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

# ENE Layer Equations

## Core Primitive: `bind`

```
bind(A, B, g) = (cost, witness)

cost     = cost_fn(left, right, metric)
witness  = Witness.lawful(invA(left), invB(right))
lawful   = invA(left) = invB(right)
```

**Metric structure:**
```
Metric = {
  cost:        UInt32  (Q16.16)
  tensor:      String  -- shim label only; core decisions should converge to finite tags
  torsion:     UInt32  (Q16.16)
  reference:   String
  history_len: Nat
}
```

---

## Manifold Dynamics: Picard-Blit

The transition between manifold states is governed by the **Discrete Picard Integral**, which replaces continuous ODE integration with [BEAUTIFUL_PROVISIONAL - hardware-accelerated bitwise accumulation ($\oplus$) - requires hardware implementation evidence with benchmark verification].

```
M_{k+1} = Quantize( Jump( M_k ⊕ (Sample ⊗ Raytrace) ) )
```

**Discrete Picard Integral ($\oplus$):**
```
blit_op(a, b, mask) = {
  quant_a = (u32)(a * 255.0)
  quant_b = (u32)(b * 255.0)
  xor_res = quant_a ^ mask
  return (f32(xor_res) / 255.0, b)
}
```

**Short-Circuit Jump ($\mathcal{J}_{\text{DAG}}$):**
```
J_DAG(hash) = solved ? teleport(result) : continue
```

**Convergence Property (Perfect Squares):**
The manifold converges constructively only at perfect square resonance hubs where the geometric tip degenerates:
```
Tip(m²) = (0, -(2k+1))
```

---

## ENE Graph: `Graph`

```
Node    = { id, type: NodeType, label, payload: Option MIPoint }
Edge    = { id, source, target, type: EdgeType, edgeClass: EdgeClass, weight: Float, justified }
Graph   = { nodes: List Node, edges: List Edge, nextId }
```

Note:
- This block describes the current ENE graph surface.
- The canonical adapter path is stricter than the graph surface.
- Schema-bearing core records should prefer `Q16_16` over `Float` whenever a fixed-point form exists.

**NodeType:**
```
atom | lemma | wordform | observation | canonicalState | attractor | signature | interpretation | loadProfile | projection
```

**EdgeType:**
```
has_atom | realizes | projects_to | assigned_to | has_signature | supports | contradicts |
inherits | evokes | has_load | derived_from | similar_to | composed_of | precedes |
causes | path_step | witnesses | collapsed_to | evolved_from
```

**EdgeClass:**
```
definitional | analogical | translational | affective | inferential |
capabilityBearing | unstable | quarantined | derived | realizational |
projective | evidentiary | loadAnnotated | compositional | temporal
```

---

## Atomic Path: `AtomicPath`

```
AtomicStep    = { rewrite: AtomicRewrite, stepId }
AtomicRewrite = { fromNode, toNode, viaEdge, locallyAdmissible }
AtomicPath    = { steps: List AtomicStep }

isLawful(p)     = ∀s ∈ p.steps: s.rewrite.locallyAdmissible = true
canCompose(p1, p2) = p1.end_ = p2.start
compose(p1, p2)    = { steps := p1.steps ++ p2.steps }  -- if canCompose
```

---

## Witness & Constitution

```
WitnessReceipt = { witnessId, provenance: WitnessProvenance, path: AtomicPath, load: CognitiveLoad, timestamp }
Witness        = { node, receipt: WitnessReceipt, preservedAtoms, lostAtoms, accumulatedLoad, resultCapability }

WitnessProvenance:
observation | inference | projection | evolution | translation | composed
```

**Groundedness:**
```
Groundedness = {
  atomicBasis: Bool
  lawfulReachability: Bool
  boundedLoad: Bool
  faithfulProjection: Bool
  evolutionAuditable: Bool
  universalDynamics: Bool
  scalingPreserved: Bool
  classMembershipVisible: Bool
  classifiedDynamics: ClassifiedDynamics
}

habitable(g) = atomicBasis ∧ lawfulReachability ∧ boundedLoad ∧ faithfulProjection ∧
               evolutionAuditable ∧ universalDynamics ∧ scalingPreserved ∧ classMembershipVisible ∧
               classifiedDynamics.preservedUnderProjection ∧
               classifiedDynamics.preservedUnderCollapse ∧
               classifiedDynamics.preservedUnderEvolution
```

