Famous Equations (Adjusted)
Curated classical forms rewritten in your Phase-Lift / Adaptive-π style, scored with the same leaderboard rubric.
Scoring Rubric (0-100)
- Tractability (0-20)
- Physical plausibility (0-20)
- Validation (0-20)
- Artifact completeness (0-10)
- Total normalized from a 70-point base
- Novelty is shown as a dated tag only
F1
Schrödinger Equation
Madelung + Phase-Lift real-valued split
Adjusted form
$$\partial_t\rho+\nabla\cdot(\rho\,\mathbf v)=0,\quad \hbar\,\partial_t\phi+\tfrac{m}{2}\mathbf v^2+V+Q[\rho]=0$$
Reference: famous-adjusted list
Description
Schrödinger's equation rewritten via the Madelung decomposition with Phase-Lift: the complex wave function splits into a continuity equation for probability density ρ and a Hamilton–Jacobi equation for lifted phase φ, eliminating the imaginary unit entirely.
Rubric
T 16/20, P 16/20, V 15/20, A 5/10, normalized to 74/100
Novelty tag
26 @ legacy
Definitions
ρ = probability density; v = probability-current velocity; φ = Phase-Lifted field; Q[ρ] = Bohm quantum potential; V = external potential.
Assumptions
- ρ > 0 everywhere so Q[ρ] is well-defined (no nodes).
- Phase-Lift variable θ_R maps bijectively to φ over the domain.
- The decomposition is exact when ψ ≠ 0.
Caveat
Equivalent only under the stated smoothness conditions; nodal regions (ρ = 0) require separate treatment.
List index
F1
Category
Famous (Adjusted)
Core refs
Repository
F2
Aharonov–Bohm Effect
Phase-Lifted geometric phase with winding sectors
Adjusted form
$$\theta_R[\gamma]=\mathrm{unwrap}\!\Big(\frac{q}{\hbar}\oint_\gamma \mathbf A\cdot d\mathbf\ell;\;\theta_{\rm ref},\pi_a\Big)=\frac{q}{\hbar}\int_S \mathbf B\cdot d\mathbf S + 2\pi_a w$$
Reference: famous-adjusted list
Description
The Aharonov–Bohm phase expressed through Phase-Lift with explicit winding-number sectors. The unwrap operator removes branch-cut ambiguity and tracks accumulated windings w ∈ ℤ around the solenoid.
Rubric
T 16/20, P 16/20, V 15/20, A 5/10, normalized to 74/100
Novelty tag
25 @ legacy
Definitions
θ_R = unwrapped (lifted) phase; γ = closed loop encircling solenoid; A = vector potential; B = magnetic field; w = integer winding number; π_a = adaptive period.
Assumptions
- Gauge region excludes singular crossings during unwrapping.
- Reference phase θ_ref is consistent along γ.
- π_a → π in the classical limit.
Caveat
Branch-handling conventions affect w assignment; the unwrap policy must be fixed before comparing experiments.
List index
F2
Category
Famous (Adjusted)
Core refs
Repository
F3
Maxwell's Equations
Phase-Lifted electromagnetic potentials with adaptive winding
Adjusted form
$$\mathbf E=-\nabla V-\partial_t\mathbf A,\quad \mathbf B=\nabla\times\mathbf A,\quad \theta_R=\mathrm{unwrap}\!\Big(\frac{q}{\hbar}\oint\mathbf A\cdot d\mathbf\ell;\;\theta_{\rm ref},\pi_a\Big)$$
Reference: famous-adjusted list
Description
Maxwell's field equations expressed through gauge potentials (A, V) with Phase-Lift applied to the electromagnetic phase. The winding structure of A around closed loops is tracked explicitly via the unwrap operator, connecting classical electrodynamics to the quantum phase framework.
Rubric
T 16/20, P 15/20, V 13/20, A 5/10, normalized to 70/100
Novelty tag
24 @ legacy
Definitions
E = electric field; B = magnetic field; A = vector potential; V = scalar potential; θ_R = lifted electromagnetic phase around a loop.
Assumptions
- Gauge freedom is fixed before applying Phase-Lift (e.g., Coulomb or Lorenz gauge).
- The loop integral is well-defined (no singularities on the path).
- π_a → π recovers standard flux quantization.
Caveat
The Phase-Lift layer adds topological tracking but does not modify Maxwell's equations themselves; it enriches the description of the gauge sector.
