Appendix

BFUT Master Symbol Guide

Every symbol used across the Big Flare-Up Theory papers, with definitions, values, and the papers where each appears. The single reference for reading the mathematics.

Vijay Shankar Sharma | ORCID: 0009-0001-9622-6121 | Updated 21 September 2026

One symbol, one meaning, across all Big Flare-Up Theory (BFUT) papers. Every paper uses the exact symbol and definition reserved in this guide.

The provenance suffix used throughout this guide is _vss, the author's initials (Vijay Shankar Sharma). This applies uniformly across all papers.

NOTATION POLICY: MANDATORY AND WITHOUT EXCEPTIONS

WARNING: CHECK EVERY SYMBOL IN EVERY BFUT PAPER, TABLE, FIGURE AND EQUATION AGAINST THIS RULE. NO ALTERNATIVE SUFFIX, CAPITALISATION, PUNCTUATION OR PROVENANCE FORM IS PERMITTED.

If BFUT derives or predicts its own value of an established standard physical quantity, write the standard notation exactly as normally written and append the literal text _vss directly to it, with no space. The suffix is always exactly _vss: lowercase, preceded by one underscore, and written as ordinary text.

If a quantity is introduced uniquely by BFUT and has no established standard physical counterpart, use its reserved BFUT symbol with no _vss suffix. Examples include R₀, A, B, C, D, ρₛ, K_s, λₛ, κₛ, μₛ, Lₛ, aₛ, R_d and R_eff.

Standard, measured, defined or externally supplied quantities use their ordinary standard symbols with no provenance suffix. No alternative BFUT suffix, capitalization, punctuation, measured-value suffix or provenance variation is permitted. When a standard quantity and its BFUT-derived counterpart appear together, distinguish them only by the presence or absence of the exact literal suffix _vss.

MASTER BFUT SYMBOL REGISTER

Symbol	Definition	Expression / Value
ρₛ	Spaticle substrate equilibrium density	BFUT-constrained/adopted paper value 7.3 × 10⁻²⁷ kg m⁻³
A, B, C, D	P16 free-energy functional coefficients	A=1/2; B=0.56308; C=−1/3; D=1
R₀	P16 dimensionless condensation minimum	1.27348220802151
ℓ_model	Physical condensation length	rₚ/R₀
m_eff	Condensation-unit effective mass	ħ_vss/(c_vss·ℓ_model)
ħ_vss	BFUT-derived reduced Planck constant	mₚ·c_vss·rₚ/(πR₀)
α_vss	BFUT-derived fine-structure constant	α_vss = e²/(4πε₀·ħ_vss·c) = e²R₀/(4ε₀mₚc²rₚ) = 1/137.037 (measured 1/137.036, 0.00048%)
αₛ_vss	BFUT-derived strong coupling constant	αₛ_vss = B·R₀⁴/(8πA) = 0.11785
sin²θ_W_vss	BFUT-derived electroweak mixing angle	sin²θ_W_vss = 1 − (m_W_vss/m_Z_vss)² = 0.23257
m_W_vss, m_Z_vss	BFUT-derived W and Z masses	m_Z_vss = π⁴mₚ = 91.396 GeV/c²; m_W_vss = 256M = (256/3)mₚ = 80.066 GeV/c²
m_H_vss	BFUT-derived Higgs mass	λ_H_vss = 2AR₀/π² = 0.12903; v_vss = 6E_unit/α_vss; m_H_vss = v_vss√(2λ_H_vss) = 124.75 GeV/c²
λₛ	BFUT quartic self-interaction coefficient	Cosmological/substrate quartic; distinct from λ_H_vss
κₛ	BFUT source-to-substrate coupling	1/c_vss²
μₛ	BFUT carrier inverse-length scale	√(3Gρₛ/c_vss²)
Lₛ	BFUT carrier response length	1/μₛ
aₛ	BFUT density-derived acceleration scale	c_vss²μₛ/3
R_d, R_eff	BFUT deformation-domain radii	Defined from ρₛ and rotation
E_unit	P16 fundamental energy unit	mₚ·c_vss²/π
(Vgap/Vq)	Three-core interstitial volume fraction	(2√3−π)/(4π/3)
K_s	BFUT substrate stiffness	ρₛ c_vss²
u_vac	BFUT substrate equilibrium energy density	ρₛ c_vss²
c_vss	BFUT-derived speed of light / substrate propagation speed	Mechanical substrate form: c_vss = √(K_s/ρₛ). Independent electromagnetic-condensation calculation with standard α: c_vss = √[e²R₀/(4ε₀mₚrₚα)] = 2.99791740 × 10⁸ m/s (0.000239% below standard c). Independent calculation with BFUT-derived α_vss: c_vss = √[e²R₀/(4ε₀mₚrₚα_vss)] = 2.99792458 × 10⁸ m/s (exact match to standard c).
h_vss, Lmin	BFUT action quantities	h_vss=2πħ_vss; Lmin=ħ_vss/2
κ_vss, λC_vss, λdB_vss, E_n_vss	BFUT forms after substitution of ħ_vss	Preserve standard symbols when provenance is unambiguous
ℓ_P_vss, m_P_vss, t_P_vss	BFUT Planck-unit forms	Expressions derived using ħ_vss
Amodel	BFUT model-unit localisation coefficient	1/2
P_restore, ρmax, J_entrain	P26 anti-singularity quantities	BFUT expressions defined in P26
m_Shankar	Shankar configuration-resonance mass	m_Shankar c² = 2.60E_unit = 776.5 MeV; BFUT-specific symbol, no _vss suffix
m_BFUT	BFUT configuration-resonance mass	m_BFUT c² = 4.70E_unit = 1403.7 MeV; BFUT-specific symbol, no _vss suffix

