Phase-Biased Geometry Theory

This section outlines the principles of PBG (Phase-Biased Geometry)—a coherence-based framework where all physical behaviour arises from modal phase structure, anchoring cost, and the minimisation of coherence tension.

A mode is a stable configuration of internal phase. It is not a particle, not a wave, and not an excitation of a field. It is a coherent entity defined solely by the alignment and persistence of phase structure across space.

Each mode is described by a complex-valued coherence function:

ψ (x) = ρ (x) e^{i ϕ (x)}

Here:

$ρ (x)$ is the coherence amplitude—the degree to which the mode is well-defined at each point.
$ϕ (x)$ is the phase surface—an internal structure that must remain continuous and stable for the mode to persist.

Modes are not located at a point. They extend across a region of space, shaped by their internal structure and external constraints.

If the phase surface $ϕ (x)$ becomes unstable or undefined, the mode ceases to exist. This breakdown of internal coherence is what we interpret as decay, destruction, or measurement.

Mode Types

Modes fall into three structural classes:

Latent modes: These do not anchor into the coherence field. They propagate with minimal phase disruption (e.g. photons).
Anchored modes: These minimise coherence cost by embedding into the surrounding field. They form persistent structures (e.g. electrons, nuclei).
Saturated or decaying modes: These cannot sustain coherence due to phase conflict, interference, or overload. They fragment, decay, or transform.

No classical "particle" concept is required. A mode is simply a stable bundle of phase, evolving to preserve its structure in a changing environment.

The modal function $ψ (x)$ is the only real quantity. All derived observables—mass, energy, charge—are consequences of how this function maintains or loses coherence.

(See Appendix A — Modal Evolution from Anchoring Cost
and Appendix X).

2. Anchoring and Bias

To persist, a mode must do more than remain internally coherent—it must also anchor into its surrounding environment.

The ambient modal medium is not empty. It contains phase gradients, coherence density, and interference patterns from other modes. For a mode to survive within it, its structure must fit with the surrounding coherence landscape.

This leads to the concept of anchoring.

Anchoring Cost

Anchoring is not a mechanical interaction. It is a structural alignment—a form of compatibility between a mode and its surrounding field. The cost of anchoring is quantified by the anchoring cost functional, which expresses how difficult it is for a mode to maintain coherence in a given environment.

A basic form of this cost is:

C [ψ] = \int (α | \nabla ψ |^{2} + β | ψ |^{2}) d^{3} x

Where:

The first term penalises rapid spatial variation in phase or amplitude—sharp phase gradients are energetically costly.
The second term penalises the existence of structure in places where coherence cannot be supported.

The constants $α$ and $β$ are modal stiffness parameters—defining how resistant the system is to change in phase and coherence amplitude, respectively.

The Principle of Bias

Modes evolve by minimising this cost. But they do not do so blindly—they respond to bias.

Bias is the gradient of the anchoring cost. It is the direction in which coherence can be preserved most efficiently. It is what we traditionally interpret as “force,” but without any field or mediator.

A mode drifts, accelerates, or transforms because remaining still would increase its anchoring cost.

This is the principle of bias:

Modes do not follow Newtonian inertia.
Modes do not respond to applied forces.
Modes follow bias: the path of least resistance in coherence cost space.

Dynamic Anchoring

Anchoring is not a static property. As the coherence field evolves—due to other modes, environmental changes, or phase saturation—a mode’s anchoring configuration must adapt.

If the cost of maintaining coherence rises beyond a critical threshold, the mode begins to decohere. This is the physical process underlying decay, instability, or transition.

Anchoring thus replaces:

Mass (as a measure of anchoring persistence),
Charge (as a distortion of anchoring field topology),
And force (as the bias gradient of coherence cost).

(See Appendix B — Anchoring Cost Functional and the Coherence Field, and Appendix L — Modal Self-Interaction and Gravitational Analogue.)

Modes exist only while their anchoring is sustainable.

3. Motion from Cost Minimisation

In modal dynamics, motion is not primitive. It is not a given, nor is it driven by external forces. It is an emergent property of coherence maintenance.

A mode will only move if doing so allows it to preserve its structure with lower cost. Remaining still, if it causes coherence strain, will trigger motion. Movement is a response to bias—a gradient in the anchoring cost landscape.

This is the central principle of dynamics in PBG:

Modes move to reduce coherence tension.

The Role of Bias

The anchoring cost $C [ψ]$ depends on both the internal structure of the mode and the surrounding coherence field. If a mode finds itself in a region where the cost of maintaining phase alignment is increasing, it will drift toward lower-cost regions.

