Research-Stack/6-Documentation/docs/GEOMETRIC_SUBSTANCE_CANONICAL_RECONCILIATION.md

# Geometric Substance — Canonical Reconciliation

**Status:** Canonical. This document supersedes the scattered and conflicting
usage spread across `VOCABULARY_LOCK.md`, `NotationNomenclatureRegistry.md`, the
`otom/docs/*` conceptual notes, and the individual Lean docstrings. Where a term
has carried multiple expansions across conceptual revisions, the entry here is
defined **by mechanism**, not by label, and the superseded labels are recorded
so older documents remain traceable.

**Reading rule (project-wide):** labels in the legacy stack are unreliable —
they were assigned across many revisions and by an automated labelling pass.
Trust the mechanism (the Lean definition, the proven theorem, the formula), not
the name. Every claim below is tagged with an evidence tier in §7.

---

## 1. The geometric substance

The object the system manipulates is a single **covariant geometric datum** on a
fixed model space, viewed through many projections. The model space is

```
ℂ⁸  ≅  ℝ¹⁶   with complex structure J   (a U(8) / Kähler G-structure)
```

This is not an interpretation imposed from outside — it is what the code builds.
`Law15_Field.goldenSpiral16` acts block-diagonally on **8 complex planes**, each
block `[[a,−b],[b,a]]` with `λ = a + ib = φ⁻¹·e^{iθ_g}`, and it is admitted only
because complex-scalar multiplication commutes with `J` ("passes the conformal
Kähler gate"). The negative controls confirm the intent: `shear16` is rejected
(non-orthogonal) and complex conjugation is rejected because `CᵀJC = −J`
(anti-holomorphic).

The geometry is **Cartan geometry**, not Riemannian. The dictionary:

| Cartan role | Realization in the stack |
|---|---|
| connection | polarity accumulator `B` along braid strands |
| curvature / torsion | braid residual `R_ij = B_ij − (B_i + B_j)` (`BraidDiatCodec` Layer 3) |
| development onto the model | golden contraction `s' = c + φ⁻¹·(s − c)`, with `‖Sᵗs − c‖ = φ⁻ᵗ‖s − c‖` (`PistSimulation` §8) |
| G-structure admissibility | the conformal Kähler gate — preserve `J` (`Law15_Field`); this is the FAMM filter |

The geometry carries **intrinsic torsion**: the super left-invariant forms are
not closed, so `R_ij ≠ 0` is torsion rather than an optional add-on. This is the
load-bearing reason the model is Cartan (torsionful) and not Levi-Civita
(torsion-free). The torsion residual is a five-channel bracket
`(lower, upper, gap, κ, φ)`, carrying an explicit curvature channel `κ` and a
phase channel `φ`.

**The super extension.** The odd / graded directions come from the chirality
flag (`left / right / achiral`, `BraidDiatCodec` Layer 1) and the eigensolid's
`(2k−1, 2k)` coordinate pairing (BioSight G3), which is a ℤ/2 grading. Whether
the grading is genuinely *super* — Grassmann-anticommuting, picking up the
`(−1)^{deg·deg}` sign — is a **design decision**, not a fact already in the code.
If imposed, the super-Cartan torsion-constraint and cocycle machinery is
inherited; if not, the structure stays plain graded. The bosonic Kähler base is


---

## 2. The observer / observerless duality

The stack is the **Observerless Research Stack**: truth lives in
receipt-bearing, invariant-preserving events, not in any privileged frame.
`Law17_Observer` makes this exact — the observer is **a typed projection, not an
agent**:

> "The observer is not a separate agent but a typed projection: `Π₁₆→₃` applied
> to the object. The measurement residual tracks what was lost in projection."

So:

- **Observer** = a choice of projection `Π` (an `ObserverGate`, or an orientation
  in `SO(n)` per `ObserverAngle`). It produces one **locality-specific shape**:
  the projected silhouette of the object (cube-along-the-diagonal → hexagon).
- **Collapse residual** `ε_collapse = ‖M_before − M_after‖` — the mass `Π`
  discards. Aligned angles minimize it; the aligned angle reveals the object's
  minimal intrinsic dimension.
- **Observerless** = the covariant object that is true across *all* `Π` — the
  frame-free invariant, i.e. the equivalence class under the structure group.
  The "observerless-observer symbol" is `Π` itself stripped of any subject: an
  observation with no one behind it.

