PBT: Dimensions

A loose constellation of faint glowing abstract symbol-like glyphs floating in space like stars, connected by thin threads. The glyphs must be invented abstract marks, not real letters, numbers, or mathematical notation.

“Dimensions” here means the physical quantities behind PBT’s notation: the symbols, their units, where they came from, and what they mean both in standard physics and in Pressure-Based Theory specifically. Physics reuses a small alphabet for an enormous number of ideas — the same letter can mean unrelated things in different subfields, and even within these papers, a few symbols get pressed into double duty. This page is a map of that notation: not a new derivation, just an index of what every recurring symbol across the papers and reference guide actually stands for.

Each entry follows the same shape: what the symbol is and where its name/letter came from, where it shows up in physics generally, what it’s measured in, how this site’s papers specifically use it (with links), and — where relevant — what else the same letter means elsewhere in physics, so this also works as a disambiguation guide.

A note on scope: the “general scope of use” and “units of measure” material below is standard textbook physics, not a PBT claim. The “PBT-specific usage” material is drawn directly from the actual equations in the papers. Where a paper’s own use of a symbol was flagged in the site’s 2026-07-21 corrigendum pass, that’s noted here too rather than silently repeated as settled.

Fundamental Constants

c — Speed of Light

Name & origin: From the Latin celeritas (“swiftness”), a notation introduced by Wilhelm Weber and Rudolf Kohlrausch in 1856 and later standardized by Einstein’s use of it in special relativity (1905).

Scope of use: The speed of light in vacuum sets the maximum speed for any causal influence in relativity, appears in mass-energy equivalence ($E=mc^2$), defines the geometry of spacetime (the metric’s null cone), and links electric and magnetic constants ($c^2 = 1/(\mu_0\varepsilon_0)$).

Units of measure: Meters per second (m/s), exactly $299{,}792{,}458$ m/s — since 1983 this is a defined exact value; the meter is now defined in terms of $c$, not the other way around. Alternate units: kilometers per second, or simply “1” in natural/geometrized unit systems used in relativity and particle physics.

PBT-specific usage: $c$ is the wave speed of light in the finer-particle medium (Paper 2) — light bending is modeled as refraction through a density gradient in the aether ($n(r) \approx 1 + 2GM/(c^2r)$), not curved spacetime directly. The theory’s “infinite speed” particles ($v \to \infty$) are explicitly not $c$ — they’re the underlying medium’s substrate particles, with $c$ emerging as the propagation speed of disturbances (light) through that medium, analogous to sound speed in a fluid. $c$ also appears throughout as the standard relativistic conversion factor (rest energy, Friedmann equation, GW speed tests in Paper 9: $v_g = c \pm 10^{-15}$).

Other representations: $c$ occasionally denotes specific heat capacity, or (with subscripts $c_1$–$c_4$) the dimensionless Einstein-aether coupling constants used throughout these same papers — a real, easy-to-confuse case of the same letter meaning two different things on the same page (see Symbol Collisions below).

G — Newtonian Gravitational Constant

Name & origin: Simply “gravitational constant.” The notation $G$ was popularized by Charles Vernon Boys around 1894, following his high-precision torsion-balance measurement.

Scope of use: Sets the strength of gravity in Newton’s law ($F=Gm_1m_2/r^2$) and appears throughout general relativity (Einstein’s field equations use $8\pi G/c^4$) and cosmology (the Friedmann equation).

Units of measure: m³ kg⁻¹ s⁻² in SI. Measured value $\approx 6.6743\times10^{-11}$ m³ kg⁻¹ s⁻² — notably the least precisely known of the fundamental constants, since gravity is too weak to measure with the interferometric precision available for $c$ or $e$. Alternate forms: often folded into $GM$ (“standard gravitational parameter”) for astronomical bodies, since that combination is measured far more precisely than $G$ or $M$ individually.

PBT-specific usage: Paper 1 treats ordinary $G$ as the large-scale limit of a scale-dependent effective coupling, $G_{eff}(l)$ (see below) — the familiar constant is recovered at planetary/galactic $l$, while nuclear and atomic scales get their own much larger effective values ($G_{strong}$, $G_{chem}$) from the same formula. Flagged in the 2026-07-21 corrigendum: the specific $G_{strong}\approx10^{29}$ figure doesn’t actually follow from plugging the paper’s own $\varepsilon(l)$ formula into that scale — it appears to have been asserted rather than derived.

Other representations: In particle physics, $G_F$ (Fermi coupling constant, see below) is a completely different constant despite the shared letter. $g$ (lowercase) is also standard notation for local gravitational acceleration (9.8 m/s² at Earth’s surface) and, unrelated again, the electron/muon g-factor discussed under $g$ below.

ħ, h — Planck Constant (Reduced and Ordinary)

Name & origin: Named for Max Planck, who introduced $h$ in 1900 to explain blackbody radiation. The “reduced” form $\hbar = h/2\pi$ (“h-bar”) was introduced by Paul Dirac because it’s the natural unit in quantum mechanics’ angular-momentum and wave-equation formulas.

Scope of use: The fundamental constant of quantum mechanics — sets the scale at which quantum effects (uncertainty, discreteness of angular momentum, photon energy $E=h\nu$) become significant.

Units of measure: Joule-seconds (J·s) in SI — $h = 6.62607015\times10^{-34}$ J·s exactly (fixed by the 2019 SI redefinition, which now defines the kilogram in terms of $h$); $\hbar \approx 1.054571817\times10^{-34}$ J·s. Alternate units: eV·s in atomic/particle physics; set to exactly 1 in “natural units.”

