Preface: How to Read and Use This Document

This handbook is not a tutorial, not a popularization, and not a speculative essay. It is the canonical axiomatic and terminological contract of Pattern Field Theory (PFT).

Its purpose is to:

• Fix the meanings of all core terms and operators

• Define the admissibility rules of the framework

• Specify what is and is not allowed to be claimed within PFT

• Serve as a stable reference for all subsequent technical and applied papers

This document should be read as a classification and consistency manual, not as a narrative.

How the document is structured

• The Executive Summary states the ontological commitments of PFT.

• The Interpretational Guardrails define what PFT explicitly does not allow.

• The Core Axioms define the non-negotiable structural foundations.

• The sections on Equilibrion, Commit–Merge–Detach (CMD), and Logarithmic Depth Scaling (LDS) define the three core mechanisms.

• The Equation Glossary defines the canonical operators and fixed-point conditions.

• The sections on Structural Constants and Discrete Attractors classify important closure landmarks.

• The Interpretation Layer explains how common physical and biological phenomena are classified within the framework.

• The Glossary and Acronyms sections fix terminology and notation.

How this document should be used

• When writing or reading any PFT paper, this handbook defines the allowed meanings of all terms.

• Any statement that contradicts the axioms or definitions in this document is not a statement in Pattern Field Theory.

• This document should be cited as the normative reference for the framework.

• This document is intentionally non-dynamical, non-narrative, and non-speculative.

What this document is not

• It is not a model of spacetime.

• It is not a theory of motion or forces.

• It is not an interpretation of existing physics.

• It is not an empirical paper.

It is a structural classification system for admissible states and identities.

One guiding principle

If a structure cannot be closed, it does not exist.

Everything in Pattern Field Theory follows from this rule.

Executive Summary

Pattern Field Theory (PFT) is a structural, non-dynamical framework in which physical reality is described as a closed, admissible state space over a fixed topological substrate called the Allen Orbital Lattice (AOL).

PFT does not describe reality in terms of motion, forces, or spacetime dynamics. Existence is defined solely through:

• Structural admissibility (closure under constraints)

• Irreversible state locking via Commit–Merge–Detach (CMD)

• Description depth via Logarithmic Depth Scaling (LDS)

• Prime-based structural addressing

Time is not a dimension. Energy is not a substance. Singularities are not physical states.

Acronyms and Notation Conventions

The following acronyms are used throughout this document:

All symbols such as \(\mathcal{C}\), \(X\), \(S\), \(G\), and \(\mathbb{P}\) are defined in the Equation Glossary.

Interpretational Guardrails

The following interpretations are explicitly disallowed:

• PFT is not a theory of spacetime.

• PFT is not a theory of motion or dynamics.

• PFT does not describe expansion, flow, or evolution.

• PFT does not treat primes as causes.

• PFT is not numerology.

The only permitted interpretation is:

Pattern Field Theory is a structural state-geometry in which existence is defined by closure, admissibility, and irreversible state locking.

Core Axioms

Structural Stack and Flow

This section fixes the ontology-to-interface pipeline used throughout this document. The stack defines where structure exists and where description begins. The flow defines how candidate configurations become committed and then accessible.

The PFT structural stack

Structural Flow in Pattern Field Theory (PFT)

Interpretation: A candidate configuration is first tested for structural closure by the Equilibrion. Only fixed points are admissible. Admissible structures are committed by CMD. Logarithmic Depth Scaling alters description resolution only. Structural interfaces project the description into regime-specific access channels.

Ontology

• The fundamental substrate is the Allen Orbital Lattice (AOL) with invariant topology.

• Only committed, admissible structures exist.

• There is no spacetime, no motion, no dynamics, and no background time parameter.

Axiom A1 — Topological Invariance

\[\delta G = 0\]

The topology of the Allen Orbital Lattice (AOL) is invariant.

Axiom A2 — Closure (Equilibrion)

There exists a closure operator: \[\mathcal{C} : X \to X\]

A configuration \(x\) is physically admissible if and only if: \[\mathcal{C}(x) = x\]

Axiom A3 — Defect

Structural defect is defined as: \[E(x) := \| \mathcal{C}(x) - x \|\]

Admissible states satisfy \(E(x) = 0\). Divergent defect implies non-existence.