This drift is governed by the bias gradient:

\frac{d^{2} x}{d t^{2}} \propto - \nabla C

This expression replaces Newton’s second law. The second derivative of position—what we call acceleration—is no longer caused by “force,” but by the need to reduce anchoring stress.

Why This Isn’t a Force

Traditional physics treats motion as a response to force, whether gravitational, electromagnetic, or otherwise. Modal dynamics rejects that view. In this framework:

There are no forces.
There are no fields applying vectors.
There is only coherence preservation in a structured environment.

A mode’s motion is a byproduct of maintaining its internal phase in an evolving landscape. No external “thing” pushes or pulls it. The bias arises from the structure itself, and motion is a side effect of coherence survival.

Consequences

This reinterpretation dissolves:

The need for gravitational curvature
The concept of inertial mass
The metaphysics of field interactions

Instead, motion becomes structural, governed entirely by the geometry of coherence cost.

Photon Motion

Latent modes, like photons, do not anchor. Yet they still respond to coherence gradients. Their path is altered not by force, but by decoherence penalty—an increase in anchoring tension that would break phase if uncorrected.

Thus, even unanchored modes curve their trajectory to avoid disintegration.

(See Appendix B — Anchoring Cost Functional and the Coherence Field, and Appendix I — Strong and Weak Interactions from Modal Anchoring.)

Motion in PBG is not the cause of change. It is the result of attempting to stay the same.

4. Coherence Fields and Saturation

Each mode, by existing, affects the coherence structure around it. It does not radiate or project force—it modifies the anchoring landscape into which other modes must embed.

This collective influence is described by a coherence field $B (x)$ —a scalar structure representing the availability and density of coherence support in space.

Emergence of the Coherence Field

The field $B (x)$ is not fundamental. It is not an entity in itself. It emerges from the collective anchoring effects of many modes. Where modes cluster, the local coherence density increases. Where coherence is sparse, $B (x)$ falls off.

A derived form of the field (from cost minimisation) is:

B (r) = \frac{A}{r} e^{- k r}

This is a Yukawa-like field:

The $\frac{1}{r}$ factor reflects geometric spreading
The exponential decay $e^{- k r}$ expresses coherence loss with distance
The decay constant $k$ is set by the stiffness of the coherence medium

The shape of this field is not arbitrary. It arises directly from a variational principle that minimises anchoring cost across space.

Role of the Field

$B (x)$ determines how easily a new mode can anchor in a given location. In regions of high coherence density, anchoring is easy. In sparse regions, the cost is higher.

Modes that enter a region with steep coherence gradients must adapt, re-anchor, or collapse.

Thus, the coherence field replaces:

Gravitational potential
Electromagnetic field strength
Classical scalar and vector potentials

It is not a force carrier. It is a constraint landscape—shaping where coherence can persist.

Saturation and Suppression

However, coherence is not unlimited.

In regions where too many modes overlap, the field saturates. Anchoring becomes destructive rather than stabilising. Modes interfere, suppress each other, or fragment.

This leads to:

Exclusion: Two anchored modes cannot coexist in the same coherence domain
Suppression: A mode entering a saturated region is repelled or forced to decay
Quantisation: Only certain modal structures survive in confined environments

Saturation is not a consequence of interaction. It is a constraint imposed by the coherence medium itself.

(See Appendix AE — Continuum Mechanics and Full Variational Derivations for full derivations)

The coherence field $B (x)$ defines not what a mode feels, but what it can become.

5. Interaction, Decay, and Structure

In a coherence-based framework, interaction is not a force-mediated event. It is the structural consequence of multiple modes trying to anchor within a shared coherence field.

Each mode reshapes the anchoring landscape. When two or more modes enter proximity, their coherence profiles overlap, distort, and interfere. This interaction is not additive—it is competitive and saturable.

Structural Interaction

Two modes do not push or pull each other. Instead, they modify each other's anchoring viability. If their phase structures are compatible, they may mutually stabilise, forming a bound system. If they conflict, they may disrupt each other’s coherence and trigger decay.

The nature of their interaction depends on:

The gradient of the shared coherence field $B (x)$
The degree of overlap between their phase profiles
The anchoring cost created by their combined presence

This explains:

Bound systems (atoms, molecules, orbits)
Scattering events (temporary phase disruption followed by separation)
Fusion or decay (irreversible structural reconfiguration)

Interaction is geometric and phase-structural, not energetic.