AMBIGUOUS LOCAL PARAMETERS RESERVED IN P16

P16 local parameters that would collide with reserved BFUT symbols must carry descriptive subscripts: α_geom is the geometric asymmetry coefficient in the five-term condensation functional; α_T is the thermal coupling coefficient; λ_imb is the circulation-imbalance penalty coefficient; μ_f is the detached-unit mass fraction. These are distinct from α (fine-structure constant), λₛ (Spaticle-field quartic self-interaction coefficient), μₛ (carrier inverse-length scale), and other reserved symbols.

R₀_recon denotes an independently reconstructed condensation scale obtained from external measured inputs. The condensation-functional minimum is R₀; it does not receive _vss because it is a BFUT-specific quantity with no standard counterpart.

STANDARD / EXTERNAL SYMBOLS

Symbol	Definition	Provenance / Role
mₚ	Independently measured proton mass	External physical anchor
rₚ	Independently measured proton charge radius	External SI length anchor
ħ	Independently measured reduced Planck constant	External validation value
α	Independently measured fine-structure constant	External validation value
c	Measured/defined SI speed of light	External reference value
G	Independently measured gravitational constant	External physical constant
mₑ	Independently measured electron mass	External comparison value
sin²θW	Independently measured electroweak mixing angle	External comparison value

Section 0: Input Constants and Functional Coefficients

Standard measured, defined and external quantities retain their ordinary standard notation. A BFUT-derived counterpart of an established standard quantity is distinguished only by appending the exact literal suffix _vss. BFUT-unique quantities retain their reserved symbols without any suffix.

Symbol	Definition	Expression / Value	Papers	Notes
ρₛ	Spaticle substrate equilibrium density	7.3 × 10⁻²⁷ kg m⁻³	P16, P18	Adopted paper value: 7.3 × 10⁻²⁷ kg m⁻³.
mₚ	Proton mass	1.67262192369 × 10⁻²⁷ kg	CODATA	Enters E_unit = mₚ·c²/π.
rₚ	Proton charge radius	0.8414 × 10⁻¹⁵ m	CODATA	CODATA 2018 value. BFUT derivation: R₀ = rₚ·mₚ·c/(π·ħ) gives R₀ = 1.27349. With the derived R₀ = 1.27348, ħ is recovered to 0.00048% from measured.
c / c_vss	Speed of light	Standard: c = 2.99792458 × 10⁸ m/s. BFUT with standard α: c_vss = √[e²R₀/(4ε₀mₚrₚα)] = 2.99791740 × 10⁸ m/s (0.000239% below standard c). BFUT with α_vss: c_vss = √[e²R₀/(4ε₀mₚrₚα_vss)] = 2.99792458 × 10⁸ m/s (exact match to standard c). Mechanical form: c_vss = √(K_s/ρₛ).	Exact SI	Also the Spaticle substrate maximum reorganisation rate.
G	Gravitational constant	6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²	CODATA	Not modified in BFUT. Enters Planck units and DDR domain equation.

P16 Free-Energy Functional Demonstrative Coefficients: E(R) = A/R² + B·R² + C·R + D/R

Symbol	Name	Value	Physical meaning
A	Localisation / kinetic coefficient	0.500000	Dimensionless coefficient in E(R).
B	Bulk displacement coefficient	0.56308	Dimensionless coefficient fixed by condensation geometry.
C	Boundary / cohesion coefficient	-0.333333	Dimensionless coefficient equal to -1/3.
D	Internal circulation coefficient	1.000000	Dimensionless circulation coefficient.

