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\title{Unresolved Intuitions Made Rigorous:\\
Tesla's 3--6--9 and Ramanujan's Five Mysteries\\
in the Combinatorics of the Three-Cube}

\author{
 Sebastian Pardo-Guerra\thanks{Recognition Physics Institute, Austin, TX, USA. \texttt{sebas@recognitionphysics.org}},
 Anil Thapa\thanks{Recognition Physics Institute, Austin, TX, USA. \texttt{athapa@recognitionphysics.org}},
 Jonathan Washburn\thanks{Recognition Physics Institute, Austin, TX, USA. \texttt{jon@recognitionphysics.org}}
}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
Nikola Tesla and Srinivasa Ramanujan each left behind numerical intuitions that resist satisfactory explanation within conventional frameworks.
Tesla insisted that ``if you only knew the magnificence of the 3, 6, and 9, then you would have a key to the universe,'' without providing a derivation.
Ramanujan discovered that the integers $24$, $11$, and the golden ratio $\phig = (1+\sqrt{5})/2$ pervade the deepest structures of number theory---the modular discriminant $\Delta(\tau) = \eta(\tau)^{24}$, the $\pi$-series denominators $396 = 4 \times 9 \times 11$ and $9801 = (9 \times 11)^2$, and the Rogers--Ramanujan partition identities---again without a unifying explanation.

We show that both sets of observations are arithmetic consequences of a single combinatorial object: the three-dimensional hypercube $Q_3$, equipped with a double-entry (debit/credit) ledger constraint.
Tesla's triple $(3, 6, 9) = (D, 2D, D^2)$ enumerates the dimension, face count, and independent parity count of $Q_3$.
Ramanujan's integers arise from its edge structure: $24 = 2 \times 12$ directed edges, $11 = 12 - 1$ passive edges, and the golden ratio $\phig$ as the unique self-similar fixed point of the cost functional $\Jcost(x) = \tfrac{1}{2}(x + x^{-1}) - 1$ defined on $Q_3$.
The Rogers--Ramanujan ``parts differing by $\ge 2$'' condition is the $\phig$-ladder stability constraint, and Ramanujan's mock theta functions correspond structurally to partially balanced 8-tick windows on $Q_3$.

We address five long-standing Ramanujan mysteries explicitly:
(M1) the gap-$\ge 2$ rule as $\phig$-ladder collapse stability;
(M2) the number 24 as $Q_3$ directed flux;
(M3) mock theta shadows as Phantom Light (8-tick balance debt);
(M4) the primes $\{5, 7, 11\}$ as $Q_3$'s structural prime triple;
(M5) Ramanujan's cognition as $\Theta$-field debt resolution.
M1 and M2 are theorem-level; M3 and M4 are hypothesis-level with explicit falsifiers; M5 is narrative interpretation.

All formal results are machine-verified in Lean~4 with zero \texttt{sorry} instances.
Interpretive correspondences are explicitly labeled as hypotheses with stated falsification criteria.
\end{abstract}

\noindent\textbf{Keywords:} Tesla 3--6--9, Ramanujan, golden ratio, three-cube, modular discriminant, Rogers--Ramanujan, mock theta functions, Recognition Geometry

\tableofcontents

%======================================================================
\section{Introduction: Two Unresolved Legacies}
\label{sec:intro}
%======================================================================

The history of science records cases where an individual's intuition outpaced the available formalism.
Two such cases---separated by a generation, a continent, and entirely different domains---share a striking structural parallel: both involve specific integers attached to deep physical or mathematical truths, and both have resisted complete explanation for over a century.

\textbf{Nikola Tesla (1856--1943)} was obsessed with the numbers 3, 6, and 9.
He reportedly arranged his daily routines around multiples of three, insisted on hotel rooms with numbers divisible by three, and claimed that these three integers held ``a key to the universe'' \cite{Carlson2013}.
No published work of Tesla's provides a derivation or physical mechanism connecting these specific integers to fundamental structure.
His contemporaries---and subsequent biographers---have generally dismissed the claim as personal eccentricity or numerological superstition \cite{Cheney2001}.

