16_STACK_ARITHMETICA/TENET_ARITHMETICA

Page Sections

# TENET ARITHMETICA — Doctrinal Invariants line 1
Dependencies: line 9
## I. Intentiones line 17
relation, and traversal become intelligible. line 22
Mandatory distinctions: line 24
articulate number. None creates number as being. line 38
## II. Numeric causality line 40
The unit is the non-relational numerical anchor: line 42
Multiplication operates over additive numerical being. It orders or groups what line 51
account. line 53
A base makes grouping or position visible. A notation expresses a number. A line 55
positional string must be decoded before its intrinsic number is known. line 56
## III. Ontologia cardinalis line 58
ARITHMETICA adopts this bounded carrier vocabulary: line 60
NUMBER_CARRIES_QUALITY line 78
NUMBER_DOES_NOT_EQUAL_QUALITY line 79
QUALITY_IS_BASE_NEUTRAL line 80
1_IS_PRE_NON_RELATIONAL line 81
2_TO_10_FORM_I_TO_IX_RELATIONAL_FIELD line 82
by analogy. line 87
## IV. Operation invariants line 89
ARITHMETICA recognizes three native operation identities: line 91
mixed OPERA does not silently replace either directional OPERA. line 101
Arithmetic truth is checked in the numeric layer before qualitative exposition. line 103
A useful qualitative reading cannot validate a false equation. line 104
the current separated artifacts. line 108
## V. Relation discipline line 110
Relations may be: line 112
future graph does not establish causality or priority. line 121
## VI. Graph and multi-OPERA preparation line 126
ARITHMETICA exports a typed carrier vocabulary for future graph generation and line 128
multi-OPERA coordination. It does not itself canonize a graph-generation law. line 129
Mandatory interface distinctions: line 131
It must preserve provenance and the five-class order. line 144
## VII. Failure vocabulary line 146
## VIII. Regulated generation law line 160
the first bounded fixture and canonical regression surface. line 163
The general object is: line 165
Rules MAY be arbitrary in content. They MUST NOT be arbitrary in declaration. line 173
Every rule exposes: line 174
ARITHMETICA_RULE: line 177
result is not a rule. line 196
## IX. State-space doctrine line 198
ARITHMETICA_STATE: line 203
domain and state schema. Controlled generation constructs a traceable subset: line 216
unvisited states are ontologically impossible. line 230
Every removed candidate requires a reason category: line 232
## X. Quantitative and qualitative directions line 244
ARITHMETICA supports two non-collapsible directions: line 246
EX_QUANTITAS: line 249
EX_QUALITAS: line 253
The reverse direction is constraint compilation and search. It is not presumed line 259
coincidence proves one qualitative interpretation. line 264
## XI. Foundation of GRAPHUS COMMUNIS line 266
TENET ARITHMETICA provides the foundational capability by which GRAPHUS line 268
COMMUNIS may represent: line 269
This dependency does not collapse the classes: line 279
GRAPHUS owns the declared graph object and instance topology. TENET line 289
Source-declared static topology may be recorded before a quantitative state line 293
## XII. Derivational substrate line 297
test and calibration substrate. They do not execute and do not override this line 301
TENET. line 302
## XIII. Extended failures line 304
## XIV. Multi-OPERA ordinal modes line 316
TENET ARITHMETICA assigns a qualitative mode to each ordinal position in an line 318
the OPERA identity. line 320
tests or disturbs an achieved stabilization so hidden asymmetry becomes line 336
complete sequence contains four admitted OPERAE: line 340
OPERA_1 / IDENTIFICATIO line 343
The OPERAE may have any registered identities compatible with the objective. line 349
Examples: line 350
ATSQ: line 353
QATS: line 359
No OPERA owns one of these modes intrinsically. line 366
## XV. Prospective continuation heuristic line 368
The assistant or coordinator SHOULD maintain the current ordinal quality even line 370
when the user has not declared a multi-OPERA plan. This is prospective planning line 371
Before proposing continuation it tests: line 374
5. whether stopping is more faithful than redundant generation. line 381
