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This document is an appendix to the Banya Framework Comprehensive Report. For each of the 118 physics equations, it records the original formula, the Banya Framework transformation, the subframe, the verdict rationale, and the derivation expectation value.

Appendix: Detailed Verification of 118 Physics Equations + Derivation Expectation Values

For each of the 118 equations: (1) original formula, (2) Banya Framework transformation, (3) subframe, (4) verdict rationale, (5) derivation expectation value.

Note: 'Expected derivations' describe potential additional derivations not yet performed, not already completed results.

Subframe assignment rule: Map the variables used in each physics equation to the 4 axes of the Banya Equation. (1) time, space only → classical subframe. (2) observer, superposition only → quantum subframe. (3) both sides used → full frame. (4) space only (time-independent) → space subframe.

A~B (Equations 1~16): Directly Listed in Chapter 9 of the Main Report

For detailed verification, see banya.html Chapter 9.

C. Electromagnetism (12/12 PASS)

Eq. 17. Coulomb's Law

Original: $F = kq₁q₂/r²$
Transform: $F ∝ 1/space²$ (inverse square between charges, space consumption intensity)
$F$: force | $k$: wave number/spring constant | $q$: charge | $r$: distance | $space$: space
  • Subframe: space
  • Verdict: r = space. Isomorphic to Newton's gravitation. Write rate decreases as inverse square of distance. $1/space²$ inverse square. PASS
  • Derivation expectation: space subframe used. time axis coupling: EM radiation from time-varying charges (Larmor formula). observer axis: EM decoherence rate from charge position disturbance. superposition axis: entanglement energy from Coulomb potential in quantum superposition. Additional: CAS cost translation shows Coulomb (Compare cost $α=1/137$) and Newton (Swap cost 1) are isomorphic ($1/r²$) due to the same spatial consumption structure in CAS.
  • Derived α = 1/137.036 (EM coupling = CAS Compare cost). D-01

    • Eq. 18. Electric Field Energy Density

    Original: $u = ½ε₀E²$
    Transform: $u = ½ε₀ ×$ (d(φ)/d(space))² (square of spatial potential gradient)
    $u$: energy density | $ε₀$: vacuum permittivity | $E$: energy/electric field | $space$: space
  • Subframe: space
  • Verdict: E = d(φ)/d(space), so $E²$ = square of spatial gradient. Space component density of δ². PASS
  • Derivation expectation: space subframe used. time axis: time variation of electric field energy density determines EM wave radiation energy (Poynting theorem). observer axis: quantum limit of E-field energy density measurement (vacuum fluctuation energy from $ΔE×Δt ≥$ ℏ/2). superposition axis: Casimir energy density from vacuum E-field superposition states.

    • Eq. 19. Magnetic Field Energy Density

    Original: $u = B²/(2μ₀)$
    Transform: $u =$ (∇×A)²/(2μ₀) (square of rotational field in space)
    $u$: energy density | $B$: magnetic field | $μ₀$: vacuum permeability | $∇$: nabla | $A$: area/vector potential
  • Subframe: space
  • Verdict: B = $∇×$A, so $B²$ = square of spatial rotation. Isomorphic to $E²$, space subframe energy density. PASS
  • Derivation expectation: space subframe used. time axis: time variation of B-field energy density yields induction EMF (inverse derivation of Faraday's law). observer axis: quantum limit of B-field measurement (squeezed light magnetic field resolution). superposition axis: Aharonov-Bohm phase shift from quantum superposition of magnetic field states.

    • Eq. 20. Electromagnetic Wave Equation

    Original: $∂²E/∂t² = c²∂²E/∂x²$
    Transform: $d²(field)/d(time)² = \|C\|² × d²(field)/d(space)²$ (time² and space² exchanged via \|C\|²)
    $E$: energy/electric field | $c$: speed of light | $C$ | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: c = $\|C\|$ = classical bracket norm. $c²$ is the exchange coefficient between $time²$ and $space²$. d'Alembertian structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of EM wave measurement (photon number vs. phase uncertainty trade-off). superposition axis: interference pattern from superposition of two EM waves (photon interference conditions, optical coherence). Additional: CAS cost translation shows EM wave propagation condition at Compare cost ($α$), relating wave energy and CAS tick (E=$ℏω$=energy per tick × frequency).

    • Eq. 21. Poynting Vector

    Original: $S ∝ E×B ∝ E²$
    Transform: $S ∝$ (d(φ)/d(space))² (square of spatial potential gradient = energy flow)
    $S$: action/entropy | $E$: energy/electric field | $B$: magnetic field | $space$: space
  • Subframe: space
  • Verdict: Energy flow = field squared. Reduces to E = d(φ)/d(space). space subframe complete. PASS
  • Derivation expectation: space subframe used. time axis: time-averaged Poynting vector determines radiation pressure. observer axis: quantum limit of radiation energy flow measurement (photon detector shot noise limit). superposition axis: quantum interference conditions for bidirectional EM energy flow (radiation pressure interferometer).

    • Eq. 22. Capacitor Energy

    Original: $E = ½CV²$
    Transform: $E = ½C ×$ (d(φ)/d(space) × space)² (potential = spatial gradient × distance)
    $E$: energy/electric field | $C$: capacitance | $V$: voltage | $space$: space
  • Subframe: space
  • Verdict: V = potential difference = spatial potential difference. $V²$ = square of potential in space. space subframe energy storage. PASS
  • Derivation expectation: space subframe used. time axis: RC circuit time constant and energy release rate (RC circuit characteristics). observer axis: quantum limit of capacitor energy measurement (minimum energy unit $e²/2C$ from charge quantization). superposition axis: Cooper pair tunneling conditions from quantum superposition of two energy states (Josephson junction).

    • Eq. 23. Joule's Law

    Original: $P = I²R$
    Transform: $P =$ (dQ/d(time))² × R (current = time rate of charge change, its square is power dissipation)
    $P$: power/pressure | $I$: current | $R$: curvature/resistance | $time$: time
  • Subframe: time-space
  • Verdict: I = dQ/d(time), so $I² = (1/time)²$ ratio. P is energy per unit time. Includes time. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of current measurement (shot noise: $ΔI ∝ √(eI/Δt)$). superposition axis: quantum resistance ($h/e²$ = Hall resistance quantum) derivation conditions.

    • Eq. 24. Inductor Energy

    Original: $E = ½LI²$
    Transform: $E = ½L ×$ (dQ/d(time))² (energy stored as square of time rate of change)
    $E$: energy/electric field | $L$: angular momentum/inductance | $I$: current | $time$: time
  • Subframe: time-space
  • Verdict: I = dQ/d(time), so $I²$ is time-dependent. Magnetic field energy stored as time-ratio squared. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of inductor energy measurement (minimum inductor energy from magnetic flux quantum $Φ₀$ = $h/2e$). superposition axis: SQUID flux quantization conditions from superposition of two inductor energy states.

    • Eq. 25. Biot-Savart Law

    Original: $dB ∝ Idl/r²$
    Transform: $dB ∝$ (dQ/d(time)) × d(space)/space² (current × distance element divided by space²)
    $B$: magnetic field | $I$: current | $r$: distance | $time$: time | $space$: space
  • Subframe: space
  • Verdict: dl/r² is a space element divided by $space²$. Basic $1/space²$ inverse square structure. PASS
  • Derivation expectation: space subframe used. time axis: magnetic radiation from time-varying current (antenna radiation pattern). observer axis: effect of magnetic field measurement on atomic magnetic moments (NMR principle). superposition axis: spin magnetic resonance from superposition of quantum current states.

    • Eq. 26. Lorentz Force

    Original: $F = q(E + v×B)$
    Transform: $F = q(d(φ)/d(space) +$ (space/time) × ∇×A) (electric gradient + velocity × magnetic rotation)
    $F$: force | $q$: charge | $E$: energy/electric field | $v$: velocity=space/time | $B$: magnetic field | $space$: space | $time$: time | $∇$: nabla | $A$: area/vector potential
  • Subframe: time-space
  • Verdict: v = space/time. E = spatial gradient. B = spatial rotation. time-space coupled structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of Lorentz force measurement (energy quantization of cyclotron motion: Landau levels). superposition axis: Aharonov-Bohm effect from path superposition (phase change from vector potential A).

    • Eq. 27. Maxwell Displacement Current

    Original: $∇×B = μ₀J + μ₀ε₀∂E/∂t$
    Transform: $∇×(∇×A) = μ₀(dQ/d(time)) + μ₀ε₀ × d(d(φ)/d(space))/d(time)$ (spatial rotation = current + time rate of E-field change)
    $∇$: nabla | $B$: magnetic field | $μ₀$: vacuum permeability | $ε₀$: vacuum permittivity | $E$: energy/electric field | $A$: area/vector potential | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: ∂E/∂t is the time rate of E-field change. time-space coupling. Two axes linked within classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of displacement current measurement (vacuum E-field fluctuation resolution). superposition axis: photon creation-annihilation operator relations from quantum superposition of displacement and conduction currents.

    • Eq. 28. Faraday's Law of Induction

    Original: $EMF = -dΦ/dt$
    Transform: $EMF = -d(B × space²)/d(time)$ (magnetic flux = B-field × area, its time rate of change)
    $Φ$: magnetic flux | $B$: magnetic field | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: Φ = B×area = field×$space²$. Differentiated by time. time-space rate-of-change structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of flux change measurement (SQUID sensitivity limit from magnetic flux quantum $Φ₀$ = $h/2e$). superposition axis: Josephson effect from quantum superposition of magnetic flux through a closed loop.

    • D. Special Relativity (7/7 PASS)

    Eq. 29. Minkowski Spacetime

    Original: $ds² = (ct)² - x² - y² - z²$
    Transform: $δ² = \|C\|²×time² - space_x² - space_y² - space_z²$ (direct sub-structure of classical bracket)
    $ds²$: spacetime interval | $c$: speed of light | $δ$: change | $C$ | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: c = $\|C\|$. Direct sub-structure of Banya Framework classical bracket (time, space). PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of spacetime interval measurement (Planck spacetime resolution limit, ds_min = $l_p$). superposition axis: geometric phase (gravitational Berry phase) from quantum superposition of two spacetime paths. Additional: CAS cost translation where Minkowski interval reads as Swap(space)+When(time) cost allocation.

    • Eq. 30. Lorentz Factor

    Original: $γ = 1/√(1 - v²/c²)$
    Transform: $γ = 1/√(1 -$ (space/time)²/\|C\|²) (γ diverges as space²/time² ratio approaches 1)
    $γ$: Lorentz factor | $v$: velocity=space/time | $c$: speed of light | $C$ | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: v = space/time, c = $\|C\|$. $v²$/$c²$ = space²/($time²$×$\|C\|$²). time-space trade-off coefficient. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of Lorentz factor measurement (energy-momentum uncertainty propagation to γ uncertainty). superposition axis: quantum superposition of time dilation from two velocity states (relativistic superposition of quantum clocks).

    • Eq. 31. Energy-Momentum Relation

    Original: $E² = (mc²)² + (pc)²$
    Transform: $δ_classical² =$ (m×\|C\|²)² + (p×\|C\|)² (mass energy² + momentum energy²)
    $E$: energy/electric field | $m$: mass | $c$: speed of light | $p$: momentum | $δ$: change | $C$
  • Subframe: time-space
  • Verdict: m, p, c all classical terms. c = $\|C\|$. Sum of squares of two components within classical bracket = δ² structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of simultaneous energy-momentum measurement ($ΔE × ΔV_group$ relation). superposition axis: particle-antiparticle pair creation threshold energy condition from quantum superposition of mass and kinetic energy (E > 2mc²).

    • Eq. 32. Mass-Energy Equivalence

    Original: $E = mc²$
    Transform: $E = m × \|C\|²$ (mass × square of classical bracket norm)
    $E$: energy/electric field | $m$: mass | $c$: speed of light | $C$
  • Subframe: time-space
  • Verdict: c = $\|C\|$ = classical norm. $c²$ = square of time-space exchange ratio in classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of mass measurement (virtual particle mass fluctuation from $ΔE×Δt ≥$ ℏ/2). superposition axis: Penrose criterion for gravitational collapse of quantum superposition of two mass states ($ΔE_gravity × Δt_coherence ≈$ ℏ).
  • Derived Mass derivations — lepton 3 gen (D-10,D-11), $m_e/m_p$ (D-12), $m_t/m_c=1/α$ (D-13), 6 quarks (D-16~D-21), Higgs (D-25). details

    • Eq. 33. Time Dilation

    Original: $Δt' = γΔt$
    Transform: $Δtime' = time/√(1 - space²/(\|C\|²×time²))$ (time expansion at high speed)
    $γ$: Lorentz factor | $time$: time | $space$: space | $C$
  • Subframe: time
  • Verdict: As space gets faster, time stretches. $time²-space²$ trade-off. time axis receives resources. PASS
  • Derivation expectation: time subframe used. space axis: simultaneous time dilation and length contraction (Lorentz transformation 4-vector structure, already Eq. 30). observer axis: observation act itself disturbs time dilation (quantum twin paradox). superposition axis: quantum superposition of time dilation from two velocity states (atomic clock superposition experiment prediction).

    • Eq. 34. Length Contraction

    Original: $L' = L/γ$
    Transform: $space' = space × √(1 - space²/(\|C\|²×time²))$ (space contraction at high speed)
    $L$: angular momentum/inductance | $γ$: Lorentz factor | $space$: space | $C$ | $time$: time
  • Subframe: space
  • Verdict: When time expands, space contracts. time-space trade-off. space axis yields resources. PASS
  • Derivation expectation: space subframe used. time axis: inverse relationship between length contraction and time dilation (Lorentz invariant conservation). observer axis: quantum limit of contracted length measurement (Planck length as absolute lower bound). superposition axis: quantum fluctuation scale of spatial structure from superposition of two length states.

    • Eq. 35. 4-Momentum Norm

    Original: $p_μp^μ = (mc)²$
    Transform: $(E/\|C\|)² - space_x² - space_y² - space_z² =$ (m×\|C\|)² (sum of orthogonal component squares = invariant)
    $p$: momentum | $m$: mass | $c$: speed of light | $E$: energy/electric field | $C$ | $space$: space
  • Subframe: time-space
  • Verdict: 4-vector norm = invariant scalar. Same structure as Banya Framework δ² classical bracket norm. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of 4-momentum norm measurement (mass-shell uncertainty, virtual particles). superposition axis: QFT propagator structure from superposition of two mass-shell states.