Fundamental Spaticle Field Constants

Symbol	Definition	Expression / Value	Papers	Notes
ρₛ	Intrinsic equilibrium density of the Spaticle field	7.3 × 10⁻²⁷ kg m⁻³	P16, P18	Adopted paper value: 7.3 × 10⁻²⁷ kg m⁻³.
ψ_vss or ψ_vss(r,t)	Spaticle carrier field / gravitational potential	ψ_vss = −(GM/r) exp(−μₛr) (static solution of F1-cov)	P17, P18, P25	Reserved ψ_vss notation for the Spaticle carrier field.
δψ_vss	Carrier perturbation	δψ_vss = ψ_vss − ψ_vss_vac	P18	Dynamical degree of freedom in F1-cov.
ψ_vss_vac	Vacuum equilibrium configuration	Vacuum equilibrium configuration of ψ_vss	P17, P18	Baseline state of the physical matter substrate.
λₛ	Quartic Spaticle-field self-interaction coefficient	Fixed by the selected field normalization	P17, P18	Reserved for the quartic field term; distinct from wavelength symbols.
κₛ	Source-to-substrate coupling coefficient	κₛ = 1/c²	P18	Reserved source coupling; plain κ remains the tunnelling decay constant.
μₛ	Carrier inverse-length response scale	μₛ² = 3Gρₛ/c² = (4.03358955 × 10⁻²⁷ m⁻¹)²	P18	Controls the carrier response and Yukawa length Lₛ = 1/μₛ.

Carrier Field Lengths and Scales

Symbol	Definition	Expression / Value	Papers	Notes
μₛ	Carrier inverse-length scale	μₛ = √(3Gρₛ/c²) = 4.03358955 × 10⁻²⁷ m⁻¹	P18	Density-derived response scale.
Lₛ	Carrier screening length	Lₛ = 1/μₛ = 2.47918135 × 10²⁶ m ≈ 26.205 Gly	P18	Inverse of μₛ.
aₛ	Density-derived acceleration scale	aₛ = c²μₛ/3 = c√(Gρₛ/3) = 1.20840317 × 10⁻¹⁰ m s⁻²	P18	Used by the organised-regime dark matter effects equation (DME).

Gravitational Domain Structure (DDR)

Symbol	Definition	Expression / Value	Papers	Notes
R_d	Intrinsic deformation-domain radius	= (3M/8πρₛ)¹ᐟ³	P18, P25	Earth: 4.87 ly. Sun: 337 ly. Milky Way at 6 × 10¹⁰ M☉: 405 kpc. Proton: 0.301 m. Electron: 0.0246 m.
R_eff	Effective domain radius with rotation	= R_d·(1 + v_rot²/c²)¹ᐟ³	P18, P25	Enlarged deformation domain due to organised rotation. This is a domain-boundary quantity. The DME equation supplies the organised-regime gravitational support.

Galactic and Stack Scale Entrainment DME

Symbol	Definition	Expression / Value	Papers	Notes
DME	Dark Matter Effects equation for organised substrate entrainment	v²(R) = v_b²(R)[1 + aₛR/v_b²(R)]¹ᐟ²; M_extra(<R) = M_b(R){[1 + aₛR²/(GM_b(R))]¹ᐟ² − 1}	P18, P25	The same derived aₛ is used for SPARC and KiDS-1000. SPARC: median outer residual 0.096, shape agreement 92.0%, flat classification 98.8%.

Carrier Field Equations

Symbol	Definition	Expression / Value	Papers	Notes
F1-cov	Fully covariant carrier field equation	gμν∇μ∇ν(δψ_vss) − μₛ²δψ_vss = κₛ gμν∇μ∇νψ_vssmatter	P18	General covariant carrier equation; μₛ² = 3Gρₛ/c² and κₛ = 1/c².

Observational Signal Decomposition

Symbol	Definition	Expression / Value	Papers	Notes
S_obs(t)	Actual measured local gravitational signal	= S_GR(t) + δS_carrier(t)	P18	General observable decomposition; not tied to any specific merger or detector.
δS_carrier(t)	Carrier reconfiguration residual	Basic form: ≈ −∫ exp(−(t−t′)/τc)·[dS_GR/dt′]dt′. Extended form: = A_mem·Θ(t−t₀) + A_tr·exp(−(t−t₀)/τc)·Θ(t−t₀)	P18	Extended form adds a memory term A_mem on top of the basic transient term; the two are physically distinct.
ρ∇	Gradient-sourced substrate density	= α_g\|∇ψ_vss\|²	P18	Rotation-curve and lensing sector; distinct from ρₛ.