PBT-specific usage: Appears throughout as the quantum of circulation/vorticity — spin’s circulation is quantized as $\Gamma \propto \hbar\sqrt{s(s+1)}$ (Paper 5), and $m(l) \approx \hbar/(lc)$ sets the effective particle mass at a given hierarchical scale $l$ in the core $G_{eff}(l)$ formula used across Papers 1, 2, and 12.

Other representations: None significant — $h$/$\hbar$ is one of the few symbols in physics that’s essentially unambiguous.

e — Elementary Charge

Name & origin: The base unit of electric charge, isolated experimentally by Robert Millikan’s oil-drop experiment (1909); the notation itself predates that, tracing to George Johnstone Stoney’s 1874 proposal of a fundamental charge unit.

Scope of use: The charge of a proton (and, with opposite sign, an electron); all observed free charges are integer multiples of $e$ (quarks carry fractional charge but are never observed free).

Units of measure: Coulombs (C) in SI — $e = 1.602176634\times10^{-19}$ C exactly (also fixed by the 2019 redefinition). Alternate units: “electron volts” use $e$ implicitly as the charge in the energy unit’s definition (1 eV = the energy gained by charge $e$ crossing 1 volt).

PBT-specific usage: Used as the specific value of charge $q$ in the worked Lorentz-force example in Paper 3: $q=e\approx1.6\times10^{-19}$ C, $F\approx qv\times B$.

Other representations: In mathematics, $e\approx2.71828$ (Euler’s number) is an unrelated constant — context always disambiguates, but the notation collision is real.

μ₀ — Vacuum Permeability

Name & origin: Greek mu for “permeability,” subscript zero for “of free space/vacuum.”

Scope of use: Sets the strength of magnetic effects for a given current, appears in the Biot-Savart and Ampère laws, and (via $c^2=1/\mu_0\varepsilon_0$) links electromagnetism to the speed of light.

Units of measure: Henries per meter (H/m) or, equivalently, N/A² (newtons per ampere squared). Historically exactly $4\pi\times10^{-7}$ N/A² by definition of the ampere; since the 2019 SI redefinition (which fixed $e$ instead), it’s now a measured quantity, $\approx1.25663706\times10^{-6}$ N/A².

PBT-specific usage: Used directly in Paper 3’s magnetic dipole field formula, $B=\frac{\mu_0}{4\pi}\left[\frac{3(\mathbf{m}\cdot\mathbf{r})\mathbf{r}}{r^5}-\frac{\mathbf{m}}{r^3}\right]$ — PBT reinterprets $B$ as a flux-gradient effect but keeps the standard electromagnetic formula and constant unchanged.

Other representations: Lowercase $\mu$ without the subscript is reused constantly elsewhere on this page (Bohr magneton $\mu_B$, and as a generic Greek-letter placeholder for a renormalization scale in Paper 8’s $\alpha_i(\mu)$).

α — Fine-Structure Constant

Name & origin: Coined by Arnold Sommerfeld in 1916; “fine structure” refers to the small splitting it governs in atomic spectral lines.

Scope of use: A dimensionless number quantifying the strength of the electromagnetic interaction between charged particles — one of the most precisely measured constants in physics and a recurring benchmark for any candidate unification theory.

Units of measure: Dimensionless (a pure ratio, $\approx 1/137.036$) — this is one of relatively few fundamental “constants” that isn’t measured in any unit system at all.

PBT-specific usage: Used in Paper 6’s electron g-factor formula, $g=2(1+\alpha/2\pi)$. Flagged in the 2026-07-21 corrigendum: this is Schwinger’s real 1948 QED one-loop result, used as-is rather than independently derived from PBT’s own aether-vorticity mechanism.

Other representations: Elsewhere in physics $\alpha$ commonly denotes angular acceleration, a generic decay/absorption coefficient, or (capitalized differently as needed) alpha particles/alpha decay. Within these papers it’s also reused as a generic RG-flow index ($\alpha_i(\mu)$ in Paper 8, unrelated to the fine-structure constant despite the same letter).

G_F — Fermi Coupling Constant

Name & origin: Named for Enrico Fermi, whose 1934 theory of beta decay first introduced a contact-interaction coupling of this kind.

Scope of use: Sets the strength of the weak nuclear interaction in Fermi’s effective (non-gauge) theory of beta decay — a low-energy approximation later superseded by, but still numerically consistent with, the full electroweak theory.

Units of measure: GeV⁻² in natural (particle-physics) units — $G_F\approx1.1663787\times10^{-5}$ GeV⁻². Because it isn’t dimensionless, its SI equivalent is awkward (J·m³) and essentially never used in practice.

PBT-specific usage: Central to Paper 10’s weak-decay-rate formula, $\Gamma_{weak}=G_F^2m^5/192\pi^3$, reinterpreted there as arising from a “flux leak” ($G_F\sim\Delta\text{flux}/l^2$) rather than a gauge coupling. Flagged in the 2026-07-21 corrigendum: applying this formula with the neutron’s full rest mass (as the paper does) gives a result about 13 orders of magnitude off the real neutron lifetime — the formula is correct (verified against the muon, where it reproduces the real 2.2 μs lifetime almost exactly) but was misapplied; real beta decay uses the much smaller proton-neutron mass difference (the Q-value), not the full rest mass.

Other representations: Not to be confused with plain $G$ (gravity) or $G_{eff}$/$G_{strong}$/$G_{chem}$ (PBT’s own scale-dependent effective gravitational couplings, below) — four different constants sharing the same root letter.

Λ_QCD — QCD Scale Parameter

Name & origin: Capital Greek lambda, subscripted for Quantum Chromodynamics (QCD), the theory of the strong force.

Scope of use: The energy scale at which the strong coupling constant becomes large (confinement sets in) — it arises via “dimensional transmutation,” where a dimensionless coupling at high energy generates a genuine mass/energy scale at low energy, a genuinely non-classical quirk of quantum field theory.

