical mode;
• exact projection;
• gauge redundancy;
• exact constrained direction;
• gapped stable mode;
• massless intended mode;
• unstable physical mode;
• boundary-localized mode;
• matching-only mode;
• not evaluated.

No unclassified entry is compatible with a complete spectrum claim.

4. Complete deformation inventory

The deformation ledger is separate from the eigenmode ledger because an infinitesimal deformation, an exact modulus, and an unstable fluctuation are different objects.

For every deformation \(\delta z_a\), record:

deformation_record:
  deformation_id:
  sector: metric|volume|radion|bundle|connection|wilson_line|boundary|actor|rulebook|observer
  mathematical_domain:
  gauge_redundancy:
  linearized_equation_status:
  nonlinear_integrability:
  obstruction_status:
  physical_norm:
  hessian_eigenvalue:
  constraint_status:
  cutoff_status:
  observer_signature:
  verdict:

4.1 Mandatory distinctions

Granularity enforces:

\[ \text{structurally absent} \neq \text{gauge redundant} \neq \text{obstructed} \neq \text{exactly constrained} \neq \text{massive} \neq \text{sub-resolution}. \]

No category may be used as a synonym for another.

5. Vacuum shifts are finite records

For a protected-sector constant shift

\[ \mathcal Lm\to\mathcal Lm-c, \]

\(c\) is a finite parameter in the renormalized effective equations.

If the shift changes a local curvature observable, the difference is physical regardless of any continuum interpretation.

Therefore:

\[ \boxed{\text{Granularity cannot dissolve a nonzero vacuum-response operator.}} \]

Granularity may remove dependence on a bare regulator convention only after the complete finite renormalized observable has been shown invariant.

6. Background and vacuum signatures

Each candidate’s operational signature includes:

• existence or nonexistence of a complete background solution;
• external curvature response;
• internal reaction fields where observable through conserved effects;
• ordinary matter response;
• vacuum constant-shift response;
• phase-transition response at the declared cosmological scope;
• residual cosmological value role;
• boundary and global-sector records.

Two candidates cannot be operationally equivalent if one has vacuum-offset protection and the other produces a finite curvature shift inside the admitted test family.

7. Exact and operational equivalence

7.1 Exact physical equivalence

Candidates \(i\) and \(j\) are exactly physically equivalent when an isomorphism preserves:

• physical state space;
• constraint and gauge quotient;
• parent Dynamics;
• observables and amplitudes;
• boundaries and global sectors;
• Scale packet;
• deformation classification;
• background and vacuum response;
• observer map.

7.2 Operational equivalence

Let \(R_i(t)\) be candidate \(i\)’s record for admitted test \(t\). Define

\[ d(i,j)=\sup{t\in\mathcal T}dt(Ri(t),Rj(t)). \]

Then

\[ Gi\sim{\rm phys}G_j \]

only if:

1. all exact discrete invariants agree;
2. all test distances lie within frozen tolerances;
3. every test available to one candidate is represented for the other;
4. background, rigidity, and vacuum-response classes agree at the claimed scope.

Tolerances must be fixed before outputs are loaded.

8. No false mathematical uniqueness

Different mathematical descriptions may represent one physical class when they differ only by:

• coordinates;
• gauge presentation;
• equivalent bundle data;
• unobservable subfloor labels;
• continuum interpolation;
• canonical representative choice.

A simplicity metric may select one representative for exposition. It does not create new physical evidence.

9. Full-tower and tail obligations

A finite enumeration supports a full-tower claim only with:

• an exact theorem;
• a certified asymptotic or positivity bound;
• or a deterministic enumeration through a physical cutoff plus a rigorous tail estimate.

Prohibited promotions include:

• zero mode to full KK tower;
• invariant deformation to all smooth deformations;
• local stability to global stability;
• finite EFT to microscopic completion;
• constant-shift response to complete cosmological history.

10. Candidate exhaustion

A grammar is exhausted through one of:

10.1 Explicit finite roster

Every candidate identifier is listed and completeness is proven under the grammar.

10.2 Bounded generator

A deterministic generator enumerates candidates under finite bounds and quotients duplicates by a frozen canonicalization map.

10.3 Class theorem

An infinite shelf is partitioned into finitely many obstruction or equivalence classes with checked representatives.

“Obvious alternatives” is not an exhaustion proof.

11. Stopping rule

Let

\[ \mathcal G{\rm surv}=\{Gi:\text{all hard constraints pass}\}. \]

Then:

• no survivors: negative closure of the frozen grammar;
• one physical equivalence class: positive scoped closure;
• multiple distinguishable classes: gate remains open;
• exhaustion unproved: gate remains open even if only one tested candidate survives.

