Phase Differential Theory
Phase Differential
A Framework for Reality
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Glossary

Working
vocabulary.

The glossary defines the terminology used throughout the Phase Differential Theory research programme. Wherever possible, established scientific terminology is retained. New terms are introduced only where the framework requires concepts not adequately described by existing language.

Current foundations

Terms from the R00 to R13 suite, each with a plain-language meaning, a formal meaning where useful, and its source. The same definitions are used on the mathematics page and in the concept explorer.

The finite-capacity “Axiom 0” of the earlier paper PDT I is a historical term and is distinct from A0 (Distinction) in R01.

Groupoid

A collection of things with reversible comparisons between them.

A small category in which every arrow has an inverse: identities, inverses and associative composition where defined.

R01Mathematics

Comparison arrow

One comparison from one element to another.

An arrow a:x→ya: x \to y of the groupoid GG with objects in XX.

R01Mathematics

Phase functor

The rule assigning a phase to every comparison, compatible with chaining and reversing.

A functor Φ:G→BU(1)\Phi: G \to BU(1) with Φ(b∘a)=Φ(b)Φ(a)\Phi(b\circ a)=\Phi(b)\Phi(a), Φ(idx)=1\Phi(\mathrm{id}_x)=1.

R01Mathematics

Loop phase

The combined phase after going round a closed chain of comparisons.

The product of edge phases around an oriented closed path; equal to 1 for pure vertex-phase assignments.

R02Mathematics

Faithful phase encoding

Writing each phase as an angle without losing information about the phase itself.

U(1)≅R/2πZU(1) \cong \mathbb{R}/2\pi\mathbb{Z}; it does not give every full transport arrow a distinct angle.

R01Mathematics

Phase carrier

The smallest real space in which phases can be represented without loss.

The minimal faithful continuous real linear representation of U(1)U(1), of real dimension two (Theorem R02.2).

R02Theorem R02.2 statementMathematics

Candidate record

One of the possible outcomes being considered in an episode.

A member of a finite, exhaustive family of mutually exclusive records specified by the model.

R01R03Mathematics

Admissibility law

The model's rule for which candidates are still allowed.

A map s↦C(s)s \mapsto C(s) with C(s2)⊆C(s1)C(s_2)\subseteq C(s_1) for s2≥s1s_2 \ge s_1, fixed from antecedent inputs.

R01R03Mathematics

Antecedent inputs

Information fixed before the outcome is known or used.

R01Mathematics

Attained singleton

The first stage at which exactly one candidate is left, when such a stage actually occurs.

s∗=min⁡{s:∣C(s)∣=1}s^* = \min\{s : |C(s)| = 1\}, which must exist, be finite and precede extinction.

R01R03Mathematics

A6 realisation

The adopted principle that the last coherent survivor becomes an actual record.

At the attained singleton s∗s^*, the sole candidate is appended to the history and the episode stops. Also called the last coherent survivor principle; this is the current account of what was earlier called phase snap.

R01Mathematics

Retained history

The list of records so far, which grows without rewriting earlier entries.

hn+1=(r1,…,rn,rn+1)h_{n+1} = (r_1,\dots,r_n,r_{n+1}) preserves the prefix hnh_n.

R01Mathematics

Episode parameter

The label ordering the stages of one selection episode.

The parameter ss, distinct from the record count nn and calibrated time tt.

R01R08Mathematics

Physical clock

A physical process matched to measured duration.

A clock map together with calibration inputs; not determined by record order alone.

R08Mathematics

FCS

Finite Canonical Selection, an optional extra assumption.

Finite Canonical Selection supplies a nonempty finite admissible set invariant under a specified symmetry. It is an additional assumption when used. It does not necessarily select one member.

R01Mathematics

Density operator

The mathematical description of a quantum state, allowing for uncertainty.

ρ≥0\rho \ge 0 with Tr⁡ρ=1\operatorname{Tr}\rho = 1 on a finite-dimensional Hilbert space H\mathcal{H}.

R04Mathematics

Effect

The part of a measurement corresponding to one outcome.

An operator 0≤Er≤I0 \le E_r \le I; a measurement satisfies ∑rEr=I\sum_r E_r = I.

R04Mathematics

Quantum instrument

A description of a measurement that gives both outcome probabilities and the state afterwards.

Maps Ir\mathcal{I}_r with p(r)=Tr⁡[Ir(ρ)]p(r) = \operatorname{Tr}[\mathcal{I}_r(\rho)].

R04R05Mathematics

Kraus operator

A building block of an instrument for one outcome.