Coherence Conflict and Decay

When the cost of maintaining coherence exceeds a threshold, the mode can no longer remain stable. This can occur due to:

Entering a saturated coherence field
Interfering destructively with another mode
Internal phase tension exceeding anchoring support

Decay is not caused by instability in an energy state. It is the structural failure of phase maintenance.

There is no collapse, no quantum jump—only the irreversible breakdown of coherence.

Decay paths are not probabilistic—they are structurally constrained:

If a mode decoheres, it fragments into lower-cost modes
The result is determined by what coherence structures are permitted by the surrounding field

Emergent Structure

Persistent structures arise when modes find local minima in the coherence cost landscape. These are not bound by potential wells or energy levels—they are coherence-stable topologies.

Examples include:

Atoms: Anchored configurations of compatible modal clusters
Orbits: Recurring phase-compatible trajectories in gradient fields
Galaxies: Large-scale coherence-stabilised envelopes of modal emitters

What appears to us as stable structure is, in modal terms, simply the least destructive arrangement of phase under bias.

(See Appendix AA — Modal Thermodynamics, Appendix AB — Modal Statistics, Appendix AC — Coherence Class, and Appendix AD — Chiral Anchoring.)

Structure is not held together by forces.
It is held together by what coherence will allow.

6. Unified Action Principle

The evolution of modes, their anchoring, motion, interaction, and decay can all be derived from a single principle:

Coherent modes evolve to minimise their total anchoring cost.

This is not a heuristic—it is a variational principle. It replaces the Lagrangian formalism of classical mechanics and quantum field theory, and the geometric minimisation of general relativity. In the coherence framework, all physical evolution arises from the minimisation of a unified action functional.

The Anchoring Action

For a single mode $ψ (x, t)$ evolving in spacetime, the action is defined as:

S [ψ] = \int (- γ | \partial_{t} ψ |^{2} + α | \nabla ψ |^{2} + β | ψ |^{2}) d^{4} x

Each term has a distinct role:

The temporal gradient term penalises rapid phase evolution over time.
The spatial gradient term penalises sharp changes in phase or coherence across space.
The magnitude term penalises unsupported coherence.

Together, they define the full anchoring cost for the mode in spacetime.

Minimising this action under appropriate boundary conditions yields the mode’s natural evolution. This principle governs:

Stationary anchoring (e.g. persistent particles)
Bias-following motion (e.g. curved trajectories)
Decay and fragmentation (e.g. coherence collapse)
Stabilisation into structures (e.g. atoms, orbits)

Comparison to Classical Formalisms

In conventional physics:

Field theory minimises a Lagrangian density $L (ϕ, \partial_{μ} ϕ)$
Relativity minimises proper time or geodesic curvature
Thermodynamics maximises entropy or free energy

All these are formalisms designed around specific metaphysical assumptions—particles, fields, spacetime curvature, statistical ensembles.

In contrast, this action minimises a single cost functional that:

Makes no assumption about particles or spacetime
Depends only on coherence, anchoring, and structure
Yields classical, quantum, and gravitational-like behaviours as emergent regimes

Beyond Fields and Geometry

There are no separate field equations. No fundamental force terms. No geometric constraints imposed from the outside.

Everything evolves from a unified drive: preserve coherence at minimal cost.

(See Appendix J — Casimir Pressure, Appendix M — Magnetic Moment, and Appendix Q — Galactic Coherence Fields.)

This is not the action of a system.
It is the action of structure itself.

7 — Derived Phenomena and Predictive Consequences

The preceding sections establish the formal structure of PBG: modal coherence, anchoring dynamics, coherence fields, and structural saturation. From these foundations, a range of physical consequences emerge naturally—without additional axioms, forces, or imposed particle identities.

This section outlines key predictive outcomes of the framework. Each arises directly from the anchoring principles and coherence field dynamics established above, and is formally derived in the appendices.

In PBG, cosmological redshift is not a consequence of spacetime expansion. Instead, it arises from the accumulated structural decoherence of phase-preserving modes as they propagate through an evolving coherence field. The photon does not stretch—it partially decoheres, losing coherence density and altering its modal structure.

This mechanism predicts a redshift–distance relation that converges with $Λ$ CDM at low redshift but diverges at high $z$ , offering a direct test. Time dilation and photon flux remain conserved, but the underlying cause is coherence tension rather than recessional motion.

(See Appendix C — Redshift from Anchoring Drift.)

7.2 Gravitational Lensing from Coherence Gradients

PBG does not treat light as following null geodesics in curved spacetime. Instead, light is a coherence-sustaining mode that follows the least-decohering path through a structured bias field. The coherence field $B (x)$ generated by matter distributions causes curved trajectories due to anchoring penalties that vary with transverse displacement.