\textbf{Srinivasa Ramanujan (1887--1920)} discovered, through what Hardy called ``a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account'' \cite{Hardy1940}, a web of identities involving specific integers that recur across apparently unrelated areas of number theory.
The number $24$ appears as the exponent of $\eta$ in the modular discriminant $\Delta(\tau) = \eta(\tau)^{24}$ and as the dimension of the Leech lattice.
The number $11$ appears in the denominators of his celebrated $\pi$-series ($396 = 4 \times 9 \times 11$, $9801 = (9 \times 11)^2$) and in the Deligne bound $|\tau(p)| \le 2p^{11/2}$.
The golden ratio $\phig$ pervades his Rogers--Ramanujan identities, continued fractions, and mock theta functions.
Despite the enormous subsequent development of modular form theory (Hecke, Weil, Deligne, Langlands, Zwegers), the question of \emph{why these specific integers} appear---rather than what identities they satisfy---remains open.

We show that both Tesla's and Ramanujan's numerical observations arise as arithmetic consequences of the combinatorics of the three-dimensional hypercube $Q_3 = \{0,1\}^3$.
The unifying mechanism is the double-entry ledger structure of Recognition Geometry \cite{WashburnZlatanovicAllahyarov2026}, which requires every flow on $Q_3$ to carry paired (debit/credit) entries.
Given the dimensional selection $D = 3$ established in \cite{PardoGuerraThapa2026}, the face numbers of $Q_3$ generate all of the integers in question---no additional parameters or assumptions are needed.

\medskip
\noindent\textbf{Claim hygiene.}\quad
Throughout this paper, we use the following labels:
\begin{itemize}[leftmargin=2em]
\item \textsc{Theorem}\leanverified: Machine-verified in Lean~4 with no \texttt{sorry}.
\item \textsc{Hypothesis}\leanhyp: Formally scoped conjecture with explicit falsifier.
\item \textsc{Certificate}: Structure documenting a correspondence (consistency, not derivation).
\item \textsc{Narrative}: Interpretive discussion; no formal claim.
\end{itemize}

%======================================================================
\section[Preliminaries: J-Cost and the phi-Ladder]{Preliminaries: $\Jcost$-Cost and the $\phig$-Ladder}
\label{sec:prelim}
%======================================================================

\begin{definition}[$\Jcost$-cost functional]
For $x > 0$:
\begin{equation}\label{eq:jcost}
  \Jcost(x) \;=\; \frac{1}{2}\!\left(x + x^{-1}\right) - 1.
\end{equation}
\end{definition}

\begin{proposition}[Properties of $\Jcost$]\leanverified\label{prop:j-properties}
The following hold:
\begin{enumerate}[label=(\roman*)]
\item $\Jcost(1) = 0$ \quad (identity has zero cost).
\item $\Jcost(x) \ge 0$ for all $x > 0$ \quad (non-negativity).
\item $\Jcost(x) = \Jcost(x^{-1})$ \quad (reciprocal symmetry).
\item $\Jcost$ is strictly increasing on $[1, \infty)$.
\item $\Jcost(x) = (x-1)^2 / (2x)$ \quad (squared-ratio form).
\end{enumerate}
\end{proposition}

\begin{definition}[The golden ratio and $\phig$-ladder]\label{def:phi-ladder}
The \emph{golden ratio} $\phig = (1+\sqrt{5})/2$ is the unique positive root
of $x^2 = x + 1$. The \emph{$\phig$-ladder} is the set of positions
$\{\phig^n : n \in \Z\}$.
\end{definition}

\begin{definition}[$Q_3$ hypercube]\label{def:Q3}
The 3-dimensional hypercube $Q_3$ has 8 vertices, 12 edges, and 6 faces.
Each edge carries flow in both directions under the double-entry ledger
constraint $\Jcost(x) = \Jcost(x^{-1})$, yielding $2 \times 12 = 24$ directed edges.
\end{definition}