P16 Condensation Functional Symbols

Symbol	Definition	Expression / Value	Papers	Notes
E(n)	Total condensation energy for n units	= −J·pairs(s) + λ_cond·Σ(s)² + (n−3)² + α_geom·(n−k) + Dₛ·cos(3φ)	P16	Full five-term functional.
Dₛ	Circulation phase reward coefficient	= 1.5 [demonstrative]. Must be positive.	P16	Encodes three-sphere topology via cos(3φ). Rewards 3+e, penalises 4+0.
φ	Circulation phase angle in P16 condensation	3+e: φ=π/3→cos(3φ)=−1 (min). 4+0: cos=+1.	P16	Scope limited to P16 functional. NOT the Spaticle field variable (which is ψ_vss throughout).
E_unit	Fundamental energy unit	= mₚ·c²/π = 298.661 MeV	P16	Proton mass formula. Anchors all P16 energy calculations.
(Vgap/Vq)	Interstitial volume fraction	= (2√3 − π)/(4π/3) = 0.0770	P16	Universal geometric constant from three-sphere close packing. Connects to electron mass.

Coupling Constants and Masses (P17, P19)

Symbol	Definition	Expression / Value	Papers	Notes
α_vss	Fine structure constant	= 1/137.036 [measured]. BFUT: α_vss = e²/(4πε₀ħ_vssc), ħ_vss = mₚcrₚ/(πR₀) → 1/137.037 (0.00048%)	P17, P19	Reserved exclusively for the fine structure constant.
αₛ_vss	Strong coupling constant	BFUT: αₛ_vss = B·R₀⁴/(8πA) = 0.11785	P17, P19	Condensation invariant; comparison scale is the Z-boson scale.
sin²θ_W / sin²θ_W_vss	Electroweak mixing angle	BFUT: sin²θ_W_vss = 1 − (m_W_vss/m_Z_vss)² = 0.23257.	P17, P19	Output of the independently derived W and Z resonance masses; not an input to either mass.
m_W_vss, m_Z_vss	W and Z boson masses	m_Z_vss = π⁴mₚ = 91.396 GeV/c²; m_W_vss = 256M = (256/3)mₚ = 80.066 GeV/c²	P19	Independent neutral core-stay and charged n=4 reconfiguration resonances.
m_H_vss	H-class radial resonance mass	λ_H_vss = 2AR₀/π² = 0.12903; v_vss = 6E_unit/α_vss = 245.565 GeV; m_H_vss = 124.75 GeV/c²	P19, P19A	Radial electroweak mode; λ_H_vss is distinct from the cosmological/substrate quartic λₛ.
λₛ	Quartic Spaticle-field self-interaction coefficient	Fixed by the selected field normalization	P19	Cosmological/substrate quartic. Do not identify λₛ with the electroweak Higgs coupling λ_H_vss.
m*_vss	BFUT particle-sector mass scale	m_e_vss/α_vss = 1.2483430 × 10⁻²⁸ kg	P19	Feeds the gravitational energy density used to derive ρₛ.
rₑ_vss	BFUT electron length scale	α_vss·ħ_vss/(m_e_vss·c) = 2.8178873 × 10⁻¹⁵ m	P19	Used with m*_vss in the ρₛ derivation chain.
a₀ / a₀_vss	Bohr radius	a₀_vss = ħ_vss/(m_e_vss·c·α_vss) = 52,916.71 fm. Measured a₀: 52,917.8 fm (0.002%)	P19	Derived from the P16 particle chain using ħ_vss, m_e_vss and α_vss; ρₛ is not the numerical parent of the atomic scale.
Kphys	Physical coefficient ħ_vss c	197.33 MeV·fm	P19	Enters the Bohr radius angular-momentum quantisation argument.
T5	Thermal disruption term in the P16 functional	= αT·T·\|Ψ\|²	P19	Fifth additive term; a disruption parameter, not an added energy constant.
Ψ	Spaticle field amplitude/configuration	Field variable used in the P19 thermal treatment	P19	Distinct from ψ_vss (carrier field) and ψ(x,t) (standard quantum wavefunction).

P22 Time Symbols

Symbol	Definition	Expression / Value	Papers	Notes
η	Propagation efficiency	η = √(1 − v²/c²) = 1/γ	P22	Fraction of substrate propagation budget available for internal processes.
t_sub	Accumulated substrate state evolution	t_sub = ∫η dt	P22	What clocks measure; not a pre-existing dimension.

