Foundations
Coherence Geometry CG
A geometric and variational framework in which organization, stability, and evolution arise from coherence constraints acting on coupled configurations, rather than from independent variables or symbolic rules.
Coherence
The property of a system in which interacting elements maintain mutual compatibility through a shared amplitude structure. Coherence is not an external condition imposed on a system but an intrinsic organizational state — one that persists where constraints are satisfied and dissolves where they are not.
In Coherence Geometry, coherence is the primary substrate from which structure, stability, and pattern formation arise.
Amplitude
The shared scalar quantity that couples interacting elements within a coherence-geometric system. Unlike phase, which may vary independently across elements, amplitude is held in common — a conserved structural resource distributed across the system.
The sharing of amplitude is the mechanism through which coherence is maintained, and its distribution determines which configurations can form and persist.
Multi-Phase Number μ-number
A generalization of the complex number in which a single conserved amplitude is shared across multiple independent phase components. Where a complex number encodes one amplitude and one phase, a μ-number encodes one amplitude and an arbitrary number of phases, each varying independently.
The space of μ-numbers, denoted M, provides the representational substrate for coherence-geometric structure and generalizes the complex plane to distributed, coherent systems.
Constraint
A condition that must be satisfied for a configuration to persist within a coherent system. Constraints are not imposed from outside — they arise from the internal structure of interactions themselves, from the requirement that elements sharing a common amplitude remain mutually compatible.
Where constraints are satisfied, patterns stabilize. Where they cannot be satisfied, configurations dissolve. The particular form of any emergent structure is determined not by prescribed rules but by the specific constraints acting within that system. Constraints in Coherence Geometry take several forms: compatibility conditions between interacting phase fields, closure conditions that govern global structure, and coherence conditions that bound how influence propagates between elements. In each case, the constraint is a structural fact about the system — not a parameter to be tuned.
Constraints in Coherence Geometry differ from optimization constraints in conventional mathematics. In standard usage, constraints are imposed externally to bound a solution space. In CG, constraints are intrinsic — conditions the system must satisfy in order to exist as a coherent configuration at all.
Representations
Geometric and Algebraic Representations
Two equivalent representations of coherence-geometric structure at the mathematical foundation (Level 0). The geometric representation emphasizes configuration, curvature, and variational organization across coupled states, while the algebraic representation emphasizes operators, transformations, and compositional relations.
These do not define distinct frameworks, but complementary descriptions of the same underlying coherence-governed structure.
Coherence-Geometric Framework
The structural and variational organization induced by coherence constraints (Level 1), prior to any physical, informational, or semantic interpretation.
Representational Substrate
A domain-specific interpretation of coherence-geometric structure that assigns meaning to states without altering the underlying geometry or constraints (Level 2).
Examples include physical and informational substrates.
Physical Representational Substrate PRS
An interpretation of coherence geometry in which coherence-governed states are treated as physically instantiated configurations (e.g., fields, amplitudes), and coherence dynamics are interpreted as physical evolution in space and time.
Coherence-Geometric Information Substrate CGIS
An interpretation of coherence geometry in which stable coherence basins correspond to informational states, perturbations correspond to noise or uncertainty, and relaxation corresponds to inference, recovery, or decoding.
Dynamics
Coherence Triad
A structural classification of the primary modes by which coherence curvature is resolved within a representational substrate (Level 3). In the Physical Representational Substrate, these modes correspond to relaxation (CDAR), activation (CDAD), and transport (CDT). In the Coherence-Geometric Information Substrate, they correspond to inference/recovery (CDIR), storage (CDIS), and transmission (CDIT).
The triad reflects distinct expressions of a single underlying coherence-governed mechanism rather than separate frameworks or independent processes.
Coherence-Governed Dynamics CGD
The general regime of evolution within the PRS in which structure, transport, and pattern formation arise from coherence constraints acting on coupled fields.
Curvature-Driven Amplitude Relaxation CDAR
A relaxation-dominated regime of CGD in which coherence curvature drives dissipative settling into stable configurations, producing basin formation, pattern stabilization, and equilibrium-like structure.
Curvature-Driven Activation Dynamics CDAD
An activation-dominated regime of CGD in which coherence interactions sustain activity, growth, or decay through feedback between coupled configurations, producing ignition, oscillation, or self-sustained dynamics.
Curvature-Driven Transport CDT
A transport-dominated regime of CGD in which coherence is redistributed through flow-like processes, giving rise to advection, circulation, and effective conservation behavior.
Information
Geometric Encoding and Decoding GED
The operational regime within CGIS governing informational representation, perturbation, and recovery through coherence-geometric structure rather than symbolic manipulation.
Coherence-Driven Information Storage CDIS
A storage-dominant role within GED in which information persists as stable configurations within coherence basins, providing a geometric basis for memory and representational stability.
Coherence-Driven Information Transmission CDIT
A transmission-dominant role within GED in which coherence configurations propagate across channels or domains, corresponding to communication and signal transfer.
Coherence-Driven Inference and Recovery CDIR
An inference-dominant role within GED in which perturbed configurations relax toward coherent basins, giving rise to decoding, error correction, and inference.
Operations
Operational Method
A procedure that operates on states within a representational substrate to produce evolution, inference, control, or organization (Level 3).
Projection
A structural operation that maps coherence-geometric states onto reduced or effective descriptions by suppressing internal coupling and geometry for tractability (Level 4).
Generator
A perturbation, forcing, initialization, or boundary condition that excites or explores a representational substrate without determining its organizing structure.
Organizer
An intrinsic coherence-governed mechanism that stabilizes, sustains, or redistributes structure within a representational substrate, independent of specific generators.
Projector
A representation or mapping that reports organized coherence-geometric states in a reduced, conventional, or externally interpretable form.