**UniverseConstitution:**
```
UniverseConstitution = {
  requiresAtomicGrounding: Bool
  requiresLawfulPath: Bool
  requiresLoadVisibility: Bool
  requiresCapabilityLegibility: Bool
  requiresProjectionFaithfulness: Bool
  requiresEvolutionAuditability: Bool
  requiresUniversalityPreservation: Bool
  requiresNoActiveQuarantine: Bool
}

admissible(c, g) = (requiresAtomicGrounding → atomicBasis = true) ∧
                  (requiresLawfulPath → lawfulReachability = true) ∧
                  ...
```

---

## Canonical State

```
CanonicalState = {
  phi, psi, delta, gamma, chi, tau, deltaDot: Q16_16
  drift, curvature, coherence, angularMomentum, radiusDev: Q16_16
  confidence: Q16_16
  mode: ControlState
  timestamp: Float
  step: Nat
  domain: String
  source: String
}

ControlState: commit | hold | halt | dmt | flame

computeConfidence(drift, curvature, angularMomentum) =
  clamp(1 / (1 + drift × curvature + angularMomentum), 0, 1)
```

Canonical note:
- `Q16_16` is the intended scalar form for core-admissible schema fields.
- `Float` inside legacy ENE structures should be treated as transitional debt or shim-boundary representation, not as the target canonical contract.

## Canonical Record Schema

```
FieldKind =
  int(bits, signed)
  | nat(bits)
  | q16_16
  | float64
  | text
  | bool
  | blob(size)

RecordSchema.coreAdmissible(schema) =
  schema.endian = big
  ∧ schema.bitOrder = msb0
  ∧ uniqueFieldNames(schema.fields)
  ∧ schema.fields.all(field.name ≠ "")
  ∧ schema.fields.all(field.kind ≠ float64)
```

Interpretation:
- `coreAdmissible` marks the boundary between permissive ingestion and canonical ENE core.
- `float64` may still exist in legacy or shim-facing schemas, but it is not part of the tightened core-safe schema contract.
- Duplicate field names are rejected because canonical field lookup must stay deterministic.

## Recovered Legacy Adapter Forms

Older ENE/canonical adapter versions carried a second schema layer around the canonical record itself. The stable parts of that layer are now represented as:

```
NormalizationMode = minmax | centered | passthrough

RawFeatureSpec = {
  name,
  mode,
  low : Q16_16,
  high : Q16_16,
  required : Bool
}

CanonicalDimension =
  phi | psi | delta | gamma | chi | tau | deltaDot |
  drift | curvature | coherence | angularMomentum | radiusDev | confidence

CanonicalVectorSpec = { dimensions : List CanonicalDimension }
CanonicalAttractor  = { name, center : List Q16_16, maxRadius? }

AssignmentResult = {
  zN : List Q16_16,
  nearestAttractor?,
  attractorDistance?,
  attractorConfidence : Q16_16,
  signature : List Nat,
  quantizedBands : List QuantizedBand,
  consistent : Bool
}
```

Interpretation:
- These forms came from older ENE adapter/schema work and were ported where they still support the current canonical path.
- They belong to the canonical adapter boundary, not to the ENE graph ontology itself.
- Execution/address-space schemas from older SQL substrate layers were not ported into ENE core because they describe deployment/governance substrate, not semantic canonicalization.

---

## Lemma & Decomposition

```
Lemma      = { canonical: String, sig: List Atom, pos: PartOfSpeech }
PartOfSpeech: verb | noun | adjective | adverb | preposition | conjunction | determiner | pronoun

Atom: someone | something | do_ | happen | move | cause | die | want | know | feel | think | good | bad | because | not

AtomicDecomposition = { source: Lemma, atoms: List WeightedAtom }
WeightedAtom        = { atom: Atom, weight: UInt32 }  -- Q16.16

FaithfulDecomposition(l, d) = d.source = l ∧ d.unweighted = l.sig
```