This reproduces gravitational lensing, including the solar deflection of starlight and galactic arcs, without invoking curvature or metric tensors. At first approximation, lensing follows the same angular predictions as general relativity, but deviations arise in systems with coherence amplification or suppression.

(See Appendix D — Lensing via Coherence Gradients, Appendix AL solar lensing.)

7.3 Fermions and Bosons from Anchoring Saturation

Standard quantum statistics classify particles by symmetry postulates. In PBG, these classes emerge structurally. The coherence class $χ$ is a measure of how readily identical modes can co-anchor. Modes with destructive interference or overlapping saturation resist duplication (fermion-like), while phase-aligned modes can reinforce (boson-like).

No antisymmetrisation rule is imposed. Instead, the exclusion principle is a geometric outcome of modal overlap, anchoring tension, and coherence instability. Partial classes and intermediate statistics emerge naturally from structural conditions.

(See Appendix AB and Appendix AC.)

7.4 Collapse and Measurement from Anchoring Instability

Measurement and wavefunction collapse are reinterpreted in PBG as anchoring transitions. A coherent mode subject to environmental saturation or phase interference will destabilise and undergo irreversible decoherence. This process is not instantaneous nor nonlocal, but structural—anchored in the geometry of coherence fields.

Classical outcomes correspond to persistent anchoring minima. Track formation, detector clicks, and localisation all arise as the mode sheds coherence under dynamic constraints. Collapse is thus not fundamental, but an emergent loss of structural reversibility.

(See Appendix T — Anchoring Instability and Measurement.)

7.5 Time and the Arrow of Decoherence

Time in PBG is not a geometric coordinate but a statistical flow of modal coherence. The arrow of time corresponds to net decoherence—a direction in which phase-coherent structures decompose. This provides a structural explanation for entropy growth, irreversibility, and the alignment of thermodynamic, cosmological, and quantum time.

No external time variable is imposed. Modal turnover, anchoring delay, and decoherence rate define temporal direction from within the system.

(See Appendix U and Appendix Q.)

Energy, temperature, and entropy in PBG are not statistical over microstates, but statistical over coherence. A hot system is one in which modal anchoring turnover is rapid. Equilibrium corresponds to structural stasis in coherence exchange. The heat death is not inevitable—recoherence is permitted.

This resolves the asymmetry of time, the low-entropy early universe, and the end-state of the cosmos without imposing a boundary condition or invoking inflation.

(See Appendix AA — Modal Thermodynamics and Appendix Q — Coherence Ensembles.)

PBG replaces classical fields with modal interference. What we call “electric” or “magnetic” behaviour arises from phase gradients and coherence flow. The photon is a coherence-sustaining phase mode, and charge is a topological asymmetry in modal structure.

The Lorentz force emerges from biased phase propagation, not interaction with a field. Maxwell’s equations are reinterpreted as modal coherence conditions.

(See Appendix E — Electromagnetism from Phase Interference, Magnetic Moment of Protons and Neutrons.)

7.8 Gravity as a Coherence Field

In PBG, gravity is not a force and not curvature. It is a consequence of mutual coherence biasing: large bodies emit structured coherence fields, and anchored modes follow the least-cost path within that structure. Geodesic motion emerges as a statistical outcome, not a geometric axiom.

The coherence field $B (x)$ evolves from the Helmholtz equation under modal saturation, producing a Yukawa-type field that governs both orbits and light bending.

(See Appendix H and Appendix AE.)

7.9 Constants from Coherence Structure

PBG allows the derivation of natural constants from anchoring conditions. The speed of light $c$ arises from a coherence propagation constraint in vacuum. Modal sensitivity constants such as $γ_{0}$ , $β$ , and the coherence kernel strength $k$ are set by phase anchoring structure, not arbitrarily.

This framework offers testable paths to derive or constrain values that are free parameters in the Standard Model.

(See Appendix K — Speed of Light from Coherence Anchoring.)

7.10 The CMB as a Coherence Fossil

The cosmic microwave background is not a relic radiation field from recombination, but a global shell of persistent modal interference. The CMB in PBG is a coherence structure still present today—an evolving harmonic envelope with embedded acoustic interference.

The power spectrum and polarisation patterns reflect modal anchoring and turnover structure, not photon–electron scattering.

(See Appendix R — CMB Shell Structure.)

Next: SM Companion — where mass, charge, and decay emerge from modal anchoring

Theory: Modal Dynamics

1. Modal Structure