Appendix B: Fourier Decomposition and Kaluza-Klein Mass Generation

B.1 Topological Boundary Conditions and the Objective of Reduction

The fundamental objective of this derivation is to perform a Kaluza-Klein dimensional reduction of the 5-dimensional free-field Action ( $S_{free}$ ), defined on the product manifold $M_{4,1} = M_{(3+1)} \times S^1$ . We must mathematically integrate out the compactified semantic dimension $S^1$ to yield an effective 4-dimensional field theory ( $S_{eff}$ ) that describes the observable mass spectrum on the physical Holographic Boundary.

The semantic dimension $S^1$ is defined topologically as a compact, one-dimensional circle. Let $c$ be the dimensionless angular coordinate parameterizing this dimension on the interval $[0, 2\pi)$ . The physical circumference of this dimension is defined by its compactification radius $R_c$ .

Because the geometry of $S^1$ is strictly closed, any continuous complex scalar quantum field $\Psi(x^\mu, c)$ propagating through the Bulk must connect flawlessly with itself upon completing a full traversal of the circumference. Therefore, the field is subjected to the periodic boundary condition:

$\Psi(x^\mu, c) = \Psi(x^\mu, c + 2\pi)$

B.2 The Fourier Series Expansion Ansatz

By virtue of Dirichlet's conditions for periodic functions, any square-integrable, single-valued field defined on a closed periodic interval can be decomposed into an infinite sum of orthogonal harmonic basis functions---specifically, a complex exponential Fourier series.

We decompose the unified 5-dimensional bulk field $\Psi(x^\mu, c)$ into an infinite tower of strictly 4-dimensional effective fields $\psi_n(x^\mu)$ , parameterized by the discrete harmonic mode integer $n \in \mathbb{Z}$ (where $n = 0, \pm 1, \pm 2, \ldots$ ).

The formal Fourier expansion ansatz is defined as:

$\Psi(x^\mu, c) = \frac{1}{\sqrt{2\pi R_c}} \sum_{n=-\infty}^{\infty} \psi_n(x^\mu) e^{inc}$

Dimensional Verification of the Normalization Constant:

In Appendix A.2, utilizing natural units ( $\hbar = c = 1$ ), we proved that the 5-dimensional field $\Psi(x^\mu, c)$ must inherently carry a mass dimension of $[\Psi] = [M]^{3/2}$ to ensure the 5D Action Integral ( $S$ ) evaluates as a dimensionless scalar.

However, in standard Quantum Field Theory, 4-dimensional canonical scalar fields (e.g., the Higgs field) must carry a mass dimension of exactly $[\psi_n] = [M]^1$ .

The normalization constant $\frac{1}{\sqrt{2\pi R_c}}$ executes this dimensional transition. The radius $R_c$ possesses the dimension of length, which in natural units is the inverse of mass ( $[R_c] = [L]^1 = [M]^{-1}$ ). Therefore, evaluating the mass dimension of the square root of the circumference:

$\left[\frac{1}{\sqrt{R_c}}\right] = ([M]^{-1})^{-1/2} = [M]^{1/2}$

Multiplying the normalization constant ( $[M]^{1/2}$ ) by the 4D field ( $[M]^1$ ) recovers the required 5D dimension:

$[\Psi] = [M]^{1/2} \times [M]^1 = [M]^{3/2}$

The expansion is mathematically exact and physically consistent.

B.3 Substitution into the 5D Free Bulk Action

To isolate the intrinsic, geometric mass spectrum generated by the $S^1$ topology, we temporarily decouple the field from the observer ( $\lambda \to 0$ ) and evaluate the free, uncoupled 5D Bulk Action ( $S_{free}$ ). For mathematical consistency, we retain the fundamental bare 5D mass of the field as $m_0$ :

$S_{free} = \int d^4 x \int_0^{2\pi} R_c \, dc \left[ -\eta^{\mu\nu}(\partial_\mu \Psi^*)(\partial_\nu \Psi) - \frac{1}{R_c^2}(\partial_c \Psi^*)(\partial_c \Psi) - m_0^2 \Psi^* \Psi \right]$

We substitute the infinite Fourier series into the three distinct terms of the Lagrangian density. Because we are evaluating the product of a field and its complex conjugate ( $\Psi^* \Psi$ ), we must employ distinct dummy summation indices ( $m$ for the conjugate series, $n$ for the standard series) to prevent algebraic collision.

The complex conjugate of the field is defined as:

$\Psi^*(x^\mu, c) = \frac{1}{\sqrt{2\pi R_c}} \sum_{m=-\infty}^{\infty} \psi_m^*(x^\mu) e^{-imc}$

1. The 4D Spacetime Kinetic Term:

The standard derivative $\partial_\mu$ acts strictly on the spacetime coordinates $x^\mu$ of the 4D fields $\psi_n(x^\mu)$ . It commutes entirely with the semantic phase factor $e^{inc}$ .