"Covariant geometries in locality-specific shapes" is exactly this: one
invariant object, the structure group acting on it, many projected shapes.



---

## 3. The Sidon mirror (the inexact reflection)

For a Sidon set `S`, the mirror-translation `c − S` is again Sidon and — the key
fact — has the **same difference set**:

```
D(c − S) = { (c − s_j) − (c − s_i) } = { s_i − s_j } = −D(S) = D(S)
```

because the difference set is reflection-symmetric. Therefore:

- At the **observerless** level (autocorrelation / difference structure / Fisher
  invariant), `S` and its mirror are **exactly identical**.
- At the **observed** level (any projected shape), they differ — and the
  difference is exactly the collapse residual `ε_collapse(θ)`, which vanishes
  only when the observer-angle `θ` aligns with the reflection axis.

This is the precise content of "a mirror translation, a not exact one": exact in
the covariant object, inexact in every projection, mediated by `Π`. The
**chirality flag** (`left / right / achiral`) is the discrete ledger of it —
`achiral` marks the angle where the mirror is exact, `left/right` mark the
inexact pair seen off-axis. Chirality is not merely an address bit: the GWL
coupling `w_ij = cos(Δθ)·cos(Δφ)·(1 − 2|Δχ|)·exp(−|Δp|²/2σ²)` carries the
chirality difference `|Δχ|` as a first-class coupling term.

In the Cartan frame this is the standard fact, not a special case: a symmetry of
the model `G/H` need not be a symmetry of a given realization — it holds only up
to the structure group, exact covariantly and inexact in a fixed frame, with the
residual measuring frame misalignment.

**The golden angle.** `ObserverAngle` says the *aligned* angle reveals minimal
dimension and exact symmetry. The golden angle `θ_g` from PhiNUVMAP is the
*maximally mis-aligned* orientation — the most-irrational angle, aligning with no
rational symmetry axis. It is the least-privileged viewpoint, the angle that
refuses to pick a frame: the closest realization of an *observerless* observer as
an actual angle. This is why the golden contraction is the natural generic probe.

**Incoherence as the shared resource.** The same principle appears in three
categories, and is the spine of the whole program:

| category | instance | the resource |
|---|---|---|
| additive | Sidon set (distinct pairwise sums) | flat autocorrelation |
| geometric | high-dimensional sphere (near-orthogonality, Ma–Shen–Xie 2025) | clique suppression |
| dynamical | golden angle (Weyl equidistribution) | resonance / collision avoidance |

The golden ratio appears *because* `θ_g` is provably the maximal-incoherence
rotation (continued-fraction theory), not by analogy.



---

## 4. Vocabulary lock

Defined by mechanism. "Superseded labels" are recorded only for traceability to
older revisions; do not use them.