Units of measure: MeV or GeV (energy units), $\approx 200$–$300$ MeV depending on the renormalization scheme used to define it precisely.

PBT-specific usage: Paper 10 proposes $\Lambda_{QCD}=\sqrt{\gamma P/\rho}$, reinterpreting it as a pressure/density ratio. Flagged in the 2026-07-21 corrigendum: this specific formula is dimensionally inconsistent — $\sqrt{P/\rho}$ is the standard fluid-dynamics formula for the speed of sound in a medium (units of velocity), not an energy, and no factor rescues it into MeV. The reference value $\Lambda_{QCD}\approx200$ MeV used elsewhere in the paper (e.g. the confinement-radius calculation) is correct as a citation; it just isn’t actually derived by this equation.

Other representations: Entirely distinct from cosmological $\Lambda$ (below) despite the identical letter — one of the more consequential symbol collisions on this page, since both appear in Paper 7 and Paper 9 in the same breath as “grand unification.”

μ_B — Bohr Magneton

Name & origin: Named for Niels Bohr, from his 1913 atomic model.

Scope of use: The natural unit of magnetic moment for an electron’s orbital or spin angular momentum — the quantity that sets the scale of atomic magnetism and spin precession (the Zeeman effect, electron spin resonance, MRI physics).

Units of measure: Joules per tesla (J/T) in SI — $\mu_B\approx9.27401\times10^{-24}$ J/T.

PBT-specific usage: Appears directly in the spin-precession equation used in Papers 5 and 6, $dS/dt=-(g\mu_B/\hbar)\,S\times B$ — a standard equation of motion for magnetic moments, reused unchanged, with PBT supplying its own mechanical account of why $g\approx2$ (vorticity/aether stiffness) rather than a new formula for precession itself.

Other representations: Not to be confused with plain $\mu$ (permeability, viscosity, a generic RG scale) or the muon, sometimes also abbreviated $\mu$ in particle-physics contexts (not used that way on this site).

M_☉ — Solar Mass

Name & origin: $M$ for mass, $\odot$ the astronomical symbol for the Sun (a circle with a central dot, in use since antiquity for the Sun itself before being adopted as an astronomical unit symbol).

Scope of use: A convenience unit for astrophysical masses — galaxies, stars, and black holes are conventionally quoted in multiples of the Sun’s mass rather than kilograms, since the numbers involved in kg are unwieldy.

Units of measure: By definition, $M_\odot\approx1.989\times10^{30}$ kg. Purely a scaled restatement of the kilogram, not an independent unit system.

PBT-specific usage: Used directly in the Milky Way rotation-curve simulations in Papers 1, 2, and 4: $M_{enc}\approx6\times10^{10}\,M_\odot$.

Other representations: None — this notation is essentially unambiguous within physics/astronomy.

Core PBT Framework Variables

These are the theory’s own load-bearing apparatus — the scale-dependent quantities that don’t exist in standard physics and are specific to how PBT’s papers construct their unification claim.

l — Hierarchical Length Scale

Name & origin: Lowercase $l$ for “length” — PBT’s central independent variable, the size scale at which a given physical regime (nuclear, atomic, planetary, galactic) is being evaluated.

Scope of use: Not a standard physics symbol in this specific role — ordinary physics uses length scales constantly (e.g. $r$ for radius, $\lambda$ for wavelength) but doesn’t typically treat “the scale itself” as a free parameter that a fundamental constant depends on.

Units of measure: Meters (m), ranging across the papers from $\sim10^{-35}$ m (Planck-scale cutoffs, Paper 9) through $\sim10^{-15}$ m (nuclear) to galactic and cosmological scales.

PBT-specific usage: The independent variable in the core hierarchical-scaling formula used across nearly every paper, $\varepsilon(l)=\varepsilon_0(l_0/l)^\gamma$ — PBT’s central claim is that a single formula, evaluated at different $l$, reproduces gravity, nuclear binding, and atomic bonding as different regimes of the same underlying pressure field. Paper 13 (unpublished draft)’s later nuclear-binding model works at fixed (nuclear) $l$ rather than testing the cross-scale claim itself.

Other representations: Elsewhere in physics, lowercase $l$ commonly denotes the orbital angular momentum quantum number — unrelated to its use here.

ε(l) — Hierarchical Energy Density

Name & origin: Lowercase Greek epsilon, standard physics notation for energy density; here written as an explicit function of scale, $\varepsilon(l)$.

Scope of use: Energy density (energy per unit volume) is an ordinary physics quantity — what’s specific to PBT is treating it as scale-dependent via a power law, rather than as a locally-measured, scale-independent field value the way energy density is normally used (e.g. in the stress-energy tensor).

Units of measure: J/m³ (joules per cubic meter) — SI energy density.

PBT-specific usage: The theory’s central equation, appearing in nearly every paper: $\varepsilon(l)=\varepsilon_0(l_0/l)^\gamma$, with $\varepsilon_0\approx7.4\times10^{35}$ J/m³ and $l_0\approx10^{-25}$ m (Paper 1). This says energy density rises as a power law toward smaller scales — the mechanism PBT uses to make gravity, nuclear binding, and atomic bonding all “the same formula” evaluated at different $l$. It also directly sets pressure ($P=\varepsilon/3$ for isotropic flux) and, at cosmological scale, dark-energy density ($\rho_{eff}=\varepsilon(l\to\infty)/c^2$ in Paper 9).

Other representations: Plain $\varepsilon$ (without the scale dependence) is standard notation for electric permittivity elsewhere in physics — a real, easy collision, since $\varepsilon_0$ specifically is also the standard symbol for vacuum permittivity in electromagnetism, a completely different constant with a completely different value ($8.854\times10^{-12}$ F/m) from PBT’s $\varepsilon_0\approx7.4\times10^{35}$ J/m³. Anyone cross-referencing this site against a standard textbook should not conflate the two.