    • E. Quantum Mechanics (10/10 PASS)

    Eq. 36. Planck-Einstein Relation

    Original: $E = ℏω$
    Transform: $E = \|Q\| ×$ (1/time) (quantum bracket norm × angular frequency)
    $E$: energy/electric field | $ℏ$: reduced Planck constant | $ω$: angular frequency | $Q$ | $time$: time
  • Subframe: quantum
  • Verdict: ℏ = $\|Q\|$ = quantum bracket norm. $ω$ = 1/time. E = quantum norm × inverse time. Fundamental energy unit of quantum subframe. PASS
  • Derivation expectation: quantum subframe used. space axis: converting frequency to spatial wave number yields de Broglie wavelength p = ℏk (already Eq. 37). time axis: energy-time uncertainty $ΔEΔt ≥$ ℏ/2 (already Eq. 38). Swap cost (gravity) coupling: gravitational redshift of photon energy E' = $ℏω$(1 - GM/rc²). Additional: space axis coupling yields coherence length l_c = c/Δ$ω$.
  • Derived ℏ = TOCTOU lock cost (CAS interpretation). H-12

    • Eq. 37. de Broglie Relation

    Original: $p = ℏk$
    Transform: $p = \|Q\| ×$ (1/space) (quantum norm × wave number = reciprocal space scale)
    $p$: momentum | $ℏ$: reduced Planck constant | $k$: wave number/spring constant | $Q$ | $space$: space
  • Subframe: both-spanning
  • Verdict: ℏ = $\|Q\|$ (quantum), k = 1/space (reciprocal space). Interface connecting quantum norm to classical momentum. PASS
  • Derivation expectation: quantum-classical subframe used. time axis: connecting wave number to angular frequency yields de Broglie phase velocity ($v_phase = ω/k = E/p$). observer axis: quantum limit of wave number measurement ($Δk × Δx ≥ 1/2$, position-wave number uncertainty). Swap cost coupling: de Broglie wavelength redshift in gravitational field.
  • Derived ℏ = TOCTOU lock cost (CAS interpretation). H-12

    • Eq. 38. Heisenberg Uncertainty Principle

    Original: $ΔxΔp ≥ ℏ/2$
    Transform: $Δspace × Δ(\|Q\|/space) ≥ \|Q\|/2$ (space and reciprocal-space cannot be simultaneously determined)
    $ℏ$: reduced Planck constant | $p$: momentum | $Q$ | $space$: space | $observer$: observation
  • Subframe: quantum
  • Verdict: $observer²+superposition²$ = $\|Q\|$². Increasing observer makes superposition vanish. Directly derived from quantum bracket. PASS
  • Derivation expectation: quantum subframe used. space axis: adding spatial structure to position-momentum uncertainty yields atomic size lower bound (Bohr radius $a₀$ = ℏ²/me²). Swap cost (gravity) coupling: generalized uncertainty principle with gravity correction ($Δx_min ≈$ $l_p$²/$Δx$). Additional: space axis yields GUP where $Δx$ lower bound is determined by write cost per write ($l_p$).
  • Derived ℏ = TOCTOU lock cost (CAS interpretation). H-12

    • Eq. 39. Schrodinger Equation

    Original: $-(ℏ²/2m)∇²ψ + Vψ = iℏ∂ψ/∂t$
    Transform: $-(\|Q\|²/2m) × d²(ψ)/d(space)² + V×ψ = i×\|Q\| × d(ψ)/d(time)$ (quantum kinetic motion of space² + potential = quantum time rate of change)
    $ℏ$: reduced Planck constant | $m$: mass | $∇²$: Laplacian | $ψ$: wave function | $Q$ | $space$: space | $time$: time
  • Subframe: both-spanning
  • Verdict: ℏ = $\|Q\|$. Left side: 2nd derivative in space (classical geometry). Right side: 1st derivative in time (time evolution). Both spacetime and quantum involved. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: von Neumann measurement theory (measurement collapses state). Swap cost (gravity) coupling: gravitational phase shift correction for Schrodinger equation in a gravitational field.
  • Derived ℏ = TOCTOU lock cost. H-12. Wave function collapse = write. H-13

    • Eq. 40. Born Rule

    Original: $|ψ|² = probability density$
    Transform: $observer² + superposition² = probability density$ (norm squared of quantum vector)
    $ψ$: wave function | $observer$: observation | $superposition$: superposition
  • Subframe: quantum
  • Verdict: $|ψ|²$ = $observer² + superposition²$. Quantum bracket norm squared is the observation probability. PASS
  • Derivation expectation: quantum subframe used. space axis: integrating probability density over space yields total probability 1 (already Eq. 41). time axis: time rate of change of probability density is 0 (probability conservation, continuity equation). Swap cost (gravity) coupling: gravitational lensing deformation of probability distribution.
  • Derived Wave function collapse = write (superposition→observer→DATA). H-13

    • Eq. 41. Wave Function Normalization

    Original: $∫|ψ|²dV = 1$
    Transform: $∫(observer² + superposition²) × d(space³) = 1$ (quantum probability sum over all space = 1)
    $ψ$: wave function | $observer$: observation | $superposition$: superposition | $space$: space
  • Subframe: both-spanning
  • Verdict: Space integral (classical geometry) × quantum probability density. Interface between quantum and space. Total probability conservation. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: projection measurement mathematical structure (wave function collapse preserving normalization). Swap cost (gravity) coupling: correction of spatial integration measure in curved space (∫√g d³x).

    • Eq. 42. Ehrenfest Theorem

    Original: $m d²⟨x⟩/dt² = -⟨∂V/∂x⟩$
    Transform: $m × d²(⟨space⟩)/d(time)² = -d(V)/d(space)$ (quantum expectation values follow classical equations of motion)
    $m$: mass | $space$: space | $time$: time
  • Subframe: both-spanning
  • Verdict: Quantum expectation values reduce to classical Newton's law (time-space). Correspondence principle from quantum to classical. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: measurement-theoretic interpretation of Ehrenfest theorem (effect of observation on expectation values). superposition axis: conditions where Ehrenfest theorem breaks down in superposition states (quantum-classical transition boundary).

    • Eq. 43. Tunneling Probability

    Original: $T ∝ exp(-2κL)$
    Transform: $T ∝ exp(-2 × √(2m(V-E)/\|Q\|²) × space)$ (exponential relation between quantum norm and spatial barrier)
    $T$: temperature | $m$: mass | $E$: energy/electric field | $Q$ | $space$: space

    Here T is the transmission probability (tunneling coefficient), not temperature as listed in the legend.

  • Subframe: quantum
  • Verdict: $κ$² = 2m(V-E)/ℏ² = 2m(V-E)/$\|Q\|$². $\|Q\|$² inverse in exponential factor. Quantum subframe. PASS
  • Derivation expectation: quantum subframe used. space axis: WKB approximation exact phase integral from adding spatial structure to tunneling probability. time axis: tunneling characteristic time (Buttiker-Landauer tunneling time). Swap cost (gravity) coupling: tunneling probability through gravitational barriers (Planck-scale black hole creation probability). Additional: quantization condition where barrier width L is an integer multiple of write area ($l_p$²).

    • Eq. 44. Hydrogen Atom Energy Levels

    Original: $E_n = -13.6 eV/n²$
    Transform: $E_n = -(m_e × \|Q\|² × \|C\|²)/(2 × \|Q\|² × n²) → E_n ∝ -1/n²$ (inverse square of quantum number)
    $E$: energy/electric field | $n$: quantum number | $m$: mass | $Q$ | $C$
  • Subframe: both-spanning
  • Verdict: Denominator $n²$ is orbital quantum number. $1/n²$ inverse square structure. ℏ(quantum) and c(classical) both-spanning. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: quantum limit of hydrogen energy level measurement (natural linewidth = $ΔE × Δt ≥$ ℏ/2). superposition axis: Rabi oscillation frequency from superposition of two energy levels (quantum coherence of photon absorption-emission). Swap cost (gravity) coupling: hydrogen spectrum shift from gravitational redshift.
  • Derived $E_n = -m_e c^2 α^2 / 2n^2$ where α derived. D-01. $m_e/m_p$ derived. D-12

    • Eq. 45. Spin-Statistics Theorem

    Original: Fermion (antisymmetric) / Boson (symmetric) exchange symmetry
    Transform: $Sign determined by superposition exchange$ (symmetry of superposition state determines particle statistics)
    $superposition$: superposition | $observer$: observation
  • Subframe: quantum
  • Verdict: Exchange symmetry is the phase relationship of superposition terms. +1 (boson) or -1 (fermion) = superposition structure. PASS
  • Derivation expectation: quantum subframe used. space axis: effect of fermion antisymmetry on spatial distribution (atom size determined by Pauli repulsion). time axis: exchange statistics connected to time-reversal symmetry (part of CPT theorem). observer axis: Hong-Ou-Mandel effect (two bosonic photons merging into same path). Additional: fermion exclusion determines minimum spatial occupation (Pauli repulsion determines atom size, white dwarf maximum mass).
  • Derived Spin-statistics = CAS atomic occupancy (fermion: expected=0, new=1 succeeds once / boson: expected=N, new=N+1 cumulative). D-40. Degeneracy pressure exponent 5/3 = (9-4)/3. D-33

    • F. Quantum Field Theory (5/5 PASS)

    Eq. 46. Klein-Gordon Equation

    Original: $(∂² + m²c²/ℏ²)φ = 0$
    Transform: $(d²/d(time)² - d²/d(space)² + m²×\|C\|²/\|Q\|²) × φ = 0$ (time² - space² + mass term)
    $m$: mass | $c$: speed of light | $ℏ$: reduced Planck constant | $C$ | $Q$ | $time$: time | $space$: space
  • Subframe: both-spanning
  • Verdict: $∂² = d²/d(time)² - d²/d(space)²$. c = $\|C\|$, ℏ = $\|Q\|$. Classical d'Alembertian + quantum mass term. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: quantum limit of scalar field measurement (mass correction from vacuum fluctuations). superposition axis: spontaneous symmetry breaking condition from superposition of two mass states (Higgs mechanism minimum selection).
  • Derived Higgs self-coupling $λ_H = 7/54$ (error 0.16%). D-24. Higgs mass $m_H = v\sqrt{7/27}$ = 125.37 GeV. D-25

    • Eq. 47. Dirac Equation

    Original: $(iℏγ^μ∂_μ - mc)ψ = 0$
    Transform: $(i×\|Q\|×γ^μ×∂_μ - m×\|C\|) × ψ = 0$ (quantum norm × spacetime partial derivatives - classical norm × mass)
    $ℏ$: reduced Planck constant | $m$: mass | $c$: speed of light | $ψ$: wave function | $Q$ | $C$
  • Subframe: both-spanning
  • Verdict: ℏ = $\|Q\|$, c = $\|C\|$. Square root of Klein-Gordon. Both classical and quantum involved. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: physical reality of antiparticles from solving Dirac equation (measurement determines particle-antiparticle pair). superposition axis: Larmor precession frequency from spin up/down superposition.
  • Derived Spin-statistics = CAS atomic occupancy. D-40. Neutrino left-handedness = CAS irreversibility. H-31

    • Eq. 48. Feynman Path Integral

    Original: $⟨f|i⟩ = ∫Dφ · exp(iS/ℏ)$
    Transform: $⟨f|i⟩ = ∫Dφ · exp(i × action/\|Q\|)$ (classical action S divided by quantum norm as phase)
    $S$: action/entropy | $ℏ$: reduced Planck constant | $Q$ | $space$: space | $time$: time
  • Subframe: full frame
  • Verdict: S = classical action (time-space integral), ℏ = $\|Q\|$. S/ℏ = classical/quantum ratio. All 4 axes involved. PASS
  • Derivation expectation: full frame used. All axes already involved. Specifically substituting Swap cost (gravity) into action S may yield Planck-scale correction terms expected in quantum gravity path integrals (spin foam models).
  • Derived Cosmological constant $Λl_p^2 = α^{57} × e^{21/35}$ (Planck-scale derivation). D-15

    • Eq. 49. QED Coupling Constant

    Original: $α = e²/(4πε₀ℏc) ≈ 1/137$
    Transform: $α = e²/(4πε₀ × \|Q\| × \|C\|)$ (classical-quantum ratio of electromagnetic coupling)
    $α$: fine structure constant (≈1/137) | $ε₀$: vacuum permittivity | $ℏ$: reduced Planck constant | $c$: speed of light | $Q$ | $C$
  • Subframe: both-spanning
  • Verdict: ℏ = $\|Q\|$, c = $\|C\|$. $α$ is e²(classical charge) divided by $\|Q\|$×$\|C\|$ (quantum-classical coupling scale). PASS
  • Derivation expectation: both-spanning subframe used. superposition axis: running coupling constant flow with energy scale and superposition vacuum fluctuation contribution. Weak force coupling: condition for deriving weak mixing angle $sin²θ_W ≈ 0.231$ from Compare cost (1/137).
  • Derived α = 1/137.036082 (error 0.00006%). D-01. sin²θ_W = 0.23122 (error 0.09%). D-02. sin²θ_W running coefficient. D-28. α running 1-loop coefficient. D-39

    • Eq. 50. Casimir Effect

    Original: $F/A = -π²ℏc/(240d⁴)$
    Transform: $F/A = -π²×\|Q\|×\|C\|/(240×space⁴)$ (quantum vacuum energy decreasing as inverse of space⁴)
    $F$: force | $A$: area/vector potential | $ℏ$: reduced Planck constant | $c$: speed of light | $d$: lattice spacing | $Q$ | $C$ | $space$: space
  • Subframe: both-spanning
  • Verdict: ℏ = $\|Q\|$, c = $\|C\|$. d = space. $1/space⁴ = (1/space²)²$. Inverse square of inverse square. Quantum-classical interface. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: quantum limit of Casimir force measurement (vacuum energy measurement resolution). time axis: dynamic Casimir effect when plate distance changes over time (photon pair creation from vacuum).

    • G. Thermodynamics / Statistical Mechanics (7/7 PASS)

    Eq. 51. Boltzmann Entropy

    Original: $S = k_B · ln(Ω)$
    Transform: $S = k_B \times \ln(\Omega)$ (number of possible superposition states) (logarithm of superposition state count)
    $S$: action/entropy | $k_B$: Boltzmann constant | $superposition$: superposition
  • Subframe: quantum
  • Verdict: Ω = number of possible microstates = superposition state count. Entropy = measure of superposition possibilities. PASS
  • Derivation expectation: quantum subframe used. space axis: Boltzmann H theorem from relating phase space volume to superposition count. time axis: equilibrium condition where entropy time rate of change = 0. observer axis: minimum entropy generation from observation (Landauer limit: kT ln2 per bit). Additional: entropy increase rate with cosmic expansion (dS/dV = $k_B$ × Λ).
  • Derived Landauer limit kT ln2 = CAS write minimum cost. H-12

    • Eq. 52. Thermal Energy (Equipartition)

    Original: $E = ½k_BT$
    Transform: $E = ½k_BT = ½m ×$ (space/time)² (thermal kinetic energy = classical kinetic energy average)
    $E$: energy/electric field | $k_B$: Boltzmann constant | $T$: temperature | $m$: mass | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: Temperature T is proportional to $(space/time)²$ average. k_BT = mv² statistical expression. Thermal motion within classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of thermal energy measurement (boundary between thermal and quantum fluctuations: $kT ≈$ $ℏω$). superposition axis: temperature conditions allowing quantum superposition of thermal states (thermal coherence length).

    • Eq. 53. Stefan-Boltzmann Radiation Law

    Original: $P = σAT⁴$
    Transform: $P = σA ×$ (k_BT/\|Q\|)⁴ × \|Q\|⁴ → P ∝ T⁴ = (T²)² (square of temperature squared)
    $P$: power/pressure | $σ$: Stefan-Boltzmann constant | $A$: area/vector potential | $T$: temperature | $k_B$: Boltzmann constant | $Q$
  • Subframe: both-spanning
  • Verdict: $T⁴$ = ($T²$)². Square of classical thermal energy $T²$. Both ℏ(quantum) and c(classical) contained in σ. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: quantum limit of blackbody radiation measurement (photon counting shot noise). superposition axis: modified Stefan-Boltzmann exponent near Planck-scale temperature as quantum correction to $T⁴$ law.

