# TENET ARITHMETICA — Doctrinal Invariants Version: `2.2.0-TENET-ARITHMETICA-CARDINAL-LAYERS` 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; - `ARITHMETICA_CARDINAL_LAYER_REGISTRY.json`. TENET states the invariants of ARITHMETICA. It does not admit input, instantiate traversal, execute arithmetic, coordinate OPERAE, or mediate output. This artifact defines one bounded cardinal regime articulated in ten layers. Each layer has an explicit numeric value and a stable, script-independent carrier identity. Glyph bindings, PRE/I..IX structural indices, graph objects, and ordinal execution positions remain distinct projections. ## 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 | Carrier ID | Phoenician candidate | Hebrew fallback | Index | Canonical quality | Structural posture | |---:|---|:---:|:---:|---|---|---| | 1 | `ARITH-CARRIER-001` | `𐤀` | `א` | `PRE` | `identity` | non-relational anchor | | 2 | `ARITH-CARRIER-002` | `𐤁` | `ב` | `I` | `distinction` | relational field | | 3 | `ARITH-CARRIER-003` | `𐤂` | `ג` | `II` | `mediation` | relational field | | 4 | `ARITH-CARRIER-004` | `𐤃` | `ד` | `III` | `stabilization` | relational field | | 5 | `ARITH-CARRIER-005` | `𐤄` | `ה` | `IV` | `asymmetry` | relational field | | 6 | `ARITH-CARRIER-006` | `𐤅` | `ו` | `V` | `synthesis` | relational field | | 7 | `ARITH-CARRIER-007` | `𐤆` | `ז` | `VI` | `rupture` | relational field | | 8 | `ARITH-CARRIER-008` | `𐤇` | `ח` | `VII` | `amplification` | relational field | | 9 | `ARITH-CARRIER-009` | `𐤈` | `ט` | `VIII` | `closure` | relational field | | 10 | `ARITH-CARRIER-010` | `𐤉` | `י` | `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. ### III.A Cardinal layer and carrier contract The normative support record is `ARITHMETICA_CARDINAL_LAYER_REGISTRY.json`. Every layer exposes: ```text ARITHMETICA_CARDINAL_LAYER: cardinal_value carrier_id script_bindings numeric_value_authority structural_index canonical_quality structural_posture prima_intentio_projection secunda_intentio_projection correlatives: agent act patient complete_correlational_expression cross_regime_refs[] lexical_status provenance unresolved_fields[] ``` The numeric value and stable ASCII carrier ID are machine-authoritative. A glyph is a reversible view and is never the primary key. ```text CARRIER_ID != NUMBER CARRIER_ID != DISPLAY_GLYPH CARRIER_ID != STRUCTURAL_INDEX CARRIER_ID != GRAPH_ID CARRIER_ID != OPERA_ID PHOENICIAN_GLYPH != HEBREW_GLYPH ALPHABETIC_CARRIER != PHOENICIAN_NUMERAL_CHARACTER 1_TO_10_FORM_ONE_LAYERED_CARDINAL_REGIME PRE_AND_I_TO_IX_REMAIN_STRUCTURAL_INDICES ``` The earlier Latin `A..I,K` mapping is retained as superseded design provenance. It is not the current carrier identity. In the provisional Semitic surface, cardinal 10 uses Yod; Kaph belongs to candidate value 20 and must not be substituted for Yod. ### III.B Complete correlative law Each cardinal layer exposes three simultaneous named roles: ```text AGENT + ACT + PATIENT ``` They form one complete arithmetic principle. They are not alternative meanings selected by the sign of a transition. The first four functional English glosses are fixed for this candidate: ```text 1 / identity: AGENT IDENTIFYING PATIENT 2 / distinction: AGENT DIFFERENTIATING PATIENT 3 / mediation: AGENT MEDIATING PATIENT 4 / stabilization: AGENT STABILIZING PATIENT ``` These are functional English glosses, not attested or canonized Latin forms. Layers `5..10` retain their registered qualities and the complete three-role shape, but their exact correlational lexemes remain `UNRESOLVED`. ```text SIGN != AGENT_OR_PATIENT_SELECTOR POSITIVE != AGENT NEGATIVE != PATIENT ``` ### III.C Prima and secunda projections ```text LAYER n UNDER PRIMA_INTENTIO: numeric being and canonical quality posture LAYER n UNDER SECUNDA_INTENTIO: relational intelligibility through agent-act-patient correlation ``` For layer 1, the prima-intentio posture remains the non-relational numerical anchor. Its secunda-intentio projection may nevertheless expose the complete identificative correlation. ```text NON_RELATIONAL_NUMERIC_POSTURE != INCAPACITY_FOR_RELATIONAL_PROJECTION ``` ### III.D Reversible script-binding policy The current display policy is provisional