γ — Hierarchical Scaling Exponent

Name & origin: Lowercase Greek gamma — here, the power-law exponent in $\varepsilon(l)=\varepsilon_0(l_0/l)^\gamma$.

Scope of use: Not a standard fixed physics quantity in this role — power-law exponents are common across physics (critical exponents in phase transitions, spectral indices in cosmology), but this specific one is unique to PBT’s own model.

Units of measure: Dimensionless (a pure exponent). PBT’s papers place it in the range $\gamma\approx2$–$4$.

PBT-specific usage: Controls how sharply energy density rises at small scales — used throughout Papers 1, 2, 5, 7, and 12 as the core scaling exponent. Flagged in the 2026-07-21 corrigendum: solving for the $\gamma$ that would actually produce Paper 1’s claimed $G_{strong}\approx10^{29}$ figure gives $\gamma\approx-0.07$ — outside the paper’s own stated 2–4 range and the wrong sign relative to its own claim that finer scales amplify $\varepsilon$.

Other representations: This is the single busiest symbol on this page. Standard physics uses lowercase $\gamma$ for the Lorentz factor in special relativity ($\gamma=1/\sqrt{1-v^2/c^2}$), for photons (gamma rays/gamma decay), and for the heat-capacity ratio in thermodynamics. Within these same papers, a second, unrelated $\gamma$ shows up as a galactic rotation-curve fit parameter in $G_{eff}(r)=G[1+k(r/r_0)^\gamma]$ (Papers 1, 2, 4) — Paper 1/2 use $\gamma=1$ there, while Paper 4 uses $\gamma=2$ for the same claimed Milky Way simulation with a different $k$ as well ($k\approx1900$ vs. $k=0.1$) — flagged in the 2026-07-21 corrigendum as two different, unreconciled parameter sets for one claimed calculation, with the original code not preserved to determine which (if either) was actually run.

σ(l) — Hierarchical Cross-Section

Name & origin: Lowercase Greek sigma — standard particle-physics notation for a cross-section (an effective interaction area).

Scope of use: Cross-sections are a standard tool across physics for quantifying interaction probability — how large a “target” a particle effectively presents to an incoming flux.

Units of measure: m² (square meters) in SI; barns ($10^{-28}$ m²) are the conventional alternate unit in nuclear/particle physics, chosen to be a convenient size for nuclear cross-sections.

PBT-specific usage: Appears in the core $G_{eff}(l)=\varepsilon(l)\sigma(l)^2/(4\pi m(l)^2)$ formula, approximated as $\sigma(l)\approx l^2$ — i.e., PBT ties the effective interaction cross-section directly to the hierarchical scale itself, rather than treating it as an independently measured quantity the way particle physics normally does.

Other representations: Elsewhere in physics $\sigma$ also denotes the Stefan-Boltzmann constant, electrical/thermal conductivity, or statistical standard deviation — none of those uses appear in these papers, but they’re common enough to flag.

G_eff(l), G_strong, G_chem — Scale-Dependent Effective Gravitational Coupling

Name & origin: $G$ for gravitational constant (see above), subscripted “eff” for “effective,” or by regime (“strong” for nuclear, “chem” for atomic/chemical).

Scope of use: Not a standard physics quantity — ordinary physics treats $G$ as a true constant; “running couplings” that change with scale are a real concept in quantum field theory (e.g. the running of $\alpha$ or the strong coupling constant), but applying that idea to gravity itself, all the way from nuclear to galactic scales via one formula, is PBT’s own proposal.

Units of measure: m³ kg⁻¹ s⁻², same dimensions as ordinary $G$, but with wildly different magnitudes claimed at different scales: $G_{strong}\approx10^{29}$ (nuclear), $G_{chem}\approx10^{32}$ (atomic), reducing to the ordinary $6.6743\times10^{-11}$ at planetary/galactic scale.

PBT-specific usage: This is the theory’s single central claim, appearing in Paper 1 as $G_{eff}(l)\approx\varepsilon(l)\sigma(l)^2/(4\pi m(l)^2)$ — one formula meant to unify gravity, the strong nuclear force, and chemical/atomic bonding as the same effective coupling evaluated at different $l$. Flagged in the 2026-07-21 corrigendum: the specific $G_{strong}\approx10^{29}$ number doesn’t actually follow from this formula with the paper’s own stated $\varepsilon_0$, $l_0$, and $\gamma$ range — computed values come out many orders of magnitude off in every case checked.

Other representations: See $G_F$ (Fermi coupling) above — a real, different constant, easily confused by the shared root letter and adjacent subject matter (both concern nuclear-scale physics).

u^μ — Aether Field

Name & origin: Lowercase $u$ with a spacetime index $\mu$, standard relativity notation for a four-velocity field; PBT borrows this directly from Einstein-aether theory (Jacobson & Mattingly, 2001), an established (if speculative) framework in the physics literature that PBT explicitly builds on.

Scope of use: In genuine Einstein-aether theory, $u^\mu$ is a dynamical, timelike unit vector field that picks out a preferred direction of time at every point in spacetime while still respecting local Lorentz invariance in its equations of motion — a real, published approach to testing whether general relativity secretly has a preferred frame.

Units of measure: Dimensionless (a unit vector, $u^\mu u_\mu=-1$ by construction, in the usual $(-,+,+,+)$ signature convention).

PBT-specific usage: The medium PBT’s forces emerge from — shadowing and flux imbalances are modeled as distortions of $u^\mu$. Appears in the action integrals of Papers 2 and 8, the charge-shadow evolution equation in Paper 3 ($\nabla_\alpha u^\mu\approx\delta_{flux}/flux_0$), and the spin/entanglement coupling terms of Paper 5.

