# LIBER X — W EX EVACUATIO STRUCTURAL PROOF

Title: LIBER X — W EX EVACUATIO STRUCTURAL PROOF

Version: 1.0.0

Official Title: REGIMEN W — HABITUS MORALES / MORAL HABITS

Status: CANONICAL SUPPORT / STRUCTURAL DERIVATION / NON_INFORMANT / NON_PROVIDER / NON_RUNTIME / RUNTIME_DISABLED

Authority: HUMAN-AUTHORIZED CANONICAL SUPPORT / NON_INFORMANT ONLY

Class: LIBER / W / COMBINATORY STRUCTURAL DERIVATION

Artifact ID: `IDX-LIBER-W-EVACUATIO-PROOF`

Registration: DISCOVERY_NON_PROVIDER / NOT_ADMITTED

Species: LIBER EX / ORDINARY NON-INFORMANT PROOF RECORD

Scope: bounded proof that normalized cross-wheel carrier alignments have the
same cardinality and carrier-pair set as the thirty-six unordered Figura
Tertia cameras

Mode: OBJECT / CANONICAL SUPPORT EVIDENCE; NOT RUNTIME INFORMANT

Source / Origin: received v0.1.0 W structural proof; B–K carrier order;
qualified v0.2.0 ROTAS semantics; v0.2.0 term matrix

Extraction Method: finite symbolic enumeration and cardinality derivation;
no OPERA G, Evacuatio, A, T, S, or W execution performed

Coverage: all nine carriers, seventy-two directed non-self alignments,
exchange normalization, and all thirty-six unordered carrier pairs

Allowed Use: structural comparison; deterministic validator fixture; future
golden evidence for a separately authorized G refactor; OPERA-object analysis

Prohibited Use: semantic generation of W terms; claiming current OPERA G as a
W upstream; runtime execution; source attestation; doctrine creation; AREPO
bypass; provider registration; authority escalation

Mutation Policy: VERSION-CONTROLLED SUCCESSORS ONLY

Depends-On: `../19_STACK_W/ROTAS_W_StructuralInstantiation.md`,
`../19_STACK_W/LIBER_X_W_TERM_MATRIX.json`, and
`../19_STACK_W/W_ROTATION_SEMANTICS_DECISION_RECORD.md`

Trace: supersedes
`_01_INTEGRATION_CANDIDATES/STACK_W/QUALIFICATION/SUCCESSORS/LIBER_X_W_EX_EVACUATIO_STRUCTURAL_PROOF_v0.2.0.md@0.2.0`,
qualified blob `87b5eaf755c2460b3f864060ab9e893b2f11b50b` at integration commit
`58ce4cec5f15a78b0263e85deea5a1e9e7412eed`; source preserved without mutation

Lineage-Origin: `_01_INTEGRATION_CANDIDATES/STACK_W/LIBER_X_W_EX_EVACUATIO_STRUCTURAL_PROOF.md@0.1.0`;
received blob `cc29b1c98de71a8bd3cb7714fda64d10e870ea0c` at intake commit
`65e2df777aed7cbada7041dfc6cbc7007959f68a`

Canonical Effect: CANONICAL SUPPORT / NON_INFORMANT

Runtime Effect: NONE / RUNTIME_DISABLED

## 1. Claim

Let:

```text
Sigma = {B,C,D,E,F,G,H,I,K}
|Sigma| = 9
```

The set of unordered two-carrier cameras has cardinality:

```text
C(9,2) = 9! / (2! 7!) = 36
```

The cross-wheel directed alignments between distinct carriers have
cardinality:

```text
9 source carriers * 8 non-self target carriers = 72
```

Under exchange normalization:

```text
(x,y) ~ (y,x), for x != y
```

each equivalence class has two directed members. Therefore:

```text
72 / 2 = 36
```

The normalized carrier-pair set is thus isomorphic as a finite combinatory
set to the Figura Tertia unordered camera set:

```text
CROSS(+V,-U, non-self) / exchange-equivalence
  ~= CAMERA_SPACE_G
```

This is a carrier-pair/cardinality claim only.

## 2. Complete normalized camera set

```text
BC BD BE BF BG BH BI BK
CD CE CF CG CH CI CK
DE DF DG DH DI DK
EF EG EH EI EK
FG FH FI FK
GH GI GK
HI HK
IK
```

The set contains exactly thirty-six unique pairs, always ordered according to
the declared carrier sequence for display.

## 3. Relation to W rotations

The eight non-zero directed relative rotations account for every non-self
target from each source carrier:

```text
9 carriers * 8 non-zero deltas = 72 directed alignments
```

Exchange normalization forgets direction. It must not be confused with the W
directed `delta mod 9`, the undirected circular distance, semantic identity,
contrariety, or OPERA traversal.

## 4. What the derivation proves

- the B–K domain generates thirty-six unordered distinct-carrier pairs;
- the normalized cross-wheel non-self alignment set has the same elements and
  cardinality as that pair set;
- a future G refactor can use this exact set as a structural golden fixture.

## 5. What the derivation does not prove

- that current OPERA G generates any W term or profile;
- that a carrier pair supplies a natural-language middle term;
- that co-carrier, counterpart, identity, or moral meaning follows from
  cardinality;
- that W depends on G;
- that any OPERA was executed;
- that the proof is a qualified informant LIBER;
- that validation creates canon, registration, runtime, or authority.

## 6. Interpretative hypothesis retained for later review

The received lineage proposes that moral Instrumentativum can later be
investigated through principles, relations, and rules and factorized into the
four W wheels. That proposal remains reconstruction-level. It belongs to the
separately queued G interpretative-Evacuatio work, not to this proof and not to
current W execution.

## 7. LIBER status report

```text
LIBER STATUS: canonical support / ordinary LIBER EX / non-informant
A COVERAGE: absent
Q COVERAGE: absent
T COVERAGE: absent
S COVERAGE: absent
36 CHAMBERS: listed structurally; not evacuated by OPERA
TABULA: carrier-pair enumeration only
ROTA FILLING ADMISSIBILITY: none at runtime
PROMOTION STATUS: canonical support / numbered placement / non-runtime
```

## Closing boundary

The derivation is complete within its declared finite combinatory scope and
strictly silent about semantic generation and runtime authority.