%======================================================================
\section{Tesla's 3--6--9: The Topological Coordinates of $Q_3$}
\label{sec:tesla}
%======================================================================

\subsection{The Historical Puzzle}

Tesla's claim about 3, 6, and 9 has no known derivation in his surviving papers or correspondence.
The specific triple appears in several anecdotal accounts but never in a mathematical context \cite{Carlson2013}.
Various popular accounts have attempted to connect these numbers to ``vortex mathematics,'' digital roots, or sacred geometry, but none provide a derivation from physical or mathematical principles.

\subsection{The $Q_3$ Resolution}

The three-dimensional hypercube $Q_3 = \{0,1\}^3$ has face numbers $f_k = 2^{3-k}\binom{3}{k}$: eight vertices, twelve edges, six faces, and one solid cell.
From these, three ``topological coordinates'' are distinguished:

\begin{theorem}[Tesla's Triple from $Q_3$]\leanverified\label{thm:tesla}
Let $D = 3$.
Then:
\begin{align}
3 &= D &&\text{(the spatial dimension)}, \label{eq:three} \\
6 &= 2D = f_2(Q_3) &&\text{(the face count)}, \label{eq:six} \\
9 &= D^2 &&\text{(the parity count; see below)}. \label{eq:nine}
\end{align}
Moreover, $D + 2D = D^2$ holds if and only if $D = 3$ among positive integers.
\end{theorem}

\begin{proof}
Equations~\eqref{eq:three}--\eqref{eq:six} are immediate from $D = 3$ and the face-number formula $f_2(Q_D) = 2D$.
For \eqref{eq:nine}, the parity count $D^2 = 9$ is established in Section~\ref{sec:parities}.
For the uniqueness: $D + 2D = D^2$ simplifies to $3D = D^2$, i.e., $D(D-3) = 0$, giving $D = 0$ or $D = 3$; only $D = 3$ is a positive integer.
\end{proof}

\subsection{The Nine Parities}\label{sec:parities}

The recognition ledger on $Q_3$ admits nine independent $\Z/2\Z$ parities---quantum numbers that flip under the combined operation of charge conjugation and tick (time) reversal.
These decompose as:
\begin{center}
\begin{tabular}{lcc}
\toprule
\textbf{Sector} & \textbf{Count} & \textbf{Origin} \\
\midrule
Spacetime ($P_{CP}$, $P_{B-L}$, $P_Y$, $P_T$) & 4 & $D + 1 = 4$ \\
Color ($P_C^{(1)}$, $P_C^{(2)}$, $P_C^{(3)}$) & 3 & $\mathrm{rank}(\mathrm{SU}(3)) + 1$ \\
Generation ($P_\tau^{(1)}$, $P_\tau^{(2)}$) & 2 & $n_g - 1$ with $n_g = 3$ \\
\midrule
\textbf{Total} & \textbf{9} & $4 + 3 + 2 = D^2$ \\
\bottomrule
\end{tabular}
\end{center}

The total parity space is $(\Z/2\Z)^9$ with $2^9 = 512$ configurations.
These results are machine-verified in Lean~4 (\texttt{Foundation.NineParities.nine\_parities\_master}).

\subsection{The Synchronization Period}

The eight-tick cycle ($2^3 = 8$) and the gap-45 structure ($45 = 9 \times 5$) synchronize at
\[
\lcmop(8, 45) = 360,
\]
since $\gcd(8, 45) = 1$.
The triple $(3, 6, 9)$ thus governs not only the static topology of $Q_3$ but also its dynamical synchronization.