Ir(ρ)=∑αKrαρKrα†\mathcal{I}_r(\rho) = \sum_\alpha K_{r\alpha}\rho K_{r\alpha}^\dagger.

R04R05Mathematics

Exact exclusion

Removing a candidate with certainty in an idealised instrument.

The construction of Theorem R05.1 using KS,r=(PS−Pr)/m−1K_{S,r} = (P_S - P_r)/\sqrt{m-1}.

R05Theorem R05.1 statementMathematics

Operational record stability

Whether a record stays reliable in a real, noisy device.

A separate claim from exact exclusion; small full-rank perturbations can remove exact zero effects (Theorem R05.4).

R05Theorem R05.4 statementMathematics

Internal metric

A way of measuring size inside the mathematical carrier.

A positive metric on an internal carrier; not by itself a physical Lorentzian spacetime.

R07Mathematics

Physical spacetime

The space and time used by physical systems, with its causal structure.

A common four-dimensional Lorentzian metric, premise G1 of R10.

R10Mathematics

Calibration

Fixing a physical scale using a measured input.

An empirical anchor such as the electron mass fixes a scale; it does not predict that value or dimensionless constants.

R08Mathematics

Conditional theorem

A result that is proved to hold whenever its stated assumptions hold.

R00Mathematics

Standard terms

Terms with established meanings in physics and mathematics, included here with the precise sense in which they are used across the research programme.

Calibration

The assignment or determination of a physical scale using specified inputs. An empirical calibration must be distinguished from an independently derived prediction.

Coherence

In standard physics, coherence describes stable phase relationships that permit interference. PDT also uses coherence in proposed mathematical and physical conditions. These uses must be identified by their source and should not be treated as interchangeable.

Decoherence Envelope

The scale over which a coherent envelope decays under interaction with its environment. In PDT, the envelope is interpreted as the loss of phase relation between system and surroundings rather than as a fundamental collapse. Related terms: Coherence, Phase Drift.

Interference

The superposition of phase-related contributions producing characteristic patterns of reinforcement and cancellation. In PDT, interference patterns are read as direct evidence of the underlying phase relations between participating components. See also: Phase Differential, Coherence.

Measurement

In standard physics, the process by which a property acquires a definite value through interaction with an apparatus. In the current foundations, quantum probabilities are established under stated operational premises (R04 to R06), specified exclusion instruments are modelled separately (R05), and A6 adopts the first attained singleton as the realisation event. Related terms: Phase Snap.

Phase difference

An angular difference between phase values, understood modulo a full turn. It does not by itself have units of mass or energy.

Physical model

A specification connecting mathematical structures to physical quantities, dynamics and observations. Its additional assumptions should be identified.

Resonance

A condition in which a system supports a stable, self-reinforcing response at a particular frequency. In PDT, resonance is treated as a regime of sustained phase locking between coupled structures. Related terms: Phase Locking, Coherence.

PDT terms

Terminology introduced by Phase Differential Theory, or used in a specific PDT sense, to name concepts the framework requires that are not adequately described by existing language.

Canonical Phase Processes

A small set of recurring phase operations, including locking, echo, inversion and drift, used as descriptive vocabulary. Snap refers to realisation, which the current foundations treat through A6. See also: Phase Locking, Phase Echo, Phase Snap.

Coherence Length

A characteristic length over which phase relations remain correlated within a given structure or model. Related terms: Coherence, Phase Locking.

Coherence Threshold

A model-specific limit on a coherence quantity used in particular earlier threshold models and visualisations. It is not a universal measured threshold, and A6 does not supply one. Related terms: Phase Snap, Coherence.

Deterministic Phase Evolution

A hypothesis explored in some earlier models that phase configurations evolve deterministically, with apparent randomness arising from coarse-graining. It is not established as universal, and quantum probabilities are not derived from coarse-graining alone. Related terms: Phase Transport.

Deterministic Phase Snap

A particular earlier PDT model in which a coherent superposition resolves into a single outcome once a model-specific coherence threshold is met. It is not an established universal mechanism. The current foundations use the A6 realisation commitment instead. See also: Phase Snap, Measurement.

Drift Tensor

A second-rank object capturing how phase drift varies with direction across a region. The drift tensor plays a central role in the recovery of effective geometric and dynamical equations within PDT. Related terms: Phase Curvature, Emergent Geometry.

Emergence Chain

The order in which topics are treated, from phase differential through coherence, geometry, matter and interactions, to time. It is not a completed derivation: each later step uses additional physical identifications and model assumptions. Related terms: Emergent Geometry, Emergent Time.