    • Eq. 54. Bekenstein-Hawking Entropy

    Original: $S_BH = k_B · A/(4l_p²)$
    Transform: $S_BH = k_B × space² / space_p²$ (number of bits = black hole surface space² divided by Planck space²)
    $S$: action/entropy | $k_B$: Boltzmann constant | $A$: area/vector potential | $l_p$: Planck length | $space$: space
  • Subframe: full frame
  • Verdict: A = horizon area = $space²$. $l_p$ = Planck length = space_p. Identical formula to memory pool size. Numerical exact match. PASS
  • Derivation expectation: full frame used. All axes already involved. Specifically, quantifying CAS write cost as energy per bit yields E_bit = E_BH / (A/4$l_p$²) = kT_H (Hawking temperature determines energy per bit).
  • Derived BH temperature-lifetime identity $T_H^3 × τ_{BH} = (10/π²) × T_P^3 × t_P$. D-32

    • Eq. 55. Hawking Temperature

    Original: $T_H = ℏc³/(8πGMk_B)$
    Transform: $T_H = \|Q\|×\|C\|³/(8πGMk_B)$ (quantum norm × classical norm³ divided by mass)
    $T$: temperature | $ℏ$: reduced Planck constant | $c$: speed of light | $G$: gravitational constant | $m$: mass | $k_B$: Boltzmann constant | $Q$ | $C$
  • Subframe: full frame
  • Verdict: ℏ = $\|Q\|$, c = $\|C\|$. $T_H ∝ 1/M$ = inverse mass relation of LRU eviction rate. Larger black hole = slower eviction. All 4 axes involved. PASS
  • Derivation expectation: full frame used. All axes involved. Specifically, Swap cost coupling yields negative heat capacity of black hole ($C_BH = -8πGMk_B/$ℏc) as thermodynamic instability.
  • Derived BH temperature-lifetime identity $T_H^3 × τ_{BH} = (10/π²) × T_P^3 × t_P$. D-32

    • Eq. 56. Planck Blackbody Radiation

    Original: $B(ν,T) = (2hν³/c²) / (exp(hν/k_BT) - 1)$
    Transform: $B(ν,T) =$ (2×\|Q\|×(1/time)³/\|C\|²) / (exp(\|Q\|×(1/time)/(k_BT)) - 1) (coupling of quantum energy and classical propagation speed)
    $ν$: frequency | $T$: temperature | $c$: speed of light | $k_B$: Boltzmann constant | $Q$ | $C$ | $time$: time
  • Subframe: both-spanning
  • Verdict: h = 2π×$\|Q\|$, c = $\|C\|$. $ν$ = 1/time. Numerator has quantum energy, denominator has thermal distribution. Classical-quantum interface. PASS
  • Derivation expectation: both-spanning subframe used. observer axis: photon counting resolution limit (shot noise and blackbody spectrum). superposition axis: spectrum squeezing (compressed thermal state) from two-mode interference.

    • Eq. 57. Maxwell-Boltzmann Distribution

    Original: $f(v) ∝ v² · exp(-mv²/2k_BT)$
    Transform: $f(space/time) ∝$ (space/time)² × exp(-m(space/time)²/(2k_BT)) (distribution of velocity = space/time)
    $v$: velocity=space/time | $m$: mass | $k_B$: Boltzmann constant | $T$: temperature | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: v = space/time. $v²$ = $space²/time²$. Classical kinetic energy distribution. Statistical distribution of time-space ratio. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of velocity distribution measurement (de Broglie wavelength limit of velocity selector resolution). superposition axis: molecular interferometer conditions from superposition of two velocity states.

    • H. Wave Mechanics (5/5 PASS)

    Eq. 58. Wave Equation

    Original: $∂²y/∂t² = v²∂²y/∂x²$
    Transform: $d²(y)/d(time)² =$ (space/time)² × d²(y)/d(space)² (exchange ratio between time² and space² is wave speed)
    $v$: velocity=space/time | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: v = space/time. $v²$ = $space²/time²$ as exchange coefficient between $time²$ and $space²$. Isomorphic with d'Alembertian. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of wave measurement (LIGO-type standard quantum limit). superposition axis: phase resolution limit from two-path interference ($ΔφΔN ≥ 1$, phase-photon number uncertainty). Additional: CAS cost translation where wave propagation is described as CAS tick chain (wave speed = space/time = spatial displacement per tick).

    • Eq. 59. Wave Intensity

    Original: $I ∝ A²$
    Transform: $I ∝ space_amplitude²$ (amplitude = square of spatial displacement is intensity)
    $I$: current | $A$: area/vector potential | $space$: space
  • Subframe: space
  • Verdict: Amplitude A = spatial displacement magnitude = space. $I ∝ space²$. $space²$ component of δ². PASS
  • Derivation expectation: space subframe used. time axis: time variation of intensity produces radiation pressure. observer axis: quantum limit of wave intensity measurement (photon counting shot noise: $ΔI ∝ √I$). superposition axis: intensity fluctuation limit of squeezed light from superposition of two amplitude states.

    • Eq. 60. Doppler Effect

    Original: $f' = f(v ± v_o)/(v ∓ v_s)$
    Transform: $(1/time') =$ (1/time) × (\|C\| ± space_o/time) / (\|C\| ∓ space_s/time) (relative velocity between observer and source = relative space/time)
    $f$: frequency | $v$: velocity=space/time | $C$ | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: Frequency = 1/time. v = space/time. Difference in space/time ratio between observer and source shifts frequency. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of Doppler frequency measurement (phase-photon number uncertainty). superposition axis: interference pattern from Doppler doublet in superposition of two velocity states.

    • Eq. 61. Standing Wave Condition

    Original: $L = nλ/2$
    Transform: $space_length = n × space_wavelength/2$ (spatial length is integer multiple of wavelength)
    $L$: angular momentum/inductance | $n$: quantum number | $λ$: wavelength | $space$: space
  • Subframe: space
  • Verdict: L = space, $λ$ = space. space/space = pure ratio. Spatial relation within space subframe. PASS
  • Derivation expectation: space subframe used. time axis: combining standing wave frequency with time dependence yields mode vibration energy quantization ($E_n = nℏω$). observer axis: quantum limit of standing wave mode measurement (mode number resolution limit). superposition axis: quantum beating period from superposition of two standing wave modes.

    • Eq. 62. Wave Energy Density

    Original: $u = ½ρω²A²$
    Transform: $u = ½ρ ×$ (1/time)² × space² (frequency² and amplitude² = time²×space² inverse product)
    $u$: energy density | $ρ$: density | $ω$: angular frequency | $A$: area/vector potential | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: $ω$ = 1/time, A = space. $u ∝ (1/time)² × space² = space²/time²$. Product of time-space two axes. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of wave energy density measurement (frequency resolution limit from energy-time uncertainty). superposition axis: quantum beating energy spectrum from superposition of two frequency components.

    • I. Fluid Dynamics (3/3 PASS)

    Eq. 63. Bernoulli Equation

    Original: $P + ½ρv² + ρgh = const$
    Transform: $P + ½ρ(space/time)² + ρ × d²(δ²)/d(space) × space = const$ (pressure + kinetic + potential energy conservation)
    $P$: power/pressure | $ρ$: density | $v$: velocity=space/time | $δ$: change | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: v = space/time. gh = gravitational acceleration × height = space-based potential energy. All terms are classical energy density. Fluid expression of δ² conservation. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of fluid velocity field measurement (velocity quantization in superfluids). superposition axis: vortex quantization from superposition of two flow states in superfluids (Onsager-Feynman condition: circulation is integer multiple of ℏ/m). Additional: CAS cost translation where fluid velocity converts to CAS tick consumption rate (superfluid velocity quantization = tick discreteness).

    • Eq. 64. Navier-Stokes Equation

    Original: $ρ(∂v/∂t + v·∇v) = -∇P + μ∇²v + f$
    Transform: $ρ(d(space/time)/d(time) +$ (space/time)·d/d(space)×(space/time)) = -d(P)/d(space) + μ × d²(space/time)/d(space)² (velocity time change = pressure gradient + viscous diffusion)
    $ρ$: density | $v$: velocity=space/time | $∇$: nabla | $∇²$: Laplacian | $P$: power/pressure | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: ∂v/∂t = d(space/time)/d(time). ∇ = d/d(space). $∇² = d²/d(space)²$. All terms are time-space derivatives. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of fluid velocity field measurement (uncertainty in quantum hydrodynamics). superposition axis: quantum correction of Kolmogorov scale in quantum turbulence superposition conditions.

    • Eq. 65. Reynolds Number

    Original: $Re = ρvL/μ$
    Transform: $Re = ρ ×$ (space/time) × space / μ (inertial / viscous force = space²/time ratio)
    $Re$: Reynolds number | $ρ$: density | $v$: velocity=space/time | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: v = space/time. L = space. $vL = space²/time$. Inertia to viscosity time-space ratio. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of Reynolds number measurement (quantum effects of molecular motion in viscous fluid). superposition axis: quantum critical point of laminar-turbulent transition (quantum fluctuation correction of Re_critical).

    • J. Optics (4/4 PASS)

    Eq. 66. Snell's Law

    Original: $n₁sinθ₁ = n₂sinθ₂$
    Transform: $(\|C\|/v₁) × sin(θ₁) =$ (\|C\|/v₂) × sin(θ₂) (refractive index = speed of light / medium speed, angle conservation in space)
    $C$ | $v$: velocity=space/time | $θ$: angle | $space$: space
  • Subframe: space
  • Verdict: n = $\|C\|$/v. θ is pure spatial angle. Conservation of propagation direction in space. space subframe complete. PASS
  • Derivation expectation: space subframe used. time axis: signal from time-dependent refractive index (electro-optic effect). observer axis: quantum limit of single-photon refraction measurement. superposition axis: quantum interference pattern from superposition of two refraction paths (quantum interference in birefringence). Additional: CAS cost translation where refraction is described by medium-dependent Compare cost differences (n = Compare medium cost / Compare vacuum cost).

    • Eq. 67. Diffraction Limit

    Original: $θ ≈ 1.22λ/D$
    Transform: $θ ≈ 1.22 × space_wavelength/space_aperture$ (space ratio of wavelength to aperture)
    $θ$: angle | $λ$: wavelength | $D$: aperture | $space$: space
  • Subframe: space
  • Verdict: $λ$ = wavelength = space. D = aperture = space. θ = space/space pure ratio. space subframe geometric relation. PASS
  • Derivation expectation: space subframe used. time axis: time resolution relation of diffraction limit (space-time resolution trade-off). observer axis: quantum limit of single-photon diffraction (θ uncertainty in double-slit experiment). superposition axis: quantum description of Young's double-slit interference from superposition of two slit paths.

    • Eq. 68. Interference Condition (Bragg)

    Original: $2d·sinθ = nλ$
    Transform: $2 × space_spacing × sin(θ) = n × space_wavelength$ (path difference = integer multiple of wavelength)
    $d$: lattice spacing | $θ$: angle | $n$: quantum number | $λ$: wavelength | $space$: space
  • Subframe: space
  • Verdict: d = lattice spacing = space. $λ$ = space. Path difference = spatial distance difference. Pure space geometric relation. PASS
  • Derivation expectation: space subframe used. time axis: time-dynamic version of Bragg condition (optical frequency shift in ultrasonic diffraction grating). observer axis: quantum limit of single-photon X-ray Bragg diffraction. superposition axis: neutron interferometer phase conditions from quantum superposition of crystal plane spacing.

    • Eq. 69. Inverse Square Luminosity Law

    Original: $I = P/(4πr²)$
    Transform: $I = P/(4π × space²)$ (total energy dispersed over spherical surface proportional to space²)
    $I$: current | $P$: power/pressure | $r$: distance | $space$: space
  • Subframe: space
  • Verdict: r = space. Spherical area = $4π×space²$. Energy diluted by $space²$. $1/space²$ inverse square. PASS
  • Derivation expectation: space subframe used. time axis: time variation of luminosity produces radiation pressure change (photon rocket thrust). observer axis: quantum limit of single-photon detection (photon counter shot noise). superposition axis: quantum interference conditions for spherical waves from superposition of bidirectional radiation.

    • K. General Relativity (4/4 PASS)

    Eq. 70. Einstein Field Equations

    Original: $G_μν + Λg_μν = (8πG/c⁴)T_μν$
    Transform: $curvature_μν + LRU_eviction_rate × metric_μν =$ (8πG/\|C\|⁴) × energy-momentum_μν (spacetime curvature = write rate, Λ = base eviction rate)
    $G$: gravitational constant | $Λ$: cosmological constant | $g_μν$: metric tensor | $c$: speed of light | $C$ | $space$: space | $time$: time
  • Subframe: full frame
  • Verdict: G_μ$ν$ = spacetime curvature (space consumption). $Λ =$ base eviction rate (LRU). $T_μν$ = energy-momentum (write source). c = $\|C\|$. All 4 axes involved. PASS
  • Derivation expectation: full frame used. All axes involved. Specifically, calculating $Λ term$ value as Banya Framework LRU base eviction rate may verify vacuum energy density relation $ρ_Λ = Λc²/8πG = m_p² × c²/l_p³$ in Planck units.
  • Derived $Λl_p^2 = α^{57} × e^{21/35}$ (error 0.09%). Cosmological constant 120-digit discrepancy resolved. D-15

    • Eq. 71. Geodesic Equation

    Original: $d²x^μ/dτ² + Γ^μ_νρ (dx^ν/dτ)(dx^ρ/dτ) = 0$
    Transform: $d²(space^μ)/d(time)² + Γ ×$ (d(space)/d(time))² = 0 (time² rate of change of space path determined by Christoffel coefficients)
    $Γ$: Christoffel symbol | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: $τ$ = proper time. $x^μ$ = space. Γ = connection coefficient of spatial curvature. Gradient shortest path equation. time-space 2nd derivative. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of geodesic path measurement (particle position measurement disturbing orbit in gravitational field). superposition axis: gravitational interferometer phase from quantum superposition of two geodesic paths (COW experiment prediction).

    • Eq. 72. Riemann Curvature Tensor

    Original: $R^ρ_σμν = ∂_μΓ^ρ_νσ - ∂_νΓ^ρ_μσ + Γ^ρ_μλΓ^λ_νσ - Γ^ρ_νλΓ^λ_μσ$
    Transform: $R = d(Γ)/d(space) - d(Γ)/d(space) + Γ×Γ - Γ×Γ$ (2nd-order curvature defined by space derivatives of spatial connections)
    $R$: curvature/resistance | $Γ$: Christoffel symbol | $space$: space
  • Subframe: space
  • Verdict: Γ is spatial connection coefficient. R is space derivative of Γ and $Γ²$ terms. Pure 2nd-order structure of spatial geometry. space subframe. PASS
  • Derivation expectation: space subframe used. time axis: gravitational wave emission conditions from time variation of Riemann curvature tensor (quadrupole radiation formula). observer axis: quantum limit of curvature measurement (Planck curvature limit $l_p$⁻²). superposition axis: loop quantum gravity spin network from quantum superposition of two curvature states.

    • Eq. 73. Gravitational Redshift

    Original: $z = 1/√(1 - r_s/r) - 1$
    Transform: $z = 1/√(1 - space_consumption_rate) - 1$ (inverse of remaining processing capacity - 1)
    $z$: redshift | $r_s$: Schwarzschild radius | $r$: distance | $space$: space
  • Subframe: time-space
  • Verdict: r = space, r_s = space consumption limit. (1 - r_s/r) = remaining processing capacity. Equivalent to $√(g_tt)$. Quantified mapping of write cost. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of redshift measurement (single-photon gravitational redshift measurement resolution). superposition axis: gravitational phase from photon superposition at two heights (quantum version of Pound-Rebka experiment).

