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: $F \propto 1/space²$ (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: $u = ½ε₀ \times$ (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: $u =$ (\nabla\timesA)²/(2μ₀) (square of rotational field in space) $u$: energy density | $B$: magnetic field | $μ₀$: vacuum permeability | $\nabla$: 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: $S \propto E\timesB \propto E²$ Transform: $S \propto$ (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: $E = ½C \times$ (d(φ)/d(space) \times space)² (potential = spatial gradient \times 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 \times 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: $E = ½L \times$ (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: $dB \propto Idl/r²$ Transform: $dB \propto$ (dQ/d(time)) \times d(space)/space² (current \times 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: $F = q(E + v\timesB)$ Transform: $F = q(d(φ)/d(space) +$ (space/time) \times \nabla\timesA) (electric gradient + velocity \times magnetic rotation) $F$: force | $q$: charge | $E$: energy/electric field | $v$: velocity=space/time | $B$: magnetic field | $space$: space | $time$: time | $\nabla$: 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: $\nabla\timesB = μ₀J + μ₀ε₀\partialE/\partialt$ Transform: $\nabla\times(\nabla\timesA) = μ₀(dQ/d(time)) + μ₀ε₀ \times d(d(φ)/d(space))/d(time)$ (spatial rotation = current + time rate of E-field change) $\nabla$: 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: $EMF = -d(B \times space²)/d(time)$ (magnetic flux = B-field \times 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: $\int|ψ|²dV = 1$ Transform: $\int(observer² + superposition²) \times d(space³) = 1$ (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: $m d²⟨x⟩/dt² = -⟨\partialV/\partialx⟩$ Transform: $m \times d²(⟨space⟩)/d(time)² = -d(V)/d(space)$ (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: $E = ½k_BT = ½m \times$ (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: $S_BH = k_B \cdot A/(4l_p²)$ Transform: $S_BH = k_B \times space² / space_p²$ (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 RLU 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: $f(v) \propto v² \cdot exp(-mv²/2k_BT)$ Transform: $f(space/time) \propto$ (space/time)² \times 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: $\partial²y/\partialt² = v²\partial²y/\partialx²$ Transform: $d²(y)/d(time)² =$ (space/time)² \times 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: $I \propto A²$ Transform: $I \propto space_amplitude²$ (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: $space_length = n \times space_wavelength/2$ (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: $u = ½ρ \times$ (1/time)² \times space² (frequency² and amplitude² = time²\timesspace² 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: $P + ½ρ(space/time)² + ρ \times d²(δ²)/d(space) \times space = const$ (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: $ρ(\partialv/\partialt + v\cdot\nablav) = -\nablaP + μ\nabla²v + f$ Transform: $ρ(d(space/time)/d(time) +$ (space/time)\cdotd/d(space)\times(space/time)) = -d(P)/d(space) + μ \times d²(space/time)/d(space)² (velocity time change = pressure gradient + viscous diffusion) $ρ$: density | $v$: velocity=space/time | $\nabla$: nabla | $\nabla²$: 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: $Re = ρ \times$ (space/time) \times 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: $θ \approx 1.22λ/D$ Transform: $θ \approx 1.22 \times space_wavelength/space_aperture$ (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: $2d\cdotsinθ = nλ$ Transform: $2 \times space_spacing \times sin(θ) = n \times space_wavelength$ (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: $I = P/(4π \times space²)$ (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: $curvature_μν + RLU_eviction_rate × metric_μν =$ (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 (RLU). $T_μν$ = 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 RLU 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: $d²(space^μ)/d(time)² + Γ \times$ (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: $R^ρ_σμν = \partial_μΓ^ρ_νσ - \partial_νΓ^ρ_μσ + Γ^ρ_μλΓ^λ_νσ - Γ^ρ_νλΓ^λ_μσ$ Transform: $R = d(Γ)/d(space) - d(Γ)/d(space) + Γ\timesΓ - Γ\timesΓ$ (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: $z = 1/\sqrt(1 - r_s/r) - 1$ Transform: $z = 1/\sqrt(1 - space_consumption_rate) - 1$ (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: $space/time = H₀ \times space$ (recession velocity = Hubble constant \times distance, observational expression of RLU 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: $T(z) = T₀ \times$ (1 + space_consumption_rate_inverse - 1) = T₀ \times (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: $(space potential difference) = (dQ/d(time)) \times R$ \to 1st order but reduces to 2nd order via P = I²R = (dQ/d(time))² \times 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: $(space/time² \times mass)₁₂ = -(space/time² \times mass)₂₁$ \to 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: $(mass \times space/time²) = -k \times space$ \to elastic potential energy U = ½kx² = ½k \times 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: $d(particle count)/d(time) = -λ \times N$ \to 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: $(induced EMF) = -d(B \times space²)/d(time)$ \to energy P = EMF \times I = EMF \times (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: $\partialρ/\partialt + \nabla\cdotJ = 0$ Transform: $\partial(density)/\partial(time) + \nabla\cdot(density \times space/time) = 0$ \to form of differentiating δ² = observer² + superposition² conservation in space-time $ρ$: density | $\nabla$: 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: $dS \geq 0$ (RLU 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 RLU 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: $T² \propto a³$ Transform: $time² \propto space³ \to time² =$ (4π²/GM) \times 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: $V_eff = -GM/space + L²/(2m \times space²)$ \to 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: $B \propto ν³/(exp(hν/kT) - 1)$ Transform: $(radiation spectral density) \propto$ (hν)³/h³ / (exp(E_photon/E_thermal) - 1) \to 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: $(occupation probability) = 1/(exp((E_state - μ_chemical_potential)/kT) + 1)$ \to 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: $H = -Σ p \cdot log(p)$ Transform: $(information content) = -Σ |ψ|² \times log(|ψ|²)$ \to 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: $R^ρ_σμν = \partial_μΓ^ρ_νσ - \partial_νΓ^ρ_μσ + Γ^ρ_μλΓ^λ_νσ - Γ^ρ_νλΓ^λ_μσ$ Transform: $(curvature) =$ (derivative of Christoffel symbol) + (Christoffel symbol)² \to 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: $T^00 = ½ρv² + ½ε₀E² + B²/(2μ₀) + ...$ \to 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: $(EM tensor)$ \to Lagrangian L \propto F_μν F^μν \to 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 RLU 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 \to 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: $E = ½m \times$ (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: $space = ½ \times d²(space)/d(time)² \times time²$ (time² \to 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: $F = m \times space/time²$ (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: $L \propto space² \times$ (1/time) (area \times 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: $d²(space)/d(time)² + (1/time)² \times space = 0$ $space$: 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: $time² = (const) \times space³ \to since GM/a = v², reduces to T² = a²/v²$ $space$: 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: $F \propto 1/space²$ (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: $U \propto -1/space$ (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) \times 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) (RLU 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 RLU base eviction rate may yield predictable vacuum energy scale.

