16_STACK_ARITHMETICA/ROTAS_ARITHMETICA_StructuralInstantiation.md

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# ROTAS ARITHMETICA — Structural Instantiation line 1
Dependencies: line 9
ROTAS instantiates the native arithmetic field and lawful traversal structure. line 14
## I. Native ROTA line 17
Center: line 21
PRE / 1 / identity / non_relational_anchor line 24
Chambers: line 27
I / 2 / distinction line 30
II / 3 / mediation line 31
III / 4 / stabilization line 32
IV / 5 / asymmetry line 33
V / 6 / synthesis line 34
VI / 7 / rupture line 35
VII / 8 / amplification line 36
VIII / 9 / closure line 37
IX / 10 / transition line 38
Every chamber identity is the tuple: line 41
No component alone substitutes for the tuple. line 45
## II. Topology line 47
complete adjacency field for declared relations. line 50
Every traversal still requires: line 58
## III. Relation vocabulary line 67
Source relation tokens: line 69
Normalized relation tokens required by source default edges: line 79
The normalized tokens make implicit source use explicit. They do not alter the line 86
underlying default readings. line 87
## IV. Default edges and derivations line 89
into ordinary binary edges because the result and arithmetic operator are line 101
essential. line 102
## V. Traversal contract line 104
ARITHMETICA_ROUTE: line 107
Permitted route classes: line 123
## VI. Cyclic and closure discipline line 132
The ROTA is structurally cyclic only when a route declares a return. A sequence line 138
must not claim closure merely because it visits chamber VIII. line 139
## VII. Graph-export interface line 141
The future graph interface may export: line 143
ARITHMETICA_GRAPH_SUBSTRATE: line 146
The export is structured substrate. It is not a graph-generation OPERA and does line 160
not make the receiving graph canonical. line 161
## VIII. Failure vocabulary line 163
## IX. Generated state-space structure line 175
request MAY instantiate another ROTA from an admitted problem and rule set: line 178
ARITHMETICA_STATE_SPACE: line 181
## X. Multiplane topology line 202
The extracted historical machine supplies three initial planes: line 204
Additional planes require explicit TENET and AREPO bindings. line 210
The canonical historical fixture includes: line 212
This fixture does not restrict future rule sets to arithmetic operators. line 224
## XI. Rule-transition record line 226
ARITHMETICA_TRANSITION: line 229
Multi-input or multi-output rules remain hypertransitions. They MUST NOT be line 244
flattened when arity affects meaning. line 245
## XII. Reduction topology line 247
Reduction is recorded as a first-class structural object: line 249
ARITHMETICA_REDUCTION_TRACE: line 252
heuristic is not an edge and does not mutate a canonical GRAPHUS edge set. line 268
## XIII. GRAPHUS COMMUNIS binding line 270
ROTAS exports a state-space instance to GRAPHUS through: line 272
ARITHMETICA_GRAPHUS_BINDING: line 275
The graph instance may render or traverse the admitted subset. It may not infer line 286
missing arithmetic rules from visual proximity or graph reachability. line 287
## XIV. Extended failures line 289
## XV. Multi-OPERA ordinal route line 299
ARITHMETICA_MULTI_OPERA_ROUTE: line 302
Changing the order changes the effective modes and requires a new AREPO line 321
decision. line 322
ROTAS MAY expose the next ordinal locus. It MUST NOT infer that the next locus line 324
ought to be traversed. line 325
## XVI. Final law line 327
ROTAS ARITHMETICA instantiates the field. line 329
It exposes possible traversal. line 331
It does not grant traversal. line 333