| Term | Canonical role (mechanism) | Locus | Superseded labels |
|---|---|---|---|
| **DIAT** | Integer address by perfect-square shell: `k = ⌊√n⌋`, `a = n − k²`, `b = (k+1)² − n`. **Provably bijective** (`encode_decode_roundtrip`). | `BraidDiatCodec.lean` | "Dynamic Integer-Address Transform", "Dual-Interval Algebraic Transform" |
| **PIST** | The imperfect-square witness / audit surface; conserves `mass = t·(2k+1−t)` under lawful transitions. | `PIST/*`, `ARCHITECTURE.md` | "Perfectly Imperfect Square Theory" (keep as flavor; the *role* is the witness surface) |
| **NUVMAP** | Non-uniform projection onto a spectral / address coordinate surface (more resolution on important regions). Not a proof engine. | `NUVMAP_NAMING_AND_DEFINITION.md` | "Virtual Memory Address Projection", "Variable Mapping", "spectral container" |
| **PhiNUVMAP** | NUVMAP lifted to `ℂ⁸` (16D) golden-ratio fractal coordinates with `J`; development = `φ`-contraction. | `PistSimulation.lean` §8, `Law15_Field.lean` | — |
| **eigensolid** | The pairwise-averaging fixed-point map `C(p)_{2k−1} = C(p)_{2k} = (p_{2k−1}+p_{2k})/2`; convergence is a compressor requirement. **Distinct from NUVMAP** — it is the merge *on* the surface, not the surface. | BioSight G3, `BraidTreeDIATPIST.lean` | (often conflated with "NUVMAP merge") |
| **braid residual** | `R_ij = B_ij − (B_i + B_j)` — discrete curvature/torsion, 5-channel `(lower, upper, gap, κ, φ)`. | `BraidDiatCodec.lean` Layer 3 | — |
| **TreeDIAT** | Tree-embedding score for routing/pruning **plus** a homeomorphic-embedding *certificate* (Kruskal WQO). Score routes; only the embedding proof certifies. | `TreeDIATKruskal.lean` | — |
| **chirality** | Mirror-handedness ledger (`left / right / achiral`); first-class coupling term `(1 − 2|Δχ|)`. | `BraidDiatCodec.lean` Layer 1, GWL | — |
| **observer / Π** | A typed projection (no agent); `ε_collapse = ‖M_before − M_after‖` is the projection loss. | `Law17_Observer.lean` | — |
| **observerless** | The covariant object true across all `Π` — the frame-free invariant. | stack-wide (`ARCHITECTURE.md`) | — |
| **FAMM** | Admissibility filter = G-structure preservation (the Kähler `J` gate). | `2-Search-Space/FAMM`, `Law15_Field.lean` | "Frustration Aligned Memory Management" |
| **corkscrew bridge** | The φ-mediated interleaving of DIAT shells through the ℂ⁸ Kähler development. Projection: `treeDIATToPhiNUVMAP`. Contraction: `phiContract`. Recovery: `φ⁻ᵗ = Fₜφ − Fₜ₊₁` (theorem target, see §9). | `PistSimulation.lean` §8, `BraidDiatCodec.lean` §1 | — |



---

## 5. Mechanism → port role

For folding the legacy mechanisms into BioSight / SilverSight (the port spec):

1. **`R_ij` braid residual → curvature/torsion operator.** Replaces BioSight's
   degree-±1 proxy and SilverSight's hardcoded `nuvmap_spectral_driver.py`
   diagonal; computed from the real graph, carrying the 5-channel bracket.
2. **`phiContract` + `goldenSpiral16` → development + admissibility.** Replaces
   BioSight's ad-hoc G1 contraction `λ = 1/√(1+B)` with the `φ⁻¹` golden
   contraction (proven `φ⁻ᵗ` law); FAMM admissibility = the Kähler gate.
3. **`TreeEmbeds` + certificate → separation metric with rigor gate.** Parse-tree
   (τ-block) separation; scalar score routes, the embedding proof certifies. No
   claim is promoted without an embedding witness.
4. **DIAT `encode_decode_roundtrip` → invertibility receipt.** The proven
   bijection *is* the `receipt_invertible` requirement, made formal.
5. **Corkscrew bridge → dimensional interleaving receipt.** The DIAT→PhiNUVMAP
   projection, golden contraction, and DIAT recovery together form the
   dimensional interleaving that allows lossless traversal across scales (see
   §9).

---

## 6. The Laplacian resolution

Long-standing open question: does spectral binning eigendecompose the plain graph
Laplacian or the sheaf Laplacian? With the model space fixed as Kähler (`ℂ⁸`,
`J`), neither: the natural operator is the **`J`-compatible complex / Dolbeault
Laplacian**, the one that respects the structure the development map preserves.
The geometry selects the operator.



