# ROTAS ARITHMETICA — Structural Instantiation

Version: `2.1.0-ROTAS-ARITHMETICA-CARDINAL-ROTAE`
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`;
- `ARITHMETICA_CARDINAL_LAYER_REGISTRY.json`;
- `CARDINAL_ROTAE/ROTA_ARITHMETICA_001.json` through
  `CARDINAL_ROTAE/ROTA_ARITHMETICA_010.json`.

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.

### I.A Plural registry and native composite

`ROTAS ARITHMETICA` indexes several distinct structural families:

```text
CARDINAL_GRAPH_ROTAS:
    ROTA-ARITHMETICA-001
    ROTA-ARITHMETICA-002
    ROTA-ARITHMETICA-003
    ROTA-ARITHMETICA-004
    ROTA-ARITHMETICA-005
    ROTA-ARITHMETICA-006
    ROTA-ARITHMETICA-007
    ROTA-ARITHMETICA-008
    ROTA-ARITHMETICA-009
    ROTA-ARITHMETICA-010

DECLARED_RELATION_OVERLAYS
DERIVATION_AND_HYPEREDGE_ROTAS
GENERATED_STATE_SPACE_ROTAS
MULTI_OPERA_ORDINAL_ROUTES
```

Each cardinal ROTA is individually addressable under `CARDINAL_ROTAE/`. The
machine-readable records are generated reproducibly by
`GENERATE_CARDINAL_ROTAE.ps1` from the cardinal-layer registry.

The current `ROTA-QUANTITAS-ET-QUALITAS` is retained as a composite. Its
complete ten-locus cardinal nucleus is `ROTA-ARITHMETICA-010`, while its
declared semantic overlays, typed derivations, historical fixture paths,
state-space structures, and multi-OPERA routes remain additional layers.

```text
ROTA-QUANTITAS-ET-QUALITAS
    CONTAINS_CARDINAL_NUCLEUS
ROTA-ARITHMETICA-010

COMPOSITE_ROTA != CARDINAL_NUCLEUS
```

No existing identifier is renamed or deleted by this containment record.

### I.B Dedicated cardinal ROTA contract

Every fixed cardinal ROTA declares:

```text
rota_id
version
status
tenet_ref
carrier_registry_ref
cardinal_amplitude
graphus_ref
edge_set_ref
edge_set_authority_status
center_or_anchor_binding
locus_layer_refs[]
difference_layer_refs[]
structural_counts
edge_classification_rule
transition_binding_rule
port_model
door_model
relation_overlays[]
derivation_refs[]
available_adjacencies[]
admitted_traversal_subset[]
provenance
unresolved_fields[]
```

The selected complete topology belongs only to these fixed local fixtures. It
does not assert that every generic Graphus is complete and does not canonize a
global `GRAPHUS n` family or edge set. Every local graph and edge-set reference
remains `CANDIDATE / NON-RUNTIME / NO_AUTHORITY_EFFECT`.

### I.C Construction and cumulative layer laws

For amplitude `n`:

```text
locus layers      = 1...n
difference layers = 1...(n-1)

V(n) = n
E(n) = n(n-1)/2
T(n) = n(n-1)
P(n) = n(n-1)
D(n) = 2n(n-1)
```

These formulas apply to the declared complete-cardinal fixtures only. The
highest layer of ROTA `n` is active as a locus quality but cannot occur as an
internal nonzero difference.

`ROTA-ARITHMETICA-001` is a declared degenerate anchor-only instance: one
locus, no edge, transition, port, door, or traversal event. Its return-complete
posture is a zero-length structural return, not a self-loop or an executed
traversal.

Under complete fixture topology:

```text
ROTA(n+1) adds:
    1 locus
    n undirected edges
    2n directed transitions
    2n geometric ports
    4n logical doors
```

Layer `n` receives its first nonzero transition-level realization in ROTA
`n+1`.