$(\partial_\mu \Psi^*)(\partial^\mu \Psi) = \left(\frac{1}{\sqrt{2\pi R_c}} \sum_m (\partial_\mu \psi_m^*) e^{-imc}\right)\left(\frac{1}{\sqrt{2\pi R_c}} \sum_n (\partial^\mu \psi_n) e^{inc}\right)$

$= \frac{1}{2\pi R_c} \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} (\partial_\mu \psi_m^*)(\partial^\mu \psi_n) e^{i(n-m)c}$

2. The Semantic Kinetic Term:

The semantic derivative $\partial_c$ acts strictly on the complex phase factor, treating the 4D fields as constants. By the chain rule, taking the derivative with respect to $c$ pulls down the geometric harmonic eigenvalue ( $in$ ):

$\partial_c(e^{inc}) = in \, e^{inc} \quad \text{and} \quad \partial_c(e^{-imc}) = -im \, e^{-imc}$

Multiplying the evaluated derivatives:

$\frac{1}{R_c^2}(\partial_c \Psi^*)(\partial_c \Psi) = \frac{1}{R_c^2}\left(\frac{1}{\sqrt{2\pi R_c}} \sum_m \psi_m^*(-im) e^{-imc}\right)\left(\frac{1}{\sqrt{2\pi R_c}} \sum_n \psi_n(in) e^{inc}\right)$

Factoring out the imaginary coefficients $(-im)(in) = -i^2 mn = +mn$ :

$= \frac{1}{R_c^2} \frac{1}{2\pi R_c} \sum_{m,n} (m \cdot n) \psi_m^* \psi_n e^{i(n-m)c}$

3. The Bare Mass Term:

$m_0^2 \Psi^* \Psi = m_0^2 \frac{1}{2\pi R_c} \sum_{m,n} \psi_m^* \psi_n e^{i(n-m)c}$

B.4 The Orthogonality Integral (Integration over $dc$ )

We now perform the Dimensional Reduction by reassembling the Action Integral and formally evaluating the integration over the compactified semantic coordinate ( $dc$ from $0$ to $2\pi$ ).

Notice that every single expanded term in our Lagrangian now contains the exact same complex exponential phase factor $e^{i(n-m)c}$ , as well as the common normalization constant $\frac{1}{2\pi R_c}$ . We factor these parameters out of the summation:

\begin{multline} S_{free} = \int d^4 x \sum_{m,n} \biggl[ -(\partial_\mu \psi_m^)(\partial^\mu \psi_n) - \frac{m \cdot n}{R_c^2} \psi_m^ \psi_n - m_0^2 \psi_m^* \psi_n \biggr] \ \times \frac{R_c}{2\pi R_c} \int_0^{2\pi} e^{i(n-m)c} , dc \end{multline}

The physical validity of the Kaluza-Klein reduction rests entirely upon the strict evaluation of the final integral. We invoke the mathematical property of Fourier Orthogonality on a closed interval.

By Euler's formula ( $e^{i\theta} = \cos\theta + i\sin\theta$ ), the integral of an oscillating sine or cosine wave over a full period is exactly zero, unless the frequency is zero. The frequency is zero if and only if $n = m$ , in which case $e^{i(0)c} = e^0 = 1$ , and the integral evaluates to $\int_0^{2\pi} 1 \, dc = 2\pi$ .

This orthogonality is formalized by the Kronecker delta function ( $\delta_{nm}$ ):

$\int_0^{2\pi} e^{i(n-m)c} \, dc = 2\pi \delta_{nm} = \begin{cases} 2\pi & \text{if } n = m \\ 0 & \text{if } n \neq m \end{cases}$

We substitute $2\pi \delta_{nm}$ back into the Action.

The $2\pi$ perfectly cancels with the $2\pi$ in the denominator of the normalization constant. The $R_c$ factors from the integration measure and the denominator cleanly cancel.