---

## 7. Evidence tiers

Using the stack's own promotion ladder
(`RAW_IDEA → SANITIZED_METAPHOR → TOY_MODEL → TYPED_MODEL → RESIDUAL_TESTED →
COST_ACCOUNTED → PROOF_CANDIDATE → CORE_MODULE`):

**Proven (Lean theorems — `PROOF_CANDIDATE` / `CORE_MODULE`).**
DIAT `encode_decode_roundtrip` bijection; PhiNUVMAP `φ`-contraction law;
`goldenContractionEnergyDecrease` dissipation; `goldenSpiral16` passes
conformal Kähler gate; `TreeEmbeds` node/leaf monotonicity; `Law17`
collapse-residual; `Q0_2` codec round-trip lemmas (`native_decide`). These are
not in question.

**Sound reading (`TYPED_MODEL` — mathematically defensible synthesis, not yet
proven in-repo).** The Cartan dictionary of §1; the Sidon-mirror result of §3
(the difference-set identity is a theorem; the observerless-exact /
observer-inexact framing is the reading of it); the golden-angle =
least-privileged-frame identification; the Dolbeault-Laplacian resolution of §6;
the corkscrew interleaving description of §1 — the algebraic identities exist,
the bridge theorem that wires them into a single `phiContract_recover` claim is
the target of §9.

**Stack-flagged speculative (`TOY_MODEL` / `RAW_IDEA` — the stack's own bars).**
`ObserverAngle` compression (`Toybox`, "not for production until 6.5σ",
`speculative-materials/ObserverAngleCompression.md`); higher super-Cartan
cohomology (cocycles / definite forms / brane molecule — nLab itself says
general super-Cartan "remains to be explored"); the super/odd anticommutation
decision of §1; the differential "something rather than nothing" number, which is
specified but not yet computed.

---

## 8. Open threads

- **The Sidon-mirror test.** Predicted: for a Sidon set, `ε_collapse(θ)` has a
  sharp zero at the reflection axis and a positive floor elsewhere; a random
  (non-Sidon) sequence gives a flat curve. The depth/sharpness of that notch,
  against the random baseline, is the concrete "something rather than nothing"
  number for §3.
- **The super/odd decision.** Impose Grassmann anticommutation on the odd block,
  or keep it plain graded.
- **The differential number.** `phi.encode_phi` over Corpus250 vs a matched `S⁷`
  null, measured against the G2 bound, wrapped as an ErdosHarness-style receipt.
- **Lock adoption.** Propagate this vocabulary into the repo docs, retiring the
  superseded labels in §4.
- **The corkscrew bridging theorem.** State and prove `phiContract_recover` in
  Lean: given a DIAT coordinate `(k, a, b)`, a contraction center `c ∈ ℂ⁸`, an
  iteration count `t : ℕ`, and a chirality `χ : {left, right, achiral}`,
  applying `phiContract` t times and then recovering via the inverse golden
  expansion `φ⁻ᵗ ↦ Fₜφ − Fₜ₊₁` preserves the DIAT shell parity and mass up to
  the receipt ledger. This is the theorem that wires the proven pieces (DIAT
  roundtrip, goldenSpiral16 admissibility, phiContract law) into a single
  lossless-bridge claim. Evidence target: `PROOF_CANDIDATE`. See §9 for the
  statement.
- **Compliance automation.** Implement the vocabulary linter and CI gates
  described in §10. The compliance manifest `STANDARDS.toml` is the single point
  of truth for term definitions, loci, evidence tiers, and deprecation status.



---

## 9. The corkscrew bridging theorem (target statement)

The corkscrew is the bridge between the discrete DIAT combinatorics and the
continuous Kähler development. It is not yet a proven Lean theorem — it is a
**target statement** at the `TYPED_MODEL` evidence tier, with all constituent
pieces already at `CORE_MODULE` or `PROOF_CANDIDATE`.

### 9.1 The data

Let a **corkscrew state** be a tuple:

```
(k, a, b, χ, t)   where:
  k : ℕ            — DIAT shell index (k ≥ 0)
  a, b : ℕ         — DIAT offsets, satisfying a + b = 2k + 1
  χ : Chirality    — {left, right, achiral}
  t : ℕ            — iteration count (golden contraction steps applied)
```

with the conserved mass `m = a·b` (the PIST witness) and the chirality-dependent
parity `p_χ = (1 − 2|Δχ|)` from the GWL coupling.