## XVI. Multi-OPERA trace line 395

# TENET ARITHMETICA — Doctrinal Invariants Version: `2.1.0-TENET-ARITHMETICA-SHARED-OPERA-INVARIANTS` Status: `FOUNDATIONAL CANDIDATE / STRUCTURALLY COMPLETE / NON-RUNTIME` Authority: `USER / HUMAN CONTROL PLANE` Class: `AGLA / TENET / ARITHMETICA` Mutation policy: `VERSION-CONTROLLED ONLY` Dependencies: - root TENET class law; - root five-class structural order. TENET states the invariants of ARITHMETICA. It does not admit input, instantiate traversal, execute arithmetic, coordinate OPERAE, or mediate output. ## I. Intentiones `PRIMA_INTENTIO` concerns the being of number. `SECUNDA_INTENTIO` concerns the mode by which number, order, quality, notation, relation, and traversal become intelligible. Mandatory distinctions: ```text PRIMA_INTENTIO != SECUNDA_INTENTIO NUMERUS != QUALITAS CARRIER != CARRIED_QUALITY RELATIO != ENS NOTATIO != NUMERUS BASIS != CAUSA_NUMERI STRUCTURA != ONTOLOGIA PROIECTIO != RES_PROIECTA ``` A relation, notation, base, diagram, graph, or traversal may express or articulate number. None creates number as being. ## II. Numeric causality The unit is the non-relational numerical anchor: ```text UNITAS = 1 NUMERUS_UT_ENS = PLURALITAS_ADDITIVA ADDITIO = CAUSA_CONSTITUENS / PRIMA_INTENTIO MULTIPLICATIO = CAUSA_ORDINANS / SECUNDA_INTENTIO ``` Multiplication operates over additive numerical being. It orders or groups what is numerically constituted; it does not replace addition as the constituting account. A base makes grouping or position visible. A notation expresses a number. A positional string must be decoded before its intrinsic number is known. ## III. Ontologia cardinalis ARITHMETICA adopts this bounded carrier vocabulary: | Number | Index | Canonical quality | Structural posture | |---:|---|---|---| | 1 | `PRE` | `identity` | non-relational anchor | | 2 | `I` | `distinction` | relational field | | 3 | `II` | `mediation` | relational field | | 4 | `III` | `stabilization` | relational field | | 5 | `IV` | `asymmetry` | relational field | | 6 | `V` | `synthesis` | relational field | | 7 | `VI` | `rupture` | relational field | | 8 | `VII` | `amplification` | relational field | | 9 | `VIII` | `closure` | relational field | | 10 | `IX` | `transition` | relational field | The mapping is bidirectional as a controlled vocabulary, not an identity claim. ```text NUMBER_CARRIES_QUALITY NUMBER_DOES_NOT_EQUAL_QUALITY QUALITY_IS_BASE_NEUTRAL 1_IS_PRE_NON_RELATIONAL 2_TO_10_FORM_I_TO_IX_RELATIONAL_FIELD ``` A number outside `1..10` remains a number but has no canonical quality in this bounded register. Absence of a registered quality must be disclosed, not filled by analogy. ## IV. Operation invariants ARITHMETICA recognizes three native operation identities: 1. `OPERA-QUALITAS-EX-QUANTITAS`: admitted quantitative structures to qualitative elaborations; 2. `OPERA-QUANTITAS-EX-QUALITAS`: admitted qualitative requirements to quantitative candidates; 3. `OPERA-QUANTITAS-ET-QUALITAS`: coupled quantitative/qualitative search and articulation. The three are distinct OPERAE. Bidirectionality does not imply identity, and the mixed OPERA does not silently replace either directional OPERA. Arithmetic truth is checked in the numeric layer before qualitative exposition. A useful qualitative reading cannot validate a false equation. The compressed-source identifiers `OPERA-EX-QUANTITAS` and `OPERA-EX-QUALITAS` are retained only as provenance aliases. They do not name the current separated artifacts. ## V. Relation discipline Relations may be: - source-declared; - ROTA-default; - numerically derived; - user-supplied; - provisionally inferred. Their provenance must remain distinct. Co-presence in a wheel, table, prompt, or future graph does not establish causality or priority. `COMPLETE_GRAPH` means structural adjacency is available in the native ROTA. It does not mean every traversal is preferred, admitted, meaningful, or free. ## VI. Graph and multi-OPERA preparation ARITHMETICA exports a typed carrier vocabulary for future graph generation and multi-OPERA coordination. It does not itself canonize a graph-generation law. Mandatory