P23 Photon Propagation Symbols

Symbol	Definition	Expression / Value	Papers	Notes
K_s	Substrate stiffness	K_s = ρₛc² = 6.5635567 × 10⁻¹⁰ Pa	P23	Determines c as propagation speed of all substrate disturbances. Analogous to elastic modulus.
u_vac	Substrate vacuum energy density	u_vac = ρₛc² = 6.5635567 × 10⁻¹⁰ J m⁻³	P23	Intrinsic vacuum energy density of the substrate; equivalent to the substrate stiffness K_s.

P22 Time and Propagation Budget Symbols

Symbol	Definition	Expression / Value	Papers	Notes
v_spatial	Spatial traversal component of propagation budget	v_spatial² + v_internal² = c²	P22	Fraction of substrate capacity committed to spatial motion.
v_internal	Internal evolution component of propagation budget	v_internal² = c² − v_spatial²	P22	Fraction available for internal clock evolution.
dτ/dt	Proper time rate	= η = √(1−v²/c²) kinematic; √(1−2GM/rc²) gravitational	P22	What clocks measure; not a pre-existing dimension.
c_s	Local effective substrate propagation speed	η = c_s/c₀; c_s < c₀ in compressed/deformed substrate regions	P17, P18, P22	Distinct from c₀ (vacuum propagation speed) and c (fundamental constant, numerically equal to c₀ in vacuum).
c₀	Vacuum substrate propagation speed	η = c_s/c₀; c₀ is the undisturbed-vacuum value, numerically equal to c	P22	Introduced in P22 to distinguish the vacuum reference speed from the local effective speed c_s in the propagation-efficiency formula.
η_min	Minimum propagation efficiency at maximally compressed core surface	η_min > 0; finite floor set by BFUT Paper 26 restoring pressure and coherence threshold	P22	Sets the finite maximum for gravitational time dilation; replaces the GR divergence at a singularity.
ψ(x,t)	Quantum mechanical wavefunction	Phase exp(−iEt/ℏ) accumulates through substrate reorganisation; Schrödinger equation derived in P19A	P19A, P22	Standard quantum-mechanics notation; distinct from ψ_vss, the BFUT carrier field.
γ	Lorentz factor	γ = 1/√(1 − v²/c²)	P22	Inverse of propagation efficiency η.
t_coord	Coordinate time used as the reference time parameter	Reference coordinate time in an undisturbed substrate region	P22	Used with dτ = ηdt_coord.
v_grav	Gravitational deformation-maintenance component of propagation budget	v_grav² = c²(2GM/rc²)f(r,R_d)	P22	Third propagation-budget channel in the unified time-dilation relation.
f(r,R_d)	Finite deformation-domain factor	f(r,R_d) = 1 inside the active domain and 0 outside the domain boundary	P22	Applies the finite DDR domain to the gravitational contribution.
η_N	Nordtvedt parameter	Empirical parameter constrained by Lunar Laser Ranging	P22	Strong-equivalence test parameter.

P23 Photon Propagation and c Consistency (additions)

Symbol	Definition	Expression / Value	Papers	Notes
c_vss	Speed of light from condensation geometry and electromagnetic coupling	Using standard measured α: c_vss = √[e²R₀/(4ε₀mₚrₚα)] = 2.99791740 × 10⁸ m/s (0.000239% below standard c). Using BFUT-derived α_vss: c_vss = √[e²R₀/(4ε₀mₚrₚα_vss)] = 2.99792458 × 10⁸ m/s (exact match to standard c).	P23, P27	Two independent numerical evaluations of the same BFUT consistency relation. R₀ is BFUT-specific and therefore has no _vss suffix. Only α_vss and c_vss receive the suffix because α and c have standard counterparts.

c_vss

Speed of light from condensation geometry and electromagnetic coupling

Using standard measured α: c_vss = √[e²R₀/(4ε₀mₚrₚα)] = 2.99791740 × 10⁸ m/s (0.000239% below standard c).

Using BFUT-derived α_vss: c_vss = √[e²R₀/(4ε₀mₚrₚα_vss)] = 2.99792458 × 10⁸ m/s (exact match to standard c).

P23, P27

Two independent numerical evaluations of the same BFUT consistency relation. R₀ is BFUT-specific and therefore has no _vss suffix. Only α_vss and c_vss receive the suffix because α and c have standard counterparts.