Axiom A4 — History Construction (Commit–Merge–Detach)

There exists a discrete history construction rule:

• Commit: accepts a closed configuration

• Merge: integrates it into the commit ledger

• Detach: restores generative capacity

Only committed states exist. There is no ontological time.

Axiom A5 — Prime-Indexed Identity

There exists a prime indexing: \[\iota : V(G) \to \mathbb{P}\]

Every configuration has a unique prime-factor encoding: \[\Phi(x) := \prod_{v \in V(G)} p_{\iota(v)}^{\alpha_v(x)}\]

Identity is unique and non-degenerate by unique factorization.

Axiom A6 — Prime Indexed Closure Equation (PICE)

\[\mathcal{C}(x) = \arg\min_{y \in X} \| y - x \| \quad \text{subject to prime-index consistency constraints}\]

Closure must preserve prime-indexed identity structure.

Axiom A7 — Universal Field Equation (PFT)

\[\boxed{ \mathcal{C}(x) = x }\]

This is the master condition for physical existence in PFT.

Axiom A8 — Description Depth (Logarithmic Depth Scaling)

There exist description regime mappings: \[\mathrm{LDS}_{j \to k} : S_j \to S_k\]

These change resolution only, not structure or topology.

Axiom A9 — Non-Existence of Singularities

\[E(x) = \infty \;\Rightarrow\; \mathcal{C}(x) \ne x\]

Non-closable structures cannot exist.

Interpretational Consequences

• Existence = closure.

• Physics = classification of admissible fixed points.

• Time = not part of ontology; only a commit ledger exists.

• Energy = structural defect.

• Gravity = structural load projection.

• Perception = interface projection.

CMD — Commit–Merge–Detach

CMD is the discrete mechanism by which admissible states are locked into history.

• Commit - accepts a closed configuration.

• Merge - integrates it into locked history.

• Detach - restores generative capacity and prevents runaway structural load.

CMD is not a dynamics. It is a discrete history construction rule.

CMD may incorporate additional admissibility constraints such as Phase Alignment Lock (PAL) in regime stabilization.

LDS — Logarithmic Depth Scaling

LDS is a coordinate transform on description depth. It changes the resolution at which structure is described. It does not change topology, space, or physical content.

Equilibrion

The Equilibrion is a closure operator \(\mathcal{C}\). It is not a force and not a process.

A structure exists physically if and only if it is a fixed point of \(\mathcal{C}\).

Structural defect is defined as distance from closure.

Prime Addressing

Prime numbers are used as a structural addressing and composition system.

• Each fundamental structural degree of freedom has a unique prime index.

• Composite structures are represented by unique products of primes.

• This guarantees unique identity and unique decomposition.

Primes do not cause anything. They encode identity and composition.

\(\pi\)-Structures

A \(\pi\)-structure is a prime-irreducible, closure-stable, structurally minimal identity.

\(\pi\)-structures act as stable identity anchors.

Singularities

A singularity corresponds to a non-closable or divergent structural encoding. It is non-admissible and therefore not a physical state.

Technical FAQ

• Pattern Field Theory is not a spacetime-based framework; it is a structural state-geometry defined over the Allen Orbital Lattice (AOL).

• Pattern Field Theory does not postulate any fundamental time-evolution equations; physical states are defined solely by structural closure and discrete commitment via Commit–Merge–Detach (CMD).

• Pattern Field Theory does not describe any metric or volumetric expansion; changes in scale and resolution are handled exclusively by Logarithmic Depth Scaling (LDS).

• Time in Pattern Field Theory is defined as the ordering of committed states.

• Gravitational phenomena in Pattern Field Theory are interpreted as projections of structural load and admissibility constraints, not as a fundamental force.

• Does PFT deny that light or sound exist?
  No. Pattern Field Theory does not deny the existence of electromagnetic or mechanical interaction regimes. It states that “light” and “sound” are interface projections, not fundamental entities. What exists are admissible structural interaction modes in the Allen Orbital Lattice (AOL). Biological systems implement regime-specific projection interfaces that map certain subsets of these interaction modes into compressed symbolic channels. “Light” and “sound” are therefore not properties of reality, but properties of how certain systems interface with reality.

Glossary

Equation Glossary and Canonical Operators

This section defines the canonical equations and operators used throughout Pattern Field Theory (PFT). Each equation is structural, not dynamical. None of these equations describe motion, force, or time evolution.

