16_STACK_ARITHMETICA/AREPO_ARITHMETICA

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# AREPO ARITHMETICA — Input Admissibility line 1
Dependencies: line 8
or coordinate the requested work. line 17
## I. Admission packet line 19
AREPO_ARITHMETICA_PACKET: line 22
Native operations: line 37
operation identities. line 45
Native layers: line 47
Output modes: line 55
## II. Global gates line 61
can be evaluated structurally in tests without simulating an OPERA result. line 81
## III. OPERA QUALITAS EX QUANTITAS line 83
Admissible carriers: line 85
Required normalization: line 91
4. check whether a canonical quality exists. line 96
Possible decisions: line 98
## IV. OPERA QUANTITAS EX QUALITAS line 105
Admissible carriers: line 107
No synonym table is currently canonized. Therefore only exact canonical line 113
Possible decisions: line 116
## V. OPERA QUANTITAS ET QUALITAS line 123
Requires: line 125
The gate must block any attempt to use qualitative fit as evidence that an line 133
equation is true. line 134
Admission to either directional OPERA does not admit the coupled OPERA. line 136
convergence policy. line 138
## VI. Route admission line 140
AREPO distinguishes: line 142
AVAILABLE_EDGE line 145
REQUESTED_EDGE line 146
ADMITTED_EDGE line 147
EXECUTED_EDGE line 148
## VII. Cross-stack and graph admission line 156
A future graph or coordinator request must declare: line 158
CROSS_STACK_ARITHMETICA_ADMISSION: line 161
Reading does not imply traversal. Traversal does not imply adaptation. line 177
Adaptation does not imply promotion. line 178
## VIII. Multi-OPERA plan admission line 180
MULTI_OPERA_PLAN: line 185
Requirements: line 198
## IX. Aggregate decisions line 207
The most restrictive unresolved gate governs execution. line 216
## X. Generative rule packet line 218
Every generative or reductive request MUST include: line 220
ARITHMETICA_GENERATION_PACKET: line 223
An arbitrary rule is admissible only as a completely declared rule. “Apply any line 253
useful relation” is not admissible. line 254
## XI. Generative gates line 256
## XII. Reduction admission line 273
AREPO admits each reduction stage independently: line 275
CANDIDATE line 278
The gate rejects: line 286
## XIII. Direction-specific admission line 295
## XIV. GRAPHUS admission line 312
AREPO distinguishes: line 314
RULE_VALID line 317
TRANSITION_MATERIALIZED line 318
GRAPH_EDGE_AVAILABLE line 319
TRAVERSAL_ADMITTED line 320
OPERA_EXECUTED line 321
No later state implies an earlier one by visual or naming resemblance. A line 324
GRAPHUS instance generated from ARITHMETICA remains candidate/non-runtime while line 325
either provider remains candidate/non-runtime. line 326
## XV. Extended decisions line 328
## XVI. Multi-OPERA stabilization admission line 340
It validates: line 346
The following are failures: line 356
## XVII. Final law line 366
AREPO distinguishes a meaningful arithmetic request from a merely suggestive line 368
representation. line 369
No admission means no execution. line 371

# AREPO ARITHMETICA — Input Admissibility Version: `2.1.0-AREPO-ARITHMETICA-SEPARATED-OPERA-ADMISSION` Status: `STRUCTURALLY COMPLETE CANDIDATE / NON-RUNTIME` Authority: `USER / HUMAN CONTROL PLANE` Class: `AGLA / AREPO / ARITHMETICA` Dependencies: - root AREPO class law; - `TENET_ARITHMETICA_DoctrinalInvariants.md`; - `ROTAS_ARITHMETICA_StructuralInstantiation.md`. Position: `TENET → ROTAS → AREPO → OPERA → SATOR`. AREPO admits or blocks a typed request. It does not calculate, traverse, render, or coordinate the requested work. ## I. Admission packet ```text AREPO_ARITHMETICA_PACKET: request_id: requested_operation: payload: declared_layer: declared_base: carrier_type: source_expression: requested_route: output_mode: cross_stack_target: multi_opera_plan: evidence_and_provenance: ``` Native operations: - `OPERA-QUALITAS-EX-QUANTITAS` - `OPERA-QUANTITAS-EX-QUALITAS` - `OPERA-QUANTITAS-ET-QUALITAS` The compressed-source names `OPERA-EX-QUANTITAS` and `OPERA-EX-QUALITAS` are provenance aliases only and are not admissible operation identities. Native layers: - `intrinsic` - `base_manifest` - `positional` - `qualitative` - `mixed` Output modes: - `minimal` - `explicit` - `full_trace` ## II. Global gates | Gate | Requirement | Failure | |---|---|---| | `G01-OPERATION` | one registered native OPERA | `operation_missing_or_unknown` | | `G02-PAYLOAD` | payload exists and matches operation | `payload_missing_or_malformed` | | `G03-LAYER` | layer explicitly declared | `layer_missing` | | `G04-BASE` | base declared for base-manifest or positional input | `base_missing_when_required` | | `G05-CARRIER` | integer, declared numeric representation, canonical quality, or admitted mixed record | `invalid_carrier` | | `G06-INTENTIO` | numeric and qualitative layers remain distinct | `number_quality_identity_collapse` | | `G07-CAUSALITY` | quality is not used as ontological cause | `quality_as_ontological_cause` | | `G08-NOTATION` | positional string is decoded, not treated as ontology | `positional_string_as_ontology` | | `G09-ROTA` | requested route exists and declares provenance | `route_not_instantiated` | | `G10-ARITHMETIC` | arithmetic statement is parseable and truth-checkable | `arithmetic_statement_unverifiable` | | `G11-RANGE` | qualitative lookup stays in registered `1..10` or returns no mapping | `mapping_out_of_register` | | `G12-RELATION` | relation is declared or marked provisional | `relation_provenance_missing` | | `G13-OUTPUT` | output mode is registered | `output_mode_invalid` | | `G14-AUTHORITY` | stack and requested OPERA have current runtime authority | `runtime_authority_absent` | During integration, `G14-AUTHORITY` blocks actual execution. The remaining gates can be evaluated structurally in tests without simulating an OPERA result. ## III. OPERA QUALITAS EX QUANTITAS Admissible carriers: - intrinsic integer; - base-manifest numeric record; - positional numeric record with explicit base. Required normalization: 1. derive intrinsic integer; 2. preserve original representation; 3. distinguish decoded number from notation; 4. check whether a canonical quality exists. Possible decisions: - `ADMIT_QUALITAS_EX_QUANTITAS` - `ADMIT_NO_REGISTERED_QUALITY` - `REQUEST_BASE` - `BLOCK_INVALID_NUMERIC_CARRIER` ## IV. OPERA QUANTITAS EX QUALITAS Admissible carriers: - exact canonical quality; - declared synonym resolvable through a versioned synonym table; - compound quality explicitly decomposed before lookup. No synonym table is currently canonized. Therefore only exact canonical qualities are stable; all other normalization requires a registered extension. Possible decisions: - `ADMIT_QUANTITAS_EX_QUALITAS` - `REQUEST_QUALITY_NORMALIZATION` - `BLOCK_UNKNOWN_QUALITY` - `BLOCK_COMPOUND_QUALITY_COLLAPSE` ## V. OPERA QUANTITAS ET QUALITAS Requires: - valid numeric layer; - valid or explicitly absent quality mapping; - arithmetically valid operation; - declared ROTA relation or derivation; - no contradiction between numeric, qualitative, and logical layers. The gate must block any attempt to use qualitative fit as evidence that an equation is true. Admission to either directional OPERA does not admit the coupled OPERA. `OPERA-QUANTITAS-ET-QUALITAS` requires its own decision and a bounded convergence policy. ## VI. Route admission AREPO distinguishes: ```text AVAILABLE_EDGE REQUESTED_EDGE ADMITTED_EDGE EXECUTED_EDGE ``` Complete adjacency establishes only `AVAILABLE_EDGE`. For a `DERIVATION`, AREPO also requires the arithmetic operator, operands, result, and truth status. ## VII. Cross-stack and graph admission A future graph or coordinator request must declare: ```text CROSS_STACK_ARITHMETICA_ADMISSION: source_stack: ARITHMETICA source_artifact: source_class: source_version: target_stack: target_artifact: intended_use: read | traverse | parameterize | coordinate selected_loci: selected_relations: selected_derivations: adaptation_required: adaptation_authority: provenance_retention: ``` Reading does not imply traversal. Traversal does not imply adaptation. Adaptation does not imply promotion. ## VIII. Multi-OPERA plan admission AREPO may admit a plan as structured routing input, not execute it. ```text MULTI_OPERA_PLAN: plan_id: objective: opera_nodes: dependency_edges: data_handoffs: entry_conditions: per_opera_arepo_decisions: failure_policy: closure_condition: sator_mediation: ``` Requirements: - every OPERA node resolves to a registered artifact; - dependencies are acyclic within the execution context; - each OPERA receives its own AREPO decision; - data handoffs are typed and provenance-preserving; - SATOR is the mediation layer, not an ordinary OPERA node; - plan admission does not authorize any blocked constituent OPERA. ## IX. Aggregate decisions - `ADMIT` - `ADMIT_WITH_WARNINGS` - `ADMIT_PROVISIONAL_RELATION` - `REQUEST_REMEDIATION` - `DEFER_CROSS_STACK` - `BLOCK` The most restrictive unresolved gate governs execution. ## X. Generative rule packet Every generative or