List index
F3
Category
Famous (Adjusted)
Repository
F4
Euler–Lagrange Equation
Variational principle with Phase-Lift action
Adjusted form
$$\frac{\partial \mathcal L_{PL}}{\partial \theta_R}-\frac{d}{dt}\frac{\partial \mathcal L_{PL}}{\partial \dot\theta_R}=0,\quad \mathcal L_{PL}=\mathcal L(\rho,\theta_R,\pi_a,\dot\rho,\dot\theta_R,\dot\pi_a)$$
Reference: famous-adjusted list
Description
The Euler–Lagrange equation applied to a Phase-Lift Lagrangian where the generalized coordinates are (ρ, θ_R, π_a). Stationarity of the Phase-Lift action yields the equations of motion for the lifted phase field and the adaptive period.
Rubric
T 16/20, P 15/20, V 13/20, A 5/10, normalized to 70/100
Novelty tag
20 @ legacy
Definitions
L_PL = Phase-Lift Lagrangian; θ_R = lifted phase; ρ = amplitude field; π_a = adaptive period; dot denotes time derivative.
Assumptions
- The Phase-Lift Lagrangian is specified (not arbitrary).
- Standard regularity conditions for the variational principle hold.
- Boundary terms vanish or are handled explicitly.
Caveat
This is the variational framework — the specific L_PL determines the physics.
List index
F4
Category
Famous (Adjusted)
Core refs
Repository
F5
Fourier Heat Equation
Diffusion with Phase-Lift curvature-salience source
Adjusted form
$$\partial_t u=\alpha\nabla^2 u + \kappa(x,t)\,u,\quad \kappa=\lambda\,\mathcal K[\theta_R]$$
Reference: famous-adjusted list
Description
Fourier's heat equation augmented with a curvature-salience source term. The coefficient κ is derived from the Phase-Lift curvature K[θ_R], making regions of high phase curvature act as local heat sources. This models phase-aware diffusion.
Rubric
T 16/20, P 15/20, V 13/20, A 5/10, normalized to 70/100
Novelty tag
18 @ legacy
Definitions
u = temperature field; α = thermal diffusivity; κ = curvature-driven source; K[θ_R] = curvature of lifted phase; λ = coupling constant.
Assumptions
- The curvature-salience coupling λ is small enough for perturbative treatment.
- θ_R is smooth enough for K[θ_R] to be computed.
- Standard boundary and initial conditions apply.
Caveat
The curvature source term is a Phase-Lift extension; standard Fourier law is recovered when λ = 0.
List index
F5
Category
Famous (Adjusted)
Repository
F6
Berry Phase
Discrete overlap product with unwrapped accumulation
Adjusted form
$$\theta_R[\gamma]=\sum_k \mathrm{unwrap}\!\big(\Arg\langle\psi_{k+1}|\psi_k\rangle;\;\theta_{\rm ref},\pi_a\big)$$
Reference: famous-adjusted list
Description
Berry's geometric phase computed as a discrete sum of unwrapped overlap arguments along a closed parameter path. Phase-Lift eliminates the multivalued Arg ambiguity at each step.
Rubric
T 15/20, P 15/20, V 13/20, A 5/10, normalized to 69/100
Novelty tag
24 @ legacy
Definitions
Arg = principal argument (before unwrapping); ⟨ψ_{k+1}|ψ_k⟩ = overlap between adjacent adiabatic states; θ_ref = reference phase for unwrapping.
Assumptions
- Adjacent-state overlaps are non-zero along the entire path.
- Sampling density is fine enough to prevent spurious 2π_a jumps.
- The adiabatic approximation holds throughout.
Caveat
Numerical stability depends on path discretization; coarse sampling can alias the accumulated phase.
List index
F6
Category
Famous (Adjusted)
Core refs
Repository
F7
Noether's Theorem
Conserved current from Phase-Lift symmetry
Adjusted form
$$\partial_\mu j^\mu_{PL}=0,\quad j^\mu_{PL}=\frac{\partial\mathcal L_{PL}}{\partial(\partial_\mu\theta_R)}\,\delta\theta_R+\frac{\partial\mathcal L_{PL}}{\partial(\partial_\mu\pi_a)}\,\delta\pi_a$$
Reference: famous-adjusted list
Description
Noether's theorem applied to the Phase-Lift Lagrangian: continuous symmetries of L_PL yield conserved currents j^μ_PL. The lifted phase θ_R and adaptive period π_a each contribute to the conserved current when the action is invariant under their transformations.
Rubric
T 15/20, P 15/20, V 13/20, A 5/10, normalized to 69/100
Novelty tag
21 @ legacy
Definitions
j^μ_PL = Phase-Lift Noether current; L_PL = Phase-Lift Lagrangian; δθ_R, δπ_a = infinitesimal symmetry variations.
Assumptions
- The Phase-Lift action admits the continuous symmetry in question.
- Fields are on-shell (equations of motion satisfied).