### I.D Structural identity and arithmetic classification

```text
GRAPH_NODE_ID != ARITHMETICA_CARRIER_ID
GRAPH_EDGE_ID != DIFFERENCE_CARRIER
GRAPH_TRANSITION_ID != SIGNED_DISPLACEMENT
GEOMETRIC_PORT_ID != LOGICAL_DOOR_ID
TRANSITION_ID != TRAVERSAL_EVENT_ID
```

An undirected edge is classified by absolute difference:

```text
edge_difference({i,j}) = |j-i|
edge_carrier({i,j}) = TENET.carrier_id(|j-i|)
```

Multiple edges may lawfully share one difference carrier.

```text
SAME_CARRIER != SAME_EDGE
DIFFERENCE_CLASS != DECLARED_RELATION_TOKEN
```

Thus the existing tokens `MEDIATES`, `STABILIZES`, `DISTURBS`, `CLOSES`, and
`TRANSITIONS` remain declared semantic overlays. They are not arithmetic
difference carriers.

### I.E Directed transition binding

Every transition `i->j` records:

```text
signed_displacement = j-i
operator_layer = abs(j-i)
source locus quality -> agent
operator-layer correlate -> act
target locus quality -> patient
sign -> ascending or descending orientation
```

Positive and negative orientations both retain agent, act, and patient.
Reversing a transition swaps the agent-side and patient-side loci, retains the
absolute operator class, and reverses orientation.

The active and passive linguistic renderings are presentation views. Their
exact grammar remains unresolved and cannot mutate the structural binding.

### I.F Port, door, and event model

```text
GEOMETRIC PORT:
    one edge-node incidence

LOGICAL DOOR:
    one transition-node-role incidence
```

Every undirected edge has two geometric ports. Every directed transition has
one source/output agent-side door and one target/input patient-side door. The
opposed transitions are distinct even where their projections share edge
geometry.

Availability does not create an event:

```text
SAME_LOCUS != SAME_EVENT
SAME_EDGE != SAME_EVENT
SAME_TRANSITION != SAME_EVENT
CONNECTION != TRAVERSAL
```

Traversal events receive separate identities only after lawful admission and
actual traversal.

### I.G ROTA 4 regression fixture

`CARDINAL_ROTAE/ROTA_ARITHMETICA_004.json` is the first complete
machine-readable regression fixture. It contains:

```text
4 loci
6 undirected edges
12 directed transitions
12 geometric ports
12 logical output doors
12 logical input doors
24 logical doors total
0 initial traversal events
```

Its difference multiplicities are:

```text
d=1: 3 edges / 6 transitions
d=2: 2 edges / 4 transitions
d=3: 1 edge  / 2 transitions
```

The old edge-enumerating `4B...4G` proposal is rejected as an arithmetic carrier
scheme. Unique structural IDs remain independent of repeated carrier classes.

### I.H Non-collapse among structural families

```text
GRAPHUS_n != ROTA_ARITHMETICA_n
ROTA_ARITHMETICA_n != NOTA_ARITHMETICA_n
FIXED_CARDINAL_ROTA != GENERATED_STATE_SPACE_ROTA
CARDINAL_GRAPH_ROTA != MULTI_OPERA_ORDINAL_ROUTE
CARDINAL_GRAPH_EDGE != ARITHMETIC_DERIVATION
```

No NOTA is created by this registry.

## 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 composite `ROTA-QUANTITAS-ET-QUALITAS`, whose cardinal nucleus is
`ROTA-ARITHMETICA-010`, remains the default bounded historical 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`
- `rotas_registry_rota_instance_collapse`
- `cardinal_rota_as_generic_graphus_ontology`
- `difference_carrier_as_edge_identity`
- `declared_relation_as_difference_class`
- `source_target_as_correlative_without_binding`
- `sign_as_agent_patient_selector`
- `geometric_port_logical_door_collapse`
- `transition_availability_as_event`
- `cardinal_rota_as_generated_state_space`
- `cardinal_rota_as_multi_opera_route`
- `degenerate_anchor_as_self_loop_execution`
- `display_glyph_as_machine_identifier`

## 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 indexes distinct ROTAE without collapsing their structural families.

It exposes possible traversal.

It does not grant traversal.