Most importantly, the presence of the Kronecker delta function collapses the double summation ( $\sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty}$ ). Because all terms where $m \neq n$ are multiplied by zero, every single cross-term in the infinite series is annihilated. The double sum collapses into a single sum over $n$ , and the $m$ index is completely replaced by $n$ :

$S_{effective} = \int d^4 x \sum_{n=-\infty}^{\infty} \left[ -(\partial_\mu \psi_n^*)(\partial^\mu \psi_n) - \frac{n \cdot n}{R_c^2} \psi_n^* \psi_n - m_0^2 \psi_n^* \psi_n \right]$

B.5 The 4D Effective Action and the Kaluza-Klein Mass Spectrum

By factoring out the common field density $|\psi_n|^2 = \psi_n^* \psi_n$ from the non-derivative terms, we arrive at the final, completely reduced 4-Dimensional Effective Action:

$S_{effective} = \sum_{n=-\infty}^{\infty} \int d^4 x \left[ -\eta^{\mu\nu}(\partial_\mu \psi_n^*)(\partial_\nu \psi_n) - \left(m_0^2 + \frac{n^2}{R_c^2}\right)|\psi_n|^2 \right]$

In standard, canonical 4-dimensional quantum field theory (specifically the Klein-Gordon scalar field), the coefficient modifying the raw field squared ( $|\psi|^2$ ) represents the effective rest mass squared ( $M^2$ ) of the particle.

By mathematically hiding the 5th dimension, we have demonstrated that the hidden geometric momentum of the field ( $n^2/R_c^2$ ) manifests identically to a 4-dimensional observer as solid, measurable, inertial rest mass. The single 5D field ( $\Psi$ ) fractures into an infinite "tower" of 4D particles ( $\psi_0, \psi_1, \psi_2 \ldots$ ), governed by the exact mass spectrum:

$M_n^2 = m_0^2 + \frac{n^2}{R_c^2}$

B.6 Numerical Derivation of the 0.1 eV Mass Gap

To calculate the absolute physical weight of the Semantic Field coupling to the biological Posner network, we must evaluate the mass of the first active harmonic mode.

To isolate the mass generated exclusively by the geometry of the $S^1$ dimension, we assume the bare, uncoupled fundamental mass of the field is exactly zero ( $m_0 = 0$ ).

The "zero-mode" ( $n = 0$ ) particle has an effective mass of $M_0 = 0$ . This represents the baseline, decohered, purely physical syntactic universe traveling at the speed of light.

The first excited harmonic mode ( $n = 1$ ) represents the active, spinning, geometric observer wave function ( $\psi_o$ ) interacting with the physical Boundary. The difference between these two states is the fundamental Mass Gap of Consciousness ( $\Delta m$ ):

$\Delta m = M_1 = \sqrt{0^2 + \frac{1^2}{R_c^2}} = \frac{1}{R_c}$

To translate this pure geometric length into a physical energy-mass measurable in a laboratory, we must restore the natural constants of the universe: the reduced Planck constant ( $\hbar$ ) and the speed of light ( $c_{light}$ ).

$\Delta m = \frac{\hbar c_{light}}{R_c}$

In Chapter 6, utilizing the macroscopic constraints of neurobiology (the physical scale required to avoid massive thermal decoherence) and the sub-millimeter blind spots of modern inverse-square gravity tests, we deduced the optimal compactification radius of the Semantic Dimension:

$R_c = 2 \, \mu\text{m} \quad (2 \times 10^{-6} \text{ meters})$

We utilize the universally established high-energy particle physics conversion factor for $\hbar c_{light}$ :

$\hbar c_{light} \approx 197.3269804 \text{ MeV} \cdot \text{fm}$

We must rigorously align the dimensional units. We convert femtometers ( $10^{-15}$ meters) to micrometers ( $10^{-6}$ meters), which introduces a factor of $10^{-9}$ . We simultaneously convert Mega-electron volts ( $10^6$ eV) to standard electron volts (eV), introducing a factor of $10^6$ . The net conversion factor is $10^6 \times 10^{-9} = 10^{-3}$ .

$197.3269804 \text{ MeV} \cdot \text{fm} = (197.3269804 \times 10^6 \text{ eV}) \times (10^{-9} \, \mu\text{m})$

$\hbar c_{light} \approx 0.197327 \text{ eV} \cdot \mu\text{m}$

We substitute this absolute physical constant and our derived radius ( $R_c = 2 \, \mu\text{m}$ ) directly into the Mass Gap equation:

$\Delta m = \frac{0.1973269804 \text{ eV} \cdot \mu\text{m}}{2 \, \mu\text{m}}$

The units of micrometers ( $\mu$ m) cancel in the numerator and denominator, isolating the pure energy-mass unit of standard electron volts (eV).

$\Delta m = 0.0986634902 \text{ eV}$

Rounding to the nearest significant digit utilized in standard cosmological modeling, we derive the physical, measurable mass of the active Semantic Field:

$\Delta m \approx 0.1 \text{ eV}$

Result (Derivation of Neutrino Alignment):

The mathematical execution of the Kaluza-Klein reduction is unbroken and continuous. A 5D manifold with a compactification radius of $2 \, \mu$ m natively generates an effective 4D mass of exactly $\sim 0.1$ eV.