- Boundary terms vanish in the variational derivation.
Caveat
The specific conserved quantities depend on which symmetry is applied; this is the structural form.
List index
F7
Category
Famous (Adjusted)
Core refs
Repository
F8
Josephson Relations
Adaptive-π phase dynamics in superconducting junctions
Adjusted form
$$I=I_c\sin\!\Big(\frac{\pi}{\pi_a}\theta_R\Big),\quad \frac{d}{dt}\Big(\frac{\pi}{\pi_a}\theta_R\Big)=\frac{2e}{\hbar}V$$
Reference: famous-adjusted list
Description
The DC and AC Josephson relations with adaptive-π scaling. The standard sin(φ) current–phase relation becomes sin(πθ_R/π_a), where π_a is an adaptive period parameter that recovers the classical limit when π_a → π.
Rubric
T 15/20, P 14/20, V 13/20, A 5/10, normalized to 67/100
Novelty tag
22 @ legacy
Definitions
I_c = critical current; θ_R = lifted phase difference across junction; π_a = adaptive period; V = junction voltage; e = electron charge.
Assumptions
- π_a varies slowly relative to junction dynamics.
- Calibration restores the classical Josephson limit when π_a → π.
- Single-junction, short-junction approximation applies.
Caveat
Non-standard phase scaling requires experimental calibration; not a replacement for the standard Josephson effect.
List index
F8
Category
Famous (Adjusted)
Core refs
Repository
F9
Dirac Equation
Spinor polar decomposition with Phase-Lift
Adjusted form
$$(i\gamma^\mu\partial_\mu - m)\psi=0,\quad \psi^{PL}=\sqrt{\rho}\,R\,e^{i\phi},\;\phi=\frac{\pi}{\pi_a}\theta_R$$
Reference: famous-adjusted list
Description
The Dirac equation for a free spin-½ particle with the four-component spinor decomposed into amplitude √ρ, an SU(2) rotation R, and a Phase-Lifted scalar phase φ. This separates the spin degree of freedom from the Phase-Lift winding structure.
Rubric
T 14/20, P 14/20, V 11/20, A 5/10, normalized to 63/100
Novelty tag
23 @ legacy
Definitions
γ^μ = Dirac gamma matrices; ψ = four-component spinor; ρ = probability density; R ∈ SU(2) = spin rotation; φ = lifted phase; m = rest mass.
Assumptions
- The polar decomposition ψ = √ρ R exp(iφ) is valid where ψ ≠ 0.
- Spin and phase degrees of freedom decouple at the decomposition level.
- Relativistic and non-relativistic limits are tracked separately.
Caveat
The spinor decomposition breaks down at nodes; this is a representation change, not a modification of Dirac physics.
List index
F9
Category
Famous (Adjusted)
Core refs
Repository
F10
Navier–Stokes (Vorticity Form)
Phase-Lift vortex tracking with adaptive winding
Adjusted form
$$\partial_t\boldsymbol\omega+(\mathbf v\cdot\nabla)\boldsymbol\omega=(\boldsymbol\omega\cdot\nabla)\mathbf v+\nu\nabla^2\boldsymbol\omega,\quad w=\frac{1}{2\pi_a}\oint\mathbf v\cdot d\mathbf\ell$$
Reference: famous-adjusted list
Description
The vorticity transport equation from Navier–Stokes with Phase-Lift applied to circulation integrals. The integer winding number w counts vortex sheet crossings using the adaptive period π_a, connecting fluid topology to the phase framework.
Rubric
T 14/20, P 14/20, V 11/20, A 5/10, normalized to 63/100
Novelty tag
22 @ legacy
Definitions
ω = ∇ × v = vorticity; v = velocity field; ν = kinematic viscosity; w = winding number of velocity circulation.
Assumptions
- Flow is incompressible (∇·v = 0).
- Viscosity is constant and isotropic.
- Circulation integral is well-defined (no singularities on the contour).
Caveat
Phase-Lift adds topological vortex tracking; it does not solve the Navier–Stokes existence/smoothness problem.
List index
F10
Category
Famous (Adjusted)
Repository
F11
Feynman Path Integral
Sector-summed partition function with Phase-Lift winding constraint
Adjusted form
$$Z=\sum_{w\in\mathbb Z}\int \mathcal D\rho\,\mathcal D\theta_R\,\mathcal D\pi_a\; e^{\frac{i}{\hbar}S_{PL}[\rho,\theta_R,\pi_a]}\;\delta\!\big(\theta_R(T)-\theta_R(0)-2\pi_a w\big)$$
Reference: famous-adjusted list
Description
Feynman's path integral reformulated with Phase-Lift coordinates. The sum over winding sectors w replaces the standard sum over paths, and the delta function enforces quantized phase returns. Each sector contributes independently.
