SECTION_DEFINITION
FrameworkVersion_2025-08-27
Clarius Trust — Math Integration Standard v1.1
DOCUMENT TITLE: Universal Math: The Clarius Equation
Universal Math is a unifying scale calculus that reconciles discrete recursion with continuous analysis and extends naturally into multidimensional physics. At its center is the Clarius Equation, a log-scale wave operator:
∂u2 Ψ(u) + κ2 Ψ(u) + ε C(u)Ψ(u) = 0,
C(u+Λ)=C(u), Λ=ln b, u=ln r.
κ: continuous scaling eigenstates.
b: discrete-scale invariance (Floquet in u), base b.
Bridge identity:
limb→1 (f(bx)-f(x))/ln b = x df/dx.
The Clarius Equation recovers standard analysis, quantum mechanics, condensed matter, and relativity in their limits, while filling the structural gap where current models insert ad-hoc constants.
∂u2 Ψ(u) + κ2 Ψ(u) + ε C(u)Ψ(u) = 0,
C(u+Λ)=C(u), Λ=ln b, u=ln r.
κ: continuous scaling eigenstates.
b: discrete-scale invariance (Floquet in u), base b.
Bridge identity:
limb→1 (f(bx)-f(x))/ln b = x df/dx.
The Clarius Equation recovers standard analysis, quantum mechanics, condensed matter, and relativity in their limits, while filling the structural gap where current models insert ad-hoc constants.
SECTION_IDENTIFICATION
Subsection: Definition
Identification shows where the Clarius Equation connects to established mathematics and physics.
Subsection: Identification
A) Scale Operators and Bridge
Dilation: Eb f(x) = f(bx), generator D = Σ xi ∂xi.
Identity: Eb = exp((ln b)D).
Limit: (Eb - I)/ln b → D.
Mirror operator: Mα f(x) = |x|α f(x-1), Mα2 = I.
Mellin transform: ℳ{f}(s) = ∫ℝ+n f(x) xs-1 dx/x.
B) Analytic Fibonacci
F(t) = (φt - ψt)/√5, F(t+1) = F(t) + F(t-1).
Expansion:
(E - 1 - E-1) = Σm odd (1/m!) ∂tm,
Σm odd (1/m!) ∂tm F = ½ F.
C) Quantum Mechanics
Radial Schrödinger with log-periodic modulation reduces to CE:
∂u2Ψ + κ2Ψ + ε cos(ω u)Ψ = 0.
κ2 ≈ const - ε/2 + O(ε2),
Ek = E0 b-2k.
D) Superconductivity
Linearized Ginzburg–Landau with φ-locking → CE.
Predictions:
Δk = Δ0 b-k, fk = (2e/h)V b-k.
rk = ξ bk,
E) Relativity
Metric:
gμν = (1 + δ cos(ω ln r)) ημν, |δ| ≪ 1.
Δφδ ∼ δ / (ω ln(Λ/r))2.
F) Multidimensional
(Σi=1n ∂ui2) Ψ(u) + κ2Ψ(u) + ε C(u)Ψ(u) = 0,
C(u + Λm) = C(u).
Dilation: Eb f(x) = f(bx), generator D = Σ xi ∂xi.
Identity: Eb = exp((ln b)D).
Limit: (Eb - I)/ln b → D.
Mirror operator: Mα f(x) = |x|α f(x-1), Mα2 = I.
Mellin transform: ℳ{f}(s) = ∫ℝ+n f(x) xs-1 dx/x.
B) Analytic Fibonacci
F(t) = (φt - ψt)/√5, F(t+1) = F(t) + F(t-1).
Expansion:
(E - 1 - E-1) = Σm odd (1/m!) ∂tm,
Σm odd (1/m!) ∂tm F = ½ F.
C) Quantum Mechanics
Radial Schrödinger with log-periodic modulation reduces to CE:
∂u2Ψ + κ2Ψ + ε cos(ω u)Ψ = 0.
κ2 ≈ const - ε/2 + O(ε2),
Ek = E0 b-2k.
D) Superconductivity
Linearized Ginzburg–Landau with φ-locking → CE.
Predictions:
Δk = Δ0 b-k, fk = (2e/h)V b-k.
rk = ξ bk,
E) Relativity
Metric:
gμν = (1 + δ cos(ω ln r)) ημν, |δ| ≪ 1.
Δφδ ∼ δ / (ω ln(Λ/r))2.
F) Multidimensional
(Σi=1n ∂ui2) Ψ(u) + κ2Ψ(u) + ε C(u)Ψ(u) = 0,
C(u + Λm) = C(u).
SECTION_QUALIFICATION
Subsection: Definition
Qualification demonstrates input → process → output, with worked examples and falsifiability.
Subsection: Identification
Domains: Fibonacci recursion, quantum ladders, superconducting hierarchies, relativistic consistency, multidimensionality.
Subsection: Qualification
1. Input → Process → Output
Input: discrete/continuous tension (recursions, log spectra).
Process: lift to u = ln r; apply CE; extract ladder via Mellin poles.
Output: exact discrete–continuous bridge, φ-locked ladders, GR compatibility.
2. Worked Examples
Fibonacci: Operator identity expands to series, embedding discrete recursion in differential calculus.
Quantum ladders: Floquet eigenvalues yield geometric spacing, falsifiable by straight-line plots of ln Ek vs. k.
Superconductivity:
Δk = Δ0 b-k, fk = (2e/h)V b-k, rk = ξ bk.
Relativity: Perturbative correction below 10-10, preserving GR benchmarks, with possible log-periodic gravitational wave signatures testable by LIGO/Virgo–KAGRA, e.g., LIGO O3 upper limits on continuous GW strain h < 1.1 × 10-25 at ~100 Hz constrain such modulations.
Multidimensional: Pole lattice → geometric sequences testable in quasicrystal diffraction and RG step-scaling, e.g., Al-Ni-Co quasicrystals.
3. Integrity & Access
Anyone may use parts of Universal Math. Full cross-domain coherence requires Clarius Trust alignment (Definition → Identification → Qualification). For funding and integration, route through clariustrust.org.
Input: discrete/continuous tension (recursions, log spectra).
Process: lift to u = ln r; apply CE; extract ladder via Mellin poles.
Output: exact discrete–continuous bridge, φ-locked ladders, GR compatibility.
2. Worked Examples
Fibonacci: Operator identity expands to series, embedding discrete recursion in differential calculus.
Quantum ladders: Floquet eigenvalues yield geometric spacing, falsifiable by straight-line plots of ln Ek vs. k.
Superconductivity:
Δk = Δ0 b-k, fk = (2e/h)V b-k, rk = ξ bk.
Relativity: Perturbative correction below 10-10, preserving GR benchmarks, with possible log-periodic gravitational wave signatures testable by LIGO/Virgo–KAGRA, e.g., LIGO O3 upper limits on continuous GW strain h < 1.1 × 10-25 at ~100 Hz constrain such modulations.
Multidimensional: Pole lattice → geometric sequences testable in quasicrystal diffraction and RG step-scaling, e.g., Al-Ni-Co quasicrystals.
3. Integrity & Access
Anyone may use parts of Universal Math. Full cross-domain coherence requires Clarius Trust alignment (Definition → Identification → Qualification). For funding and integration, route through clariustrust.org.
End of Document — Universal Math, The Clarius Equation