interface distinctions: ```text ARITHMETIC_LOCUS != GRAPH_NODE ROTA_EDGE != EXECUTION_PERMISSION DERIVATION != TRAVERSAL OPERA_TRACE != OPERA TASK_GRAPH != FIVE_CLASS_ORDER SATOR != ORDINARY_OPERA_NODE ``` A later graph stack may consume registered ARITHMETICA loci, qualities, relations, derivations, and traces only through explicit cross-stack admission. It must preserve provenance and the five-class order. ## VII. Failure vocabulary - `intentio_collapse` - `number_quality_identity_collapse` - `quality_as_ontological_cause` - `multiplication_as_constituent_cause` - `base_as_number_cause` - `notation_as_number` - `false_equation_qualitatively_ratified` - `unregistered_quality_invented` - `complete_graph_as_free_traversal` - `graph_projection_as_ontology` - `task_graph_as_class_reordering` ## VIII. Regulated generation law ARITHMETICA is not limited to the fixed `1..10` vocabulary. That vocabulary is the first bounded fixture and canonical regression surface. The general object is: ```text ARITHMETICA_PROBLEM := <DOMAIN, STATE_SCHEMA, RULE_SET, CONSTRAINTS, OBJECTIVE, REDUCTION_POLICY, TERMINATION_POLICY, PROJECTION_POLICY> ``` Rules MAY be arbitrary in content. They MUST NOT be arbitrary in declaration. Every rule exposes: ```text ARITHMETICA_RULE: rule_id version source_ref domain codomain arity direction input_types preconditions transition_or_derivation truth_or_validity_check output_types reversibility cost_or_priority termination_effect ``` An undeclared function, intuition, analogy, model inference, or convenient result is not a rule. ## IX. State-space doctrine A state is a typed possibility, not merely a number: ```text ARITHMETICA_STATE: state_id quantitative_carriers[] qualitative_assignments[] active_plane rule_history[] constraint_status provenance parent_state_refs[] depth ``` The initial space `Ω0` contains every possibility admitted by the declared domain and state schema. Controlled generation constructs a traceable subset: ```text Ω0 -> TYPE_FILTER -> RULE_APPLICABILITY -> CONSTRAINT_FILTER -> EQUIVALENCE_DEDUPLICATION -> GOAL_OR_RELEVANCE_FILTER -> RESOURCE_BOUND -> Ω* ``` `Ω*` is a reduction of the considered state space, not a claim that rejected or unvisited states are ontologically impossible. Every removed candidate requires a reason category: - type-invalid; - rule-inapplicable; - constraint-violating; - equivalent duplicate; - dominated; - out of scope; - over budget; - nonterminating under the declared policy; - unresolved. ## X. Quantitative and qualitative directions ARITHMETICA supports two non-collapsible directions: ```text EX_QUANTITAS: quantitative states or relations -> admitted qualitative elaborations EX_QUALITAS: qualitative requirements or patterns -> quantitative constraints -> candidate quantitative states or relations ``` The reverse direction is constraint compilation and search. It is not presumed to be a mathematical inverse, unique, total, or lossless. `QUANTITAS_ET_QUALITAS` maintains both surfaces and their derivational bridge. No qualitative adequacy proves a quantitative statement, and no quantitative coincidence proves one qualitative interpretation. ## XI. Foundation of GRAPHUS COMMUNIS TENET ARITHMETICA provides the foundational capability by which GRAPHUS COMMUNIS may represent: - typed states as carriers; - rules as candidate transitions or hypertransitions; - quantitative weights, costs, ranks, and bounds; - state deltas; - equivalence and deduplication; - admissible state-space reduction; - derivational provenance. This dependency does not collapse the classes: ```text TENET_ARITHMETICA != GRAPHUS ARITHMETICA_RULE != CANONICAL_GRAPH_EDGE STATE_SPACE != GRAPH_TOPOLOGY RULE_APPLICATION != TRAVERSAL_PERMISSION STATE_REDUCTION != ONTOLOGICAL_DELETION ``` GRAPHUS owns the declared graph object and instance topology. TENET ARITHMETICA owns the invariants governing quantitative states, rule-derived relations, and reduction traces consumed by those instances. Source-declared static topology may be recorded before a quantitative state model is bound. Any generated, scored, pruned, or state-mutating GRAPHUS instance MUST resolve