E0 — AOL Topology Invariance

\[\begin{equation} \delta G = 0 \end{equation}\]

Meaning: The topology of the Allen Orbital Lattice (AOL) is invariant. No physical process changes the adjacency or connectivity structure. All variation occurs in configuration and admissibility, not topology.

—

E1 — Equilibrion (Closure) Operator

\[\begin{equation} \mathcal{C} : X \to X \end{equation}\]

Meaning: The Equilibrion is the structural closure operator. It maps any configuration to its nearest structurally consistent (closed) form.

It is neither a force nor a dynamical law, but a structural consistency operator.

—

E2 — Admissibility (Fixed Point Condition)

\[\begin{equation} A(x) = 1 \iff \mathcal{C}(x) = x \end{equation}\]

Meaning: A configuration exists physically if and only if it is a fixed point of the Equilibrion. Existence equals closure.

—

E3 — Structural Defect Functional

\[\begin{equation} E(x) := \| \mathcal{C}(x) - x \| \end{equation}\]

Meaning: Structural defect is the distance from closure. Zero defect means fully admissible. Divergent defect means non-admissible boundary.

This replaces the notion of energy as a substance.

—

E4 — CMD History Order (Time)

\[\begin{equation} s_1 \prec s_2 \prec \cdots \prec s_n \end{equation}\]

Meaning: Time is the ordering of committed states. There is no background time parameter.

Time has no ontological status in Pattern Field Theory; only committed states exist. The apparent ordering of states is a representational indexing of the commit record, not a structure that exists in reality itself.

—

E5 — LDS Regime Mapping

\[\begin{equation} \mathrm{LDS}_{j \to k} : S_j \to S_k \end{equation}\]

Meaning: Logarithmic Depth Scaling is a coordinate transform between description regimes. It changes resolution, not structure.

—

E6 — Prime Indexing of Structure

\[\begin{equation} \iota : V(G) \to \mathbb{P} \end{equation}\]

Meaning: Each fundamental structural degree of freedom is assigned a unique prime index. This provides a non-degenerate identity system.

—

E7 — Prime Encoding of a Configuration

\[\begin{equation} \Phi(x) := \prod_{v \in V(G)} p_{\iota(v)}^{\,\alpha_v(x)} \end{equation}\]

Meaning: Every configuration has a unique prime-factor representation. Composite structure is unique factorization.

—

E8 — \(\pi\)-Structure (Prime-Stable Fixed Point)

\[\begin{equation} \mathcal{C}(x_\pi) = x_\pi \quad \wedge \quad \Phi(x_\pi)\ \text{is irreducible} \end{equation}\]

Meaning: A \(\pi\)-structure is a prime-irreducible, closure-stable identity. It is a minimal, stable structural anchor.

—

E9 — PICE (Prime-Indexed Closure Equation)

\[\begin{equation} \mathcal{C}(x) = \arg\min_{y \in X} \| y - x \| \quad \text{subject to prime-index consistency constraintss} \end{equation}\]

Name: PICE — Prime-Indexed Closure Equation

Meaning: The Equilibrion selects the nearest configuration that restores prime-consistent structural closure. This is the core admissibility selection rule.

—

E10 — Universal Field Equation (PFT Form)

\[\begin{equation} \boxed{ \mathcal{C}(x) = x } \end{equation}\]

Name: Universal Field Equation (PFT)

Meaning: This is the master condition for physical existence in Pattern Field Theory.

There is no force law, no evolution equation, and no time parameter.

A structure exists if and only if it is closed.

All of physics in PFT is a classification of the fixed points and boundary failures of this equation.

—

E11 — Non-Existence of Singularities

\[\begin{equation} E(x) = \infty \;\Rightarrow\; \mathcal{C}(x) \ne x \end{equation}\]

Meaning: Any configuration with divergent defect cannot be closed and therefore cannot exist as a physical state.

Singularities are non-admissible boundaries, not objects.

—

Structural Summary

• Existence is defined by closure: \(\mathcal{C}(x) = x\)

• Physics is the study of admissible fixed points

• “Time” is the representational index of the commit ledger, not a physical structure

• Energy is defect

• Identity is prime factorization

• Singularities are non-closable boundaries

Structural Constants, Invariants, and Attractor Landmarks

The following mathematical constants and constructs are not treated as causes, forces, or generators. In Pattern Field Theory, they are treated as structural invariants, attractors, or closure landmarks that appear as stable features of admissible structure, coding, or projection.