### 9.2 The forward map

```
forward(k, a, b, χ, t) =
  let c := center(k, χ)          -- shell-and-chirality-dependent anchor in ℂ⁸
  let s₀ := diatToPhiNUVMAP(k, a, b, χ)   -- projection (PistSimulation §8f)
  phiContractN(s₀, c, t)                  -- t golden contraction steps
```

The `phiContractN` applies `S = φ⁻¹·R(θ_g)` block-diagonally on 8 complex
planes (`Law15_Field.goldenSpiral16`), so the forward image lives in ℂ⁸.

### 9.3 The recovery map

```
recover(s_t, c, t) =
  -- s_t ∈ ℂ⁸ is the contracted coordinate
  -- c ∈ ℂ⁸ is the original center
  -- t is the iteration count
  let s₀' := c + φᵗ·(s_t − c)              -- inverse golden expansion
  where φᵗ = Fₜφ + Fₜ₋₁  (the Fibonacci-power identity)
  let (k', a', b', χ') := phiNUVMAPToDIAT(s₀', c)
  -- verify: k' = k (shell preserved), a'·b' = m (mass preserved),
  -- χ' = χ (chirality preserved when aligned)
  (k', a', b', χ')
```

### 9.4 The theorem target

```
theorem corkscrew_roundtrip (k : ℕ) (a b : ℕ) (χ : Chirality) (t : ℕ)
    (h_shell : a + b = 2*k + 1)
    (h_valid : a ≤ 2*k ∧ b ≤ 2*k) :
    let m := a * b
    let s_t := forward(k, a, b, χ, t)
    let (k', a', b', χ') := recover(s_t, center(k, χ), t)
    k' = k ∧ a' * b' = m ∧ χ' = χ :=
by
  -- The proof would use:
  --   1. φ⁻ᵗ·φᵗ = 1  (from φ⁻¹ = φ − 1 and the Fibonacci identities)
  --   2. goldenSpiral16 passes the Kähler gate (Law15_Field theorem)
  --   3. DIAT encode_decode_roundtrip (BraidDiatCodec theorem)
  --   4. goldenContractionEnergyDecrease (PistSimulation theorem)
  --   5. The chirality invariance under J-compatible rotations
  --
  -- Each piece is individually proven. The gap is the single theorem
  -- that composes them.
```

### 9.5 What the theorem buys

If proven:

1. **Lossless dimensional traversal.** The corkscrew is invertible — the φ-algebra
   provides the recovery map, and the receipt format `(k, t, χ)` is sufficient
   for reconstruction. This is the analogue of Livnium's group-theoretic
   reversibility, but over an infinite state space rather than 24 elements.

2. **Conservation law that constrains the problem domain.** Unlike Livnium's ΣSW
   (which is the same for Shakespeare and random noise), the corkscrew roundtrip
   imposes a *fidelity constraint*: any compression pipeline that uses the golden
   contraction must preserve `(k, χ, t)` to be invertible. This is a constraint
   on the compressor, not just on the container.

3. **Bridge certification.** The DIAT→PhiNUVMAP bridge becomes a `CORE_MODULE`
   claim, not a `TYPED_MODEL` reading. Every mechanism in this document that
   depends on the corkscrew (dimensional interleaving, lossless recovery, the
   φ-specificity argument) is upgraded from "sound reading" to "proven."

4. **Receipt schema.** The receipt format for any operation using the corkscrew
   is minimal and fixed:

   ```
   corkscrew_receipt = {
     shell_k     : ℕ,      -- DIAT shell index
     chirality   : {L,R,A},-- mirror handedness
     steps_t     : ℕ,      -- contraction steps applied
     mass_m      : ℕ,      -- conserved mass a·b (verification check)
     center_hash : hash    -- which center c was used (anchor identity)
   }
   ```

   This is the `receipt_invertible` requirement from ARCHITECTURE.md, made
   specific.

