# STACK ARITHMETICA — Local System Index

Version: `2.2.0-STACK-ARITHMETICA-CARDINAL-ROTAE`
Status: `INTEGRATION / NON-RUNTIME / NOT CANONICAL`
Authority: `USER / HUMAN CONTROL PLANE`

Purpose: provide controlled, traceable generation under admitted rules,
state-space reduction, quantitative/qualitative projection, and the arithmetic
foundation used by GRAPHUS COMMUNIS ARTIS.

## Binding order

1. `TENET_ARITHMETICA_DoctrinalInvariants.md`
2. `ROTAS_ARITHMETICA_StructuralInstantiation.md`
3. `AREPO_ARITHMETICA_InputAdmissibility.md`
4. `OPERA_QUALITAS_EX_QUANTITAS_ExecutionMechanism.md`
5. `OPERA_QUANTITAS_EX_QUALITAS_ExecutionMechanism.md`
6. `OPERA_QUANTITAS_ET_QUALITAS_ExecutionMechanism.md`
7. `SATOR_ARITHMETICA_OutputRequirements.md`
8. `LIBER_EX_OPERAE_ARITHMETICA.md`
9. `ARITHMETICA_OPERAE_DERIVATION_DATASET.json`
10. `ARITHMETICA_SOURCE_REGISTER.md`
11. `TESTS_ARITHMETICA.md`
12. `VALIDATE_ARITHMETICA.ps1`

Cardinal-layer support resolved by TENET and ROTAS:

13. `ARITHMETICA_CARDINAL_LAYER_REGISTRY.json`
14. `GENERATE_CARDINAL_ROTAE.ps1`
15. `CARDINAL_ROTAE/ROTA_ARITHMETICA_001.json`
16. `CARDINAL_ROTAE/ROTA_ARITHMETICA_002.json`
17. `CARDINAL_ROTAE/ROTA_ARITHMETICA_003.json`
18. `CARDINAL_ROTAE/ROTA_ARITHMETICA_004.json`
19. `CARDINAL_ROTAE/ROTA_ARITHMETICA_005.json`
20. `CARDINAL_ROTAE/ROTA_ARITHMETICA_006.json`
21. `CARDINAL_ROTAE/ROTA_ARITHMETICA_007.json`
22. `CARDINAL_ROTAE/ROTA_ARITHMETICA_008.json`
23. `CARDINAL_ROTAE/ROTA_ARITHMETICA_009.json`
24. `CARDINAL_ROTAE/ROTA_ARITHMETICA_010.json`

Archived source substrates:

- `../00_ARCHIVED/INTEGRATED_CANDIDATES/STACK_ARITHMETICA_SUBSTRATES/`
  `ARITHMETICA STACK COMPRESSED.json`.

The Google Doc relocation is user-managed and is not performed by this slice.
Its stable document ID remains registered independently of local pointer
location.

Archived substrate is provenance-only and is not part of operative stack
resolution.

Structural order:

`TENET -> ROTAS -> AREPO -> OPERA -> SATOR`

## Native identities

- composite ROTA: `ROTA-QUANTITAS-ET-QUALITAS`;
- cardinal nucleus of the composite: `ROTA-ARITHMETICA-010`;
- fixed cardinal family: `ROTA-ARITHMETICA-001..010`;
- OPERA: `OPERA-QUALITAS-EX-QUANTITAS`
- OPERA: `OPERA-QUANTITAS-EX-QUALITAS`
- OPERA: `OPERA-QUANTITAS-ET-QUALITAS`

The first two source names without an explicit result class are legacy aliases
only. Each current identity resolves to one artifact.

## Cardinal carrier and ROTA family

`TENET_ARITHMETICA_DoctrinalInvariants.md` defines one bounded `1..10`
cardinal regime. `ARITHMETICA_CARDINAL_LAYER_REGISTRY.json` gives every layer
an explicit numeric value and stable ASCII carrier ID. Its Phoenician surface
is a preferred candidate view; Square Hebrew is
`TECHNICAL_FALLBACK_ONLY`.

```text
CARRIER_ID != DISPLAY_GLYPH
PHOENICIAN_ARITHMETICA != HEBREW_KABBALAH
HEBREW_FALLBACK != KABBALAH_BINDING
```

The ten files under `CARDINAL_ROTAE/` are individually addressable local
complete-cardinal fixtures. Their local `GRAPHUS n` and edge-set references are
provisional, non-runtime, and scoped to the corresponding ROTA. They do not
canonize a generic Graphus family.

`ROTA-ARITHMETICA-001` is an anchor-only degenerate ROTA with a zero-length
structural return and no traversal event. `ROTA-ARITHMETICA-004` is the first
full regression fixture. The generator can reproduce all ten files and check
their byte-equivalence without changing authority.

Every structural node, edge, transition, port, door, and event retains an ID
independent of arithmetic carrier notation. Edges are classified by absolute
cardinal difference; repeated difference carriers do not collapse edge
identity.

## Operative scope

ARITHMETICA accepts a declared rule set, constructs or references a typed state
space, admits only lawful transitions, reduces possibilities traceably, and
returns both result and decision history. It supports:

- quantitative computation followed by qualitative elaboration;
- qualitative constraints followed by quantitative search;
- coupled quantitative/qualitative generation;
- arbitrary contextual rules only after AREPO admission;
- single-OPERA and multi-OPERA histories;
- reusable historical fixtures without treating them as universal laws.

## GRAPHUS COMMUNIS ARTIS binding

`00_TRANSVERSAL_ARTIFACTS/TENET/TENET_GRAPHUS_COMMUNIS_ARTIS.md` consumes TENET
ARITHMETICA as its quantitative foundation. GRAPHUS articulates states,
possibilities, constraints, rejected alternatives, selected paths, deltas, and
stop/continue decisions.

Traversal is emulated through self-binding and coherence pressure. It requires
neither a visual skill nor an invocation-local graph file. Coherence may reduce
the admitted possibility set; it cannot create an edge, rule, authority, or
permission.

SATOR discloses GRAPHUS before interpretive elaboration.

## Multi-OPERA ordinal modes

Sequence positions carry the operative quality:

1. identification;
2. differentiation;
3. mediation;
4. stabilization;
5. destabilization/asymmetry;
6. synthesis;
7. rupture;
8. amplification;
9. closure;
10. transition.

Thus `ATSQ` binds Q to stabilization, while `QATS` binds Q to
identification. A declared desire for conceptual stabilization normally
requires at least four admitted OPERAE unless an equivalent closure is
formally demonstrated. Prospective ordinal reasoning guides, but never
authorizes, continuation.

## Authority boundary

The archived compressed JSON's internal `ACTIVE_RUNTIME_KERNEL` label is not
inherited.
This stack is materially written and structurally testable, but remains:

- non-executable;
- non-deployable;
- non-canonical;
- not mirror-authorized;
- not Sigma-authorized.

No artifact, graph reachability, ordinal heuristic, or coherence result grants
permission to execute an additional OPERA.

The Phoenician/Hebrew display policy, decimal transposition records, generic
`GRAPHUS n` authority, exact correlational Latin, bidi presentation grammar,
and values beyond the bounded `1..10` regime remain unresolved or explicitly
provisional. None is a deployment or Kabbalah bridge.

## Promotion gate

Runtime consideration requires human doctrinal review, full validator pass,
tests beyond the historical fixtures, graph-interface compatibility tests,
explicit authority promotion, and later Sigma generation from accepted textual
artifacts.