Collapse-λ, Entangled Bayesian Collapse, and Orch OR Integration

Collapse-λ is our experimental framework for simulating quantum-aware, recursive computation and modeling proto-conscious behavior. Built using Lisp (symbolic recursion), Haskell (typed monadic interpretation), and Python/Qiskit (quantum tensor simulation), Collapse-λ goes beyond standard quantum computing. It treats collapse, not computation, as the fundamental operation.

In this post, we introduce support for Bayesian-weighted collapse, entangled qubit groups, and incorporate key ideas from Orchestrated Objective Reduction (Orch OR)—a leading quantum theory of consciousness.

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I. Background: From Collapse to Consciousness

Most quantum computing systems rely on unitary evolution followed by measurement. However, in Collapse-λ, we adopt a different approach: collapse is primary. It is not just a measurement artifact, but a decision—the resolution of potential into actuality.

In cognitive terms, collapse is the moment a thought "becomes" conscious, when ambiguity is resolved. This parallels Penrose and Hameroff's Orch OR theory, which suggests that conscious events are orchestrated collapses occurring in the brain's microtubule structures.

Collapse-λ aims to simulate this concept algorithmically, combining symbolic structure, probability, and timing to model consciousness-like processes.

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II. Collapse-λ Tri-Layer Architecture

Collapse-λ operates across three layers:

1. Lisp DSL – The Symbolic Layer

This layer defines recursive, symbolic collapse operations like:

lisp
(⊕ alive dead superposition)
(collapse-until ψ predicate)
(λ (s) (return (collapsed-to s)))

Lisp's homoiconicity makes it ideal for modeling self-referential computation, symbolic recursion, and abstract cognitive processes.

2. Haskell Core – The Monadic Engine

Haskell interprets the symbolic structure as monadic computations. It evaluates collapse expressions like:

haskell
data CollapseExpr
  = Collapse CollapseExpr
  | Superpose [String]
  | Return String

interpretCollapse :: CollapseExpr -> IO String

This layer models deterministic reasoning, control flow, and collapse orchestration—like a proto-cognitive scheduler.

3. Python + Qiskit – The Quantum Runtime

The bottom layer simulates quantum entanglement, real amplitude evolution, and now includes Bayesian filters and Orch OR timing constraints. It outputs observed collapse outcomes, updated posteriors, and timing data.

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III. Bayesian Collapse: Prior-Guided Resolution

In standard quantum mechanics, collapse outcomes are determined by the Born rule, based on amplitude squared. In Collapse-λ, we allow Bayesian priors to guide collapse. This allows a system to "prefer" certain outcomes based on memory, context, or informational geometry.

Priors

python
priors = {
    'alive': 0.7,
    'dead': 0.2,
    'superposition': 0.1
}

These represent the system's prior beliefs about each possible outcome.

Bayesian Collapse Function

python
def bayesian_collapse(priors):
    states = list(priors.keys())
    weights = list(priors.values())
    return random.choices(states, weights=weights, k=1)[0]

This models an intentional selection from superposed outcomes, capturing something like cognitive bias or expectation.

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IV. Entangled Collapse Groups

To model distributed superpositions like those proposed in Orch OR (across microtubules), we allow multiple qubits to be grouped into entangled units.

Defining Entangled Groups

python
entangled_map = {
    "alive+ready": [0, 1],
    "dead+waiting": [2, 3]
}

entangled_priors = {
    "alive+ready": 0.6,
    "dead+waiting": 0.4
}

This allows us to treat multi-qubit states as single logical units, with semantic meaning (e.g., readiness, awareness).

Quantum Circuit Simulation

We use Qiskit to build the entanglement:

python
def build_entangled_circuit():
    qc = QuantumCircuit(4)
    qc.h(0)
    qc.cx(0, 1)
    qc.h(2)
    qc.cx(2, 3)
    return qc

This creates two Bell pairs: qubits (0,1) and (2,3) are entangled, representing distinct cognitive structures.

Extracting Group Amplitudes

We aggregate amplitudes based on the groupings:

python
def group_amplitudes(statevec, entangled_map):
    amp_dict = statevec.probabilities_dict()
    group_probs = {k: 0.0 for k in entangled_map}

    for bitstring, prob in amp_dict.items():
        for group, qubits in entangled_map.items():
            positions = ''.join([bitstring[-(q+1)] for q in qubits])
            if positions == '11' or positions == '00':
                group_probs[group] += prob
    return group_probs

This gives us the raw amplitudes (probabilities) for each entangled cognitive state.

Bayesian Filtering

We multiply prior belief with amplitude likelihood to get a posterior:

python
def bayesian_filter(amp_probs, priors):
    bayes = {k: amp_probs[k] * priors[k] for k in priors}
    total = sum(bayes.values())
    return {k: v / total for k, v in bayes.items()}

This implements:

$P(h|D) = \frac{P(D|h) \cdot P(h)}{\sum P(D|h) \cdot P(h)}$

where:

• (P(h)) is the prior
• (P(D|h)) is the amplitude-based likelihood
• (P(h|D)) is the resulting posterior

Final Collapse

python
def sample_collapse(posterior):
    states = list(posterior.keys())
    weights = list(posterior.values())
    return random.choices(states, weights=weights, k=1)[0]

This selects one state, completing the collapse.

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V. Orch OR Timing Integration

According to Penrose, collapse happens when the gravitational self-energy (E_G) of the superposition reaches a critical threshold:

$\tau \sim \frac{\hbar}{E_G}$

We simulate this in Python:

python
def objective_collapse_time(E_G):
    hbar = 1.0545718e-34
    return hbar / E_G

In our collapse system, we delay collapse until the Orch OR time (\tau) is reached.

python
if elapsed_time >= objective_collapse_time(E_G):
    collapsed_state = sample_collapse(posterior)

This models spontaneous, gravitationally-induced collapse without external measurement—core to Orch OR.

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VI. Conscious Collapse Loop

We now define a conscious decision loop in Lisp:

lisp
(loop for ψ in *thoughts*
      for entangled = (build-structure ψ)
      for τ = (calc-orch-collapse-time entangled)
      until (time > τ)
      finally (collapse ψ via entanglement))

This structure models recursive, delayed, and structured thought. Collapse becomes:

• Time-bound
• Structure-aware
• Intention-guided

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VII. Sample Output

Given the following:

python
priors = {
    "alive+ready": 0.7,
    "dead+waiting": 0.3
}

entangled_map = {
    "alive+ready": [0, 1],
    "dead+waiting": [2, 3]
}

Output:

json
{
  "collapsed": "alive+ready",
  "posterior": {
    "alive+ready": 0.82,
    "dead+waiting": 0.18
  },
  "amplitudes": {
    "alive+ready": 0.6,
    "dead+waiting": 0.4
  }
}

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VIII. Summary

We have extended Collapse-λ to model the full computational machinery of Orch OR:

With these additions, Collapse-λ simulates:

• Thought as symbolic recursion
• Collapse as intentional resolution
• Consciousness as structured quantum reduction

This work represents the computational scaffolding of an emergent mind—one built not from bits alone, but from collapse, probability, and form.