Phase-Biased Geometry ~ concept

This page states the irreducible substrate, axioms, and boundaries of PBG. All derivations, results, and applications elsewhere depend on and must refer back to these principles.

1. Ontological Commitment

PBG asserts:

The only real substrate is a continuous, biasable phase structure $Φ (t, x)$ .
All observable phenomena—objects, interactions, apparent forces, and structure—are emergent from the coherence, bias, and anchoring of modes: localized, phase-coherent regions within $Φ$ .

There is no spacetime, no point particle, no underlying quantum or classical field, and no probability amplitude at the substrate level.

Modes are discrete, localized regions of phase coherence with integer phase winding $n_{ϕ}$ and a defined envelope (spatial extent).
The persistence, motion, and transformation of modes are governed by anchoring cost: the “price” paid to maintain a given configuration of phase relationships, both locally (spatial coherence) and temporally (phase stability).
Coherence bias is the fundamental cause of all apparent interactions—there are no “forces” in the usual sense.

3. The Universal Anchoring Cost Principle

All physical law in PBG arises from the minimization of the total anchoring (coherence) cost associated with sustaining phase structure:

C [Φ, {ψ_{i}}] = \int d^{3} x [α | \nabla Φ |^{2} + β \sum_{i} | ψ_{i} |^{2}] + \int d^{3} x d t \frac{1}{2} γ (\partial_{t} Φ)^{2}

$α$ quantifies spatial phase anchoring—the cost to sustain a phase gradient (spatial coherence).
$β$ quantifies envelope (mode) coherence anchoring—the cost of localizing phase structure (mode “mass” or stability).
$γ$ quantifies temporal phase anchoring—the cost to maintain phase stability in time.

No other terms or “forces” appear; all observable effects arise from this principle.

4. No Spacetime, No Classicality, No Quantum Operators

Apparent space, time, and causality emerge as relationships among modes, not as primitives.
There is no Hilbert space, no wavefunction, no probabilistic superposition at the substrate level.
Decoherence, collapse, and “measurement” are effective descriptions of mode anchoring and phase bias at large scale, not fundamental processes.

All physical “laws” and apparent interactions—such as gravity, electromagnetism, or nuclear effects—arise as gradients in anchoring cost and coherence bias between overlapping modes.
Discrete “shells,” quantized levels, and large-scale structures (e.g., atomic, planetary, galactic) are the inevitable outcome of modal phase winding, coherence minima, and cost minimization—not postulated structures or imposed quantization rules.
The “coherence kernel” (e.g., $\propto e^{- k r} / r$ ) is a derived measure of how phase bias propagates between modes, not a field propagator.

6. Calibration and Falsifiability

The only free parameters are the three anchoring costs: $α$ , $β$ , and $γ$ .
Each is calibrated once (to a distinct, cross-domain observation: e.g., light bending, speed of light, Lamb shift).
All other relations and phenomena are predictions; any mismatch is a falsification.

7. What PBG Is Not

Not a field theory (classical or quantum): there is no spacetime, field excitation, operator algebra, or background metric.
Not a quantum mechanics variant: there is no substrate Hilbert space, superposition, or probabilistic ontology.
Not a re-labeling of GR or hidden-variable models: all geometry, causality, and “forces” are emergent, not assumed.

8. Boundaries and Methodological Rules

No additional structures (dimensions, potentials, energies, backgrounds) are permitted.
All claims and predictions must reduce to phase structure, coherence, and anchoring cost.
“Force,” “field,” and “shell” are always emergent or descriptive, never substrate-level.

9. Further Reading

For mathematical derivations and empirical validation, see: Primer, Sample Physical Results, 4 random foundational derivations, PBG Concept, Appendix Master.
For a rigorous discussion of field theory differences, see: This Is Not A Field Theory.

Summary

PBG is a modal, phase-anchoring substrate: only coherence, bias, and anchoring cost are real.
All observable structure—atomic, gravitational, cosmological—emerges as phase coherence phenomena, testable and falsifiable through three calibrated anchoring costs.
No further substrate, geometry, or probabilistic mechanics is permitted; all else is emergent or illusory.

Phase-Biased Geometry: Ontological and Modal Foundations