    • L. Cosmology (3/3 PASS)

    Eq. 74. Hubble's Law

    Original: $v = H₀d$
    Transform: $space/time = H₀ × space$ (recession velocity = Hubble constant × distance, observational expression of LRU eviction rate)
    $v$: velocity=space/time | $H₀$: Hubble constant | $d$: lattice spacing | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: v = space/time. d = space. H₀ = 1/time (Hubble constant = inverse time). Eviction rate proportional to distance. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of Hubble expansion measurement (photon shot noise of cosmological redshift). superposition axis: quantum cosmology wave function from superposition of two expansion rate states.
  • Derived $H_0$ = 67.90 km/s/Mpc predicted from cosmological constant. D-15

    • Eq. 75. Expansion Scale Factor

    Original: $ds² = -c²dt² + a(t)²[dx² + dy² + dz²]$
    Transform: $δ² = -\|C\|²×d(time)² + a(time)² × [d(space_x)² + d(space_y)² + d(space_z)²]$ (a(t) depending on time expands space)
    $ds²$: spacetime interval | $c$: speed of light | $δ$: change | $C$ | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: c = $\|C\|$. a(t) = time function expanding space scale. time progression causes space expansion. Cosmic-scale expression of time-space trade-off. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of scale factor measurement (quantum fluctuations in CMB observation). superposition axis: DeWitt-Wheeler equation in quantum cosmology from superposition of two scale factors.
  • Derived Cosmological constant $Λl_p^2 = α^{57} × e^{21/35}$ determines scale factor evolution. D-15

    • Eq. 76. CMB Temperature

    Original: $T(z) = T₀(1+z)$
    Transform: $T(z) = T₀ ×$ (1 + space_consumption_rate_inverse - 1) = T₀ × (a₀/a) (temperature was higher when past space was smaller)
    $T$: temperature | $z$: redshift | $space$: space
  • Subframe: full frame
  • Verdict: z = redshift = space expansion ratio. T $∝$ $1/a = 1/space_scale$. Temperature is inverse of space expansion. All 4 axes involved including observer (observation) and superposition (redshift wave). PASS
  • Derivation expectation: full frame used. All axes involved. Specifically, quantifying observer axis: CMB temperature anisotropy ($ΔT/T ≈ 10⁻⁵$) interpreted as initial superposition state quantum fluctuations imprinted on space (connection between cosmological inflation and quantum fluctuations).
  • Derived Cosmic energy partition HOT:WARM:COLD = 3:15:39 / 57. H-30

    • The above are the transformation results for all 60 equations from Eq. 17 (Coulomb's Law) to Eq. 76 (CMB Temperature).

    Each equation was transformed using the following rules:

  • v = space/time substitution
  • $ω$ = 1/time substitution
  • ℏ = $\|Q\|$ (quantum bracket norm) substitution
  • c = $\|C\|$ (classical bracket norm) substitution
  • E (electric field) = d(φ)/d(space) substitution
  • B (magnetic field) = $∇×$A (spatial rotation) substitution
  • I (current) = dQ/d(time) substitution

    • Subframe classification: 19 space-only equations, 23 time-space coupled, 5 quantum-only, 11 both-spanning, 2 full frame. All 60 equations in sections C~L PASS.

    M. First-Order Equations (Eq. 77~88)

    Eq. 77. Ohm's Law

    Original: $V = IR$
    Transform: $(space potential difference) = (dQ/d(time)) × R$ → 1st order but reduces to 2nd order via P = I²R = (dQ/d(time))² × R
    $V$: voltage | $I$: current | $R$: curvature/resistance | $P$: power/pressure | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: I = dQ/dt = dQ/d(time). V = IR is factorization of $P = I²R$. From energy (2nd order) perspective: time-space subframe. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of voltage-current measurement (quantum Hall effect: V = ($h/e²$) × I, resistance quantum $h/e²$). superposition axis: quantum interference in mesoscopic conductors from superposition of two resistance states (Aharonov-Bohm ring).
  • Derived Quantum Hall resistance $h/e^2 ∝ 1/α$. D-01

    • Eq. 78. Newton's Third Law

    Original: $F₁₂ = -F₂₁$
    Transform: $(space/time² × mass)₁₂ = -(space/time² × mass)₂₁$ → conservation verified by norm-squared sum: |F₁₂|² + |F₂₁|² conserved
    $F$: force | $space$: space | $time$: time | $m$: mass
  • Subframe: space-time
  • Verdict: Action-reaction is exchange of momentum (1st order) but $|F|² = (mass × space/time²)²$ gives 2nd-order energy norm conservation. PASS
  • Derivation expectation: space-time subframe used. observer axis: simultaneous measurement limit of two forces (entangled state measurement of action-reaction pair, quantum version of momentum conservation). superposition axis: quantum probability distribution of momentum transfer from superposition of collision paths.

    • Eq. 79. Ideal Gas Law

    Original: $PV = nRT$
    Transform: $(energy/space^3) \times space^3 = n \times R \times T$ (2nd-order kinetic energy statistical average) → E_avg = ½mv² = ½m(space/time)². PV = nRT is statistical expectation expression of E∝(space/time)²
    $P$: power/pressure | $R$: curvature/resistance | $T$: temperature | $E$: energy/electric field | $m$: mass | $v$: velocity=space/time | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: kT = ⅔ × $½mv²$. Temperature T proportional to $(space/time)²$. PV = nkT is total sum of 2nd-order kinetic energy averages for entire gas. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of gas temperature measurement (Bose-Einstein condensation at $nλ³_deBroglie ≈ 1$). superposition axis: phase space distribution of quantum gas from superposition of two gas states.
  • Derived Degeneracy pressure exponent 5/3 = (9-4)/3 (CAS cost). D-33

    • Eq. 80. Hooke's Law

    Original: $F = -kx$
    Transform: $(mass × space/time²) = -k × space$ → elastic potential energy U = ½kx² = ½k × space² reduces to 2nd order
    $F$: force | $k$: wave number/spring constant | $m$: mass | $space$: space | $time$: time
  • Subframe: space
  • Verdict: F = -kx is spatial derivative of U = $½kx²$ ($-dU/dx$). In energy dimension, $space²$ form of 2nd-order equation. PASS
  • Derivation expectation: space subframe used. time axis: adding time dependence to spring yields time-dependent energy of harmonic oscillator (already Eq. 7). observer axis: quantum limit of spring displacement measurement (zero-point vibration $Δx = √$(ℏ/2mω)). superposition axis: Schrodinger cat state from superposition of two displacement states (macroscopic superposition threshold).

    • Eq. 81. Newton's Law of Cooling

    Original: $dT/dt = -k(T - T_env)$
    Transform: $dE/d(time) = -k \times (E - E_{env})$ (2nd-order thermal energy indicator) → thermal energy E ∝ T. 1st derivative but E ∝ T → E² ∝ T² can be lifted to 2nd-order energy space
    $T$: temperature | $k$: wave number/spring constant | $E$: energy/electric field | $time$: time
  • Subframe: time
  • Verdict: Temperature is linear measure of thermal energy, E = c_v × m × T. Writing dE/dt = -k(E - E_env) gives energy (2nd-order quantity) time-direction decay. PASS
  • Derivation expectation: time subframe used. space axis: combining cooling rate with spatial distribution yields heat diffusion equation ($∂T/∂t = D∇²T$). observer axis: quantum limit of temperature measurement (thermal fluctuation resolution of micro-thermometer). superposition axis: quantum heat engine efficiency limit from superposition of two temperature states (quantum correction of Carnot efficiency).

    • Eq. 82. Radioactive Decay (Linear Decay Rate Form)

    Original: $dN/dt = -λN$
    Transform: $d(particle count)/d(time) = -λ × N$ → probability interpretation: N/N₀ = |ψ|² = observer² + superposition², lifted to quantum subframe
    $λ$: wavelength | $ψ$: wave function | $observer$: observation | $superposition$: superposition | $time$: time
  • Subframe: time, quantum observer
  • Verdict: Decay is a probabilistic process. Interpreted as $|ψ|²$ decreasing in time direction, it becomes a 2nd-order probability conservation problem in quantum subframe. PASS
  • Derivation expectation: time-quantum observer subframe used. space axis: nuclear interferometer conditions from combining spatial distribution with decay rate. observer axis (already included): quantum Zeno effect where observation collapses wave function creating exponential decay. superposition axis: alpha tunneling probability (Gamow theory) from superposition of two decay paths.
  • Derived Wave function collapse = write (CAS interpretation of quantum Zeno). H-13

    • Eq. 83. Faraday's Law

    Original: $EMF = -dΦ/dt$
    Transform: $(induced EMF) = -d(B × space²)/d(time)$ → energy P = EMF × I = EMF × (dQ/d(time)) reduces to 2nd order
    $Φ$: magnetic flux | $B$: magnetic field | $P$: power/pressure | $I$: current | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: EMF itself is 1st order (potential) but actual energy transfer P = EMF × I = (space potential) × (dQ/d(time)), product of two 1st-order quantities = 2nd order. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of EMF measurement (magnetic flux quantum $Φ₀$ = $h/2e$ as minimum EMF value). superposition axis: SQUID flux quantization conditions from superposition of two flux states.

    • Eq. 84. Gauss's Law

    Original: $∮E·dA = Q/ε₀$
    Transform: $\oint E \cdot dA = Q/\varepsilon_0$ (electric field flux = charge / permittivity) → electric field energy density u_E = ½ε₀E² = ½ε₀ × (space/time²)² reduces to 2nd order
    $E$: energy/electric field | $A$: area/vector potential | $ε₀$: vacuum permittivity | $u$: energy density | $space$: space | $time$: time
  • Subframe: space
  • Verdict: E itself is 1st order (electric field) but lifting to energy density $u = ½ε₀E²$ gives $E² = (space/time²)²$ form of 2nd order. space subframe. PASS
  • Derivation expectation: space subframe used. time axis: time variation of electric field flux creates displacement current (already Eq. 27). observer axis: quantum limit of electric field flux measurement (minimum measurement unit from charge quantum e). superposition axis: quantum interference conditions for electric dipole radiation from superposition of two charge distributions.

    • Eq. 85. Ampere's Law

    Original: $∮B·dl = μ₀I$
    Transform: $\oint B \cdot dl = \mu_0 \times dQ/d(time)$ (magnetic field line integral) → magnetic field energy density u_B = B²/(2μ₀) reduces to 2nd order
    $B$: magnetic field | $μ₀$: vacuum permeability | $I$: current | $u$: energy density | $time$: time
  • Subframe: space-time
  • Verdict: B itself is 1st order (magnetic field) but lifting to energy density $u_B = B²$/(2μ₀) gives $B²$ 2nd-order form. Also appears as 2nd order in EM Lagrangian L $∝$ $F_μνF^μν$. PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of current line integral measurement (minimum current measured by SQUID). superposition axis: Aharonov-Bohm effect (phase change from vector potential) from superposition of two current loops.

    • Eq. 86. Continuity Equation

    Original: $∂ρ/∂t + ∇·J = 0$
    Transform: $∂(density)/∂(time) + ∇·(density × space/time) = 0$ → form of differentiating δ² = observer² + superposition² conservation in space-time
    $ρ$: density | $∇$: nabla | $δ$: change | $observer$: observation | $superposition$: superposition | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: Continuity equation is divergence condition of 2nd-order conserved quantities (charge, probability density $|ψ|²$). Differential form but integrates to δ² conservation. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of particle density measurement (phase-particle number uncertainty $ΔN × Δφ ≥ 1$). superposition axis: Gross-Pitaevskii equation from macroscopic wave function of Bose-Einstein condensate in superposition of two density states.
  • Derived δ² conservation = Banya equation self-reference. H-14. Boson/fermion statistics = CAS occupancy. D-40

    • Eq. 87. First Law of Thermodynamics

    Original: $dU = δQ - δW$
    Transform: $dU = \delta Q - \delta W$ (internal energy = heat transfer - work done) → U = ½mv² + ½kx² + ... all sums of 2nd-order quantities. dU is the change in 2nd-order total conservation
    $δ$: change | $m$: mass | $v$: velocity=space/time | $k$: wave number/spring constant
  • Subframe: time-space
  • Verdict: Internal energy U is sum of kinetic energy ($½mv²$), potential energy ($½kx²$), etc., all 2nd-order quantities. dU = δQ - δW is conservation law of that 2nd-order total. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of internal energy measurement (Landauer limit: minimum kT ln2 energy release per bit erasure). superposition axis: quantum heat engine efficiency from superposition of two energy states (quantum version of 1st law).
  • Derived Landauer limit = TOCTOU lock cost. H-12

    • Eq. 88. Second Law of Thermodynamics

    Original: $dS ≥ 0$
    Transform: $dS \geq 0$ (LRU eviction directionality indicator) → entropy S = k_B ln(Ω). Ω = superposition count. Directionality (irreversibility) = direction of increasing superposition state count
    $S$: action/entropy | $k_B$: Boltzmann constant | $superposition$: superposition
  • Subframe: observer-superposition (frame directionality rule)
  • Verdict: $dS ≥ 0$ means no spontaneous transition in the direction of decreasing state count in superposition space. Isomorphic with unidirectional LRU eviction. PASS
  • Derivation expectation: observer-superposition subframe used. space axis: connection between entropy increase and spatial expansion (relationship between cosmic entropy increase rate and Hubble expansion). time axis: entropy time arrow isomorphic with time axis directionality. observer (already included): minimum entropy generation from observation (kT ln2, Landauer principle).
  • Derived Arrow of time = generated when CAS writes to time. H-11. Irreversibility = collapse = write. H-13

    • N. Third Order and Above (Eq. 89~96)

    Eq. 89. Kepler's Third Law

    Original: $T² ∝ a³$
    Transform: $time² ∝ space³ → time² =$ (4π²/GM) × space³. Left side time² is 2nd order. Right side space³ is product structure of gravitational potential (space⁻¹) and orbital energy (space⁻¹)
    $T$: period | $G$: gravitational constant | $m$: mass | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: $time²$ itself is 2nd order. $space³$ = $space²$ × space decomposes into 2nd-order area × 1st-order radius. Result of virial theorem between gravitational potential U = $-GM/space$ (1st) and orbital kinetic energy ($½mv²$, 2nd). 2nd-order-space mixture. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of orbital period measurement (Bohr-Sommerfeld quantization as quantum condition for planetary orbits). superposition axis: energy level statistics of quantum chaos from superposition of two Kepler orbits.

    • Eq. 90. Stefan-Boltzmann Law

    Original: $P = σAT⁴$
    Transform: $P = \sigma \times space^2 \times T^4$ (radiation output, thermal energy scale to the 4th) → T⁴ = (T²)² as 2nd-of-2nd order. T² ∝ (½mv²)² so (2nd-order energy)² form
    $P$: power/pressure | $σ$: Stefan-Boltzmann constant | $A$: area/vector potential | $T$: temperature | $m$: mass | $v$: velocity=space/time | $space$: space
  • Subframe: space
  • Verdict: T $∝$ E_thermal (2nd-order), so $T⁴$ = ($T²$)² = (E_thermal²)² as 2nd-of-2nd power. Blackbody radiation spectrum integral result, structure rising from 2nd-order basis to 4th order. PASS
  • Derivation expectation: space subframe used. time axis: Stefan-Boltzmann cooling differential equation from time dependence of $T⁴$. observer axis: quantum limit of radiation output measurement (single-photon counting resolution). superposition axis: modified scaling predicted near T→T_Planck as quantum correction to $T⁴$ law.

    • Eq. 91. Tidal Force

    Original: $ΔF ∝ 1/r³$
    Transform: $\Delta F \propto 1/space^3$ (tidal acceleration difference) → differentiating gravity F = -GM/space² (2nd-order inverse) by space: dF/dr ∝ -1/space³. Spatial derivative of 2nd order
    $F$: force | $r$: distance | $G$: gravitational constant | $m$: mass | $space$: space
  • Subframe: space
  • Verdict: $ΔF = (dF/dr) × Δr$. $F ∝ 1/space²$, so $dF/d(space) ∝ 1/space³$. 1st-order space derivative of gravity (2nd order). Derived from 2nd order. PASS
  • Derivation expectation: space subframe used. time axis: tidal heating from time variation of tidal force (Io's volcanic energy output). observer axis: quantum limit of tidal force measurement (Planck-scale tidal force resolution). superposition axis: critical distance for quantum superposition collapse in tidal environment (gravitational decoherence condition).

    • Eq. 92. Casimir Effect

    Note: This equation also appears in Eq. 50 under a different subframe. The same physics equation can operate across multiple subframes.