12. Negative controls

12.1 Hidden modulus relabeled sub-resolution

Include an exact massless radion but declare its displacement smaller than the measurement floor.

Expected result: fail. A massless field is a physical degree of freedom even when a particular displacement is small.

12.2 Vacuum shift relabeled regulator artifact

Change the renormalized protected-sector action by a finite constant and observe finite curvature response.

Expected result: fail. The response is not dissolved.

12.3 Duplicate-coordinate candidate

Include two coordinate descriptions of one physical branch.

Expected result: equivalence quotient merges them.

12.4 Omitted-rival test

Remove a known grammar-generated candidate.

Expected result: exhaustion certificate fails.

12.5 Isolated solution with negative mode

Use a candidate whose solution moduli space is locally a point but whose Hessian is negative.

Expected result: distinct records OBSTRUCTED/ISOLATED and UNSTABLE; no category collapse.

13. Certificates

13.1 Finite completion certificate

finite_completion_certificate:
  candidate_id:
  floor:
  cutoff:
  object_inventory:
  primitive_moves:
  constraints:
  physical_measure:
  boundaries:
  zero_modes:
  modes_below_cutoff:
  deformation_inventory:
  background_signature:
  vacuum_response_signature:
  tail_bound:
  unresolved_items:
  evidence_hashes:
  verdict:

13.2 Operational equivalence certificate

operational_equivalence_certificate:
  grammar_version:
  test_family_version:
  exact_invariants:
  numerical_distances:
  tolerances:
  rigidity_classes:
  vacuum_response_classes:
  survivor_ids:
  equivalence_classes:
  unresolved_pairs:
  evidence_hashes:

13.3 Exhaustion certificate

exhaustion_certificate:
  grammar_version:
  method:
  bounds_or_classes:
  canonicalization_map:
  generated_count:
  unique_count:
  duplicate_controls:
  omitted_rival_control:
  coverage_proof:
  roster_hash:
  verdict:

14. Falsifiers and reopen triggers

This block fails or reopens if:

• a spectrum or deformation sector is omitted;
• a finite vacuum response is dissolved by Granularity;
• two distinct categories are conflated;
• full-tower scope lacks a theorem or bound;
• tolerances are changed after outputs;
• an exact discrete difference is hidden by tolerance;
• duplicate candidates are counted separately;
• the grammar is not exhausted;
• a known rival is missing;
• one survivor is called unique without coverage proof;
• a future admitted record separates a previously equivalent class.

15. Terminal

BB-GRN-3.3

DELIVERS:
  Finite physical completion, complete mode and deformation ledgers,
  physical equivalence classes, candidate exhaustion, and stopping rules.

FIREWALL:
  Granularity never dissolves finite instability, missing equations,
  or vacuum-energy response.

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 — finite stratified anomaly completion

1. Fixed sets are finite physical records

For an orbifold with \(NF\) connected fixed components, the anomaly ledger has \(NF\) separately addressable records.

Granularity may not identify two fixed sets merely because their contributions cancel after integration.

Define the finite record vector

\[ \mathbf A{\rm fixed} = (\mathcal A1,\ldots,\mathcal A{NF}). \]

The pass record is the zero vector, not merely

\[ \suma\mathcal Aa=0. \]

2. Stratum inventory

Every candidate must enumerate:

• regular bulk strata;
• fixed sets;
• boundaries;
• corners/intersections;
• localized field sectors;
• edge modes;
• inflow channels;
• anomaly lines.

No unenumerated stratum is permitted below the declared cutoff.

3. Exact versus operational equivalence

Anomaly classes, global phases, charge lattices, and parity/domain assignments are exact discrete records. They cannot be hidden inside numerical tolerance.

Two candidates are not operationally equivalent if they differ by a nontrivial anomaly line even when their perturbative spectra agree.

4. Finite fixed-set certificate

stratified_finite_completion:
  candidate_id:
  strata:
    - stratum_id:
      type: bulk|fixed_set|boundary|corner
      isotropy:
      local_fields:
      physical_modes:
      gauge_group:
      anomaly_record:
      inflow_record:
      gluing_record:
  integrated_anomaly:
  local_anomaly_vector:
  exact_global_phase_records:
  omitted_strata_test:
  evidence_hashes:
  verdict:

5. Negative controls

The development suite must include:

1. two fixed-set anomalies equal and opposite, so the integrated anomaly vanishes while each local entry is nonzero;
2. a parity assignment with no zero mode but nonzero localized anomaly;
3. a coordinate/covering duplicate that must be merged;
4. an omitted fixed set that must fail exhaustion;
5. a globally nontrivial determinant line with zero local polynomial.

