Instancology: A Formal Framework for Quantum-Cosmological Resolution

1. Foundational Axioms: The Instance-Field

Let the instance-field ℐ be a dynamical, layered network structured by the issuance of discrete units called instances. Each instance iₖ ∈ ℐ is defined as a tuple:

iₖ = (eₖ, τₖ, ℛₖ)

Where:

- eₖ: An eventual emergence (e.g., a quantum transition, a spacetime articulation).

- τₖ: A temporal span (the persistence duration of the instance).

- ℛₖ: A relation set (directed and weighted links to other instances, possibly defined via sheaf cohomology).

Reality stabilizes when dℛₖ/dτₖ ≤ Θ_decay.

2. Quantum Mechanics: Collapse via Instance Saturation

Modified Born Rule:

P(j) = |⟨ψⱼ | ψ⟩|² / Σₖ S(iₖ, iⱼ), where S(iₖ, iⱼ) is instance-overlap.

Prediction: Decoherence deviates from unitary predictions when relational entanglement is low.

Test: Weak measurement setups detect non-Markovian decoherence spikes near Θ_decay.

3. Spacetime as a Relational Instance-Network

The spacetime metric g_μν(x) emerges from instance-density ρ(ℐ):

g_μν(x) ∝ log(∫_{x-Δx}^{x+Δx} ρ(ℐ) dτ)

When ρ(ℐ) → ∞, instance-field fractures (no point-singularity).

Test: Gravitational wave detectors should observe non-Poissonian holographic noise.

4. Consciousness as Instance-Binding Dynamics

C = {iₖ | ℛₖ ∩ ℛ_self ≠ ∅}, i.e., instances resonating with self-binding structure.

Implication for AI: Consciousness requires relational instancing, not mere computation.

5. Experimental Forecasts and Deviations

Prediction                         | Experiment                          | Expected Signature

Instance-collapse decoherence      | Weak measurement + tomography       | Non-Markovian spikes at Θ_decay

Spacetime granularity              | Holometer/Fermi-GBM reanalysis      | Log-fractal holographic noise

Entanglement as relational loss    | Bell tests w/ instance suppression  | Visibility decay ∝ S(iₖ, iⱼ)

6. Philosophical Leap, Scientific Anchor

- Not metaphysics alone: Falsifiable by relational-temporal perturbation.

- Unified substrate: QM, GR, and consciousness within the instance-field.

- Mathematical novelty: Sheaves, stochastic decay, topological fragmentation.

7. Concluding Frame: Instancology as Post-Theory Epistemics

Instancology identifies the conditions under which a universe appears as an instance.

In the deepest layer, not things, but issuance—not causes, but coherence—not emergence, but instance.

Future science must trace relational issuance across scales, minds, and machines.