P24 Quantum Computing Symbols

Symbol	Definition	Expression / Value	Papers	Notes
U(t)	Quantum gate unitary evolution operator	U(t) = exp(−iHt/ħ_vss) = exp(−iHπR₀t/(mₚ·c·rₚ))	P24, P27	Gate speed set by condensation action scale mₚ·c·rₚ/π.
tmin	Minimum gate time for rotation θ	t_min = θ·mₚ·c·rₚ/(R₀·π·H_max)	P24, P27	H_max is maximum control field energy.
C(θ_A,θ_B)	Bell correlation function	C(θ_A,θ_B) = −cos(θ_A − θ_B)	P24	Derived from Born rule and half-integer spin topology.
CHSH	Bell inequality parameter	S ≤ 2 (classical); S ≤ 2√2 (quantum, Tsirelson bound)	P24

P26 Anti-Singularity Symbols

Symbol	Definition	Expression / Value	Papers	Notes
P_restore	Substrate restoring pressure	P_restore = (ρₛ/4)(ρ − ρₛ)	P26	From T4 term of P17 Lagrangian. Prevents singularity formation.
ρmax	Maximum finite collapse density	ρ_max ~ ρₛ·c²/(G·r_s²). Finite by substrate physics.	P26	Replaces GR singularity with finite organised compression.
J_entrain	Outward entrainment flux during collapse	∝ ∇(ρ − ρₛ)	P26	Redistribution mechanism preventing unlimited density divergence.

P27 Planck Constant Consequences

Symbol	Definition	Expression / Value	Papers	Notes
h_vss	Planck original constant	h_vss = 2πħ_vss = m_eff c_vss 2πℓ_model (action of one condensation circulation)	P27	Algebraic identity; physical content: h is the condensation action quantum.
Lmin	Minimum condensation circulation quantum	Lmin = ħ_vss/2 = (1/2)·m_eff c·ℓ_model	P27	Factor 1/2 from 720° topology (P19A). Same origin as spin-1/2.
κ_vss (tunnelling form)	Quantum tunnelling decay constant	κ_vss = πR₀√(2m(V−E))/(mₚ·c_vss·rₚ)	P27, P19A	Penetration depth 1/κ_vss = ħ_vss/√(2m(V−E)).
λC_vss	Compton wavelength (BFUT form)	λC_vss = mₚ·rₚ/(π·R₀·m) = ħ_vss/(mc). Universal: all particles 0.00048% agreement.	P27	Condensation length scaled by mass ratio mₚ/m.
λdB_vss	de Broglie wavelength (BFUT form)	λdB_vss = mₚ·c·rₚ/(π·R₀·p) = ħ_vss/p	P27, P19A	Wave-particle duality as condensation-momentum ratio.
E_n_vss (HO form)	Harmonic oscillator energy levels	E_n_vss = (n+1/2)·mₚ·c_vss·rₚ·ω/(π·R₀) = (n+1/2)ħ_vssω	P27, P19A	Ground state at ω=c/rₚ: 117.5 MeV.
ℓ_P_vss	Planck length (BFUT form)	ℓ_P = √(mₚ·rₚ·G/(π·R₀·c_vss²)) = 1.6163×10⁻³⁵ m (0.0002%)	P27
m_P_vss	Planck mass (BFUT form)	mP = √(mₚ·c_vss²·rₚ/(π·R₀·G)) = 2.1764×10⁻⁸ kg (0.0002%)	P27
t_P_vss	Planck time (BFUT form)	tP = √(mₚ·rₚ·G/(π·R₀·c_vss⁴)) = 5.3912×10⁻⁴⁴ s (0.0002%)	P27
u_vac	Substrate vacuum energy density	u_vac = ρₛc² = 6.5635567 × 10⁻¹⁰ J m⁻³	P27	One field at equilibrium. Not the QFT mode sum.
ℓ_model	Physical condensation length	ℓ_model = rₚ/R₀ = 6.607×10⁻¹⁶ m	P16, P27	SI length anchor. Enters all P16/P27 derivations.
R₀	Dimensionless condensation minimum	R₀ = 1.27348. Derived from observed rₚ, mₚ, c, ħ: R₀ = rₚ·mₚ·c/(π·ħ). Also the minimum of the P16 free-energy functional with derived coefficients. Approximate closed form: 4/π = 1.27324 (0.02%).	P16, P27	Not fitted to reproduce ħ. Independent output of functional.
Amodel	Model-unit localisation coefficient	Amodel = 1/2 exactly. Proved: A = ħ_vss²/(2m_eff), in model units = 1/2 by definition m_eff = ħ_vss/(c·ℓ_model). Exact, not approximate.	P16, P27	Recovers Schrödinger kinetic coefficient T=p²/(2m).