Emergent Geometry

An effective geometric description, including notions of distance, curvature, and locality, recovered within PDT from coarse-grained behaviour of the underlying phase structure. Related terms: Relational Geometry, Phase Curvature.

Emergent Time

The current foundations distinguish record order, episode development and physical clocks (R08). A physical clock requires identification and calibration beyond record order. See also: Emergent Geometry.

Phase

In the foundational phase assignment, phase is represented by a point on the unit circle. It is dimensionless. A physical interpretation requires the appropriate model and mapping.

Phase Curvature

A measure of the failure of phase transport to close around small loops in the phase structure. In PDT, phase curvature provides the relational quantity from which effective geometric curvature is reconstructed. See also: Phase Transport, Emergent Geometry.

Phase Differential

The directed difference of phase between two relations, written ΔΦ. PDT takes ΔΦ as the fundamental primitive of the framework: a minimal relational quantity that presupposes no background metric, geometry, or coordinate system. Most other quantities in the theory are constructed from operations on ΔΦ. Related terms: Phase Transport, Phase Curvature.

Phase Drift

A continuous, generally small change of phase relations across a system over time or space. In PDT, drift is the mechanism through which coherent structure gradually loses or reorganises its phase alignment. See also: Phase Shear, Decoherence Envelope.

Phase Echo

A canonical process in which a phase configuration partially reconstitutes itself after dispersal, due to underlying time-symmetric phase dynamics. Phase echoes serve as diagnostic signatures in PDT analyses of memory-like behaviour in coherent systems. Related terms: Canonical Phase Processes.

Phase Field

A continuous assignment of phase relations across a region of the underlying relational structure. In PDT, the phase field is the working object on which propagation, curvature, and locking are defined. See also: Phase Structure, Phase Propagator.

Phase Hierarchy

A layered organisation of phase structures, with each level supporting effective dynamics that feed into the next. Phase hierarchies provide the bridge in PDT between microscopic phase relations and macroscopic phenomena. Related terms: Stage, Emergence Chain.

Phase Inversion (Anti-Flip)

A canonical phase operation in which the sign of a phase relation is reversed, mapping a configuration into its anti-aligned counterpart. PDT uses phase inversion in the description of conjugate states and certain conserved relational quantities. See also: Canonical Phase Processes.

Phase Locking

A regime in which the phase differential between two or more relations is held stable over time, so that the participating structures evolve in coordinated fashion. Phase locking is central to the formation and persistence of coherent matter and macroscopic structure in PDT. Related terms: Resonance, Coherence.

Phase Network

A connected set of phase relations, often represented graph-theoretically, used in PDT to analyse propagation, locking, and information flow within complex structures. See also: Phase Structure, Phase Propagator.

Phase Pinning

A configuration in which a phase relation becomes locally fixed by the surrounding structure, resisting drift even under perturbation. Phase pinning underlies the stability of localised excitations within PDT. Related terms: Phase Locking, Phase Saturation.

Phase Propagator

The object describing how a phase configuration propagates between relations under the basic dynamics of PDT. Effective propagators in different regimes give rise to gravitational-like, electromagnetic-like, and short-range behaviours within a single framework. See also: Phase Transport, Phase Network.

Phase Saturation

The regime in which the local update rate of phase relations approaches its maximum admissible value, beyond which further coherent change cannot be supported. Phase saturation features in PDT discussions of horizons, extreme densities, and limiting dynamics. See also: Coherence Threshold.

Phase Shear

A spatially varying gradient of phase drift across neighbouring relations, producing directional differences in how phase evolves. Phase shear enters PDT analyses of decoherence, anisotropy, and structure formation. See also: Phase Drift.

Phase Snap

The earlier name for realisation. The current account is A6, the last coherent survivor principle: when a specified admissibility process first reaches a finite stage with exactly one candidate remaining, that candidate is taken to be realised and recorded. See A6 realisation under Current foundations.

Phase Structure

The full relational network of phases, drifts, and couplings on which PDT dynamics are defined. Phase structures range from minimal local configurations to extended cosmological assemblies. Related terms: Phase Network, Phase Hierarchy.

Phase Transport

The rule that specifies how a phase value is carried from one relation to another along a path through the phase structure. Phase transport plays a role in PDT analogous to that of parallel transport in differential geometry. Related terms: Phase Curvature, Relational Geometry.

Primitive

A quantity, object, or relation treated as fundamental within a given framework, not derived from anything more basic. In PDT, the primitive is phase differential (ΔΦ). See also: Phase Differential.