6. New Granularity firewall

\[ \text{integrated cancellation} \neq \text{local cancellation}, \]

and

\[ \text{zero local polynomial} \neq \text{zero global anomaly}. \]

Block-Specific Standalone Interface — Granularity and Exhaustion

Owned questions

This block alone owns:

• what finite physical records distinguish candidates;
• complete mode, deformation, stratum, and observer inventories;
• exact versus operational equivalence;
• cutoff and tail scope;
• candidate grammar exhaustion;
• stopping rules after survivor quotienting.

Required inputs

• complete candidate manifests;
• all block certificate outputs;
• frozen test family and tolerances;
• candidate roster/generator/class theorem;
• observer map and cutoff.

Required outputs

• finite-completion certificate;
• complete mode and stratum inventories;
• exact-invariant and operational-distance matrices;
• physical equivalence classes;
• exhaustion certificate;
• omitted-rival and duplicate controls.

Non-ownership boundary

Granularity cannot dissolve:

• a finite measured contradiction;
• a physical negative mode;
• a zero mode;
• a finite vacuum-energy response;
• a local or global anomaly;
• a missing finite calculation.

Complete Reconstruction Procedure — Granularity and Exhaustion

1. Inventory reconstruction order

A new agent reconstructing this block must proceed in this order:

1. load the frozen candidate grammar;
2. generate or load the complete candidate roster;
3. canonicalize exact duplicates before physics evaluation;
4. load every candidate’s complete parent manifest;
5. enumerate all regular strata, boundaries, fixed sets, corners, and defect sectors;
6. enumerate all physical fields, gauge redundancies, exact constraints, boundary modes, and observer records;
7. build the complete spectrum through the frozen cutoff;
8. attach the full-tail theorem, certified bound, or explicit scope limit;
9. record every exact discrete invariant;
10. compute operational signatures only after normalizations and observer maps are aligned;
11. form the equivalence relation;
12. prove candidate-grammar coverage;
13. apply the stopping rule.

Skipping canonicalization can falsely create multiple survivor classes. Skipping exact discrete invariants can falsely merge anomalously distinct candidates. Skipping the exhaustion proof can falsely promote “only tested survivor” to “selected survivor.”

2. Required finite tables

The reconstruction must emit at least these tables.

2.1 Candidate table

| Field | Meaning | |---|---| | candidate ID | stable unique identifier | | branch hash | complete parent identity | | grammar production | generator rule or roster row | | duplicate representative | canonical representative if equivalent by exact presentation | | completion level | deepest completed evaluation stage | | hard-failure stage | first hard failure, if any |

2.2 Mode table

Every mode receives:

• sector;
• stratum;
• eigenvalue or exact zero status;
• representation;
• global charge;
• chirality/parity;
• gauge/constraint status;
• observer accessibility;
• classification;
• evidence;
• cutoff/tail ownership.

No mode may be omitted because it appears inconvenient, heavy, boundary-localized, or mixed.

2.3 Exact-record table

Exact records include:

• topology;
• global gauge form;
• charge lattice;
• index;
• anomaly class;
• determinant-line phase;
• parity/domain class;
• number of fixed components;
• quantized flux or inflow coefficient;
• exact constraint rank.

Exact records are compared before numerical distances.

2.4 Operational-distance table

For each admitted test \(t\), record:

\[ dt(Ri,Rj), \qquad \varepsilont, \qquad \text{pass/fail}. \]

The overall distance uses the frozen aggregation rule. Tolerances are never chosen after seeing the candidate separation.

3. Exhaustion proof requirements

An explicit roster must explain why no in-scope item is missing.

A bounded generator must provide:

• finite bounds;
• generation algorithm;
• duplicate canonicalization;
• deterministic seed/version;
• total generated count;
• total unique count;
• omitted-rival negative control.

A class theorem must provide:

• hypotheses;
• partition into obstruction/equivalence classes;
• representative for every class;
• proof that no residual class remains.

The statement “the natural candidates were checked” is never an exhaustion proof.

4. Full-tail discipline

A finite mode list can support only one of:

• an exact all-mode theorem;
• a cutoff claim with rigorous tail bound;
• an explicitly finite operational claim.

The block must reject any promotion from a zero-mode calculation to the full Kaluza–Klein tower, from an invariant sector to all smooth perturbations, or from local records to global equivalence.

5. Final reconstruction test

A fresh agent must be able to remove one known candidate and watch the exhaustion certificate fail; add an unobservable label and watch equivalence remove it; duplicate a coordinate presentation and watch canonicalization merge it; and add a nontrivial anomaly-line phase and watch exact equivalence keep the candidates distinct.