P7 CMB Equilibrium Symbols

Symbol	Definition	Expression / Value	Papers
u_CMB	CMB radiation energy density	= 4.17 × 10⁻¹⁴ J/m³ (measured by COBE)	P7
σ_SB	Stefan-Boltzmann constant	= 5.670 × 10⁻⁸ W/m²/K⁴	P7
j_L	Cosmic luminosity density	≈ 2.6 × 10⁻³³ W/m³ (measured from galaxy surveys)	P7
T_CMB	CMB equilibrium temperature	T = (u_CMB · c / 4σ_SB)¹ᐟ⁴ = 2.725 K	P7

P7A Acoustic Structure Symbols

Symbol	Definition	Expression / Value	Papers
σ_T	Thomson scattering cross-section	= 6.6524 × 10⁻²⁹ m²; σ_T = (8π/3)(e²/mₑ c²)²	P7A
Γ_T	Photon-electron coupling rate	= n_e · σ_T · c	P7A
λ_mfp	Photon mean free path	= 1/(n_e · σ_T)	P7A
D_diff	Photon diffusion coefficient	= c/(3 n_e σ_T) = λ_mfp · c/3	P7A
P(k)	Steady-state matter power spectrum	P(k) = I(k)/D(k) in statistical steady state	P7A
I(k)	Power injection rate at wavenumber k	From shell-like and ripple-like structure generation	P7A
D(k)	Power damping rate at wavenumber k	D(k) = k² · D_diff (diffusion damping)	P7A

P9 Cosmic Rotation Symbols

Symbol	Definition	Expression / Value	Papers	Notes
TI	Tension Index	TI = T_rot / 13.8 Gyr; TI > 1 implies rotational period exceeds standard cosmic age	P9
T_rot	Characteristic rotational period of a structure	T_rot = 2πR/v_rot; measured in Gyr	P9

P10 Sunyaev-Zel'dovich Symbols

Symbol	Definition	Expression / Value	Papers	Notes
y	Compton y-parameter (thermal SZ)	y = ∫(k_B T_e / mₑ c²) n_e σ_T dl; dimensionless	P10
T_e	Intracluster electron temperature	~ 10 keV ≈ 10⁸ K for massive clusters	P10

P12 ISW Effect Symbols

Symbol	Definition	Expression / Value	Papers
Φ	Gravitational potential (ISW context)	ΔT/T = 2∫(∂Φ/∂η)dη where η is conformal time	P12
A_ISW	ISW amplitude relative to Λ-CDM prediction	A_ISW = 1 is the Λ-CDM expectation; observed 4-10 in stacked analyses	P12
η_ISW	Conformal time (ISW context)	Distinct from Lorentz factor η used in P22; context distinguishes usage	P12

P13 Weak Lensing Symbols

Symbol	Definition	Expression / Value	Papers
S₈	Weak lensing structure amplitude parameter	S₈ = σ₈ × (Ω_m/0.3)⁰·⁵; not a direct observable	P13
σ₈	RMS matter fluctuation amplitude in 8 h⁻¹ Mpc spheres	Planck reference: 0.832 ± 0.013	P13
Ω_m	Matter density parameter	Ω_m = ρ_m/ρ_crit; standard measured value ~0.315	P13

P3 Nucleosynthesis Symbols

Symbol	Definition	Expression / Value	Papers
[Li-7]	Lithium-7 abundance relative to hydrogen	Spite plateau: 1.6 × 10⁻¹⁰; BBN prediction: 5.6 × 10⁻¹⁰	P3
k_dest	Lithium-7 destruction rate coefficient	From spallation cross-sections; measured in laboratory	P3
R_prod	Lithium-7 production rate	From cosmic-ray spallation and stellar processes	P3

P21 Consciousness Index Symbols

Symbol	Definition	Expression / Value	Papers	Notes
CI	Consciousness Index (effective)	CI = CI₀ × S; scalar; CIfloor = 1.0	P20, P21
CI₀	Intrinsic consciousness capacity	CI₀ = C × I × D (multiplicative); human average calibrated at 100	P21
S	Survival condition modifier	0 < S ≤ 1; modifies effective CI without changing CI₀	P21
C	Channel capacity component of CI₀	Sensory bandwidth, environmental interaction, social communication, manipulation, internal sensing	P21
ICI	Network integration density component of CI₀	Structural connectivity, dynamic coordination, hierarchical processing; nonlinear amplifier.	P21	NOT same as I(k) in P7A
DCI	Control depth component of CI₀	Autonomy, memory depth, adaptive flexibility; nonlinear.	P21	NOT same as D(k) in P7A

P1 Gravitational Sorting Symbols

Symbol	Definition	Expression / Value	Papers	Notes
rPearson	Pearson correlation coefficient for velocity-distance relationship	r = 0.675 achieved in N-body simulation after gravitational sorting; 84% of survivors receding	P1