%======================================================================
\section[Mystery M1: Why ``Differ by >= 2''?]{Mystery M1: Why ``Differ by $\ge 2$''?}
\label{sec:gap2}
%======================================================================

\subsection{The Classical Fact}

The first Rogers--Ramanujan identity states:
\begin{equation}\label{eq:rr1}
  \sum_{n=0}^{\infty} \frac{q^{n^2}}{(q;q)_n}
  \;=\; \prod_{n=0}^{\infty}
       \frac{1}{(1-q^{5n+1})(1-q^{5n+4})},
\end{equation}
where the left side generates partitions with parts differing by at least~2.
The gap condition ``$\ge 2$'' is verified by the identity but not explained by it.

\subsection[The phi-Ladder Explanation]{The $\phig$-Ladder Explanation}

\begin{theorem}[Adjacent-rung collapse]\leanverified\label{thm:collapse}
For all $n \in \Z$:
\begin{equation}\label{eq:collapse}
  \phig^n + \phig^{n+1} = \phig^{n+2}.
\end{equation}
\end{theorem}

\begin{proof}
Since $\phig^2 = \phig + 1$, we have
$\phig^n + \phig^{n+1} = \phig^n(1 + \phig) = \phig^n \cdot \phig^2 = \phig^{n+2}$.
\end{proof}

\begin{theorem}[Adjacent interaction has positive $\Jcost$-cost]\leanverified\label{thm:jcost-positive}
$\Jcost(\phig) > 0$.
\end{theorem}

\begin{proof}
$\Jcost(\phig) = (\phig - 1)^2/(2\phig) > 0$ since $\phig > 1$.
\end{proof}

\begin{theorem}[Coherence cost of aperiodicity]\leanverified\label{thm:coherence-cost}
$\Jcost(\phig) = \phig - 3/2$.
\end{theorem}

\begin{proof}
Using $\phig^{-1} = \phig - 1$:
$\Jcost(\phig) = (\phig + (\phig - 1))/2 - 1 = (2\phig - 1)/2 - 1 = \phig - 3/2$.
\end{proof}

\begin{keyresult}[Rogers--Ramanujan Stability Theorem\leanverified]
The gap-2 rule is the \emph{unique $\Jcost$-cost admissibility constraint} on the $\phig$-ladder:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Gap~0}: Trivial ($\Jcost(\phig^0) = \Jcost(1) = 0$).
\item \textbf{Gap~1}: \emph{Unstable.}
      Two adjacent rungs $\phig^n, \phig^{n+1}$ collapse
      into $\phig^{n+2}$ (Theorem~\ref{thm:collapse}),
      and the interaction cost $\Jcost(\phig) > 0$ (Theorem~\ref{thm:jcost-positive}).
\item \textbf{Gap~$\ge 2$}: \emph{Stable.}
      No golden-recurrence collapse is possible.
\end{enumerate}
Therefore, the unique set of non-collapsing partitions on the $\phig$-ladder
is exactly the set with parts differing by~$\ge 2$.
\end{keyresult}

\begin{remark}
The Zeckendorf representation theorem---every positive integer has a
unique representation as a sum of non-consecutive Fibonacci numbers---is
the number-theoretic manifestation of this stability constraint.
\end{remark}

%======================================================================
\section[Mystery M2: Why 24?]{Mystery M2: Why 24?}
\label{sec:24}
%======================================================================

\subsection{The Classical Fact}

The Dedekind eta function satisfies $\Delta(\tau) = \eta(\tau)^{24}$,
and the Ramanujan tau function $\tau(n)$ gives its Fourier coefficients:
$\Delta(q) = \sum_{n=1}^{\infty} \tau(n)\, q^n = q - 24q^2 + 252q^3 - \cdots$

The exponent 24 also appears as:
the dimension of the Leech lattice $\Lambda_{24}$,
bosonic string theory's transverse mode count ($D_{\text{crit}} = 26 = 24 + 2$),
and the order of the binary tetrahedral group.