—

E12 — Primes (Irreducible Identity Basis)

\[\begin{equation} \mathbb{P}= \{2,3,5,7,11,13,\dots\} \end{equation}\]

Meaning: Primes form the irreducible identity basis for structural addressing and composition. They guarantee non-degenerate identity and unique decomposition. They do not generate structure. They encode identity.

—

E13 — \(\pi\) (Curvature and Closure Resonance Constant)

\[\begin{equation} \pi = \lim_{n \to \infty} \frac{P_n}{D_n} \end{equation}\]

Meaning: \(\pi\) appears as a universal curvature and closure resonance constant. In PFT, \(\pi\) is not an object and not a substance. It is a geometric and spectral invariant that appears in closed rotational, orbital, and lattice-consistent structures.

—

E14 — \(\varphi\) (Golden Ratio, Self-Similar Stability Ratio)

\[\begin{equation} \varphi = \frac{1+\sqrt{5}}{2} \end{equation}\]

Meaning: \(\varphi\) appears as a self-similarity and subdivision stability ratio. In PFT it marks scale-invariant partitioning and stable recursive proportions in admissible structures.

—

E15 — Fine Structure Constant \(\alpha\) (Structural Coupling Index)

\[\begin{equation} \alpha \approx \frac{1}{137} \end{equation}\]

Meaning: In PFT, \(\alpha\) is interpreted as a dimensionless structural coupling index between admissible regimes, not as a fundamental force constant. It measures relative structural stiffness between description layers.

—

E16 — Perfect Numbers (Closure-Symmetric Integers)

\[\begin{equation} N = \sum_{d \mid N,\, d < N} d \end{equation}\]

Meaning: Perfect numbers are integers equal to the sum of their proper divisors. In PFT they represent closure-symmetric arithmetic structures: they are discrete examples of exact internal balance under composition.

—

E17 — Kaprekar’s Constant (Base-10 Attractor)

\[\begin{equation} 6174 \end{equation}\]

Meaning: Kaprekar’s constant is a finite-state digit-rearrangement attractor. In Pattern Field Theory, it is a complete and explicit discrete closure attractor: the entire state space collapses into a single non-trivial fixed point under a constrained admissibility mapping.

Definition of the transformation: Given any four-digit number with at least two distinct digits:

• Sort its digits in descending order to form \(D\)

• Sort its digits in ascending order to form \(A\)

• Compute \(D - A\)

• Repeat the process

Example: \[3524 \rightarrow 5432 - 2345 = 3087\] \[3087 \rightarrow 8730 - 0378 = 8352\] \[8352 \rightarrow 8532 - 2358 = 6174\] \[6174 \rightarrow 7641 - 1467 = 6174\]

Once 6174 is reached, the transformation remains at 6174.

Structural interpretation in PFT: The space of four-digit numbers under this constrained transformation contains a single non-trivial closure-stable fixed point. All admissible states in this space collapse into this attractor basin.

This demonstrates, in a finite and explicit form, how:

A large state space can collapse into a single closure-stable identity under a rule-constrained mapping.

In Pattern Field Theory, Kaprekar’s constant is therefore classified as a discrete closure attractor, not as a physical constant and not as a causal mechanism. Closure under constrained transformation produces attractors without dynamics.

Classification Note: Kaprekar’s constant and the Collatz mapping are not physical laws. They are exact, formal examples of closure, attractor basins, and terminal stability in discrete state spaces. They demonstrate, in mathematically complete and finite form, the same structural phenomena that Pattern Field Theory classifies in general state spaces.

—

E18 — Collatz Mapping (Termination Attractor)

\[\begin{equation} T(n) = \begin{cases} n/2, & n \text{ even} \\ 3n+1, & n \text{ odd} \end{cases} \end{equation}\]

Meaning: The Collatz process is interpreted as a discrete structural descent toward a terminal attractor. In Pattern Field Theory, it is an example of how large discrete state spaces can collapse into a small attractor basin under a simple rule-constrained mapping.