and reversible: ```text preferred candidate view: Phoenician letter technical fallback only: Square Hebrew letter diagnostic fallback: ASCII name or explicit numeric value ``` The Hebrew glyph is never a second carrier, a coequal notation, a semantic join key, or evidence for Kabbalistic meaning. ```text PHOENICIAN_ARITHMETICA != HEBREW_KABBALAH HEBREW_FALLBACK != KABBALAH_BINDING GLYPH_FALLBACK != SEMANTIC_PROJECTION ``` Rendering may select a fallback without changing carrier ID, numeric value, quality, graph binding, or event identity. Right-to-left presentation and bidi isolation belong to the view layer; semantic storage preserves explicit code points and contains no invisible bidi controls. Glyphs are forbidden as sole machine identifiers and in canonical filenames. The alphabetic Phoenician carrier is an AGLA-designed alphanumeric convention. It must not be described as a separately encoded Phoenician numeral character or as an attested complete Phoenician numeral system. ### III.E Provisional decimal transposition `ARITHMETICA_CARDINAL_LAYER_REGISTRY.json` records a candidate relation among different carriers at different decimal scales. Every record remains `PROVISIONAL_NOT_ADMITTED`. ```text DECIMAL_TRANSPOSITION != SAME_GLYPH_WITH_NEW_PLACE_VALUE SOURCE_CARRIER != TARGET_CARRIER ``` The initial evidence surface includes `1→10`, `2→20`, `3→30`, `4→40`, and `10→100`. Only carrier 10 is inside the current bounded `1..10` registry as a target. Carriers 20, 30, 40, and 100 remain explicitly unregistered. Values 500..900, thousands, and composite syntax are unresolved. ## 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. Dedicated cardinal ROTAE consume this explicit specialization contract: ```text ARITHMETICA_GRAPH_APPLICATION: graphus_ref cardinal_amplitude locus_layer_refs[] difference_layer_refs[] source_agent_binding transition_act_binding target_patient_binding orientation_rule provenance ``` For a local complete-cardinal fixture: ```text locus i -> ARITHMETICA_LAYER(i).quality edge {i,j} -> ARITHMETICA_LAYER(|j-i|) transition i->j: source -> agent layer |j-i| -> act target -> patient sign(j-i) -> arithmetic orientation ``` This is an ARITHMETICA specialization. It does not redefine generic Graphus ontology, canonize a `GRAPHUS n` family, or grant traversal. ```text SIGNED_DISPLACEMENT(i->j) = j-i positive = ascending arithmetic orientation negative = descending arithmetic orientation ``` Source and target become agent-side and patient-side projections only through this declared binding: ```text SOURCE != AGENT TARGET != PATIENT ARITHMETICA_BINDING(SOURCE) -> AGENT ARITHMETICA_BINDING(TARGET) -> PATIENT ``` The user-authorized semantic concordance is retained without notational or artifact collapse: ```text T:B -> Difference ARITHMETICA:cardinal 2 -> distinction/difference ``` The provisional relation label is `SHARED_PRINCIPIAL_ROOT`. Its provisional vocabulary status does not weaken the fixed non-collapse: ```text SHARED_PRINCIPIAL_ROOT != ARTIFACT_IDENTITY ``` ## 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` - `carrier_id_as_number_identity` - `display_glyph_as_carrier_identity` - `carrier_as_structural_index` - `cardinal_layer_as_ordinal_position` - `correlatives_reduced_to_alternatives` - `transition_sign_as_agent_patient_selector` - `graph_source_as_agent_without_binding` - `arithmetic_binding_as_graph_ontology` - `shared_semantic_root_as_artifact_identity` - `unresolved_latin_form_presented_as_attested` - `cardinal_quality_as_graph_transition_type` - `phoenician_letter_as_historical_numeral_sign` - `hebrew_fallback_as_second_carrier` - `rtl_rendering_as_storage_order` ## 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 := ``` 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. The table is an ordinal projection of cardinal qualities, not an identity between the two axes: ```text CARDINAL_LAYER n != ORDINAL_EXECUTION_POSITION n ORDINAL_EXECUTION_POSITION n MAY RECEIVE CARDINAL_QUALITY n AS A LOCAL MODE ``` A cardinal-2 transition operator may occur at any execution ordinal. The second OPERA may receive `DIFFERENTIATIO` without becoming carrier 2. ## 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.