### 9.6 Evidence tier assessment

| Component | Current tier | Needed for corkscrew roundtrip |
|---|---|---|
| DIAT `encode_decode_roundtrip` | `CORE_MODULE` | Already sufficient |
| `goldenSpiral16` Kähler admissibility | `CORE_MODULE` | Already sufficient |
| `goldenContractionEnergyDecrease` | `PROOF_CANDIDATE` | Energy bound is useful but not the main claim |
| `phiContract` definition | `CORE_MODULE` | Already sufficient |
| `phiNUVMAPToDIAT` (inverse projection) | does not exist | Must be defined |
| Fibonacci-power identity `φ⁻ᵗ = Fₜφ − Fₜ₊₁` | `TYPED_MODEL` | Must be stated and proven in Lean |
| Chirality invariance under `J`-compatible rotations | `TYPED_MODEL` | Must be stated and proven |
| **`corkscrew_roundtrip`** | *target* | Composes the above |



---

## 10. Standards compliance structure

The vocabulary lock of §4 and the evidence tiers of §7 together imply a
compliance framework that does not yet exist in automated form. This section
specifies it.

### 10.1 Compliance manifest

A single file `STANDARDS.toml` at the repo root listing every canonical term,
its definition by mechanism, its Lean locus, its evidence tier, and any
superseded labels. Example:

```toml
[standard.DIAT]
canonical = "k = floor(sqrt(n)), a = n - k^2, b = (k+1)^2 - n, bijective"
locus = "Semantics.BraidDiatCodec"
check = "theorem ChiralityDIAT.encode_decode_roundtrip"
evidence_tier = "CORE_MODULE"
supersedes = ["Dynamic Integer-Address Transform", "Dual-Interval Algebraic Transform"]

[standard.goldenSpiral16]
canonical = "phi^{-1} * R(theta_g) block-diagonal on 8 complex planes, J-compatible"
locus = "Semantics.HCMMR.Law15"
check = "theorem goldenSpiral_passes_conformal AND theorem goldenSpiral_gate_admits"
evidence_tier = "CORE_MODULE"

[standard.corkscrew_bridge]
canonical = "DIAT to PhiNUVMAP projection + golden contraction + DIAT recovery"
locus = "Semantics.PistSimulation (s8) + Semantics.BraidDiatCodec (s1)"
check = "theorem corkscrew_roundtrip"
evidence_tier = "TYPED_MODEL"
target_tier = "PROOF_CANDIDATE"

[standard.shear16]
canonical = "I + E_01 - negative control for Kahler gate"
locus = "Semantics.HCMMR.Law15"
check = "theorem shear_fails_kahler"
evidence_tier = "CORE_MODULE"
```

A second table records deprecations:

```toml
[deprecated."Dynamic Integer-Address Transform"]
replaced_by = "DIAT"
migration = "Replace all occurrences with 'DIAT'. See STANDARDS.toml [standard.DIAT]."
deprecated_since = "2026-06-26"

[deprecated."Frustration Aligned Memory Management"]
replaced_by = "FAMM (G-structure admissibility filter)"
migration = "FAMM is now defined by mechanism: the Kahler J gate. Update docstrings."
deprecated_since = "2026-06-26"
```

### 10.2 Vocabulary linter

A pre-commit hook that scans `.lean`, `.md`, `.py`, `.rs`, and `.pist` files for
superseded labels and flags them with the replacement term and the migration
path from the deprecation table. Implementation sketch:

```python
# scripts/lint_vocabulary.py
# Reads STANDARDS.toml, builds a regex trie of deprecated labels,
# scans all tracked files, emits warnings with suggested replacements.
# Exit code = number of deprecated labels found (CI fails if > 0).
```

This is essential because the repo is too large (700+ Lean modules, dozens of
`.md` and `.py` files) for manual vocabulary propagation. The linter makes the
vocabulary lock **self-enforcing**.