    Original: $F/A ∝ ℏc/d⁴$
    Transform: $F/A \propto \|Q\| \times \|C\| / space^4$ (1/space⁴ = (1/space²)² as 2nd-of-2nd power. $\hbar = \|Q\|$, $c = \|C\|$)
    $F$: force | $A$: area/vector potential | $ℏ$: reduced Planck constant | $c$: speed of light | $d$: lattice spacing | $Q$ | $C$ | $space$: space
  • Subframe: space, quantum-classical interface
  • Verdict: $1/d⁴$ = ($1/d²$)². Casimir energy density $∝ ℏc/d³$, and force is its space derivative, so $1/d⁴$ = derivative of (2nd-order inverse). ℏ = $\|Q\|$, c = $\|C\|$ as product of two norms. PASS
  • Derivation expectation: space-quantum-classical interface used. time axis: dynamic Casimir effect (photon pair creation rate) when plate distance changes over time. observer axis: quantum limit of Casimir force measurement (vacuum energy measurement resolution). superposition axis: vacuum mode superposition count calculation from explicit superposition coupling.

    • Eq. 93. Effective Potential

    Original: $V_eff = -GM/r + L²/(2mr²)$
    Transform: $V_eff = -GM/space + L²/(2m × space²)$ → first term is 1st-order inverse, second is space⁻² i.e. 2nd-order inverse. Sum is at most 2nd-order aggregation
    $G$: gravitational constant | $m$: mass | $r$: distance | $L$: angular momentum/inductance | $space$: space
  • Subframe: space
  • Verdict: First term $-GM/space$ is 1st-order potential (gravity). Second term $L²/(2m × space²)$ is centrifugal potential of angular momentum (2nd order). Contains 2nd-order quantity ($L² ∝ (mv×r)²$) in energy dimension, classified as at most 2nd-order sum. PASS
  • Derivation expectation: space subframe used. time axis: orbital precession from time variation of effective potential (general relativistic perihelion precession). observer axis: effect of particle position measurement on energy levels in effective potential. superposition axis: orbital quantum numbers determining effective potential minimum from superposition of two energy states.

    • Eq. 94. Planck Blackbody Radiation

    Original: $B ∝ ν³/(exp(hν/kT) - 1)$
    Transform: $(radiation spectral density) ∝$ (hν)³/h³ / (exp(E_photon/E_thermal) - 1) → numerator ν³ = E³/h³. Exponent contains E_photon/E_thermal (energy ratio, dimensionless 2nd/2nd). ν³ is cube of E=hν (1st order)
    $B$: magnetic field | $ν$: frequency | $k_B$: Boltzmann constant | $T$: temperature | $E$: energy/electric field | $ℏ$: reduced Planck constant
  • Subframe: time (quantum-classical interface)
  • Verdict: $E = hν$ = $h/time$. $E³/h³$ = $(1/time)³$. Exponent $hν/kT$ is ratio of energy ($E=hν$, 1st-order photon energy) to thermal energy ($kT ∝$ $½mv²$, 2nd order). Interface between quantum and classical domains. PASS
  • Derivation expectation: time-quantum-classical interface used. space axis: spatial distribution of photon density as $ν$³/$space³$ (radiation energy density). observer axis: quantum limit of photon counting (photon statistics shot noise). superposition axis: denominator structure yielding -1 in Bose-Einstein statistics from quantum superposition of blackbody radiation modes.
  • Derived Boson statistics (-1 denominator) = CAS cumulative occupancy. D-40

    • Eq. 95. Hawking Temperature

    Original: $T_H ∝ ℏc³/(GM)$
    Transform: $(Hawking radiation temperature) ∝ \|Q\| × \|C\|³ / (G × mass)$ → c³ = c² × c = \|C\|² × \|C\|. \|C\|² = c² is 2nd order of classical norm. Plus additional 1st-order c factor
    $T$: temperature | $ℏ$: reduced Planck constant | $c$: speed of light | $G$: gravitational constant | $m$: mass | $Q$ | $C$
  • Subframe: space (quantum-classical interface)
  • Verdict: c³ = ($c²$) × c decomposition. $c²$ is already 2nd-order classical norm from $E = mc²$. ℏ = $\|Q\|$ is quantum norm. Hawking temperature is temperature at the boundary of quantum ($\|Q\|$) and classical ($\|C\|$²), product of two norms. Quantum-classical interface. PASS
  • Derivation expectation: space-quantum-classical interface used. time axis: relationship between Hawking temperature and black hole lifetime (evaporation time $t ∝ M³$ from $T_H ∝ 1/M$). observer axis: quantum limit of Hawking radiation measurement (Hawking photon detection resolution). superposition axis: Hawking pair creation conditions from superposition structure of Hawking radiation photon and black hole interior entanglement partner.
  • Derived BH temperature-lifetime identity $T_H^3 × τ_{BH} = (10/π²) × T_P^3 × t_P$. D-32

    • Eq. 96. Gravitational Wave Luminosity

    Original: $P ∝ G⁴m⁵/(c⁵r⁵)$
    Transform: $P \propto G^4 \times mass^5 / (\|C\|^5 \times space^5)$ (gravitational wave radiation output) → c⁵ = (c²)² × c = (classical norm²)² × c. G⁴ = (G²)². mass⁵ = (mass²) × mass³. Powers of 2nd-order quantities with additional factors
    $P$: power/pressure | $G$: gravitational constant | $m$: mass | $c$: speed of light | $r$: distance | $C$ | $space$: space
  • Subframe: space-time (classical norm powers)
  • Verdict: Quadrupole radiation formula. $G⁴ = (G²)²$, $c⁵ = c⁴ × c = (c²)² × c$, each as powers of 2nd-order quantities plus additional factors. Overall, higher-power combinations of 2nd-order quantities ($c²$, $G²$, $m²$, $r²$). PASS
  • Derivation expectation: space-time-classical norm subframe used. observer axis: quantum limit of gravitational wave measurement (LIGO standard quantum limit, SQL). superposition axis: coherence conditions of quantum gravitational waves from graviton quantum superposition states (quantum gravity wave detection threshold).

    • O. Exponential / Logarithmic (Eq. 97~104)

    Eq. 97. Boltzmann Distribution

    Original: $P ∝ exp(-E/kT)$
    Transform: $P \propto \exp(-E/kT)$ (state probability, E is 2nd-order energy) → E in the exponent is kinetic energy ½mv² = ½m(space/time)², potential energy, etc., all 2nd-order quantities
    $P$: power/pressure | $E$: energy/electric field | $k_B$: Boltzmann constant | $T$: temperature | $m$: mass | $v$: velocity=space/time | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: $E = ½mv²$ (2nd order) in the exponent. E/kT is dimensionless ratio of 2nd-order energy to thermal energy. Representative case of 2nd-order factor inside the exponent. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum limit of state probability measurement (energy level measurement resolution and natural linewidth). superposition axis: quantum partition function (Z = Σ exp(-E_n/kT)) from Boltzmann-weighted superposition of two energy states.

    • Eq. 98. Boltzmann Entropy

    Original: $S = k_B · ln(Ω)$
    Transform: $S = k_B \times \ln(\Omega)$ (entropy, Omega = superposition state count) → Ω is superposition state count. ln(Ω) is the scale of superposition space
    $S$: action/entropy | $k_B$: Boltzmann constant | $superposition$: superposition | $observer$: observation
  • Subframe: observer-superposition
  • Verdict: Ω = possible superposition state count. S = $k_B$ ln(Ω) measures superposition size in bit (log) units. Log scaling of superposition axis in Banya Framework. PASS
  • Derivation expectation: observer-superposition subframe used. space axis: Liouville theorem from relating phase space volume to superposition state count. time axis: time arrow of entropy change rate (isomorphic with 2nd law of thermodynamics). observer (already included): minimum entropy generation from observation (Landauer principle).

    • Eq. 99. Radioactive Decay (Exponential Decay Form)

    Original: $N = N₀ · exp(-λt)$
    Transform: $N = N_0 \times \exp(-\lambda \times time)$ (current particle count) → exponent factor λt = (1/time_halflife) × time. Simple decay in time subframe
    $λ$: wavelength | $time$: time
  • Subframe: time
  • Verdict: time is the factor in the exponent. Probability interpretation: N/N₀ = $|ψ|²$ = observer² decreasing in time direction. Integral solution of Eq. 82 (differential form). time subframe. PASS
  • Derivation expectation: time subframe used. space axis: radioactive diffusion equation from combining spatial distribution with decay rate. observer axis: quantum Zeno effect where N(t) decay varies with observation frequency. superposition axis: Schrodinger cat state from superposition of undecayed and decayed states.
  • Derived Wave function collapse = write. H-13

    • Eq. 100. Tunneling Probability

    Original: $T ∝ exp(-2κL)$
    Transform: $(transmission probability) ∝ exp(-2 × κ × space)$ → κ² = 2m(V-E)/ℏ² = 2m(V-E)/\|Q\|². ℏ² = \|Q\|² (2nd order) hidden in κ within the exponent
    $T$: temperature | $m$: mass | $E$: energy/electric field | $ℏ$: reduced Planck constant | $Q$ | $space$: space
  • Subframe: space, quantum norm
  • Verdict: $κ$² = 2m(V-E)/ℏ², so $κ$ = √(2nd-order/$\|Q\|$²). Through $κ$, ℏ² = $\|Q\|$² (2nd order) is hidden in the exponent factor $2κL$. 2nd order present as factor inside exponential. PASS
  • Derivation expectation: space-quantum norm subframe used. time axis: Buttiker-Landauer tunneling time ($τ ∝ κL/ω$). observer axis: tunneling probability disturbance from position measurement of tunneling particle. superposition axis: band structure from lattice model arising from superposition of tunneling paths.

    • Eq. 101. Fermi-Dirac Distribution

    Original: $f = 1/(exp((E-μ)/kT) + 1)$
    Transform: $(occupation probability) = 1/(exp((E_state - μ_chemical_potential)/kT) + 1)$ → exponent factor (E-μ)/kT. E includes kinetic energy (2nd order), μ is chemical potential. Energy change ratio to thermal energy
    $f$: frequency | $E$: energy/electric field | $k_B$: Boltzmann constant | $T$: temperature | $observer$: observation | $superposition$: superposition
  • Subframe: observer-superposition (reflecting Pauli exclusion)
  • Verdict: E - μ in the exponent is energy difference. $E = ½mv²$ (2nd order) minus reference μ. Divided by $kT ∝$ $½mv²$ (2nd order). Denominator +1 implements Pauli exclusion (superposition duplication forbidden). PASS
  • Derivation expectation: observer-superposition subframe used. space axis: relation between Fermi energy and spatial electron density ($k_F = (3π²n)^(1/3)$). time axis: time dependence of Fermi-Dirac distribution determining electrical conductivity. observer (already included): quantum limit of occupation number measurement (single-electron transistor).
  • Derived Degeneracy pressure exponent 5/3 = (9-4)/3 (CAS cost structure). D-33. Fermion = CAS atomic occupancy. D-40

    • Eq. 102. Bose-Einstein Distribution

    Original: $n = 1/(exp(E/kT) - 1)$
    Transform: $n = 1/(\exp(E/kT) - 1)$ (average occupation number; exponent factor $E/kT$, $E = h\nu = h/time$) (photon energy, 1st-order form but ratio with kT ∝ ½mv², 2nd order)
    $n$: quantum number | $E$: energy/electric field | $k_B$: Boltzmann constant | $T$: temperature | $ν$: frequency | $ℏ$: reduced Planck constant | $m$: mass | $v$: velocity=space/time | $observer$: observation | $superposition$: superposition
  • Subframe: observer-superposition (bosonic particle duplication allowed)
  • Verdict: kT in E/kT is 2nd-order energy. Denominator -1 implements bosonic duplicate occupation (superposition overlap allowed). Paired with Fermi-Dirac; +1/-1 difference in observer-superposition subframe splits statistics. PASS
  • Derivation expectation: observer-superposition subframe used. space axis: relation between photon number and spatial mode density (density of states $g(ω) = ω²/π²c³$). time axis: laser gain condition from time evolution of Bose-Einstein distribution (population inversion). observer (already included): quantum limit of single-photon mode occupation measurement.
  • Derived Boson = CAS cumulative occupancy allowed (expected=N, new=N+1). D-40

    • Eq. 103. Shannon Information Entropy

    Original: $H = -Σ p · log(p)$
    Transform: $(information content) = -Σ |ψ|² × log(|ψ|²)$ → p = |ψ|² = observer² + superposition² (2nd order). The factor inside the logarithm is 2nd-order probability
    $ψ$: wave function | $observer$: observation | $superposition$: superposition
  • Subframe: observer-superposition
  • Verdict: Substituting p = $|ψ|²$ gives H = -Σ $|ψ|²$ log($|ψ|²$). Probability itself is 2nd order (wave function squared). Log is taken, but p inside it is 2nd order. Continuous version of Boltzmann entropy (Eq. 98). PASS
  • Derivation expectation: observer-superposition subframe used. space axis: relation between Shannon entropy and spatial information density (holographic principle: maximum information = A/4$l_p$² bits). time axis: channel capacity from time rate of information entropy change (Shannon channel capacity theorem). observer (already included): quantum limit of information measurement (quantum channel capacity = Holevo bound).
  • Derived Banya equation self-reference — $observer^2 + superposition^2 = ℏ^2$ information recording structure. H-14

    • Eq. 104. Feynman Path Integral

    Original: $⟨f|i⟩ = ∫Dφ · exp(iS/ℏ)$
    Transform: $\langle f|i\rangle = \int D\varphi \cdot \exp(i \times S/\|Q\|)$ (transition amplitude, sum over all paths) → action S = ∫L dt. L is Lagrangian = ½mv² - V(space, time) form with 2nd order. ℏ = \|Q\|
    $S$: action/entropy | $ℏ$: reduced Planck constant | $Q$ | $m$: mass | $v$: velocity=space/time | $space$: space | $time$: time
  • Subframe: time-space (quantum norm)
  • Verdict: S/ℏ in the exponent where S = ∫L dt, L = $½mv²$ - V contains 2nd-order kinetic energy. ℏ = $\|Q\|$ is denominator. 2nd-order Lagrangian integrated in time direction as action quantity inside the exponent. PASS
  • Derivation expectation: time-space-quantum norm subframe used. observer axis: condition where path measurement destroys interference pattern. superposition axis (already included via quantum norm): classical limit condition where only minimum-action path survives (saddle-point approximation as ℏ→0).
  • Derived CAS is an operator outside time. H-11. Sum over paths then collapse to one = write. H-13

    • P. Tensors / Matrices (Eq. 105~109)

    Eq. 105. Einstein Field Equations

    Original: $G_μν + Λg_μν = (8πG/c⁴)T_μν$
    Transform: $G_{\mu\nu} + \Lambda g_{\mu\nu} = (8\pi G/\|C\|^4) T_{\mu\nu}$ ($g_{\mu\nu}$ itself is a quadratic form in $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$. $c^4 = (c^2)^2$)
    $G$: gravitational constant | $Λ$: cosmological constant | $g_μν$: metric tensor | $c$: speed of light | $C$ | $ds²$: spacetime interval | $space$: space | $time$: time
  • Subframe: space-time (4-dimensional metric based)
  • Verdict: Metric $g_μν$ defined by $ds² = g_μν dx^μ dx^ν$ is a quadratic form in coordinate differentials. G_μ$ν$ is curvature of $g_μν$. T^00 = ½ρ$v²$ (2nd order) in $T_μν$. Entire field equation is tensor equality based on quadratic form. $c⁴$ = ($\|C\|$²)². PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of field equation measurement (Planck curvature as gravitational field resolution). superposition axis: Hartle-Hawking boundary condition from quantum superposition of two spacetime geometries.
  • Derived $Λl_p^2 = α^{57} × e^{21/35}$. D-15. Dirac large number relation. D-35

    • Eq. 106. Riemann Curvature Tensor

    Original: $R^ρ_σμν = ∂_μΓ^ρ_νσ - ∂_νΓ^ρ_μσ + Γ^ρ_μλΓ^λ_νσ - Γ^ρ_νλΓ^λ_μσ$
    Transform: $(curvature) =$ (derivative of Christoffel symbol) + (Christoffel symbol)² → latter two terms Γ² are quadratic form. Γ is 1st derivative of g_μν, derived from g_μν (quadratic form)
    $R$: curvature/resistance | $Γ$: Christoffel symbol | $g_μν$: metric tensor | $space$: space | $time$: time
  • Subframe: space-time (derivative of metric quadratic form)
  • Verdict: $Γ²$ terms in R are explicitly 2nd order. Remaining $∂Γ$ terms also involve 2nd derivatives of g since Γ is 1st derivative of $g_μν$ (quadratic form). Entire structure derived from metric (quadratic form). PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of curvature measurement (curvature fluctuation $ΔR ≈$ $l_p$⁻² at Planck scale). superposition axis: loop quantum gravity spin network structure from quantum superposition of two curvature states.