Shared Physical Constants

Symbol	Definition	Expression / Value	Papers	Notes
G	Gravitational constant	6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²	All	Standard value; not modified in BFUT.
c / c_vss	Speed of light	Standard: c = 2.99792458 × 10⁸ m/s. BFUT, standard α: c_vss = √[e²R₀/(4ε₀mₚrₚα)] = 2.99791740 × 10⁸ m/s (0.000239% below standard c). BFUT, α_vss: c_vss = √[e²R₀/(4ε₀mₚrₚα_vss)] = 2.99792458 × 10⁸ m/s (exact match to standard c). Mechanical form: c_vss = √(K_s/ρₛ).	All	Also the Spaticle substrate propagation speed.
ħ / ħ_vss	Reduced Planck constant	1.054572 × 10⁻³⁴ J·s. BFUT: mₚ·c·rₚ/(π·R₀) = 1.054577×10⁻³⁴ J·s. Agreement: 0.00048%. Inverted: R₀ = rₚ·mₚ·c/(π·ħ) = 1.27349.	P16, P19	All four functional coefficients now derived from first principles (P16 Appendix).
mₚ	Proton mass	938.272 MeV/c²	P16, P19	Enters E_unit = mₚ c²/π.
mₑ / m_e_vss	Electron mass	0.510999 MeV/c². BFUT: mₚ/(6π⁵) = 0.511009 MeV (0.002%); E_unit/(6π⁴) = 0.511009 MeV (0.002%).	P16	Derived from P16 interstitial geometry.
rₚ	Proton charge radius	0.8414 fm [CODATA 2018]. BFUT: R₀·ℓ_model = rₚ by construction.	P16, P19	Single measured input anchoring the SI conversion.

P23-Specific Symbols

Symbol	Definition	Expression / Value	Papers	Notes
λ_emit	Emitted photon wavelength	Wavelength at emission	P23	Photon propagation and redshift.
λ_obs	Observed photon wavelength	λ_obs = λ_emit(1+z)	P23	Observed wavelength after propagation.
ν_emit	Emitted photon frequency	Frequency at emission	P23
ν_obs	Observed photon frequency	ν_obs/ν_emit = √[(1+β)/(1−β)] for approach	P23	Relativistic Doppler relation.
β	Velocity ratio	β = v/c	P23	Dimensionless source velocity.
m_ν	Neutrino mass	Small non-zero rest mass in the BFUT substrate-ripple description	P23
ΔE_imbalance	Residual reaction-energy imbalance	m_ν ~ ΔEimbalance/c²	P23	Neutrino mass scale.
K_eff(ω)	Frequency-dependent effective substrate stiffness	K_eff(ω) = K′(ω) + iK″(ω)	P23	Internal-relaxation response.
K′(ω)	Reactive stiffness component	Real part of K_eff(ω)	P23
K″(ω)	Dissipative stiffness component	Imaginary part of K_eff(ω)	P23
α_att	Energy attenuation coefficient	E(x) = E₀e^(−2α_att x)	P23	Distinct from α, the fine-structure constant.

Symbol	Definition	Expression / Value	Papers	Notes
L_att(ω)	Energy attenuation length	L_att(ω) ≈ (c/ω)[K′(ω)/K″(ω)]	P23
u_def	Deformation energy density	E = u_def·A_cs·Δx	P23
A_cs	Disturbance cross-sectional area	Cross-sectional area of a directed disturbance	P23	Subscript distinguishes it from coefficient A.
Δx	Coherent longitudinal extent	Longitudinal extent of a directed coherent disturbance	P23
ε_att	Fractional energy transfer per traversed element	dE/dx = −(ε_att/Δx)E	P23	Distinct from η, the propagation efficiency.
dV	Traversed volume element	dV = A_cs·dx	P23
τ_relax	Internal substrate relaxation time	Relaxation timescale of the internal substrate mode	P23	Distinct from carrier response time τ_c.
ΔK	Substrate stiffness change	Stiffness change associated with internal relaxation	P23
q(x,t)	KdV soliton amplitude	Wave-amplitude function used for the KdV soliton illustration	P23
κ_GW	Gravitational-wave attenuation parameter	Empirical attenuation parameter constrained by GW propagation observations	P23	Distinct from κ, the quantum tunnelling decay constant.

Canonical density display rule

In papers, write ρₛ = 7.3 × 10⁻²⁷ kg m⁻³. The exact particle-sector derivation value is 7.30294170 × 10⁻²⁷ kg m⁻³. This exact value is for numerical derivation only and is not to be written in papers. Use ρₛ for the Spaticle Field density. It is a BFUT-specific quantity and therefore does not receive _vss.