Other representations: Plain $u$ (without the index) is also generic notation for velocity or a potential-energy function elsewhere in physics — not used that way here.

c₁, c₂, c₃, c₄ — Aether Coupling Constants

Name & origin: Standard notation from the Einstein-aether literature (Jacobson & Mattingly) for the four dimensionless coefficients in the aether kinetic term.

Scope of use: In genuine Einstein-aether theory, these four numbers parametrize how strongly the aether field couples to spacetime curvature; observational bounds (solar-system tests, gravitational-wave speed measurements) constrain them to be very small.

Units of measure: Dimensionless.

PBT-specific usage: Paper 5 quotes specific tuned values ($c_1\approx10^{-16}$, $c_2\approx10^{-8}$, $c_3\approx-10^{-16}$, $c_4\approx10^{-6}$), stated as satisfying real observational constraints from GW170817 and pulsar timing.

Other representations: Not to be confused with the speed of light $c$ (no subscript) — a real risk given how visually similar $c_1$ and $c$ look, and both appear throughout the same equations.

ξ, η — Aether-Coupling Parameters (Spin, Entanglement, Damping)

Name & origin: Lowercase Greek xi and eta — used here as generic small-coupling-constant placeholders, following a common physics convention of reaching for the next unused Greek letter when introducing a new phenomenological parameter.

Scope of use: Neither letter has one single fixed meaning in physics generally; both are commonly recruited as free/fitted parameters in whatever context needs one (viscosity, damping coefficients, generic coupling strengths).

Units of measure: Dimensionless in every use on this site, though their real numeric role differs by paper (see below) — PBT’s papers keep both $<10^{-15}$ wherever they’re constrained by real data.

PBT-specific usage: $\xi$ appears as the spin-aether coupling in Paper 5’s Lagrangian term ($\xi\,\bar\psi\gamma^5\gamma^\mu u_\mu\psi$), the damping term in Paper 6’s precession-stability fix ($\xi\,u^\mu S_\mu=0$, $\xi=10^{-15}$), and the higher-spin coupling in Paper 9. $\eta$ appears specifically as the entanglement-coupling constant in Paper 5’s $L_{ent}=\eta\,\bar\psi_1\gamma^\mu u_\mu\psi_2$.

Other representations: $\eta$ is also standard notation for the flat-spacetime (Minkowski) metric and for viscosity elsewhere in physics — neither use appears on this site, but both are common enough to flag for anyone cross-referencing textbooks.

Λ — Cosmological Constant

Name & origin: Capital Greek lambda, introduced by Einstein in 1917 (originally to force a static universe; later reinterpreted, post-Hubble, as the source of accelerating expansion).

Scope of use: The standard term in general relativity’s field equations representing the energy density of empty space itself — the leading (though still unexplained) candidate mechanism for dark energy in mainstream cosmology.

Units of measure: m⁻² in SI (inverse length squared) as it appears in the field equations; often re-expressed as an energy density (J/m³) via $\rho_\Lambda=\Lambda c^2/8\pi G$.

PBT-specific usage: Appears in Paper 9’s Friedmann equation, $H^2=(8\pi G/3)\rho_{eff}+\Lambda c^2/3$, with PBT proposing $\Lambda\sim P_{residual}/c^2\sim10^{-52}$ m⁻² — i.e., reinterpreting the cosmological constant as the residual pressure of the hierarchical medium at $l\to\infty$, rather than an independent free parameter of the field equations.

Other representations: Entirely distinct from $\Lambda_{QCD}$ (above) despite the identical capital letter — see Symbol Collisions.

Γ — Circulation / Decay Rate / Decay Width

Name & origin: Capital Greek gamma — in fluid dynamics, standard notation for circulation (a measure of a flow field’s rotation); in particle physics, standard notation for a decay rate or resonance width.

Scope of use: These are two genuinely unrelated physics quantities in mainstream use that happen to share the capital-Gamma notation — fluid circulation (used in vortex dynamics, e.g. around an airfoil) and particle-physics decay rates (how quickly an unstable particle decays, related to its lifetime by $\tau=1/\Gamma$).

Units of measure: Circulation: m²/s (SI). Decay rate: s⁻¹, or equivalently an energy width via $\Gamma\hbar$ in natural units.

PBT-specific usage: PBT uses both meanings, on different papers, with the same letter: circulation $\Gamma\propto\hbar\sqrt{s(s+1)}$ quantizes spin as vorticity in Paper 5, while $\Gamma_{weak}=G_F^2m^5/192\pi^3$ is a particle-physics decay rate in Papers 7 and 10 — the same symbol, doing two unrelated jobs, within the same overall paper series. (The weak-decay use is also where the corrigendum’s most severe single error was found — see $G_F$ above.)

Other representations: Also standard notation for the mathematical Gamma function (generalized factorial) and for Christoffel symbols in differential geometry — neither used explicitly here, but both are common enough in adjacent physics literature to cause confusion.

ω — Vorticity / Angular Frequency

Name & origin: Lowercase Greek omega — standard notation in fluid dynamics for vorticity (the curl of a velocity field) and, separately, standard notation across all of physics for angular frequency.

Scope of use: Vorticity measures local rotation in a fluid flow; angular frequency measures how fast something oscillates or rotates, in radians per second — genuinely different quantities that happen to share notation because both describe “how fast something is spinning,” in different senses.

Units of measure: Both are measured in s⁻¹ (or rad/s for angular frequency specifically) — the units actually do match, unlike most of the collisions on this page.

