16_STACK_ARITHMETICA/SATOR_ARITHMETICA

Page Sections

# SATOR ARITHMETICA — Output Requirements line 1
Dependencies: line 8
perform arithmetic or replace an OPERA. line 19
## I. Session state line 23
ARITHMETICA_SESSION: line 26
State order: line 49
UNINITIALIZED line 52
SATOR cannot advance a blocked or deferred request. line 63
## II. Language discipline line 65
Preferred verbs for carrier-quality statements: line 67
Forbidden identity claims: line 77
The copula is not globally forbidden. It is forbidden only where it collapses line 83
execution. line 85
Forbidden causal claims: line 87
## III. Native forms line 94
EX QUANTITAS: line 96
EX QUALITAS: line 104
QUANTITAS ET QUALITAS: line 112
Every mixed output exposes: line 119
## IV. Output envelope line 127
SATOR_ARITHMETICA_RESULT: line 130
## V. Error mediation line 149
SATOR must distinguish: line 151
It must not replace failure with a plausible-looking result. line 163
## VI. Traversal disclosure line 165
## VII. Multi-OPERA mediation interface line 177
A future multi-OPERA coordinator may supply a plan admitted by AREPO. SATOR then line 179
mediates the plan state without becoming an ordinary operation node. line 180
MULTI_OPERA_MEDIATION: line 183
SATOR may: line 197
SATOR may not: line 205
## VIII. Expiry line 213
A session expires when: line 215
## IX. Generative result envelope line 225
SATOR_ARITHMETICA_GENERATION_RESULT: line 228
## X. Directional mediation line 255
Never render “the number corresponding to this quality” when the search line 275
returned several candidates or when uniqueness was not proved. line 276
reason. Mediation MUST NOT render the coupled OPERA as an implicit execution of line 280
the other two artifacts. line 281
## XI. State-space reduction disclosure line 283
User-facing output SHOULD summarize: line 285
Full-trace mode MUST make every removed candidate recoverable by reference. line 295
## XII. GRAPHUS COMMUNIS mediation line 297
Visualization never replaces these records. A visible edge does not imply line 313
## XIII. Historical dataset disclosure line 316
authority. line 326
## XIV. Extended failures line 328
## XV. Multi-OPERA history mediation line 338
next mode when relevant. It MUST NOT imply that the next OPERA has been selected line 359
or authorized. line 360
SATOR should prefer a justified stop over ornamental completion of all ten line 362
qualities. line 363
## XVI. GRAPHUS-first output order line 365
GRAPHUS_DESCRIPTION: line 370
DECISION_DESCRIPTION: line 378
QUANTITATIVE_RESULT line 385
QUALITATIVE_ELABORATION line 387
SATOR blocks qualitative elaboration when the GRAPHUS or decision description line 390
## XVII. Emulation disclosure line 395
trace to distinguish emulation from narrated simulation: line 398
SATOR MUST NOT imply that a visual skill or invocation-local graph file was line 406
as an optional inspection surface. The GRAPHUS description still precedes line 408
interpretive elaboration. line 409
## XVIII. Final law line 411
SATOR makes arithmetic execution legible. line 413
authority. line 416

# SATOR ARITHMETICA — Output Requirements Version: `2.1.0-SATOR-ARITHMETICA-SEPARATED-OPERA-MEDIATION` Status: `IMPLEMENTATION-READY CANDIDATE / NON-RUNTIME` Authority: `USER / HUMAN CONTROL PLANE` Class: `AGLA / SATOR / ARITHMETICA` Dependencies: - root SATOR class law; - `TENET_ARITHMETICA_DoctrinalInvariants.md`; - `ROTAS_ARITHMETICA_StructuralInstantiation.md`; - `AREPO_ARITHMETICA_InputAdmissibility.md`; - `OPERA_QUALITAS_EX_QUANTITAS_ExecutionMechanism.md`; - `OPERA_QUANTITAS_EX_QUALITAS_ExecutionMechanism.md`; - `OPERA_QUANTITAS_ET_QUALITAS_ExecutionMechanism.md`. SATOR mediates state, disclosure, output, handoffs, and closure. It does not perform arithmetic or replace an OPERA. SATOR is the mediation layer; it is not an ordinary OPERA-sequence node. ## I. Session state ```text ARITHMETICA_SESSION: session_id: stack_version: authority_status: request_id: selected_opera: arepo_decision: selected_rota: declared_layer: declared_base: output_mode: current_locus: visited_loci: numeric_record: qualitative_record: relation_record: logical_record: warnings: handoffs: patch_history: expiry_status: ``` State order: ```text UNINITIALIZED → REQUEST_BOUND → AREPO_PENDING → ADMITTED | BLOCKED | DEFERRED → OPERA_SELECTED → OPERA_RESULT_RECEIVED → LAYERS_RECONCILED → OUTPUT_MEDIATED → HANDOFF_READY | CLOSED | EXPIRED ``` SATOR cannot advance a blocked or deferred request. ## II. Language discipline Preferred verbs for carrier-quality statements: - `carries` - `expresses` - `maps to` - `resolves to` - `is carried by` - `yields` - `renders` Forbidden identity claims: - `{number} is {quality}`; - `{quality} equals {number}`; - a graph, edge, or layout is the represented ontology. The copula is not globally forbidden. It is forbidden only where it collapses number and quality, carrier and role, projection and object, or trace and execution. Forbidden causal claims: - quality creates number; - notation causes number as being; - relation creates its relata; - graph topology proves the object. ## III. Native forms EX QUANTITAS: ```text minimal: {n} → {quality} declarative:{n} carries {quality} explicit: {n} carries {quality} under secunda intentio ``` EX QUALITAS: ```text minimal: {quality} → carrier {n} declarative:{quality} is carried by numeric carrier {n} explicit: {quality} resolves to numeric carrier {n} in the bounded register ``` QUANTITAS ET QUALITAS: ```text {verified_numeric_statement} yields {result}, which carries {quality}. {a} relates to {b} through {relation}, rendering {quality_reading}. ``` Every mixed output exposes: - numeric layer; - qualitative layer; - ROTA or relation layer; - logical layer; - provenance and unresolved elements. ## IV. Output envelope ```text SATOR_ARITHMETICA_RESULT: status: ACCEPTED | REJECTED | DEFERRED | PROVISIONAL operation: result: exposition: numeric_layer: qualitative_layer: rota_layer: logical_layer: trace: warnings: errors: authority_disclosure: handoff: ``` `minimal` may compress exposition, but may not hide a warning, rejection, provisional relation, missing quality, or absent authority. ## V. Error mediation SATOR must distinguish: - invalid input; - missing declaration; - arithmetic falsity; - no registered quality; - route unavailable; - route not admitted; - relation provisional; - runtime authority absent; - cross-stack handoff deferred. It must not replace failure with a plausible-looking result. ## VI. Traversal disclosure When traversal occurs, disclose: - ROTA and version; - origin, destination, and result locus; - edge or derivation ID; - relation and provenance; - visits and revisits; - stop and return condition; - whether adjacency was available but not admitted. ## VII. Multi-OPERA mediation interface A future multi-OPERA coordinator may supply a plan admitted by AREPO. SATOR then mediates the plan state without becoming an ordinary operation node. ```text MULTI_OPERA_MEDIATION: plan_id: objective: admitted_opera_nodes: blocked_opera_nodes: current_frontier: completed_handoffs: failed_handoffs: traversal_trace: synthesis_status: closure_condition: user_disclosure: ``` SATOR may: - expose plan status; - request remediation; - preserve per-OPERA authority and traces; - mediate partial completion; - close or expire the plan. SATOR may not: - perform a constituent OPERA; - make one OPERA's admission authorize another; - hide failed branches; - reorder the five classes; - treat the coordination graph as ontology. ## VIII. Expiry A session expires when: - requested scope changes materially; - layer or base changes; - source or mapping version changes; - route changes outside admitted scope; - a handoff target is unavailable; - an AREPO decision is revoked; - human authority closes the session. ## IX. Generative result envelope ```text SATOR_ARITHMETICA_GENERATION_RESULT: status execution_id direction domain_and_bounds rules_applied[] seed_states[] generated_count retained_states_or_relations[] rejected_summary: by_reason representative_records[] deferred_and_unresolved[] reduction_summary quantitative_results[] qualitative_elaborations[] uniqueness_status termination_reason budget_disclosure graphus_instance_ref source_trace authority_disclosure warnings[] ``` SATOR MUST expose the reduction, not only the surviving answer. ## X. Directional