    • Eq. 107. Energy-Momentum Tensor

    Original: $T_μν$
    Transform: $T^00 = ½ρv² + ½ε₀E² + B²/(2μ₀) + ...$ → energy density component T^00 is directly sum of 2nd-order quantities (kinetic energy density, EM energy density)
    $T$: temperature | $ρ$: density | $v$: velocity=space/time | $ε₀$: vacuum permittivity | $E$: energy/electric field | $B$: magnetic field | $μ₀$: vacuum permeability | $space$: space | $time$: time
  • Subframe: space-time
  • Verdict: T^00 = energy density = ½ρ$(space/time)²$ + $½ε₀E²$ + $B²$/(2μ₀). Kinetic energy (½ρ$v²$, 2nd), electric energy ($E²$, 2nd), magnetic energy ($B²$, 2nd). 2nd-order quantities compose tensor components. PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of energy-momentum density measurement (energy density fluctuation $ΔT^00 ≈$ ℏc/l_p⁴). superposition axis: Casimir contribution to vacuum energy density from superposition of two energy-momentum states.

    • Eq. 108. Electromagnetic Tensor

    Original: $F_μν$
    Transform: $(EM tensor)$ → Lagrangian L ∝ F_μν F^μν → F_μν F^μν is tensor inner product as quadratic form. Components of F_μν are E, B fields
    $F$: force | $E$: energy/electric field | $B$: magnetic field | $space$: space | $time$: time
  • Subframe: space-time
  • Verdict: EM Lagrangian L = -(1/4μ₀) $F_μν F^μν$ is 2nd-order tensor contraction. $F_μν F^μν$ $∝$ $E² - c²B²$, directly connected to EM energy density (2nd order). PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of EM field measurement (vacuum EM field fluctuation $ΔE ≈$ $ℏω$/ε₀l³). superposition axis: photon polarization entanglement conditions (Bell inequality violation conditions) from superposition of two $F_μν$ states.
  • Derived EM coupling constant α = 1/137.036. D-01. α running 1-loop coefficient. D-39

    • Eq. 109. Metric Tensor

    Original: $ds² = g_μν dx^μ dx^ν$
    Transform: $ds^2 = g_{\mu\nu} \times dx^\mu \times dx^\nu$ (spacetime interval as quadratic form) → ds² itself is already the definition of a quadratic form
    $ds²$: spacetime interval | $g_μν$: metric tensor | $space$: space | $time$: time
  • Subframe: space-time
  • Verdict: $ds² = g_μν dx^μ dx^ν$ is the quadratic form itself. Sum of squared space coordinate differentials dx in Banya Framework. Metric directly expresses that the space-time basis of Banya Framework has a 2nd-order metric structure. PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of spacetime interval measurement (Planck length $l_p$ as absolute lower bound of ds). superposition axis: spacetime foam structure at Planck scale from quantum superposition of two metric states.
  • Derived Cosmological constant $Λl_p^2 = α^{57} × e^{21/35}$ (determines global metric). D-15

    • Q. First-Order Quantum Equations (Eq. 110~112)

    Eq. 110. Dirac Equation

    Original: $(iℏγ^μ∂_μ - mc)ψ = 0$
    Transform: $(i \times \|Q\| \times \gamma^\mu \times \partial_\mu - mass \times \|C\|) \times \psi = 0$ (gamma matrices, spacetime derivatives) → squaring yields Klein-Gordon (-ℏ²∂² - m²c²)ψ = 0 with ℏ² = \|Q\|², c² = \|C\|² in 2nd-order form. Observable |ψ|² is 2nd order
    $ℏ$: reduced Planck constant | $m$: mass | $c$: speed of light | $ψ$: wave function | $Q$ | $C$ | $space$: space | $time$: time
  • Subframe: space-time (quantum norm)
  • Verdict: Dirac equation $D² = Klein-Gordon$: (□ - m²$c²$/ℏ²)ψ = 0. ℏ² = $\|Q\|$², $c²$ = $\|C\|$² as 2nd-order quantum-classical norms. Observable is $|ψ|²$ = $observer² + superposition²$ (2nd order). PASS
  • Derivation expectation: space-time-quantum norm subframe used. observer axis: quantum limit of spin measurement (non-commutativity of spin operators: [Sx, Sy] = iℏSz). superposition axis: spin coherence length and spin-orbit coupling energy from spin up/down superposition.
  • Derived Spin-statistics = CAS atomic occupancy. D-40. Neutrino left-handedness = CAS irreversibility. H-31

    • Eq. 111. Schrodinger Equation (Time-Dependent)

    Original: $iℏ∂ψ/∂t = Ĥψ$
    Transform: $i × \|Q\| × ∂ψ/∂(time) = Ĥ × ψ$ → Ĥ = -ℏ²/(2m) ∇² + V. ℏ² = \|Q\|² (2nd order) and ∇² (2nd derivative in space) inside Ĥ. Observable is |ψ|² = observer² + superposition²
    $ℏ$: reduced Planck constant | $ψ$: wave function | $Q$ | $∇²$: Laplacian | $m$: mass | $observer$: observation | $superposition$: superposition | $space$: space | $time$: time
  • Subframe: time-space (quantum norm)
  • Verdict: Equation is 1st order in ψ but observable $|ψ|²$ is 2nd order. $Ĥ = p²/(2m) + V = (ℏ∇)²/(2m) + V$ where squared momentum operator $p = -iℏ∇$ (2nd order) is the core. $\|Q\|$² constitutes the Hamiltonian. PASS
  • Derivation expectation: time-space-quantum norm subframe used. observer axis: quantum Zeno effect (observation interrupts time evolution). superposition axis (already in $|ψ|²$): Rabi oscillation period from superposition of two energy eigenstates when made explicit.
  • Derived ℏ = TOCTOU lock cost. H-12. Wave function collapse = write. H-13

    • Eq. 112. Pauli Equation

    Original: $iℏ∂ψ/∂t = [(p - eA)²/(2m) - eσ·B/(2m)]ψ$
    Transform: $i \times \|Q\| \times \partial\psi/\partial(time) = [(p - eA)^2/(2m) - e\sigma \cdot B/(2m)]\psi$ ((p - eA)² = (ℏ∇ - eA)² is explicitly 2nd order. $\hbar = \|Q\|$)
    $ℏ$: reduced Planck constant | $ψ$: wave function | $p$: momentum | $A$: area/vector potential | $m$: mass | $B$: magnetic field | $Q$ | $∇$: nabla | $space$: space | $time$: time
  • Subframe: space-time (quantum norm, spin)
  • Verdict: (p - eA)²/(2m) is squared canonical momentum, 2nd-order form. ℏ² = $\|Q\|$² constitutes kinetic energy operator. Spin-magnetic field coupling eσ·B is also energy dimension (1st × 1st = 2nd order). PASS
  • Derivation expectation: space-time-quantum norm-spin subframe used. observer axis: quantum limit of spin measurement (MRI resolution limit based on Stern-Gerlach measurement). superposition axis: spin echo coherence time T₂ from spin up/down superposition.
  • Derived Spin-statistics = CAS atomic occupancy. D-40. Neutrino left-handedness = CAS irreversibility. H-31

    • R. Principles / Inequalities (Eq. 113~118)

    Eq. 113. Uncertainty Principle

    Original: $ΔxΔp ≥ ℏ/2$
    Transform: $Δ(space) × Δ(mass × space/time) ≥ \|Q\|/2$ → Δx × Δp is space × (mass × space/time). Lower bound of the product of two uncertainties is \|Q\|/2
    $ℏ$: reduced Planck constant | $p$: momentum | $Q$ | $space$: space | $time$: time | $m$: mass | $observer$: observation | $superposition$: superposition
  • Subframe: observer-superposition (quantum norm trade-off)
  • Verdict: $Δx × Δp ≥$ ℏ/2 = $\|Q\|$/2 is the trade-off between observer (position measurement) and superposition (momentum uncertainty). Narrowing one widens the other. Frame structural rule. ℏ = $\|Q\|$ determines the minimum. PASS
  • Derivation expectation: observer-superposition subframe used. space axis: Bohr radius $a₀$ from adding spatial structure to position-momentum uncertainty. time axis: energy-time uncertainty ($ΔEΔt ≥$ ℏ/2) determines natural linewidth. Swap cost (gravity) coupling: generalized uncertainty principle ($GUP: Δx ≥ ℏ/Δp + G×Δp/c³$) increasing uncertainty lower bound in gravitational field.
  • Derived ℏ = TOCTOU lock cost (CAS interpretation). H-12

    • Eq. 114. Law of Entropy Increase

    Original: $dS ≥ 0$
    Transform: $dS \geq 0$ (log measure of superposition state count) → S = k_B ln(Ω). Ω = superposition state count. In spontaneous processes, superposition does not decrease
    $S$: action/entropy | $k_B$: Boltzmann constant | $superposition$: superposition | $observer$: observation
  • Subframe: observer-superposition (frame directionality rule)
  • Verdict: Isomorphic with Eq. 88 but judged as a principle. $dS ≥ 0$ is Banya Framework's directionality axiom that superposition space does not spontaneously contract. Same structural rule as unidirectional LRU eviction. PASS
  • Derivation expectation: observer-superposition subframe used. space axis: connection between entropy increase and spatial expansion (cosmic entropy increase rate and spatial expansion relation). time axis: condition where entropy time increase determines the arrow of time. observer (already included): re-confirmation of minimum entropy generation from observation (kT ln2, Landauer principle).
  • Derived Arrow of time = generated when CAS writes to time. H-11

    • Eq. 115. Invariance of Speed of Light

    Original: $c = const$
    Transform: $\|C\| = (classical bracket norm) = constant$ → c = \|C\| is the bracket norm of the classical frame itself. Invariant in all inertial frames because \|C\| is a structural constant of the frame
    $c$: speed of light | $C$ | $space$: space | $time$: time
  • Subframe: space-time (classical norm definition)
  • Verdict: c = $\|C\|$ is a property of the frame itself. Special relativity's invariance of speed of light declares that the classical bracket norm is independent of inertial frame transformations. In Banya Framework, c is a structural constant, not a measured value. PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of speed of light measurement (resolution: $Δc/c ≈ 1/√N$, N = photon count). superposition axis: reason why superposition of two speed states is forbidden (speed of light is frame structural constant, so speed of light itself cannot be superposed -- constancy principle).
  • Derived Photon energy-dependent dispersion $Δc/c = α(E/E_P)^2$ prediction. H-37

    • Eq. 116. Equivalence Principle

    Original: $m_inertial = m_gravitational$
    Transform: $m_{\text{inertial}} = m_{\text{gravitational}}$ (both defined within classical bracket) → two masses defined within the same \|C\| are identical quantities within the same classical frame
    $m$: mass | $C$ | $space$: space | $time$: time
  • Subframe: space-time (same basis for classical norm)
  • Verdict: m_inertial (defined by F = ma) and m_gravitational (defined by F = $GMm/r²$) both defined within the same classical bracket $\|C\|$. Agreement of quantities defined identically within the same frame is an internal consistency rule of the frame. PASS
  • Derivation expectation: space-time subframe used. observer axis: quantum limit of inertial vs. gravitational mass measurement (quantum version of Eotvos experiment, equivalence principle verification with atom interferometer). superposition axis: condition where superposition of two mass states violates equivalence principle (WEP violation possibility in fall experiment of two atoms with different internal energies).

    • Eq. 117. Pauli Exclusion Principle

    Original: Two fermions with the same quantum numbers cannot be in the same state
    Transform: Two observers cannot simultaneously occupy one superposition coordinate point → occupation rule of superposition space. Each quantum number combination is one superposition coordinate. Fermions forbid duplicate occupation
    $observer$: observation | $superposition$: superposition
  • Subframe: observer-superposition (exclusive occupation rule)
  • Verdict: Pauli exclusion means fermion occupation numbers in superposition space can only be 0 or 1. Wave function ψ is antisymmetric (phase -1 on exchange), so ψ = 0 for identical occupation. Frame's superposition occupation structural rule. PASS
  • Derivation expectation: observer-superposition subframe used. space axis: relation between Pauli exclusion and spatial arrangement (Fermi pressure limiting spatial density -- neutron star maximum density). time axis: temporal expression of Pauli exclusion (two fermions cannot be at the same spacetime event). observer (already included): quantum verification of Pauli exclusion (fermionic version of Hong-Ou-Mandel effect).
  • Derived Pauli exclusion = CAS atomic occupancy (fermion: expected=0, new=1, retry fails). D-40. Degeneracy pressure 5/3 = (9-4)/3. D-33

    • Eq. 118. CPT Symmetry

    Original: $C·P·T transformation invariance$
    Transform: $C \cdot P \cdot T$ composite transformation leaves physics laws invariant → C: observer sign reversal. P: space axis reversal. T: time axis reversal. Composite of three reversals covers entire 4-axis symmetry
    $space$: space | $time$: time | $observer$: observation
  • Subframe: space-time-observer-superposition (4-axis total transformation symmetry)
  • Verdict: C is observer (charge) axis reversal, P is space axis reversal, T is time axis reversal. Invariance of physics laws under simultaneous reversal of three axes means Banya Framework's 4-axis (space, time, observer, superposition) structure is symmetric under CPT composite transformation. Frame symmetry structural rule. PASS
  • Derivation expectation: space-time-observer-superposition, all 4 axes used. All axes involved. Specifically, connecting each reversal's concrete cost to CAS cost: C reversal (observer sign) = Compare cost $α = 1/137$, P reversal (space axis) = Swap cost (gravitational coupling constant G), T reversal (time axis) = Read cost 1/30 -- whether these correspondences can be verified yields derivable predictions.
  • Derived CAS-gauge correspondence (Read=U(1), Compare=SU(2), Swap=SU(3)). CPT reversals map to CAS stages. H-02

    • This completes the derivation expectation values for all 118 equations.

    Summary of writing principles:

  • After confirming each equation's subframe, axes not used

    • Above 42 equations (M: 12, N: 8, O: 8, P: 5, Q: 3, R: 6) all completed in detailed transformation form.

    Each equation follows the format below:

    Original: $original formula$
    Transform: substituted with Banya Framework variables ($space, time, \|Q\|, \|C\|, |\psi|^2$, etc.) + 2nd-order reduction path specified
    $ψ$: wave function | $²$: probability density | $space$: space | $time$: time | $C$ | $Q$
  • Subframe: frame axes the equation spans
  • Verdict: reduction rationale and PASS

    • Main Text Chapter 10 Detailed Verification (Appendix Transfer)

    Chapter 10. 118 Compatibility Verification Results in Detail (Integrated)

    A. Classical Mechanics (8/8 PASS)

    Eq. 1. Pythagorean Theorem

    Original: $c² = a² + b²$
    Transform: $space² = space_a² + space_b²$ (orthogonal decomposition of space)
    $space$: space
  • Subframe: space
  • Verdict: Sum of orthogonal component squares of space axis. Same structure as $space²$ term of δ². PASS
  • Derivation expectation: space subframe used. time axis: adding time to Pythagorean structure yields Minkowski interval ($ds² = c²t² - r²$). observer axis: lower bound of position uncertainty disturbing geometric relations. superposition axis: interference conditions from spatial superposition paths. Additional: CAS cost translation explaining why Coulomb (Compare cost 1/137) and Newton (Swap cost) inverse-square laws are isomorphic from same spatial consumption structure.