Worked example: \[27 \rightarrow 82 \rightarrow 41 \rightarrow 124 \rightarrow 62 \rightarrow 31 \rightarrow 94 \rightarrow 47 \rightarrow 142 \rightarrow 71 \rightarrow 214 \rightarrow 107 \rightarrow \dots \rightarrow 4 \rightarrow 2 \rightarrow 1\]

Once the sequence reaches: \[4 \rightarrow 2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow \dots\] it enters a small terminal loop.

Structural interpretation in PFT: This shows that a very large discrete configuration space, under a fixed admissibility-like rule, can collapse into a small closure-stable attractor cycle.

In Pattern Field Theory, this is classified as a rule-constrained attractor basin in a discrete state space, not as a physical process and not as a temporal dynamics.

Classification Note: Kaprekar’s constant and the Collatz mapping are not physical laws. They are exact, formal examples of closure, attractor basins, and terminal stability in discrete state spaces. They demonstrate, in mathematically complete and finite form, the same structural phenomena that Pattern Field Theory classifies in general state spaces.

—

E19 — Basel Constant

\[\begin{equation} \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \end{equation}\]

Meaning: The Basel result shows a deep link between discrete summation and continuous curvature invariants. In PFT this is classified as a bridge identity between discrete structure and continuous closure geometry.

—

Structural Interpretation Summary

• Primes = irreducible identity basis

• \(\pi\) = curvature and closure resonance invariant

• \(\varphi\) = self-similarity stability ratio

• \(\alpha\) = structural coupling index between regimes

• Perfect numbers = exact arithmetic closure symmetry

• Kaprekar = finite-state closure attractor

• Collatz = terminal descent attractor in rule-constrained space

• Basel identity = discrete–continuous closure bridge

None of these constants or constructs are treated as causes. They are classification landmarks of admissible structure.

Interpretation Layer: How the Allen Orbital Lattice (AOL) Explains Structure in Reality

This section does not introduce any new theoretical primitives. It is a classification and interpretation guide showing how real phenomena are expressed as admissible structure in the Allen Orbital Lattice (AOL) under the Equilibrion (closure operator), the Commit–Merge–Detach (CMD) mechanism, and Logarithmic Depth Scaling (LDS).

Nothing in this section introduces dynamics, forces, motion, or spacetime processes.

Recurring Hexagonal and Lattice Structures

Examples include:

• Jupiter’s hexagonal atmospheric structure

• Crystals

• Honeycomb and lattice growth

• Certain atmospheric cell patterns

Structural explanation:

These are not flows and not self-organization processes. They are minimum-defect tilings under adjacency and closure constraints in the Allen Orbital Lattice (AOL).

Hexagonal packing is a closure-optimal local topology for two-dimensional adjacency with minimal structural load.

Such structures appear whenever a regime projects a two-dimensional slice of the Allen Orbital Lattice (AOL) under closure constraints.

DNA, Chromosomes, and Biological Structure

Deoxyribonucleic acid (DNA) and chromosomes are not “information carriers” in an abstract sense. They are:

• High-stability structures

• Low-defect configurations

• Highly redundant

• Prime-addressable

Structural explanation:

They are long-lived closure-stable structures in the configuration manifold. They persist because they occupy deep admissibility basins with very low structural defect.

Life does not optimize them. They are simply what remains admissible for a long time.

Neutron Stars, Planets, and Orbital Structures

Neutron stars and planetary orbital structures are not balanced by “forces” in Pattern Field Theory (PFT).

They are:

• Extremely tight admissibility envelopes

• High-load but still closable structures

• Configurations near the boundary of non-admissibility

Structural explanation:

The GW170817 compactness ceiling is an example of an admissibility boundary. Beyond this boundary, closure fails. No singularity forms because non-closable structure cannot be committed by Commit–Merge–Detach (CMD).

Cosmic Microwave Background Radiation (CMBR)

The Cosmic Microwave Background Radiation (CMBR) is not “relic explosion light” in Pattern Field Theory (PFT) terms.

It is:

A large-scale uniform regime projection of early closure-stable structure.

It reflects the earliest globally admissible locked regimes, not a dynamical event.

Dominions and Regimes

A dominion is:

A region of state space with its own admissibility grammar and closure rules.