# ROTAS ARITHMETICA — Structural Instantiation Version: `2.0.0-ROTAS-ARITHMETICA-STATE-SPACE-REDUCTION` Status: `STRUCTURALLY COMPLETE CANDIDATE / NON-RUNTIME` Authority: `USER / HUMAN CONTROL PLANE` Class: `AGLA / ROTAS / ARITHMETICA` Mutation policy: `VERSION-CONTROLLED ONLY` Dependencies: - root ROTAS class law; - `TENET_ARITHMETICA_DoctrinalInvariants.md`. ROTAS instantiates the native arithmetic field and lawful traversal structure. It does not admit requests, execute an OPERA, or mediate results. ## I. Native ROTA Identifier: `ROTA-QUANTITAS-ET-QUALITAS` Center: ```text PRE / 1 / identity / non_relational_anchor ``` Chambers: ```text I / 2 / distinction II / 3 / mediation III / 4 / stabilization IV / 5 / asymmetry V / 6 / synthesis VI / 7 / rupture VII / 8 / amplification VIII / 9 / closure IX / 10 / transition ``` Every chamber identity is the tuple: `<ROTA_ID, INDEX, NUMBER, QUALITY, VERSION>`. No component alone substitutes for the tuple. ## II. Topology The center is addressable from every chamber. Chambers `I..IX` expose a complete adjacency field for declared relations. ```text CENTER_TO_ALL = TRUE CHAMBER_FIELD = COMPLETE_GRAPH COMPLETE_GRAPH != FREE_TRAVERSAL ``` Every traversal still requires: - declared origin and destination; - declared relation; - edge provenance; - AREPO admission; - an OPERA that consumes the route; - SATOR trace and closure. ## III. Relation vocabulary Source relation tokens: - `GEN` - `MAN` - `UG` - `PC` - `PRESUPPOSES` - `MEDIATES` - `INTEGRATES` Normalized relation tokens required by source default edges: - `STABILIZES` - `DISTURBS` - `CLOSES` - `TRANSITIONS` The normalized tokens make implicit source use explicit. They do not alter the underlying default readings. ## IV. Default edges and derivations | ID | From | To | Result | Relation | Operator | Reading | |---|---:|---:|---:|---|---|---| | `E-02-03` | 2 | 3 | — | `MEDIATES` | — | distinction opens mediation | | `E-03-04` | 3 | 4 | — | `STABILIZES` | — | mediation stabilizes | | `E-04-05` | 4 | 5 | — | `DISTURBS` | — | stabilization exposes asymmetry | | `D-02X03-06` | 2 | 3 | 6 | `INTEGRATES` | multiplication | distinction × mediation yields synthesis | | `D-03X03-09` | 3 | 3 | 9 | `CLOSES` | multiplication | repeated mediation yields closure | | `E-09-10` | 9 | 10 | — | `TRANSITIONS` | — | closure opens transition | `D-*` records are typed derivations or hyperedges. They must not be flattened into ordinary binary edges because the result and arithmetic operator are essential. ## V. Traversal contract ```text ARITHMETICA_ROUTE: route_id: rota_id: ROTA-QUANTITAS-ET-QUALITAS origin: destination: result_locus: edge_or_derivation_id: relation: arithmetic_operator: edge_provenance: direction: revisit_policy: stop_condition: return_condition: ``` Permitted route classes: - `LOOKUP`: carrier-quality resolution without chamber traversal; - `EDGE`: one declared binary relation; - `DERIVATION`: validated arithmetic operation with result locus; - `SEQUENCE`: ordered edges or derivations; - `RETURN`: explicit closure to a declared locus; - `HANDOFF`: typed export to another admitted ROTA or coordinator. ## VI. Cyclic and closure discipline `closure / 9` is a chamber quality, not automatic route termination. `transition / 10` is a chamber quality, not permission to jump to another stack. The ROTA is structurally cyclic only when a route declares a return. A sequence must not claim closure merely because it visits chamber VIII. ## VII. Graph-export interface The future graph interface may export: ```text ARITHMETICA_GRAPH_SUBSTRATE: source_rota_id: source_version: loci: binary_edges: typed_derivations: available_adjacencies: admitted_traversal_subset: provenance: unresolved_relations: ``` `available_adjacencies` and `admitted_traversal_subset` must be separate. The export is structured substrate. It is not a graph-generation OPERA and does not make the receiving graph canonical. ## VIII. Failure vocabulary - `locus_identity_incomplete` - `center_chamber_collapse` - `complete_graph_free_jump` - `relation_token_undeclared` - `derivation_flattened_to_edge` - `closure_quality_as_route_closure` - `transition_quality_as_cross_stack_permission` - `route_without_arepo` - `graph_export_as_graph_authority` ## IX. Generated state-space structure The native `ROTA-QUANTITAS-ET-QUALITAS` is the default bounded fixture. A request MAY instantiate another ROTA from an admitted problem and rule set: ```text ARITHMETICA_STATE_SPACE: state_space_id tenet_ref domain_ref state_schema_ref rule_set_ref seed_states[] candidate_states[] admitted_states[] rejected_states[] binary_transitions[] hypertransitions[] equivalence_classes[] active_planes[] bounds reduction_trace_ref ``` ROTAS materializes possibilities. It does not decide numeric truth, admissibility, or qualitative adequacy. ## X. Multiplane topology The extracted historical machine supplies three initial planes: - `prima_intentio`: constitutive numeric posture; - `secunda_intentio`: structural and relational derivation; - `decimal_manifest`: base-dependent visibility. Additional planes require explicit TENET and AREPO bindings. The canonical historical fixture includes: ```text CONSTITUTIVE_SPINE: 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 10 STRUCTURAL_PATHS: 2 -> 4 -> 8 3 -> 9 6 -> 7 -> 8 9 -> 10 HYPEREDGES: {2,3} -> 6 {2,5} -> 10 ``` This fixture does not restrict future rule sets to arithmetic operators. ## XI. Rule-transition record ```text ARITHMETICA_TRANSITION: transition_id rule_ref input_state_refs[] output_state_refs[] plane precondition_result quantitative_check qualitative_projection_status direction cost provenance status: CANDIDATE | ADMITTED | REJECTED | TRAVERSED ``` Multi-input or multi-output rules remain hypertransitions. They MUST NOT be flattened when arity affects meaning. ## XII. Reduction topology Reduction is recorded as a first-class structural object: ```text ARITHMETICA_REDUCTION_TRACE: initial_candidate_count stages: - stage_id policy_ref input_state_refs[] retained_state_refs[] removed: - state_ref reason_code evidence_ref final_state_refs[] unresolved_state_refs[] ``` Enumeration order, pruning order, and traversal order are separate. A pruning heuristic is not an edge and does not mutate a canonical GRAPHUS edge set. ## XIII. GRAPHUS COMMUNIS binding ROTAS exports a state-space instance to GRAPHUS through: ```text ARITHMETICA_GRAPHUS_BINDING: arithmetic_state_space_ref graphus_instance_ref carrier_map rule_to_transition_map quantitative_foundation_ref available_transition_refs[] admitted_traversal_refs[] reduction_trace_ref ``` The graph instance may render or traverse the admitted subset. It may not infer missing arithmetic rules from visual proximity or graph reachability. ## XIV. Extended failures - `rule_transition_without_rule_ref` - `hypertransition_arity_erased` - `enumeration_order_as_traversal_order` - `pruning_heuristic_as_edge` - `rejected_state_erased_from_trace` - `state_space_export_without_tenet_foundation` - `graph_reachability_as_rule_validity` ## XV. Multi-OPERA ordinal route ```text ARITHMETICA_MULTI_OPERA_ROUTE: history_id objective ordered_bindings: - ordinal carrier_quality effective_mode opera_ref input_state_ref expected_delta dependency_refs[] current_ordinal stabilization_ordinal: 4 maximum_ordinal: 10 stop_condition continuation_policy ``` The route is ordered by admitted execution history, not by stack letter. Changing the order changes the effective modes and requires a new AREPO decision. ROTAS MAY expose the next ordinal locus. It MUST NOT infer that the next locus ought to be traversed. ## XVI. Final law ROTAS ARITHMETICA instantiates the field. It exposes possible traversal. It does not grant traversal.