Rubric
T 14/20, P 14/20, V 11/20, A 5/10, normalized to 63/100
Novelty tag
23 @ legacy
Definitions
Z = partition function; S_PL = Phase-Lift action; w = winding number; δ = Dirac delta enforcing boundary condition.
Assumptions
- The Phase-Lift measure D[ρ,θ_R,π_a] is well-defined.
- The sector sum converges or is regularized consistently.
- S_PL is specified concretely for the system.
Caveat
The action S_PL must be given explicitly for any concrete calculation; this is the structural framework.
List index
F11
Category
Famous (Adjusted)
Repository
F12
U(N) Gauge Holonomy
Determinant-phase splitting with Phase-Lift winding
Adjusted form
$$U(\gamma)=e^{i\varphi}V,\; V\in SU(N),\quad \theta_R=\mathrm{unwrap}(\varphi;\theta_{\rm ref},\pi_a),\quad w_{\det}=\frac{\theta_R(T)-\theta_R(0)}{2\pi_a}$$
Reference: famous-adjusted list
Description
A U(N) holonomy (parallel transport around a loop γ) factored into an SU(N) part V and a U(1) determinant phase. Phase-Lift tracks the determinant phase through unwrapping, yielding an integer winding number w_det that classifies the topological sector.
Rubric
T 14/20, P 14/20, V 11/20, A 5/10, normalized to 63/100
Novelty tag
21 @ legacy
Definitions
U(γ) = holonomy matrix; φ = det-phase; V = SU(N) component; w_det = determinant winding number.
Assumptions
- Holonomy path admits stable determinant-phase unwrapping.
- Global phase convention is fixed over the loop.
- The U(1) ⊗ SU(N) factorization is non-degenerate.
Caveat
Winding assignment can shift under different gauge conventions; this is a frame-dependent decomposition.
List index
F12
Category
Famous (Adjusted)
Core refs
Repository
F13
Klein–Gordon Equation
Relativistic scalar field with polar Phase-Lift decomposition
Adjusted form
$$\Big(\Box+\frac{m^2c^2}{\hbar^2}\Big)\psi=0,\quad \psi^{PL}=\sqrt{\rho}\,e^{i\phi},\;\phi=\frac{\pi}{\pi_a}\theta_R$$
Reference: famous-adjusted list
Description
The Klein–Gordon equation for a relativistic scalar field, with the complex field ψ decomposed into amplitude √ρ and Phase-Lifted scalar phase φ = πθ_R/π_a. This separates amplitude dynamics from phase winding in a Lorentz-covariant setting.
Rubric
T 14/20, P 14/20, V 11/20, A 5/10, normalized to 63/100
Novelty tag
20 @ legacy
Definitions
□ = d'Alembertian (∂²/∂t² − c²∇²); ψ = complex scalar field; ρ = |ψ|²; φ = lifted phase; m = particle mass.
Assumptions
- Polar decomposition is valid where ψ ≠ 0.
- Relativistic and non-relativistic limits are not mixed in derived reductions.
- π_a → π recovers the standard Klein–Gordon phase.
Caveat
Nodal regions (ψ = 0) require separate treatment; this is a representation, not a modification.
List index
F13
Category
Famous (Adjusted)
Core refs
Repository
F14
Einstein Field Equations
Conformal-metric Phase-Lift with adaptive curvature scale
Adjusted form
$$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu},\quad \tilde g_{\mu\nu}=e^{2\sigma(\pi_a)}g_{\mu\nu},\;\sigma(\pi_a)=\ln\!\Big(\frac{\pi_a}{\pi}\Big)$$
Reference: famous-adjusted list
Description
Einstein's field equations with a Phase-Lift conformal rescaling of the metric. The conformal factor σ(π_a) is determined by the adaptive period, linking spacetime curvature to the phase framework. When π_a → π, the standard metric is recovered.
Rubric
T 13/20, P 13/20, V 9/20, A 5/10, normalized to 57/100
Novelty tag
25 @ legacy
Definitions
G_μν = Einstein tensor; Λ = cosmological constant; T_μν = stress-energy tensor; g̃_μν = conformally rescaled metric; σ = conformal factor from π_a.
Assumptions
- The conformal rescaling is smooth and non-degenerate.
- π_a varies on scales much larger than the Planck length.
- The rescaled equations reduce to standard GR when π_a = π.
Caveat
Highly speculative application; conformal rescaling changes the physical content unless carefully constrained. This is a framework suggestion, not established physics.
List index
F14
Category
Famous (Adjusted)
Repository