PBT-specific usage: Central to PBT’s account of quantum spin: $\omega=\nabla\times v$ defines vorticity in the aether flow (Paper 5), which is then quantized to represent particle spin. The same symbol also does standard duty as angular frequency in the spin-precession equation, $\omega=g\mu_BB/\hbar\approx1.76\times10^{11}$ rad/s (Paper 6) — here the two meanings are actually linked, not just coincidentally shared, since PBT’s account treats precession frequency as a consequence of the same underlying vorticity.

Other representations: Capital $\Omega$ (not used explicitly on this site) commonly denotes solid angle or angular velocity of a rotating reference frame elsewhere in physics.

Standard Physics Variables PBT Borrows or Reinterprets

These are ordinary, widely-used physics symbols that PBT’s papers keep (often with their standard formulas intact) while proposing a different underlying mechanism for what they represent.

E — Energy (Mass-Energy Equivalence)

Name & origin: Simply “energy” — the notation is old and not tied to a specific coiner, though its most famous appearance is Einstein’s 1905 $E=mc^2$.

Scope of use: The equation applies to any physical process where mass and energy interconvert: nuclear fission and fusion (reactors, stars), particle-accelerator collisions creating new matter from kinetic energy, the early universe’s matter formation from pure energy, and antimatter annihilation in PET medical imaging.

Units of measure: Joules (J) in SI, $1\text{ J}=1\text{ kg}\cdot\text{m}^2/\text{s}^2$. Alternates: electronvolts (eV, MeV, GeV) in particle/nuclear physics; ergs in the older CGS system ($1\text{ J}=10^7$ erg).

PBT-specific usage: PBT doesn’t invoke $E=mc^2$ directly as a named equation anywhere in the papers, but energy in general appears throughout as the quantity carried by hierarchical pushes — e.g. binding energies (8 MeV/nucleon nuclear, 4–6 eV atomic in Paper 1), and as the basis of Paper 13 (unpublished draft)’s binding-energy-per-nucleon fits (the “$B/A$” in its semi-empirical-mass-formula re-derivation).

Other representations: $E$ also denotes electric field strength in electromagnetism (force per unit charge, units N/C or equivalently V/m) — not used explicitly in these papers, but standard enough to flag; total relativistic energy is more precisely written $E^2=(mc^2)^2+(pc)^2$, with $p$ momentum.

F — Force

Name & origin: Standard notation since Newton, simply “force.”

Scope of use: The central quantity of classical mechanics — anything that changes a body’s momentum, from gravity and electromagnetism to contact forces.

Units of measure: Newtons (N) in SI, $1\text{ N}=1\text{ kg}\cdot\text{m/s}^2$. Alternates: dynes in CGS ($1\text{ N}=10^5$ dyn); pounds-force in imperial/US customary units.

PBT-specific usage: PBT’s central conceptual move is treating all forces (gravity, electromagnetic, weak, strong) as varieties of the same underlying mechanical push — Paper 3 works the Lorentz force explicitly as $F=q(v\times B)\approx qv\times(\text{particle flow direction})$, reinterpreting the standard formula’s origin without changing the formula itself.

Other representations: $F$ also denotes the Fermi coupling’s namesake context loosely (not a notational overlap, just the same physicist), and — unrelated — the Faraday constant in electrochemistry. Neither use appears here.

v — Velocity

Name & origin: Standard notation for velocity across all of physics.

Scope of use: Rate of change of position — arguably the most universally used variable in physics, appearing in every subfield.

Units of measure: m/s in SI; km/s or km/h commonly used for astronomical or everyday speeds respectively.

PBT-specific usage: Two distinct roles: (1) the substrate particles’ own speed, taken as $v\to\infty$ in the idealized limit (Paper 1), the mechanism meant to eliminate classical push-gravity’s drag and heating problems; (2) ordinary orbital/rotational velocity in the galactic rotation-curve formula $v(r)=\sqrt{G_{eff}(r)M_{enc}(r)/r}$, used across Papers 1, 2, and 4 as the theory’s flagship dark-matter-free prediction.

Other representations: None significant, though $v_r$ (radial component) and $v_\theta$ (azimuthal component) are used as ordinary vector-component subscripts in Paper 5’s vorticity derivation.

H, H₀ — Hubble Parameter

Name & origin: Named for Edwin Hubble, who established the expansion of the universe observationally in 1929 (the underlying relation was also derived theoretically by Georges Lemaître slightly earlier).

Scope of use: Quantifies the universe’s expansion rate; $H_0$ specifically denotes its present-day value, one of the most actively debated numbers in cosmology (the “Hubble tension” between different measurement methods).

Units of measure: Conventionally km/s/Mpc (kilometers per second per megaparsec) — a mixed unit reflecting how it’s measured (recession velocity per unit distance); equivalent to an inverse time when unit-converted (s⁻¹).

PBT-specific usage: Appears in Paper 9’s Friedmann equation, $H^2=(8\pi G/3)\rho_{eff}+\Lambda c^2/3$, with PBT deriving $H_0\sim70$ km/s/Mpc from its own residual energy density term. Flagged in the 2026-07-21 corrigendum: the real Planck (CMB) value is 67.4 km/s/Mpc and the real local/SH0ES value is ~73.0 km/s/Mpc — the well-known Hubble tension between two disagreeing real measurements. PBT’s $70$ sits between them (3.86% off Planck, 4.11% off SH0ES) and doesn’t distinctly match either, contrary to how it was originally described.

Other representations: Capital $H$ is also standard notation for the Hamiltonian (total energy operator) in mechanics and quantum mechanics, and for magnetic field intensity (distinct from $B$, magnetic flux density) in electromagnetism — neither use appears in these papers.

M, M_enc — Mass

Name & origin: Standard notation for mass; “enc” subscript for “enclosed,” standard astrophysics shorthand for the mass contained within a given radius.

Scope of use: The fundamental measure of a body’s inertia and gravitational source strength.