Eq. 15. Escape Velocity

Original: $v_esc = \sqrt(2GM/r)$ Transform: $(space/time)² = 2 \times GM/space$ (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: $ΔF \propto 1/r³$ 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: $F \propto 1/space²$ (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: $u = ½ε₀ \times$ (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: $u =$ (\nabla\timesA)²/(2μ₀) (square of rotational field in space) $u$: energy density | $B$: magnetic field | $μ₀$: vacuum permeability | $\nabla$: 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: $S \propto E\timesB \propto E²$ Transform: $S \propto$ (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: $E = ½C \times$ (d(φ)/d(space) \times space)² (potential = spatial gradient \times 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 \times 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: $E = ½L \times$ (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: $dB \propto Idl/r²$ Transform: $dB \propto$ (dQ/d(time)) \times d(space)/space² (current \times 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: $F = q(E + v\timesB)$ Transform: $F = q(d(φ)/d(space) +$ (space/time) \times \nabla\timesA) (electric gradient + velocity \times magnetic rotation) $F$: force | $q$: charge | $E$: energy/electric field | $v$: velocity=space/time | $B$: magnetic field | $space$: space | $time$: time | $\nabla$: 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: $\nabla\timesB = μ₀J + μ₀ε₀\partialE/\partialt$ Transform: $\nabla\times(\nabla\timesA) = μ₀(dQ/d(time)) + μ₀ε₀ \times d(d(φ)/d(space))/d(time)$ (spatial rotation = current + time rate of E-field) $\nabla$: 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: $EMF = -d(B \times space²)/d(time)$ (magnetic flux = B \times 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. 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. 35. 4-Momentum Norm | Subframe: time-space | (E/$\|C\|$)²-$space²$ = (m$\|C\|$)² | PASS

E. Quantum Mechanics (10/10 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. 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. 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. 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. 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. 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. 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. 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 | $dS≥0$, RLU 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

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. 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

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

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

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

Eq. 114. Entropy Increase | Subframe: observer-superposition | $dS≥0$ directionality axiom | 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 (Banya Framework)
Inventor: Han Hyukjin (Hyukjin Han)
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

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