Only after all four controls pass is the Granularity/Exhaustion record reconstructible.

Enumerated exhaustion implementation

Granularity owns the canonical candidate table, but CSE owns the freeze. The two blocks share the same enumeration hash.

The exhaustion certificate must list every canonical candidate explicitly, including early failures. A candidate removed by canonical equivalence remains visible as an alias rather than disappearing from the audit trail.

Required machine counters and identities are defined in QUANTITATIVESTOPPINGANDEXHAUSTIONRULEV33_2.md.

The minimum executable controls are:

1. delete one known canonical row—the certificate must fail;
2. duplicate one coordinate presentation—the canonical count must remain unchanged;
3. add one physically null label—the physical equivalence classes must remain unchanged;
4. add one exact discrete anomaly or charge difference—the candidates must remain distinct;
5. change the generator bound—the branch and enumeration hash must change.

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-C08 | Deterministic spectrum enumerator + mode ledger + tail certificate | A script enumerates every physical eigenmode with eigenvalue below the frozen cutoff, labels representation/stratum/status, and matches an analytic count or certified tail bound. | A below-cutoff sector is omitted, duplicated, or inferred from heat-kernel asymptotics without kernel enumeration. | | GEN-C10 | Enumeration artifact or finiteness theorem + exhaustion certificate | The grammar generator terminates and lists every canonical candidate, or a theorem partitions the grammar into finitely many checked classes; the omitted-rival control fails when a row is removed. | The shelf is described only as ‘natural’ or ‘representative,’ or a known in-grammar candidate is absent. | | SHP-C06 | Cross-sector zero-mode inventory | A merged mode table covers gauge, fermion, scalar, metric, boundary, edge, and topological sectors below cutoff. | An undeclared massless or parametrically light mode remains. | | RIG-C09 | Tower/boundary/tunnelling scope certificate | The claim states the highest KK level, boundary sectors, and nonperturbative channels covered, with a bound or honest OPEN rows. | A finite homogeneous calculation is promoted to full stability. | | BND-C12 | Machine reconstruction + negative-control suite | A script rebuilds the fixed-set ledger and correctly classifies integrated-zero/local-nonzero, wrong-sign inflow, unquantized inflow, global-phase, domain, and gluing controls. | Any control is misclassified or a missing row defaults to zero. | | CLS-C04 | Survivor equivalence-class report | After all hard tests, exactly one operational equivalence class survives under the frozen map and tolerance. | Multiple distinguishable classes survive or equivalence was not computed. | | OBS-C06 | Kernel/image enumeration | The map’s kernel, image, degeneracies, inaccessible sectors, and information loss are explicitly listed through the claimed cutoff. | A null direction is treated as absent physics or an inaccessible sector is silently discarded. | | CLS-C07 | Explicit enumeration + quantitative completion counters | The grammar emits Ngenerated, Ncanonical, Nevaluated, Nfailed, N_survivors, and every canonical candidate ID; completion requires generator termination or a finiteness theorem, zero unclassified candidates, and a passing omitted-rival control. | The process stops because no new rivals were recently proposed or because one tested candidate remains. |

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.

Granularity V4.0 — Agent-Ready Gate-Closure Package

Granularity remains one physical layer. It converts certified Shape and Dynamics outputs into finite physical records, exact/operational classes, candidate and test-family exhaustion, and stopping decisions.

BB-GRN-4.0 repairs the nontransitive tolerance-quotient problem, makes lawful test-family saturation independent from candidate exhaustion, and adds histories, regional records, effective/composite sectors, refinement lattices, and Dynamics-V4.2 integration.

Use this package together with shape.zip and dynamics.zip. Granularity may not manufacture missing candidates, operators, measurements, or anomaly calculations, and it may not collapse OPEN or inaccessible evidence to zero.

Start Here — Granularity V4.0

Read in this order:

1. 01CORE/BBGRN40MAXRIGORALLGATE_GRANULARITY.md
2. 02REGISTRIES/GRANULARITYCONSTRAINTREGISTRYV4_0.json
3. 02REGISTRIES/GRANULARITYGATECROSSWALKV4_0.json
4. 03EXECUTION/TRIADINTEGRATION_CONTRACT.md
5. 03EXECUTION/GATEEXECUTION_PROMPT.md

For each gate, build the finite-completion packet, saturate both candidate and test grammars, compare exact invariants first, construct operational classes only through equality of frozen canonical record-cell signatures, run omitted-rival and omitted-test controls, and keep all missing evidence OPEN.

The exact BB-GRN-3.3 source is retained in 06_SOURCES/ for preservation review.