Units of measure: Kilograms (kg) in SI; solar masses ($M_\odot$, above) for astrophysical bodies; MeV/$c^2$ or GeV/$c^2$ for particle masses (via mass-energy equivalence).

PBT-specific usage: $M_{enc}$ is the enclosed galactic mass in the rotation-curve formula (above); plain $m$ (lowercase) appears separately as nucleon mass, muon/electron mass, and the scale-dependent effective particle mass $m(l)\approx\hbar/(lc)$ in the core $G_{eff}(l)$ formula.

Other representations: Lowercase $m$ is also the standard quantum-mechanical magnetic quantum number, and (as $\mathbf{m}$, boldface) the magnetic dipole moment vector used in Paper 3’s dipole field formula — three different uses of the same root letter within adjacent equations on the same page.

q, e — Electric Charge

Name & origin: Generic notation for charge (the letter has no special historical significance beyond being available); $e$ specifically denotes the elementary charge (see above).

Scope of use: The source quantity for all electromagnetic phenomena.

Units of measure: Coulombs (C) in SI.

PBT-specific usage: Used generically in Paper 3’s Lorentz-force and dipole formulas, and as the charge-shadow source term in the aether-evolution equation ($\delta_{flux}\propto q/r^2$).

Other representations: Lowercase $q$ is also frequently used elsewhere in physics for generalized coordinates in Lagrangian mechanics — an entirely unrelated use, not present here but worth flagging for cross-reference.

B — Magnetic Field

Name & origin: Standard electromagnetism notation; the choice of letter itself has no deep etymology (it simply followed $E$ for electric field alphabetically in 19th-century notation conventions).

Scope of use: The field responsible for magnetic forces on moving charges and magnetic materials — central to electromagnetism, MRI, particle accelerators, and countless technologies.

Units of measure: Tesla (T) in SI; gauss (G, unrelated to the gravitational constant $G$ despite the identical letter) in the older CGS system, $1\text{ T}=10^4$ G.

PBT-specific usage: Reinterpreted throughout Papers 3, 5, and 6 as a flux-gradient effect ($B\approx\mu_0\times\text{flux gradient}\times\sigma_{charge}$) rather than a fundamental field — but the standard dipole formula, Lorentz force law, and precession equation are all kept as-is, with PBT supplying a different account of their mechanical origin rather than new formulas.

Other representations: As already noted, “gauss” (unit) and “$G$” (gravitational constant) are unrelated to $B$ and to each other, despite superficial notational proximity — a real source of confusion worth flagging explicitly.

ψ, φ — Field Symbols (Spinor and Scalar)

Name & origin: Lowercase Greek psi and phi — standard quantum-mechanics/field-theory notation, with $\psi$ conventionally used for fermionic (spin-1/2, matter-particle) fields and wavefunctions, and $\phi$ for scalar (spin-0) fields.

Scope of use: $\psi$ is the symbol Schrödinger used for the quantum wavefunction (1926) and that Dirac later used for his relativistic spinor field describing electrons; $\phi$ is the standard symbol for scalar fields like the Higgs field.

Units of measure: Depends on normalization convention; in natural units, both are typically dimensionless or carry mass-dimension factors set by the specific Lagrangian — not usually expressed in SI units directly.

PBT-specific usage: Paper 5’s Lagrangian uses $\psi$ for the spin/entanglement field (standard Dirac-field notation, $\bar\psi(i\gamma^\mu D_\mu-m)\psi$); Paper 8’s full QFT action explicitly labels $\phi$ “for pushes” and $\psi$ “for spin” — i.e., PBT maps its own two central concepts (mechanical pushes, spin) onto the standard scalar/spinor field slots of an otherwise fairly conventional-looking QFT Lagrangian.

Other representations: Capital $\Psi$ is used once in Paper 5 for the Rarita-Schwinger higher-spin field (distinct from lowercase $\psi$); $\phi$ is also standard notation for the azimuthal angle in spherical/cylindrical coordinates and for the golden ratio in mathematics — neither use appears here.

θ — Angle

Name & origin: Standard geometric notation for an angle, used throughout physics and mathematics without a specific attributed origin.

Scope of use: Ubiquitous — any angular quantity, from geometric angles to phase angles to mixing angles between particle states (e.g. the Weinberg angle in electroweak theory).

Units of measure: Radians (rad) in SI/physics convention; degrees or arcseconds in observational astronomy.

PBT-specific usage: Specifically the light-deflection angle in Papers 1, 2, and 12: $\theta\approx4GM/(c^2b)\approx1.75’’$ for the Sun — the classic solar-eclipse light-bending test, reproduced here via aether refraction rather than spacetime curvature but landing on the same real, historically confirmed number.

Other representations: None flagged — $\theta$’s meaning is essentially always “an angle,” context-dependent but not genuinely ambiguous the way several symbols above are.

ρ — Density

Name & origin: Standard Greek-letter notation for density (mass or charge per unit volume) across physics.

Scope of use: Mass density (kg/m³) in mechanics and fluid dynamics; charge density in electromagnetism; energy density in cosmology (often written $\rho_{eff}$ or $\rho_\Lambda$ for specific components).

Units of measure: kg/m³ for mass density in SI. In cosmology, often re-expressed via $c^2$ as an energy density (J/m³) instead.

PBT-specific usage: Appears as effective cosmological density $\rho_{eff}=\varepsilon(l\to\infty)/c^2\sim10^{-26}$ kg/m³ (Paper 9) and as nuclear matter density $\rho\sim10^{18}$ kg/m³ in the (dimensionally flagged) $\Lambda_{QCD}$ formula of Paper 10.

Other representations: Also standard notation for electrical resistivity in condensed-matter physics — not used that way here.