an admitted `ARITHMETIC_STATE_SPACE_CAPABILITY`. ## XII. Derivational substrate `LIBER_EX_OPERAE_ARITHMETICA.md` and `ARITHMETICA_OPERAE_DERIVATION_DATASET.json` preserve the historical OPERAE as test and calibration substrate. They do not execute and do not override this TENET. ## XIII. Extended failures - `rule_content_used_without_rule_declaration` - `state_space_unbounded_without_policy` - `candidate_pruned_without_reason` - `qualitative_projection_as_quantitative_proof` - `reverse_projection_claimed_unique_without_proof` - `state_reduction_as_ontological_deletion` - `arithmetic_rule_as_canonical_graph_edge` - `graph_topology_as_numeric_truth` - `graphus_without_arithmetic_foundation_for_generated_state` ## XIV. Multi-OPERA ordinal modes TENET ARITHMETICA assigns a qualitative mode to each ordinal position in an OPERA history. The mode belongs to the execution position, not permanently to the OPERA identity. | Position | Carrier quality | Multi-OPERA mode | |---:|---|---| | 1 | identity | `IDENTIFICATIO` | | 2 | distinction | `DIFFERENTIATIO` | | 3 | mediation | `MEDIATIO` | | 4 | stabilization | `STABILIZATIO` | | 5 | asymmetry | `DESTABILIZATIO_ASYMMETRIAE` | | 6 | synthesis | `SYNTHESIS` | | 7 | rupture | `RUPTURA` | | 8 | amplification | `AMPLIFICATIO` | | 9 | closure | `CLAUSURA` | | 10 | transition | `TRANSITIO` | `DESTABILIZATIO_ASYMMETRIAE` operationalizes the fifth carrier's asymmetry: it tests or disturbs an achieved stabilization so hidden asymmetry becomes visible. It does not change the registered carrier quality from `asymmetry`. When the user explicitly requests conceptual stabilization, the minimal complete sequence contains four admitted OPERAE: ```text OPERA_1 / IDENTIFICATIO -> OPERA_2 / DIFFERENTIATIO -> OPERA_3 / MEDIATIO -> OPERA_4 / STABILIZATIO ``` The OPERAE may have any registered identities compatible with the objective. Examples: ```text ATSQ: A = IDENTIFICATIO T = DIFFERENTIATIO S = MEDIATIO Q = STABILIZATIO QATS: Q = IDENTIFICATIO A = DIFFERENTIATIO T = MEDIATIO S = STABILIZATIO ``` No OPERA owns one of these modes intrinsically. ## XV. Prospective continuation heuristic The assistant or coordinator SHOULD maintain the current ordinal quality even when the user has not declared a multi-OPERA plan. This is prospective planning state, not authority to execute another OPERA. Before proposing continuation it tests: 1. what state delta the next ordinal mode can produce; 2. whether that delta is already present; 3. whether another OPERA can realize the mode lawfully; 4. whether the next mode clarifies, differentiates, mediates, stabilizes, destabilizes, synthesizes, ruptures, amplifies, closes, or transitions; 5. whether stopping is more faithful than redundant generation. After stabilization, additional OPERAE require explicit benefit: - position 5 exposes asymmetry or destabilizes the current settlement; - position 6 synthesizes what survived; - position 7 introduces a declared rupture; - position 8 amplifies without merely repeating; - position 9 closes; - position 10 transitions to a new scale or cycle. The heuristic MUST reduce degeneration by excess operations. Reachability, available context, or unused OPERAE are not reasons to continue. ## XVI. Multi-OPERA trace ```text ARITHMETICA_OPERA_HISTORY: history_id objective stabilization_requested entries: - ordinal arithmetic_carrier inherited_quality effective_mode opera_ref arepo_decision_ref input_state_ref output_state_ref state_delta redundancy_check continuation_reason current_quality next_mode_candidate stop_reason ``` Every later OPERA receives the history as typed input. It MUST NOT erase or reinterpret earlier ordinal assignments. ## XVII. Shared OPERA request and result invariants Every ARITHMETICA OPERA receives the same typed envelope: ```text OPERA_ARITHMETICA_REQUEST: operation_id request_id arepo_decision_id payload declared_layer declared_base route_id output_mode session_id rule_set_ref state_space_ref ordinal_mode_binding_ref graphus_binding_ref ``` Every OPERA MUST: 1. verify its exact identity and artifact version; 2. verify that AREPO admission names that same