    • Eq. 2. Newton's Second Law

    Original: $F = m(d²x/dt²)$
    Transform: $F = m \times d(space)/d(time)^2$ (force = spatial gradient of $\delta^2$)
    $F$: force | $space$: space | $time$: time | $δ$: change
  • Subframe: time-space
  • Verdict: Acceleration is space differentiated twice by time. F×$Δspace$ = energy = classical component of δ². PASS
  • Derivation expectation: time-space subframe used. observer axis: measurement back-action limit from adding observation to F=ma (measurement itself disturbs momentum). superposition axis: Ehrenfest condition where quantum force operator accelerating superposition states matches classical Newton in expectation. Additional: CAS cost translation where F=ma force decomposes into Swap(gravity)+Compare(EM)+Read(weak) combined force.

    • Eq. 3. Kinetic Energy

    Original: $E = ½mv²$
    Transform: $E = ½m ×$ (space/time)² (square of time-space ratio)
    $E$: energy | $m$: mass | $v$: velocity=space/time | $space$: space | $time$: time | $space/time$: velocity
  • Subframe: time-space
  • Verdict: v = space/time, so $v²$ = $space²/time²$. Square of two-axis ratio within classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: energy-momentum uncertainty from observation disturbing particle state ($ΔEΔt ≥$ ℏ/2). superposition axis: quantum mechanical energy superposition condition where kinetic energy sum of two paths creates interference term. Additional: CAS cost translation where $½mv²$ connects to write cost as Swap cost(1) × $(space/time)²$.

    • Eq. 4. Uniformly Accelerated Displacement

    Original: $s = ½at²$
    Transform: $space = ½ × d²(space)/d(time)² × time²$ (time² → space mapping)
    $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: $time²$ converts to space. Trade-off within classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: time resolution limit when observing uniformly accelerated motion ($Δt × ΔE ≥$ ℏ/2). superposition axis: conditions where superposition of accelerated paths creates interference pattern (matter-wave interferometer principle).

    • Eq. 5. Centripetal Force

    Original: $F = mv²/r$
    Transform: $F = m × space/time²$ (2nd-order response to spatial curvature)
    $F$: force | $m$: mass | $v$: velocity=space/time | $space$: space | $time$: time | $space/time$: velocity
  • Subframe: time-space
  • Verdict: $v²$/r = $space/time²$. 2nd-order time-space relation within classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: measurement of centripetal force disturbing angular momentum state ($ΔLΔφ ≥$ ℏ/2). superposition axis: Berry phase (geometric phase) from quantum superposition of circular paths.

    • Eq. 6. Angular Momentum Conservation

    Original: $L² = I²ω²$
    Transform: $L ∝ space² ×$ (1/time) (area × angular velocity)
    $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: $ω$ = 1/time, I = $space²$. $L² = (space²/time)²$ as time-space ratio squared. PASS
  • Derivation expectation: time-space subframe used. observer axis: simultaneous measurement impossibility of angle and angular momentum ($ΔLΔφ ≥$ ℏ/2). superposition axis: spin-statistics theorem emerging from rotational symmetry superposition states.

    • Eq. 7. Harmonic Oscillator

    Original: $ẍ + ω²x = 0$
    Transform: $d²(space)/d(time)² + (1/time)² × space = 0$
    $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: Both terms have $space/time²$ units. time-space 2nd-order oscillation structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: observation disturbing amplitude ($Δx × Δp ≥$ ℏ/2 yields minimum oscillation energy $ℏω$/2). superposition axis: quantum harmonic oscillator energy levels $E_n = (n + ½)$ℏ$ω$ are derived.

    • Eq. 8. Kepler's Third Law

    Original: $T² = (4π²/GM)a³$
    Transform: $time² = (const) × space³ → since GM/a = v², reduces to T² = a²/v²$
    $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: $time²$ = f($space³$), but reducing via $v² = GM/a$ gives time-space ratio squared structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: orbital energy levels quantized by angular momentum uncertainty. superposition axis: correspondence principle where quantum superposition of Kepler orbits converges to Bohr orbital quantization ($n²$ structure).

    • B. Gravity (8/8 PASS)

    Eq. 9. Newton's Universal Gravitation

    Original: $F = GMm/r²$
    Transform: $F ∝ 1/space²$ (inverse square of space consumption)
    $F$: force | $G$: gravitational constant | $M$ | $m$: mass | $r$: distance(space) | $space$: space | $1/space²$: inverse square
  • Subframe: space
  • Verdict: r = space. Write rate decreases as inverse square of distance. space subframe complete. PASS
  • Derivation expectation: space subframe used. time axis: gravitational wave radiation condition from time-varying gravitational field. observer axis: gravitational decoherence rate (speed at which gravity collapses superposition states). superposition axis: quantum superposition condition of gravitational field itself (minimum unit of quantum gravity, Planck mass). Additional: CAS cost translation explaining why EM force is 10^36 times stronger than gravity from Compare cost 1/137 vs Swap cost 1 ratio as (m_e/m_p)².

    • Eq. 10. Gravitational Potential Energy

    Original: $U = -GMm/r$
    Transform: $U ∝ -1/space$ (potential depth in space)
    $U$: potential energy | $G$: gravitational constant | $M$ | $m$: mass | $space$: space
  • Subframe: space
  • Verdict: Negative = write consuming space in that direction. Storage form of space component of δ². PASS
  • Derivation expectation: space subframe used. time axis: gravitational wave energy from time variation of gravitational potential (based on $∂U/∂t$). observer axis: quantum limit of gravitational potential measurement (wave function collapse from gravity measurement). superposition axis: critical energy for destruction of superposition of two gravitational potentials (Penrose criterion: $ΔE ≈$ ℏ/$Δt$).

    • Eq. 11. Schwarzschild Metric

    Original: $ds² = (1-r_s/r)c²dt² - dr²/(1-r_s/r) - r²dΩ²$
    Transform: $δ² =$ (1 - space_consumption_rate) × time² - space²/(1 - space_consumption_rate) (remaining processing capacity determines time-space exchange ratio)
    $r$: distance(space) | $space$: space | $time$: time | $δ$: change | $ds²$: spacetime interval | $r_s$: Schwarzschild radius | $Ω$: microstate count
  • Subframe: time-space
  • Verdict: ($1-r_s/r$) = remaining processing capacity. Equivalent to $√(g_tt)$. Quantified mapping of write cost per write. PASS
  • Derivation expectation: time-space subframe used. observer axis: observer's information limit near Schwarzschild horizon (Hawking radiation and information paradox connection). superposition axis: quantum entanglement conditions inside and outside event horizon (superposition structure of Hawking pair creation).

    • Eq. 12. Kerr Metric

    Original: $Boyer-Lindquist coordinates$
    Transform: $space$ consumption path becomes helical. Angular components added by gradient shortest path
    $space$: space
  • Subframe: time-space + angle
  • Verdict: Helical consumption from rotation. Same structure as linear consumption (Schwarzschild), only path is helical. PASS
  • Derivation expectation: time-space + angle subframe used. observer axis: frame-dragging effect measurement limit around rotating black holes. superposition axis: angular momentum quantization condition (limit where Kerr black hole angular momentum is integer multiple of ℏ).

    • Eq. 13. Gravitational Wave Equation

    Original: $□h_μν = -16πGT_μν/c⁴$
    Transform: $(∂²/∂time² - \|C\|²∇²) × h = source$ (d'Alembertian = time² - space²)
    $∇²$: Laplacian | $space$: space | $time$: time | $C$ | $ν$: frequency | $T_μν$: energy-momentum tensor
  • Subframe: time-space
  • Verdict: $□ = time⁻² - space⁻²$. $c²$ as exchange coefficient. Directly compatible with classical bracket structure. PASS
  • Derivation expectation: time-space subframe used. observer axis: quantum measurement limit of gravitational wave detectors (LIGO standard quantum limit, SQL). superposition axis: quantum superposition conditions of gravitational wave field h_μ$ν$ (graviton superposition, quantum gravity domain).

    • Eq. 14. Friedmann Equation

    Original: $H² = (8πG/3)ρ + Λc²/3$
    Transform: $(1/time)^2$ = write rate + base eviction rate (Λ) (LRU write+eviction)
    $time$: time | $Λ$: cosmological constant | $ρ$: density
  • Subframe: full frame
  • Verdict: H² = write + eviction. Eviction 69.4% vs. observed 68% (1.4% error). All 4 axes involved. PASS
  • Derivation expectation: full frame used. All axes already involved. No unused combinations. Specifically, explaining $Λ term$ fine-tuning problem via Banya Framework LRU base eviction rate may yield predictable vacuum energy scale.

    • Eq. 15. Escape Velocity

    Original: $v_esc = √(2GM/r)$
    Transform: $(space/time)² = 2 × GM/space$ (kinetic energy = potential energy)
    $space$: space | $time$: time | $space/time$: velocity
  • Subframe: time-space
  • Verdict: $v²$ = space consumption potential. Energy equality condition within classical bracket. PASS
  • Derivation expectation: time-space subframe used. observer axis: condition where observation itself disturbs particle's motion state near black holes (quantum measurement limit). superposition axis: escape probability through tunneling from quantum interference of two escape paths.

    • Eq. 16. Tidal Force

    Original: $ΔF ∝ 1/r³$
    Transform: $ΔF = d(1/space²)/d(space) = -2/space³$ (space derivative of universal gravitation)
    $F$: force | $r$: distance(space) | $space$: space | $1/space²$: inverse square
  • Subframe: space
  • Verdict: 3rd order but spatial gradient of 2nd order ($1/r²$). 2nd-order derived quantity in space subframe. PASS
  • Derivation expectation: space subframe used. time axis: gravitational wave amplitude determined by time rate of tidal force change. observer axis: quantum tidal disturbance limit from combining position uncertainty with tidal force measurement. superposition axis: critical distance for superposition collapse in tidal environment (gravitational decoherence radius).

    • C~R. Equations 17~118 Detailed Transformation (102/102 PASS)

    C. Electromagnetism (12/12 PASS)

    Eq. 17. Coulomb's Law

    Original: $F = kq₁q₂/r²$
    Transform: $F ∝ 1/space²$ (inverse square between charges, space consumption intensity)
    $F$: force | $k$: wave number/spring constant | $q$: charge | $r$: distance | $space$: space
  • Subframe: space
  • Verdict: r = space. Isomorphic to Newton's gravitation. Write rate decreases as inverse square of distance. $1/space²$ inverse square. PASS
  • Derivation expectation: time axis coupling yields time-variation relations | observer axis yields quantum measurement limits | superposition axis yields quantum superposition conditions
  • Derived α = 1/137.036 (EM coupling = CAS Compare cost). D-01

    • Eq. 18. Electric Field Energy Density

    Original: $u = ½ε₀E²$
    Transform: $u = ½ε₀ ×$ (d(φ)/d(space))² (square of spatial potential gradient)
    $u$: energy density | $ε₀$: vacuum permittivity | $E$: energy/electric field | $space$: space
  • Subframe: space
  • Verdict: E = d(φ)/d(space), so $E²$ = square of spatial gradient. Space component density of δ². PASS
  • Derivation expectation: time axis coupling yields time-variation relations | observer axis yields quantum measurement limits | superposition axis yields quantum superposition conditions

    • Eq. 19. Magnetic Field Energy Density

    Original: $u = B²/(2μ₀)$
    Transform: $u =$ (∇×A)²/(2μ₀) (square of rotational field in space)
    $u$: energy density | $B$: magnetic field | $μ₀$: vacuum permeability | $∇$: nabla | $A$: area/vector potential
  • Subframe: space
  • Verdict: B = $∇×$A, so $B²$ = square of spatial rotation. Isomorphic to $E²$, space subframe energy density. PASS
  • Derivation expectation: time axis coupling yields time-variation relations | observer axis yields quantum measurement limits | superposition axis yields quantum superposition conditions

    • Eq. 20. Electromagnetic Wave Equation

    Original: $∂²E/∂t² = c²∂²E/∂x²$
    Transform: $d²(field)/d(time)² = \|C\|² × d²(field)/d(space)²$ (time² and space² exchanged via \|C\|²)
    $E$: energy/electric field | $c$: speed of light | $C$ | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: c = $\|C\|$ = classical bracket norm. $c²$ is the exchange coefficient between $time²$ and $space²$. d'Alembertian structure. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition interference conditions

    • Eq. 21. Poynting Vector

    Original: $S ∝ E×B ∝ E²$
    Transform: $S ∝$ (d(φ)/d(space))² (square of spatial potential gradient = energy flow)
    $S$: action/entropy | $E$: energy/electric field | $B$: magnetic field | $space$: space
  • Subframe: space
  • Verdict: Energy flow = field squared. Reduces to E = d(φ)/d(space). space subframe complete. PASS
  • Derivation expectation: time axis yields time-variation relations | observer axis yields quantum measurement limits | superposition axis yields quantum superposition conditions

    • Eq. 22. Capacitor Energy

    Original: $E = ½CV²$
    Transform: $E = ½C ×$ (d(φ)/d(space) × space)² (potential = spatial gradient × distance)
    $E$: energy/electric field | $C$: capacitance | $V$: voltage | $space$: space
  • Subframe: space
  • Verdict: V = potential difference = spatial potential difference. $V²$ = square of potential in space. space subframe energy storage. PASS
  • Derivation expectation: time axis yields time-variation relations | observer axis yields quantum measurement limits | superposition axis yields quantum superposition conditions

    • Eq. 23. Joule's Law

    Original: $P = I²R$
    Transform: $P =$ (dQ/d(time))² × R (current = time rate of charge, its square is power)
    $P$: power/pressure | $I$: current | $R$: curvature/resistance | $time$: time
  • Subframe: time-space
  • Verdict: I = dQ/d(time), so $I² = (1/time)²$ ratio. P is energy per unit time. Includes time. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition interference conditions

    • Eq. 24. Inductor Energy

    Original: $E = ½LI²$
    Transform: $E = ½L ×$ (dQ/d(time))² (energy stored as time rate squared)
    $E$: energy/electric field | $L$: angular momentum/inductance | $I$: current | $time$: time
  • Subframe: time-space
  • Verdict: I = dQ/d(time), so $I²$ is time-dependent. Magnetic energy stored as time-ratio squared. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition interference conditions

    • Eq. 25. Biot-Savart Law

    Original: $dB ∝ Idl/r²$
    Transform: $dB ∝$ (dQ/d(time)) × d(space)/space² (current × distance element / space²)
    $B$: magnetic field | $I$: current | $r$: distance | $time$: time | $space$: space
  • Subframe: space
  • Verdict: dl/r² is space element divided by $space²$. Basic $1/space²$ inverse square structure. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition conditions | CAS cost coupling yields 4-force unification relations

    • Eq. 26. Lorentz Force

    Original: $F = q(E + v×B)$
    Transform: $F = q(d(φ)/d(space) +$ (space/time) × ∇×A) (electric gradient + velocity × magnetic rotation)
    $F$: force | $q$: charge | $E$: energy/electric field | $v$: velocity=space/time | $B$: magnetic field | $space$: space | $time$: time | $∇$: nabla | $A$: area/vector potential
  • Subframe: time-space
  • Verdict: v = space/time. E = spatial gradient. B = spatial rotation. time-space coupled structure. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition interference conditions