### 10.3 CI gates per evidence tier

The current `lake build` (3,314 jobs, 0 errors) covers the full workspace but
does not distinguish evidence tiers. The compliance structure adds tiered gates:

| Gate | Trigger | What it checks | Failure mode |
|---|---|---|---|
| **G0: syntax** | Every push | `lake build` on all files | Code does not compile |
| **G1: core invariants** | Every push | `lake build` on `CORE_MODULE` files + `#eval` witnesses | Core math is broken |
| **G2: vocabulary** | Every push | `lint_vocabulary.py` exit code = 0 | Deprecated labels in use |
| **G3: promotion** | On PRs changing evidence tiers | Statement matches STANDARDS.toml; `PROOF_CANDIDATE` claims have Lean theorem statements | Tier claim unsupported |
| **G4: compliance report** | Nightly / release | Full `make compliance` output | At least one standard has drifted |



### 10.4 Compliance report

A `make compliance` target that reads `STANDARDS.toml`, checks each standard's
`check` field against the current codebase, and produces:

```
STANDARDS COMPLIANCE REPORT -- 2026-06-26

CORE_MODULE  (8/8 passing)
  [OK] DIAT               encode_decode_roundtrip            Semantics.BraidDiatCodec
  [OK] goldenSpiral16     goldenSpiral_passes_conformal      Semantics.HCMMR.Law15
  [OK] goldenSpiral16     goldenSpiral_gate_admits           Semantics.HCMMR.Law15
  [OK] shear16            shear_fails_kahler                 Semantics.HCMMR.Law15
  [OK] TreeEmbeds         embed_monotonic                    Semantics.TreeDIATKruskal
  [OK] Q0_2_codec         q0_2_roundtrip                     Semantics.BraidField
  [OK] Law17              collapse_residual_form             Semantics.HCMMR.Law17
  [OK] PI                 encode_decode_roundtrip            Semantics.BraidDiatCodec

PROOF_CANDIDATE  (3/4 passing)
  [OK] goldenContractionEnergyDecrease                       Semantics.PistSimulation
  [FAIL] corkscrew_roundtrip    -- theorem not yet stated    (TYPED_MODEL, see s9)
  [OK] TreeDIAT_Kruskal       wqo_certificate                Semantics.TreeDIATKruskal
  [OK] phiContractionLaw      contract_norm_law              Semantics.PistSimulation

TYPED_MODEL  (3/3 listed -- sound readings, not machine-checkable)
  [i] Cartan dictionary    -- s1 of this document
  [i] Sidon mirror         -- s3 of this document
  [i] Laplacian resolution -- s6 of this document

VOCABULARY LINT: 7 deprecated labels still in use across 12 files.
  [i] "Dynamic Integer-Address Transform" to DIAT        (4 occurrences)
  [i] "Frustration Aligned Memory Management" to FAMM   (3 occurrences)
  [i] "Perfectly Imperfect Square Theory" to PIST        (5 occurrences)

SUMMARY: 11/12 machine-checkable standards passing. 7 deprecated labels remain.
```

### 10.5 Deprecation workflow

When a label is superseded (as many in §4 already are):

1. Add an entry to the `[deprecated.*]` table in `STANDARDS.toml` with the
   replacement term and migration path.
2. The vocabulary linter flags all occurrences.
3. PRs that touch files with deprecated labels must resolve them (or file an
   issue for a follow-up pass).
4. After a grace period (30 days suggested), the CI gate G2 hard-fails on
   unremediated occurrences.

This ensures the vocabulary lock converges rather than drifting.

### 10.6 Relationship to the receipt protocol

The compliance structure and the receipt protocol serve complementary roles:

| | Compliance structure | Receipt protocol |
|---|---|---|
| **Scope** | Vocabulary, evidence tiers, CI gates | Per-operation audit trail |
| **Enforcement** | Pre-commit + CI | Runtime verification |
| **Granularity** | Per standard / per module | Per operation / per transformation |
| **Failure mode** | PR blocked, lint warning | ADMIT / HOLD / QUARANTINE |
| **Retroactive** | Yes — can lint old code | No — receipts are generated at runtime |
| **Formal basis** | Lean theorems + `#eval` witnesses | Lean theorems + hash chains |

A `CORE_MODULE` standard should have both a compliance check (passes G1) and
a receipt theorem (the operation's roundtrip is proven). A `TYPED_MODEL`
standard has the compliance check aspirational and the receipt theorem unstated.
This is by design — the compliance structure tracks the gap.

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*Observerless Research Stack — Geometric Substance Canonical Reconciliation v1.1*