Quantum, atomic, biological, and macroscopic descriptions are different dominions. They are connected by Logarithmic Depth Scaling (LDS) mappings, not by scale transitions in space.

Life, Perception, and Structural Interfaces

Light, Sound, Color, and Surfaces as Projection Interfaces

Hearing, vision, taste, and even consciousness are:

Regime-specific structural compression interfaces.

They are not “windows to reality”. They are interfaces that discard most structure and retain only what is survival-relevant.

Other species have different interfaces because they occupy different admissibility niches.

In Pattern Field Theory, there are no sensory qualities in reality.

There are only admissible structural interaction regimes in the Allen Orbital Lattice (AOL).

What biological systems call “light”, “sound”, “color”, “surface”, or “texture” are not properties of reality. They are regime-specific projection interfaces and structural compression grammars.

Electromagnetic interaction modes exist. Visual systems map certain subsets of these modes into spatial contrast and symbolic color categories. Photosynthetic systems map overlapping subsets of the same modes into chemical bond reconfiguration pathways.

Mechanical and elastic interaction cascades in matter exist. Auditory systems map certain subsets of these cascades into low-dimensional symbolic channels called “sound”.

Surfaces do not “reflect”. They impose boundary constraints under which certain interaction patterns are re-admitted. “Reflection” is a geometric interpretation produced by visual projection interfaces.

Color does not exist. Only interaction spectra exist. Color is a categorical compression imposed by biological interfaces.

Different species inhabit different admissibility niches and therefore implement different projection grammars. There is no privileged interface.

In Pattern Field Theory, perception is not access to reality. It is a regime-specific structural interface optimized for survival-relevant compression.

Consciousness itself is not an observer and not a substance.

It is a high-level regime control interface that:

• Coordinates which structural information is admitted into the organism’s internal regime

• Controls which states are committed to internal history

• Regulates compression, attention, and action selection

In Pattern Field Theory, consciousness is a commit-coordination and regime-selection interface, not an entity and not a field.

History, Zeno, and Irreversibility

In Pattern Field Theory (PFT):

• History is the commit order defined by Commit–Merge–Detach (CMD).

• When a pattern is no longer committed, it no longer exists.

• There is no continuous becoming.

Zeno-type paradoxes vanish because:

There is no motion. There are only discrete committed states.

Dimensional Explanation

In Pattern Field Theory (PFT), “dimension” is not a container and not a coordinate volume. It is a classification of how identities can relate and compose.

• One-dimensional (1D) structure corresponds to pure adjacency chains: identities can only relate in sequence.

• Two-dimensional (2D) structure corresponds to surface tiling and closure: identities can relate in two independent adjacency directions and form closed loops.

• Two-dimensional plus one-dimensional (2D + 1D) structure corresponds to layered surfaces with stacking order: identities relate within surfaces and between surfaces via ordered stacking relations.

Three-dimensional (3D) structure is not primitive in PFT. It is a composed structure consisting of:

Two-dimensional relational closure plus one-dimensional history and stacking relations.

In Pattern Field Theory, dimension is therefore a statement about relational degrees of freedom between identities, not about space as a container.

Logarithmic Depth Scaling Example

Logarithmic Depth Scaling (LDS) does not move anything.

It means:

The same structure is described with more internal resolution.

This is equivalent to refining a mesh without changing the object.

Riemann Hypothesis and the Millennium Problems

In Pattern Field Theory (PFT), these problems are not “about numbers”.

They are:

• Questions about admissibility spectra

• Questions about closure stability

• Questions about structural density and boundary behavior

For example:

The Riemann zeta function becomes a structural boundary scanner for prime-indexed admissibility.

The Millennium Problems are not physics problems. They are questions about deep closure and structural stability of formal systems.

Why This Functions as a “Guide to the Universe”

Because Pattern Field Theory (PFT) does not describe:

• How things move

• How things push

• How things evolve

It describes:

What is allowed to exist, what can persist, and what cannot be closed.

This makes it a classification theory of reality.

References

All results in this document are internally defined within the Pattern Field Theory framework. External references are used only for empirical comparison in applied papers.

Document Timestamp and Provenance

This document is part of Pattern Field Theory (PFT) and the Allen Orbital Lattice (AOL). Pattern Field Theory™ (PFT™) and related marks are claimed trademarks. Any research, derivative work, or commercial use requires an explicit license from the author.