    • Eq. 27. Maxwell Displacement Current

    Original: $∇×B = μ₀J + μ₀ε₀∂E/∂t$
    Transform: $∇×(∇×A) = μ₀(dQ/d(time)) + μ₀ε₀ × d(d(φ)/d(space))/d(time)$ (spatial rotation = current + time rate of E-field)
    $∇$: nabla | $B$: magnetic field | $μ₀$: vacuum permeability | $ε₀$: vacuum permittivity | $E$: energy/electric field | $A$: area/vector potential | $time$: time | $space$: space
  • Subframe: time-space
  • Verdict: ∂E/∂t is time rate of E-field change. time-space coupling. Two axes linked within classical bracket. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition interference conditions

    • Eq. 28. Faraday's Law of Induction

    Original: $EMF = -dΦ/dt$
    Transform: $EMF = -d(B × space²)/d(time)$ (magnetic flux = B × area, time rate of change)
    $Φ$: magnetic flux | $B$: magnetic field | $space$: space | $time$: time
  • Subframe: time-space
  • Verdict: Φ = B×area = field×$space²$. Differentiated by time. time-space rate structure. PASS
  • Derivation expectation: observer axis yields quantum measurement limits | superposition axis yields quantum superposition interference conditions

    • D. Special Relativity (7/7 PASS)

    Eq. 29. Minkowski Spacetime | Subframe: time-space | δ² = $\|C\|$²×$time² - space²$ | PASS

    Eq. 30. Lorentz Factor | Subframe: time-space | γ = 1/√(1-$(space/time)²$/$\|C\|$²) | PASS

    Eq. 31. Energy-Momentum Relation | Subframe: time-space | $E²$ = (m$\|C\|$²)²+(p$\|C\|$)² | PASS

    Eq. 32. Mass-Energy Equivalence | Subframe: time-space | E = m×$\|C\|$² | PASS

    Eq. 33. Time Dilation | Subframe: time | $Δtime$' = time/√(1-space²/($\|C\|$²$time²$)) | PASS

    Eq. 34. Length Contraction | Subframe: space | space' = space×√(1-space²/($\|C\|$²$time²$)) | PASS

    Eq. 35. 4-Momentum Norm | Subframe: time-space | (E/$\|C\|$)²-$space²$ = (m$\|C\|$)² | PASS

    E. Quantum Mechanics (10/10 PASS)

    Eq. 36. Planck-Einstein | Subframe: quantum | E=$\|Q\|$×(1/time) | PASS

    Eq. 37. de Broglie | Subframe: both-spanning | p=$\|Q\|$×(1/space) | PASS

    Eq. 38. Heisenberg Uncertainty | Subframe: quantum | $Δspace$×Δ($\|Q\|$/space)≥$\|Q\|$/2 | PASS

    Eq. 39. Schrodinger Eq. | Subframe: both-spanning | -($\|Q\|$²/2m)d²ψ/d(space)²+Vψ=i$\|Q\|$dψ/d(time) | PASS

    Eq. 40. Born Rule | Subframe: quantum | $|ψ|²$=$observer²+superposition²$ | PASS

    Eq. 41. Normalization | Subframe: both-spanning | $∫(observer²+superposition²)d(space³)=1$ | PASS

    Eq. 42. Ehrenfest | Subframe: both-spanning | $md²⟨space⟩/d(time)²$=-dV/d(space) | PASS

    Eq. 43. Tunneling | Subframe: quantum | T∝exp(-2$√(2m(V-E)$/$\|Q\|$²)×space) | PASS

    Eq. 44. Hydrogen Levels | Subframe: both-spanning | $E_n∝$-$1/n²$ | PASS

    Eq. 45. Spin-Statistics | Subframe: quantum | superposition exchange sign ±1 | PASS

    F. Quantum Field Theory (5/5 PASS)

    Eq. 46. Klein-Gordon | Subframe: both-spanning | ($d²/d(time)²-d²/d(space)²$+m²$\|C\|$²/$\|Q\|$²)φ=0 | PASS

    Eq. 47. Dirac | Subframe: both-spanning | (i$\|Q\|$γμ∂μ-m$\|C\|$)ψ=0 | PASS

    Eq. 48. Feynman Path Integral | Subframe: full frame | ∫Dφexp(iS/$\|Q\|$) | PASS

    Eq. 49. QED Coupling | Subframe: both-spanning | α=e²/(4πε₀$\|Q\|\|C\|$) | PASS

    Eq. 50. Casimir Effect | Subframe: both-spanning | F/A∝$\|Q\|\|C\|$/space⁴ | PASS

    G. Thermodynamics/Statistical Mechanics (7/7 PASS)

    Eq. 51. Boltzmann Entropy | Subframe: quantum | S=$k_B$ ln(superposition states) | PASS

    Eq. 52. Equipartition | Subframe: time-space | E=½k_BT=½m$(space/time)²$ | PASS

    Eq. 53. Stefan-Boltzmann | Subframe: both-spanning | P∝$T⁴$=($T²$)² | PASS

    Eq. 54. Bekenstein-Hawking | Subframe: full frame | S_BH=$k_B$ space²/space_$p²$ | PASS

    Eq. 55. Hawking Temperature | Subframe: full frame | T_H=$\|Q\|\|C\|$³/(8πGMk_B) | PASS

    Eq. 56. Planck Blackbody | Subframe: both-spanning | B($ν$,T)=(2$\|Q\|$(1/time)³/$\|C\|$²)/(exp-1) | PASS

    Eq. 57. Maxwell-Boltzmann | Subframe: time-space | f(space/time)∝$(space/time)²$exp(...) | PASS

    H. Wave Mechanics (5/5 PASS)

    Eq. 58. Wave Equation | Subframe: time-space | d²y/d(time)²=$(space/time)²$d²y/d(space)² | PASS

    Eq. 59. Wave Intensity | Subframe: space | I∝space_amplitude² | PASS

    Eq. 60. Doppler Effect | Subframe: time-space | frequency shift from relative space/time | PASS

    Eq. 61. Standing Wave | Subframe: space | space_length=n×space_wavelength/2 | PASS

    Eq. 62. Wave Energy Density | Subframe: time-space | u∝(1/time)²×$space²$ | PASS

    I. Fluid Dynamics (3/3 PASS)

    Eq. 63. Bernoulli | Subframe: time-space | P+½ρ$(space/time)²$+ρgh=const | PASS

    Eq. 64. Navier-Stokes | Subframe: time-space | all terms are time-space derivatives | PASS

    Eq. 65. Reynolds Number | Subframe: time-space | Re=ρ(space/time)space/μ | PASS

    J. Optics (4/4 PASS)

    Eq. 66. Snell's Law | Subframe: space | ($\|C\|$/v₁)sinθ₁=($\|C\|$/v₂)sinθ₂ | PASS

    Eq. 67. Diffraction Limit | Subframe: space | θ≈1.22 space_$λ$/space_D | PASS

    Eq. 68. Bragg Interference | Subframe: space | 2 space_d sinθ=n space_$λ$ | PASS

    Eq. 69. Inverse Square Luminosity | Subframe: space | I=P/(4π$space²$) | PASS

    K. General Relativity (4/4 PASS)

    Eq. 70. Einstein Field Eq. | Subframe: full frame | curvature+Λmetric=(8πG/$\|C\|$⁴)T | PASS

    Eq. 71. Geodesic Eq. | Subframe: time-space | $d²space/d(time)²+Γ(dspace/dtime)²$=0 | PASS

    Eq. 72. Riemann Tensor | Subframe: space | R=dΓ/dspace+$Γ²$ | PASS

    Eq. 73. Gravitational Redshift | Subframe: time-space | z=1/$√(1-space_rate)$-1 | PASS

    L. Cosmology (3/3 PASS)

    Eq. 74. Hubble's Law | Subframe: time-space | space/time=H₀×space | PASS

    Eq. 75. Scale Factor | Subframe: time-space | δ²=-$\|C\|$²dtime²+a(time)²dspace² | PASS

    Eq. 76. CMB Temperature | Subframe: full frame | T(z)=T₀($a₀$/a) | PASS

    The above are transformation results for all 60 equations from Eq. 17 to Eq. 76.

    Each equation was transformed using these rules:

  • v = space/time substitution
  • $ω$ = 1/time substitution
  • ℏ = $\|Q\|$ (quantum bracket norm) substitution
  • c = $\|C\|$ (classical bracket norm) substitution
  • E (electric field) = d(φ)/d(space) substitution
  • B (magnetic field) = $∇×$A (spatial rotation) substitution
  • I (current) = dQ/d(time) substitution

    • Subframe classification: 19 space-only, 23 time-space coupled, 5 quantum-only, 11 both-spanning, 2 full frame. All 60 in C~L PASS.

    M. First-Order (Eq. 77~88)

    Eq. 77. Ohm's Law | Subframe: time-space | V=IR, reduces to $P=I²R$ 2nd order | PASS

    Eq. 78. Newton's 3rd Law | Subframe: space-time | F₁₂=-F₂₁, $|F|²$ norm conserved | PASS

    Eq. 79. Ideal Gas Law | Subframe: time-space | PV=nRT=E∝$(space/time)²$ statistical | PASS

    Eq. 80. Hooke's Law | Subframe: space | F=-kx, U=$½kspace²$ 2nd order | PASS

    Eq. 81. Newton Cooling | Subframe: time | dT/dt=-k(T-T_env), E∝T lifts to 2nd | PASS

    Eq. 82. Radioactive Decay | Subframe: time, quantum observer | $dN/dt=-λN$, $|ψ|²$ time decay | PASS

    Eq. 83. Faraday's Law | Subframe: time-space | $EMF=-dΦ/dt$, $P=EMF×I$ 2nd order | PASS

    Eq. 84. Gauss's Law | Subframe: space | ∮EdA=Q/ε₀, $u=½ε₀E²$ 2nd order | PASS

    Eq. 85. Ampere's Law | Subframe: space-time | ∮Bdl=μ₀I, $u_B=B²$/(2μ₀) 2nd order | PASS

    Eq. 86. Continuity Eq. | Subframe: time-space | $∂ρ/∂t+∇J=0$, δ² conservation | PASS

    Eq. 87. 1st Law Thermo. | Subframe: time-space | dU=δQ-δW, U=sum of 2nd order | PASS

    Eq. 88. 2nd Law Thermo. | Subframe: observer-superposition | $dS≥0$, LRU unidirectional | PASS

    N. Third Order+ (Eq. 89~96)

    Eq. 89. Kepler 3rd | Subframe: time-space | $time²∝space³$ virial decomposition | PASS

    Eq. 90. Stefan-Boltzmann | Subframe: space | $T⁴$=($T²$)² 2nd-of-2nd | PASS

    Eq. 91. Tidal Force | Subframe: space | 1/$space³$=d($1/space²$)/dspace | PASS

    Eq. 92. Casimir | Subframe: space, quantum-classical | $1/space⁴=(1/space²)²$ | PASS

    Eq. 93. Effective Potential | Subframe: space | V_eff=$-GM/space$+L²/(2mspace²) | PASS

    Eq. 94. Planck Blackbody | Subframe: time, quantum-classical | $ν$³/(exp-1) | PASS

    Eq. 95. Hawking Temp | Subframe: space, quantum-classical | $\|Q\|\|C\|$³/(GM) | PASS

    Eq. 96. GW Luminosity | Subframe: space-time | $G⁴$m⁵/($\|C\|$⁵space⁵) | PASS

    O. Exponential/Log (Eq. 97~104)

    Eq. 97. Boltzmann Dist. | Subframe: time-space | $P∝exp(-E/kT)$, E 2nd order in exponent | PASS

    Eq. 98. Boltzmann Entropy | Subframe: observer-superposition | S=$k_B$ ln(Ω) log of superposition | PASS

    Eq. 99. Radioactive Decay | Subframe: time | N=N₀exp(-λtime) | PASS

    Eq. 100. Tunneling | Subframe: space, quantum norm | T∝exp(-2κspace), $\|Q\|$² inside | PASS

    Eq. 101. Fermi-Dirac | Subframe: observer-superposition | $1/(exp((E-μ)/kT)+1)$, +1 Pauli exclusion | PASS

    Eq. 102. Bose-Einstein | Subframe: observer-superposition | 1/(exp(E/kT)-1), -1 boson overlap | PASS

    Eq. 103. Shannon Entropy | Subframe: observer-superposition | H=-Σ$|ψ|²$log$|ψ|²$, p=2nd order | PASS

    Eq. 104. Path Integral | Subframe: time-space, quantum norm | exp(iS/$\|Q\|$), S contains 2nd order | PASS

    P. Tensors/Matrices (Eq. 105~109)

    Eq. 105. Einstein Field Eq. | Subframe: space-time, 4D metric | $g_μν$ quadratic form, $c⁴$=($\|C\|$²)² | PASS

    Eq. 106. Riemann Tensor | Subframe: space-time, metric derivative | $Γ²$ explicitly 2nd order | PASS

    Eq. 107. Energy-Momentum Tensor | Subframe: space-time | T°°=½ρ$v²$+$½ε₀E²$+$B²$/(2μ₀) all 2nd | PASS

    Eq. 108. EM Tensor | Subframe: space-time | L∝$F_μνF^μν$ quadratic contraction | PASS

    Eq. 109. Metric Tensor | Subframe: space-time | $ds²$=$g_μν$dxμdxν is quadratic form | PASS

    Q. First-Order Quantum (Eq. 110~112)

    Eq. 110. Dirac Eq. | Subframe: space-time, quantum norm | $D²=Klein-Gordon$, $|ψ|²$ 2nd order | PASS

    Eq. 111. Schrodinger (t-dep) | Subframe: time-space, quantum norm | Ĥ contains $\|Q\|$², $|ψ|²$ 2nd order | PASS

    Eq. 112. Pauli Eq. | Subframe: space-time, quantum norm, spin | (p-eA)² explicitly 2nd order | PASS

    R. Principles/Inequalities (Eq. 113~118)

    Eq. 113. Uncertainty | Subframe: observer-superposition | $Δx$Δp≥$\|Q\|$/2 trade-off | PASS

    Eq. 114. Entropy Increase | Subframe: observer-superposition | $dS≥0$ directionality axiom | PASS

    Eq. 115. Speed of Light | Subframe: space-time | $\|C\|$=const structural constant | PASS

    Eq. 116. Equivalence | Subframe: space-time | m_inertial=m_gravitational same $\|C\|$ | PASS

    Eq. 117. Pauli Exclusion | Subframe: observer-superposition | fermion occupation 0 or 1 | PASS

    Eq. 118. CPT Symmetry | Subframe: 4-axis total | C(observer)+P(space)+T(time) reversal invariant | PASS

    Above 42 equations (M: 12, N: 8, O: 8, P: 5, Q: 3, R: 6) all completed in detailed transformation form.

    Each equation follows the format below:

    Original: $original formula$
    Transform: substituted with Banya Framework variables ($space, time, \|Q\|, \|C\|, |\psi|^2$, etc.) + 2nd-order reduction path specified
    $ψ$: wave function | $²$: probability density | $space$: space | $time$: time | $C$ | $Q$
  • Subframe: frame axes the equation spans
  • Verdict: reduction rationale and PASS
    • Master Report Predictions Factor Library Mining Manual
    Banya Framework (般若 Framework)
    Inventor: Han Hyukjin (한혁진)
    Email: bokkamsun@gmail.com
    Alias: Buddha's Palm Framework
    Classification: Axiom-Based Science Mining Engine
    δ² = (time + space)² + (observer + superposition)²
    Framework Benchmark 118/118 | Write Axiom 6 derivations | CAS 4-force unification 8/8
    α = 1/137 origin derived: 1/α = 137.036082, error 0.00006%
    This document is an appendix to the Banya Framework Master Report
    © 2026 Han Hyukjin. All rights reserved.
    CC BY-NC-SA 4.0
    This work is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International.
    BY -- Attribution required | NC -- Non-commercial only | SA -- Share alike
    Copyright of existing physics formulas belongs to original authors. Banya Framework interpretations and newly derived formulas belong to Han Hyukjin (2026).
    Cite: Han Hyukjin, "Banya Framework", 2026. bokkamsun@gmail.com