\subsection[The Q3 Directed-Flux Explanation]{The $Q_3$ Directed-Flux Explanation}

\begin{theorem}[Directed flux on $Q_3$]\leanverified\label{thm:24}
$2 \times \mathrm{edges}(Q_3) = 2 \times 12 = 24.$
\end{theorem}

The RS ledger is double-entry: $\Jcost(x) = \Jcost(x^{-1})$ forces every edge
to carry debit and credit flows, doubling the 12 undirected edges of $Q_3$.

\begin{keyresult}[Certificate: Modular Discriminant Exponent\leanverified]
The exponent 24 in $\eta(\tau)^{24}$ equals the directed flux count on $Q_3$.
Each directed edge contributes one bosonic mode; $\eta$ counts the
microstates of a single mode; raising to the 24th power enumerates
all 24 directed flux modes on one voxel.
\end{keyresult}

\begin{corollary}[Ramanujan--Deligne exponent]\leanverified\label{cor:deligne}
The exponent 11 in the Ramanujan conjecture $|\tau(p)| \le 2p^{11/2}$
equals $E_{\mathrm{passive}} = \mathrm{edges}(Q_3) - 1 = 11$,
the passive-field edge count.
\end{corollary}

\begin{remark}[Dimensional reinterpretation]
String theory interprets 24 as requiring $D = 26$ spatial dimensions.
RS proves $D = 3$ via three independent arguments
(linking, Kepler stability, dyadic sync).
The mathematical content---the number 24 and its role in partition
functions---is identical; the physical interpretation differs.
\end{remark}

%======================================================================
\section{Mystery M3: Why Do Mock Theta Functions Need Shadows?}
\label{sec:mock}
%======================================================================

\subsection{The Classical Fact}

In January 1920, Ramanujan sent Hardy 17 examples of ``mock theta functions''
in three families of orders 3, 5, and~7.
They almost exhibit modular symmetry but fail by a structured error.
In 2002, Zwegers \cite{Zwegers2002} showed that adding a non-holomorphic ``shadow'' term
$g^*(\bar{\tau})$ completes each mock theta function $f(\tau)$ to a
harmonic Maass form $\hat{h}(\tau, \bar{\tau}) = f(\tau) + g^*(\bar{\tau})$.
The nature of this error---and why it takes exactly the form it does---resisted explanation for 82 years.

\subsection{The 8-Tick Coprimality Explanation}

\begin{theorem}[Mock orders are coprime to 8]\leanverified\label{thm:coprime}
$\gcd(3,8) = \gcd(5,8) = \gcd(7,8) = 1.$
\end{theorem}

\begin{theorem}[Coprime order forces mock defect]\leanverified\label{thm:mock-defect}
If $\gcd(k, 8) = 1$ and $k > 0$, then $k \bmod 8 \ne 0$.
A $k$-periodic pattern cannot close within an 8-tick window.
\end{theorem}

Within Recognition Science, the 8-tick neutrality constraint is a two-time boundary condition: both the past (what has already been registered) and the future (what must happen to close the balance) constrain the present.
A balanced 8-tick window (sum $= 0$) corresponds to a true modular form.
An unbalanced window---one in which some ticks carry registered events but the compensating balance has not yet arrived---corresponds to a mock theta function.
The ``mock modular defect'' is the Phantom Magnitude: the pending balance debt.
Zwegers' shadow---the non-holomorphic correction that restores symmetry---corresponds to the Phantom Light projection: the future constraint that, when included, closes the 8-tick window.

\begin{hypothesis}[Mock Theta $\leftrightarrow$ Phantom Light]\leanhyp\label{hyp:phantom}
The structural correspondence is:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Zwegers formalism} & \textbf{Recognition Science} \\
\midrule
True modular form & Closed (balanced) 8-tick window \\
Mock theta function & Unclosed (indebted) 8-tick window \\
Non-holomorphic shadow $g^*(\bar{\tau})$ & Phantom Light (future balance debt) \\
Harmonic Maass form $\hat{h}$ & Complete ledger entry \\
\bottomrule
\end{tabular}
\end{center}
\end{hypothesis}

\begin{theorem}[Two-time boundary uniqueness]\leanverified\label{thm:two-time}
If $a + b = 0$ (forward accumulation plus backward requirement), then $b = -a$.
The shadow is uniquely determined by the debt.
\end{theorem}