reductive request MUST include: ```text ARITHMETICA_GENERATION_PACKET: request_id direction: QUANTITATIVE_TO_QUALITATIVE | QUALITATIVE_TO_QUANTITATIVE | BIDIRECTIONAL domain: state_schema: seed_states[] rule_set: - rule_id version source_ref arity domain codomain preconditions transition_or_derivation truth_or_validity_check constraints[] equivalence_policy reduction_policy objective termination_policy resource_budget projection_policy requested_rota requested_graphus_instance output_mode ``` An arbitrary rule is admissible only as a completely declared rule. “Apply any useful relation” is not admissible. ## XI. Generative gates | Gate | Requirement | Failure | |---|---|---| | `GG01-DOMAIN` | typed and bounded domain, or explicit lazy bound | `generation_domain_unbounded` | | `GG02-STATE` | versioned state schema | `state_schema_missing` | | `GG03-RULE` | every rule fully declared | `rule_declaration_incomplete` | | `GG04-SEED` | valid seed state or seed generator | `seed_state_missing` | | `GG05-TRUTH` | quantitative checks independent from qualitative fit | `truth_check_missing` | | `GG06-EQUIVALENCE` | deduplication relation declared | `equivalence_policy_missing` | | `GG07-PRUNING` | every pruning stage supplies reason codes | `opaque_state_pruning` | | `GG08-TERMINATION` | goal, fixed point, depth, time, count, or human stop | `termination_policy_missing` | | `GG09-BUDGET` | resource bounds and over-budget posture | `resource_budget_missing` | | `GG10-DIRECTION` | projection direction explicit | `projection_direction_missing` | | `GG11-INVERSE` | reverse uniqueness not presumed | `inverse_uniqueness_unproved` | | `GG12-GRAPHUS` | generated graph binds TENET ARITHMETICA capability | `graphus_foundation_missing` | ## XII. Reduction admission AREPO admits each reduction stage independently: ```text CANDIDATE -> RETAIN -> REJECT_WITH_REASON -> DEFER_UNRESOLVED -> PRUNE_DOMINATED -> STOP_BY_POLICY ``` The gate rejects: - silent beam truncation; - hidden model ranking; - deduplication without an equivalence relation; - removal solely because a result is inconvenient; - qualitative preference used to falsify quantitative validity; - post-hoc rule changes without a new request version. ## XIII. Direction-specific admission `QUANTITATIVE_TO_QUALITATIVE` requires: - quantitatively valid states or relations; - a declared projection vocabulary; - provenance and confidence for every qualitative elaboration; - explicit multiplicity when several elaborations remain admissible. `QUALITATIVE_TO_QUANTITATIVE` requires: - typed qualitative constraints; - a compilation rule from qualitative terms to quantitative predicates; - a search domain; - disclosure of zero, one, or many candidates; - no claim of inverse uniqueness without proof. ## XIV. GRAPHUS admission AREPO distinguishes: ```text RULE_VALID TRANSITION_MATERIALIZED GRAPH_EDGE_AVAILABLE TRAVERSAL_ADMITTED OPERA_EXECUTED ``` No later state implies an earlier one by visual or naming resemblance. A GRAPHUS instance generated from ARITHMETICA remains candidate/non-runtime while either provider remains candidate/non-runtime. ## XV. Extended decisions - `ADMIT_GENERATION` - `ADMIT_GENERATION_WITH_BOUNDS` - `ADMIT_REDUCTION_STAGE` - `ADMIT_QUALITATIVE_PROJECTION` - `ADMIT_QUANTITATIVE_SEARCH` - `DEFER_UNBOUNDED_SPACE` - `BLOCK_OPAQUE_RULE` - `BLOCK_OPAQUE_PRUNING` - `BLOCK_GRAPHUS_FOUNDATION` ## XVI. Multi-OPERA stabilization admission If conceptual stabilization is explicitly requested, AREPO requires a plan capable of filling the four modes `IDENTIFICATIO`, `DIFFERENTIATIO`, `MEDIATIO`, and `STABILIZATIO`. It validates: - the ordered OPERA identities; - one effective ordinal mode per admitted step; - compatibility between OPERA capability and assigned mode; - typed handoff and state delta; - independent admission for every OPERA; - a stop decision after stabilization; - a nonredundancy proof for any fifth or later step. The following are failures: - `stabilization_plan_shorter_than_four_without_equivalent_closure`; - `opera_identity_used_as_fixed_ordinal_mode`; - `multi_opera_order_changed_without_readmission`; - `continuation_without_state_delta`; - `unused_opera_as_continuation_reason`; - `amplification_as_repetition`; - `prospective_heuristic_as_execution_authority`. ## XVII. Final law AREPO distinguishes a meaningful arithmetic request from a merely suggestive representation. No admission means no execution.

16_STACK_ARITHMETICA/AREPO_ARITHMETICA_InputAdmissibility.md