S — Spin / Action

Name & origin: Capital $S$ is used in physics for two genuinely unrelated central quantities: spin angular momentum, and the action integral in Lagrangian/path-integral mechanics.

Scope of use: Spin is the intrinsic angular momentum of a particle (with no classical analog, despite the “spinning” name); action is the integral that Lagrangian and Hamiltonian mechanics are built around, minimized/extremized along a system’s actual physical path (the principle of least action).

Units of measure: Spin: J·s (same units as angular momentum, often expressed in units of $\hbar$). Action: also J·s — the two quantities share dimensions, which is not a coincidence, since both trace back to the same underlying phase-space structure in quantum mechanics.

PBT-specific usage: Both meanings appear: $S$ as spin in the precession equation $dS/dt=-(g\mu_B/\hbar)S\times B$ (Papers 5, 6), and $S$ as the action integral in the full QFT Lagrangian of Papers 2 and 8 ($S=\int\sqrt{-g}[\ldots]d^4x$).

Other representations: As noted, this is a genuine same-page collision — both uses appear across papers that reference each other, though rarely in the same single equation, which limits practical confusion.

P — Pressure

Name & origin: Standard notation for pressure across physics and engineering.

Scope of use: Force per unit area — foundational across fluid mechanics, thermodynamics, and (in PBT’s case specifically) the entire premise of a “pressure vessel” universe.

Units of measure: Pascals (Pa) in SI, $1\text{ Pa}=1\text{ N/m}^2$ — named for Blaise Pascal in 1971. Alternates: atmospheres (atm), bar, psi (pounds per square inch) in engineering contexts.

PBT-specific usage: The theory’s namesake quantity — $P=\varepsilon/3$ for isotropic flux (Paper 1) directly ties pressure to the hierarchical energy density, and “residual pressure” ($P_{residual}$) is PBT’s proposed mechanism for the cosmological constant (Paper 9). Paper 13 (unpublished draft)’s later “equilibrium bubble” nuclear-binding model is built entirely on nucleons producing outward pressure balanced against ambient inward pressure.

Other representations: Capital $P$ is also standard notation for momentum in some (especially older or non-English-language) texts, and for power (energy per unit time) — not used that way here, but genuinely common enough elsewhere to flag; PBT’s papers use lowercase $p$ for momentum in the one place it appears ($E^2=(mc^2)^2+(pc)^2$-style relations aren’t explicitly used here, but the convention is standard).

Z, N, A — Proton Number, Neutron Number, Mass Number

Name & origin: Standard nuclear-physics notation: $Z$ for atomic/proton number (from German Zahl, “number”), $N$ for neutron number, $A$ for mass number ($A=Z+N$).

Scope of use: The basic bookkeeping variables of nuclear physics — every isotope is fully specified by its $(Z,N)$ pair (or equivalently $(Z,A)$), and nearly every nuclear formula (including the standard semi-empirical mass formula) is written in terms of these three.

Units of measure: Dimensionless counts (integers) — not physical quantities with units, just particle counts.

PBT-specific usage: Central to Paper 13 (unpublished draft)’s nuclear-binding-energy re-derivation, which fits a four-term model $B/A=a_V-a_SA^{-1/3}-a_CZ(Z-1)/A^{4/3}-a_A(N-Z)^2/A^2$ against real isotope data — this paper works entirely in standard nuclear-physics notation, reinterpreting the mechanism behind each term (proton-proton Coulomb repulsion as “extra outward pressure,” etc.) without changing the standard formula’s structure.

Other representations: None of real concern — this is standard, stable nuclear-physics notation used consistently across the literature.

Symbol Collisions Within PBT’s Own Notation

Because these papers were written across many months and by design reuse familiar physics letters, a few symbols carry two unrelated meanings within the site’s own papers — worth listing plainly in one place, since it’s easy to import a formula from one paper while assuming a definition that only applies in another:

$\gamma$ — the hierarchical scaling exponent in $\varepsilon(l)=\varepsilon_0(l_0/l)^\gamma$ (used with $\gamma\approx2$–$4$), and an unrelated rotation-curve fit exponent in $G_{eff}(r)=G[1+k(r/r_0)^\gamma]$ (used as $\gamma=1$ in Papers 1/2, $\gamma=2$ in Paper 4, for the same claimed simulation — itself flagged as unreconciled in the 2026-07-21 corrigendum).
$\Lambda$ / $\Lambda_{QCD}$ — the cosmological constant (Paper 9) versus the QCD confinement scale (Paper 10) — different physics entirely, sharing only the capital letter and subscript-free/subscripted distinction.
$\Gamma$ — fluid circulation, quantizing spin (Paper 5) versus a particle-decay rate (Papers 7, 10) — see the $\Gamma$ entry above.
$c$ / $c_1$–$c_4$ — the speed of light versus the four dimensionless Einstein-aether coupling constants — visually adjacent, physically unrelated.
$G$ / $G_{eff}$ / $G_{strong}$ / $G_{chem}$ / $G_F$ — four distinct “$G$-family” constants (ordinary gravity, PBT’s scale-dependent effective gravity at two different regimes, and the unrelated Fermi weak-coupling constant) — see the individual entries above.
$\varepsilon$ / $\varepsilon_0$ — PBT’s hierarchical energy density versus the standard electromagnetic vacuum permittivity — same symbol, same subscript convention, completely different quantities and values (PBT’s $\varepsilon_0\approx7.4\times10^{35}$ J/m³ vs. electromagnetism’s $\varepsilon_0\approx8.854\times10^{-12}$ F/m).

None of these are errors by themselves — physics as a field reuses its alphabet constantly, and every collision above also exists somewhere in mainstream physics notation. They’re listed here because a reader moving between papers (or between this site and a standard textbook) can otherwise silently import the wrong definition.