Relational Geometry

A geometric description constructed entirely from relations between phases, without presupposing a background metric or coordinate system. PDT works in a relational-geometric setting from which familiar geometric notions are subsequently recovered. See also: Phase Differential, Emergent Geometry.

Rosette

A six-fold phase configuration used as a model or visualisation. It is not derived from the minimal phase axioms alone. Related terms: Six-Fold Symmetry.

Six-Fold Symmetry

The discrete rotational symmetry of the rosette model or visualisation. A unique sixfold structure does not follow from the minimal phase axioms alone. See also: Rosette.

Stage

A successive layer of construction within PDT, in which higher-order structures, units, or effective theories are recovered from lower-level phase relations. See also: Emergence Chain.

Vacuum Transition

A reorganisation of the underlying phase structure between distinct low-energy configurations, treated within PDT as a structured transition rather than a featureless change of state. Related terms: Phase Structure, Phase Hierarchy.

Research programme

Terms describing how the research programme itself is structured, communicated, and held open to independent scrutiny.

Active Research

Lines of investigation currently being pursued within the research programme, distinguished from results that are already well established. Active research is reported openly, including provisional findings and unresolved questions. See also: Framework Status, Open Questions.

Conditional result

A conclusion established under explicitly stated assumptions. Whether those assumptions apply physically is a separate question.

Experimental Programme

The set of proposed and ongoing experimental investigations associated with PDT, including precision interferometry, short-range gravity tests, and entangled-photon studies. See also: Falsifiable Prediction, Independent Verification.

Falsifiability Contract

The public commitment that each component of the framework is paired with quantitative predictions whose failure would withdraw or revise that component. The falsifiability contract is recorded openly and not adjusted retroactively. Related terms: Falsifiable Prediction, Framework Status.

Falsifiable Prediction

A specific, quantitative prediction whose failure at stated precision would refute the corresponding part of the framework. PDT lists its falsifiable predictions explicitly across the experimental programme. See also: Experimental Programme, Falsifiability Contract.

Framework Status

The continuously maintained record of which parts of the framework are well developed, which are under active investigation, and which have been revised or withdrawn in response to evidence. Related terms: Active Research, Open Questions.

Independent Verification

Replication, scrutiny, and critique of PDT results by researchers outside the original collaboration. The programme is structured to invite and support such verification rather than depend on internal consensus. Related terms: Falsifiability Contract.

Laboratory

The public working environment in which simulations, visualisations, experimental proposals, and software tools associated with the programme are developed and shared. See also: Simulation, Software Platform.

Mathematical Development

The ongoing formal work that derives, extends, and audits the mathematical structures of PDT, including consistency proofs, limit theorems, and connections to established formalisms. See also: Research Papers.

Open Questions

Areas where the programme does not yet provide a complete account, including aspects of the Standard Model spectrum, detailed cosmological predictions, and the full treatment of many-body measurement. Open questions are stated explicitly rather than concealed. See also: Active Research, Framework Status.

PTG

A programme identifier used by both current and historical paper series. Its general expansion has not been author-confirmed. Refer to each paper by its verified full title.

QM Reformulation

The strand of the programme concerned with quantum behaviour. The current foundations establish quantum probability and dynamics under explicitly stated operational premises (R04 to R06). Related terms: Measurement.

Reader Summaries

Plain-language companions to the research papers, intended to make the structure of each result accessible to readers outside the immediate specialism without weakening the underlying technical claims. See also: Research Papers.

Research Papers

The formal, citable record of the framework, published openly with versioned updates and explicit changelogs so that the historical development of the programme remains inspectable. See also: Reader Summaries, The Book.

Simulation

A computational model used to investigate phase dynamics or candidate phenomena within a chosen model. Simulations illustrate models; they do not establish experimental confirmation. Related terms: Software Platform, Laboratory.

Software Platform

The set of software tools developed in support of the research programme, including simulations, analysis utilities, and prototypes of applied technologies inspired by phase concepts. Related terms: Simulation, Laboratory.

The Book

A connected, narrative presentation of the programme that integrates the formal papers into a single argument, written for working physicists and serious general readers, and revised as the research evolves. Related terms: Research Papers.

A living vocabulary

The vocabulary grows with the research.

The terminology of Phase Differential Theory develops alongside the research programme. As new mathematical structures, experimental concepts, software tools, and applications are introduced, the glossary will continue to expand and be refined so that terminology remains consistent across the papers, the laboratory, the book, and the wider research programme.