# LIBER EX OPERAE ARITHMETICA

Version: `1.0.0-LIBER-EX-OPERAE-ARITHMETICA`
Status: `DERIVATIONAL MEMORY / AUXILIARY DATASET GUIDE / NON-RUNTIME`
Authority: `SOURCE PROVENANCE / HUMAN CONTROL PLANE`

## I. Source identity

This LIBER flattens the historical OPERAE preserved in the multi-tab Google
Doc `Numeric ontology`.

- Document ID: `1Eb3zAtQ4oZ_F9f4gewRe14qdBIT5TUM4r6It8Z8x5IU`
- Source revision:
  `AIroW34y3EvqwFC9p01FqpsvxJvUu7G-LEBaBfXP5qzqxsuV6Bl0teku2B21UdEyEnizsG1Mj6EsnuCLE-9Dtg`
- Observed tab count: `19`
- Google document location: user-managed; stable identity is the Document ID
  above.
- Machine dataset: `ARITHMETICA_OPERAE_DERIVATION_DATASET.json`

Flattening removes the Google Docs tab topology from the auxiliary dataset. It
does not remove tab identity, original order, historical OPERA identity, or
source revision from individual records.

## II. Derivational sequence

The historical sprint used the following OPERA sequence:

| Phase | Historical OPERA | Derivational role |
|---:|---|---|
| 0 | `CAPUT NUMERORUM` | establish the numeric subject field and admissibility axes |
| 1 | `OPERA A — DECIMAL SYSTEM` | describe representation without identifying it with numeric being |
| 2 | `OPERA Q — NUMERIC SUBJECT` | define the numeric subject under controlled questions |
| 3 | `OPERA S — NUMERI` | formalize numbers `1..10` as subjects |
| 4 | `OPERA E — NUMERI DOSSIERS` | interrogate each number through the E/Q question cycle |
| 5 | `OPERA A — NUMERIC DIGNITIES` | apply dignities while separating intrinsic and representational claims |
| 6 | `OPERA T — PAIRWISE MATRIX` | compute the pairwise relation field |
| 7 | `OPERA T — SYSTEM POSITIONS` | compare each number with the complete bounded system |
| 8 | `NUMERIC RELATIONAL LATTICE` | consolidate the relational-computational lattice |

These historical OPERAE are evidence of derivation. Their source labels such as
`ACTIVE` do not grant current runtime or canonical authority.

## III. Extracted corpora

### Foundational doctrine

- `TENET ONTOLOGIA CARDINALIS`
- `TENET NUMERIC CAUSALITY`
- `TENET NUMERIC DOMAIN BRIDGE`

These tabs supply the distinctions between numeric being, qualitative carrier,
base, notation, relation, and domain-specific translation.

### Intermediate machines

- `CARCER BASE TEN EXPLICIT ONTOLOGY`
- `CARCER QUALITAS PROJECTION`
- `CARCER NUMERIC LADDER`
- `POPULUS NUMERIC LADDER`

Their historically named classes are normalized by function:

- invariant content routes to TENET;
- topology and planes route to ROTAS;
- admissibility content routes to AREPO;
- execution phases route to OPERA;
- presentation requirements route to SATOR.

No `CARCER` or `POPULUS` class is introduced into the active five-class order.

### Result corpora

- ten per-number S/E/A dossiers;
- forty-five unordered pair analyses and their T matrix;
- scale-family comparisons for tens, hundreds, and thousands;
- repeated-digit and successive-pair families;
- ascending-edge and reversed-edge comparison;
- expression/identity comparison for `10 + 2 = 12`;
- three generativity/reduction machines.

## IV. Generalized machine

The historical artifacts instantiate a common process:

```text
LOAD
  -> BIND DOMAIN AND RULE SET
  -> ENUMERATE CANDIDATE STATES OR RELATIONS
  -> APPLY ADMISSIBILITY AND TRUTH GATES
  -> REDUCE THE STATE SPACE
  -> TRAVERSE OR DERIVE
  -> PROJECT QUANTITATIVE AND QUALITATIVE RESULTS
  -> RETURN TRACE, REJECTIONS, AND LIMITS
```

The content of a rule is extensible. Its declaration is not optional. Every
rule requires identity, version, domain, arity, direction, preconditions,
transition semantics, output type, termination conditions, and provenance.

## V. Historical fixtures

The dataset retains the following bounded fixtures:

- additive generation: `1 -> all members of 1..10`;
- multiplicative generation:
  `2 -> 4,6,8,10`, `3 -> 6,9`, `5 -> 10`;
- subtractive reduction: every larger member reaches admitted lower members;
- divisive reduction:
  `8 -> 4 -> 2 -> 1`, `9 -> 3 -> 1`,
  `10 -> {5,2} -> 1`, `6 -> {3,2} -> 1`;
- power generation: `2 -> {4,8}`, `3 -> 9`, `1 -> 1`;
- radical return: `4 -> 2`, `9 -> 3`, `1 -> 1`;
- typed hyperedges including `{2,3} -> 6` and `{2,5} -> 10`;
- constitutive spine `1 -> 2 -> ... -> 10`.

These are validation fixtures, not a closed list of operators.

## VI. Authority boundary

This LIBER and its dataset:

- preserve historical execution memory;
- support tests, calibration, regression, and explanation;
- may seed a declared ARITHMETICA state space;
- do not admit a request;
- do not execute an OPERA;
- do not define GRAPHUS topology by themselves;
- do not acquire authority from the source document's historical labels.

The active five-class artifacts govern every proposed execution.