OPERA; 3. preserve original carrier, notation, layer, base, and provenance; 4. use only declared rules and admitted transitions; 5. separate quantitative, qualitative, relational, logical, and graph layers; 6. retain accepted, rejected, deferred, and deduplicated branches; 7. return typed result, state delta, limits, and trace to SATOR. These requirements are shared doctrine. An individual OPERA artifact specifies only its native transformation and OPERA-specific failures. ## XVIII. Shared normalization, generation, and trace law Normalization is common: ```text normalize_number(intrinsic, n) -> n normalize_number(base_manifest, carriers, base) -> validate manifest -> derive intrinsic integer normalize_number(positional, digits, base) -> validate every digit against base -> decode positional value ``` Decoded value, original notation, and base remain separate. All three OPERAE instantiate this regulated engine: ```text GENERATIO_ARITHMETICA_REGULATA(problem, admission): bind admitted domain, state schema, seeds, rules, constraints, and budgets initialize accepted, rejected, deferred, deduplicated, and frontier registers while frontier is non-empty and termination policy permits: select a state by the declared policy enumerate applicable declared rules materialize each candidate verify quantitative truth or declared rule validity apply type, constraint, equivalence, dominance, and resource gates retain, reject, defer, or deduplicate with an explicit reason expand only retained states whose expansion remains admitted project retained states in the OPERA's native direction return result, state delta, reduction trace, and unresolved set ``` Every rule adapter conforms to: ```text ARITHMETICA_RULE_ADAPTER: adapter_id operator_or_relation input_arity output_arity parse applicable apply verify canonicalize inverse_or_reverse_search cost provenance ``` Every OPERA returns: ```text ARITHMETICA_GENERATION_TRACE: execution_id operation_id request_id arepo_decision_id problem_version rule_set_version domain_ref seed_state_refs[] generated_state_count accepted_state_refs[] rejected[]: state_ref stage reason evidence_ref deduplicated[]: removed_ref representative_ref equivalence_ref deferred_state_refs[] quantitative_proofs[] qualitative_projection_records[] actual_state_delta termination_reason budget_consumed graphus_binding_ref warnings[] status ``` This is a decision and transformation trace, not hidden chain-of-thought. ## XIX. Shared handoff, ordinal, and GRAPHUS binding Any native OPERA may emit: ```text OPERA_HANDOFF: producer_opera producer_trace typed_result admissible_consumers[] required_revalidation expiry_condition ``` A handoff does not schedule, admit, or execute another OPERA. When a multi-OPERA history participates, each artifact receives the `OPERA_ORDINAL_MODE_BINDING` defined by Sections XIV-XVI. The mode constrains the purpose of that execution without changing its OPERA identity. A zero or equivalent delta triggers redundancy review and cannot justify automatic continuation. When GRAPHUS participates, every OPERA emulates traversal by session self-binding: ```text prior state -> admitted rule or edge -> active constraints and coherence test -> accepted delta -> resulting state as the next binding ``` The trace retains materially relevant rejected alternatives. It neither narrates imaginary movement nor requires a visual skill or invocation-local graph file. Coherence pressure may reduce admitted possibilities; it cannot invent topology, rules, authority, or permission. Shared failures: - `operation_identity_admission_mismatch` - `undeclared_rule_application` - `candidate_branch_untraced` - `pruning_reason_missing` - `termination_budget_ignored` - `qualitative_fit_as_numeric_truth` - `cyclic_derivation_as_progress` - `historical_fixture_as_closed_operator_law` - `zero_delta_as_automatic_continuation` - `coherence_pressure_invents_transition` ## XX. Final law Number is constituted numerically. Quality is carried under a declared mode of intelligibility. Structure exposes order without creating being.

16_STACK_ARITHMETICA/TENET_ARITHMETICA_DoctrinalInvariants.md