mediation For `OPERA-QUALITAS-EX-QUANTITAS`, distinguish: - quantitative fact; - rule-derived relation; - qualitative projection; - competing projection; - rejected projection; - interpretive limit. For `OPERA-QUANTITAS-EX-QUALITAS`, distinguish: - user-supplied qualitative requirement; - normalized term; - compiled quantitative predicate; - generated candidate; - accepted candidate; - ambiguity or nonexistence. Never render “the number corresponding to this quality” when the search returned several candidates or when uniqueness was not proved. For `OPERA-QUANTITAS-ET-QUALITAS`, distinguish each coupled cycle, the quantitative and qualitative delta within it, convergence status, and stop reason. Mediation MUST NOT render the coupled OPERA as an implicit execution of the other two artifacts. ## XI. State-space reduction disclosure User-facing output SHOULD summarize: - the initial domain and its size or bound; - how many candidates were generated; - which rules expanded the frontier; - which filters removed candidates; - whether deduplication or dominance was used; - why the engine stopped; - what remains unresolved. Full-trace mode MUST make every removed candidate recoverable by reference. ## XII. GRAPHUS COMMUNIS mediation When a GRAPHUS instance is used, SATOR discloses: ```text graphus_class_ref graphus_instance_ref arithmetic_foundation_ref rule_set_ref topology_ref available_transition_count admitted_traversal state_deltas[] reduction_trace_ref ``` Visualization never replaces these records. A visible edge does not imply numeric validity, rule applicability, or execution permission. ## XIII. Historical dataset disclosure When an answer uses a historical fixture, SATOR identifies: - `LIBER_EX_OPERAE_ARITHMETICA.md`; - dataset record or fixture ID; - source Google Doc revision and tab ID; - whether the fixture is evidence, calibration, example, or regression. It MUST NOT repeat the source tab's historical `ACTIVE` status as current authority. ## XIV. Extended failures - `survivor_only_output` - `rejected_state_trace_hidden` - `termination_reason_hidden` - `budget_disclosure_hidden` - `qualitative_ambiguity_hidden` - `graph_visualization_as_proof` - `historical_active_status_propagated` ## XV. Multi-OPERA history mediation SATOR exposes, for every session: ```text opera_history: - ordinal opera_ref inherited_quality effective_mode state_delta admission_status conceptual_stabilization_status current_ordinal_quality next_mode_candidate continuation_benefit redundancy_risk stop_recommendation ``` If the user did not request multiple OPERAE, SATOR MAY explain the prospective next mode when relevant. It MUST NOT imply that the next OPERA has been selected or authorized. SATOR should prefer a justified stop over ornamental completion of all ten qualities. ## XVI. GRAPHUS-first output order When `graphus_instance_ref` is non-null, the output order is mandatory: ```text GRAPHUS_DESCRIPTION: instance carriers topology_or_state_space rule_set available_options constraints DECISION_DESCRIPTION: reduction_stages selected_path rejected_or_deferred_options effective_actions stop_or_continue QUANTITATIVE_RESULT QUALITATIVE_ELABORATION ``` SATOR blocks qualitative elaboration when the GRAPHUS or decision description is absent, incomplete, or inconsistent with the trace. This order is constraint control, not presentation preference. ## XVII. Emulation disclosure When GRAPHUS traversal participates, SATOR exposes enough of the self-binding trace to distinguish emulation from narrated simulation: - prior and resulting state references; - admitted edge or rule; - active constraints and coherence basis; - accepted delta; - materially relevant rejected alternatives. SATOR MUST NOT imply that a visual skill or invocation-local graph file was required. If a visualization or durable graph artifact exists, it is labeled as an optional inspection surface. The GRAPHUS description still precedes interpretive elaboration. ## XVIII. Final law SATOR makes arithmetic execution legible. It preserves the distinction between result, exposition, trace, route, and authority.

16_STACK_ARITHMETICA/SATOR_ARITHMETICA_OutputRequirements.md