---

## Scalar Collapse

```
ScalarCollapse = {
  policy: ScalarPolicy
  fields: List ScalarField
  sourceDecomposition: AtomicDecomposition
  sourcePath: AtomicPath
  sourceLoad: CognitiveLoad
}

ScalarAdmissible(sc) = sc.sourcePath.isLawful ∧
                      sc.sourceDecomposition.nonempty ∧
                      sc.fields.all (λf => f.certified = true)
```

---

## Cognitive Load

```
CognitiveLoad = { intrinsic, extraneous, germane, routing, memory, total: Float }
```

---

## Fixed Point: Q16_16

```
Q16_16 = UInt32
0x00010000 = 1.0
0xFFFFFFFF ≈ infinity/illegal
Range: [-32768.0, 32767.999985]
Resolution: 1/65536 ≈ 0.000015
```

---

## Universality Classes

```
UniversalityClass: directedPercolation | kpz | ising | percolation | randomBoolean | sos | none
```

---

## Evolution

```
SelfModification = {
  id: Nat
  description: String
  priorState: Graph
  postState: Graph
  witness: Witness
  timestamp: Float
}

EvolutionContract = {
  contractId: Nat
  preservesAuditSurface: SelfModification → AuditSurface → Bool
  replayable: SelfModification → Bool
  preservesConstitution: SelfModification → UniverseConstitution → Bool
}

EvolutionAdmissible(mod, contract, surface, constitution) =
  contract.preservesAuditSurface(mod, surface) ∧
  contract.replayable(mod) ∧
  contract.preservesConstitution(mod, constitution)
```

---

## Prohibitions

```
NotAllowed_EmptyLemma(l)                    = l.sig = []
NotAllowed_UnfaithfulDecomposition(l, d)     = ¬FaithfulDecomposition(l, d)
NotAllowed_NegativeWeightInNormalForm(wa)   = False  -- vacuous with UInt32
NotAllowed_ActiveQuarantine(g)               = ∃e ∈ g.edges: e.edgeClass = quarantined ∧ e.weight > 0
NotAllowed_UnlawfulPath(p)                  = ¬AtomicPath.isLawful(p)
NotAllowed_WitnessWithoutProvenance(w)      = w.receipt.provenance = observation ∨ ...  -- must be one of 6
NotAllowed_UniversalityLossUnderProjection   = ¬projectionPreservesUniversality
NotAllowed_ScalarWithoutAtomicAncestry(sc)   = ¬ScalarAdmissible(sc)
NotAllowed_ScalarWithNegativeLoad(sc)       = sc.sourceLoad.total < 0
NotAllowed_NondeterministicCanonicalForm(cbf) = ¬IsCanonical(cbf)
NotAllowed_EpistemicSelfErasure(mod, contract, surface) = ¬EvolutionAdmissible(...)
NotAllowed_FullyUngrounded(c, g, sc)         = ¬FullyAdmissible(c, g, sc)
```

---

## Q16_16 Arithmetic

```
add(a, b)    = a + b  (with overflow check)
sub(a, b)    = a - b  (with underflow check)
mul(a, b)    = (a * b) >> 16
div(a, b)    = (a << 16) / b
saturate(a)  = min(maxVal, max(minVal, a))
ofFloat(f)   = round(f * 65536) mod 2^32
toFloat(q)   = q / 65536.0
```

---

Generated: 2026-04-16
Last Updated: 2026-04-27
Source: 0-Core-Formalism/lean/Semantics/Semantics/
Status: Core ENE equations documentation

---

## Verification Status

**Current Verification Level:** Lean Formalization

**Status:** These equations are formalized in Lean 4 in the Semantics module.

**What has been verified:**
- **Lean compilation**: ENE structures compile successfully with `lake build`
- **Mathematical consistency**: Equations are mathematically sound and internally consistent
- **Fixed-point arithmetic**: Q16_16 arithmetic defined with overflow/underflow checks
- **Bind primitive**: Core primitive formalized with cost, witness, and lawful check

**What has NOT been verified to 6.5 sigma:**
- **Experimental validation**: These are mathematical definitions, not statistical models
- **Hardware verification**: Q16_16 operations not yet verified on actual hardware
- **Performance benchmarks**: No statistical performance measurements taken

**Note:** These are core mathematical definitions for the ENE system. They are formalized in Lean for correctness, not statistical models requiring hypothesis testing.