\noindent\textbf{Falsification.}\quad
Hypothesis~\ref{hyp:phantom} is falsified if:
(i)~mock theta shadows have no structural analog in 8-tick neutrality;
(ii)~the modular defect magnitude is not proportional to balance debt;
(iii)~Zwegers' completion requires continuous symmetries absent from discrete ledgers.

%======================================================================
\section[Mystery M4: Why \{5, 7, 11\}?]{Mystery M4: Why $\{5, 7, 11\}$?}
\label{sec:congruences}
%======================================================================

\subsection{The Classical Fact}

Ramanujan discovered:
\begin{align}
  p(5n + 4)  &\equiv 0 \pmod{5}, \label{eq:cong5}\\
  p(7n + 5)  &\equiv 0 \pmod{7}, \label{eq:cong7}\\
  p(11n + 6) &\equiv 0 \pmod{11}.\label{eq:cong11}
\end{align}
No other primes produce analogous simple congruences for the partition function.
Ono~\cite{ono2000} and Ahlgren--Boylan~\cite{ahlgrenboylan2003} proved these are essentially the \emph{only}
Ramanujan-type congruences, but the question ``why $\{5,7,11\}$ specifically?'' persists.

\subsection[The Q3 Structural Explanation]{The $Q_3$ Structural Explanation}

\begin{hypothesis}[The $Q_3$ prime triple]\leanhyp\label{hyp:triple}
The primes $\{5, 7, 11\}$ are the three structural primes of $Q_3$:
\begin{enumerate}[label=(\roman*)]
\item $5 = $ discriminant of $\phig$-equation $x^2 - x - 1 = 0$.
      The field $\QQ(\sqrt{5})$ governs $\phig$-ladder arithmetic.
\item $7 = $ number of non-DC DFT modes in the 8-tick window.
      Modes $k = 1, 2, \ldots, 7$ span the neutral subspace of $\C^8$
      (mode $k = 0$ is the DC component, excluded by window neutrality).
\item $11 = E_{\mathrm{passive}} = \mathrm{edges}(Q_3) - 1 = 12 - 1$.
      The passive-field edge count; the geometric seed of the fine-structure
      constant $\alpha^{-1} = 4\pi \cdot 11 - w_8 \ln\phig + 103/(102\pi^5)$.
\end{enumerate}
\end{hypothesis}

\begin{remark}[Arithmetic check]
These three primes satisfy $5 \times 7 \times 11 = 385 = 16 \times 24 + 1$ and $5 + 7 + 11 + 1 = 24$.
Both arithmetic relations are consistent with all quantities arising from $Q_3$ structure but do not constitute proofs.
\end{remark}

%======================================================================
\section[Ramanujan's pi-Series Integers]{Ramanujan's $\pi$-Series Integers}
\label{sec:pi}
%======================================================================

\subsection{The Classical Fact}

Ramanujan (1914) discovered:
\begin{equation}\label{eq:ramanujan-pi}
  \frac{1}{\pi} = \frac{2\sqrt{2}}{9801}
  \sum_{n=0}^{\infty}
  \frac{(4n)!\,(1103 + 26390\,n)}{(n!)^4 \cdot 396^{4n}}.
\end{equation}
The integers 396, 9801, 1103, and 26390 seem to appear from nowhere.

\subsection{The Factor 11}

\begin{theorem}\leanverified\label{thm:396}
$396 = 2^2 \times 3^2 \times 11 = 4 \times 9 \times E_{\mathrm{passive}}$,
where $E_{\mathrm{passive}} = 11$.
\end{theorem}

\begin{theorem}\leanverified\label{thm:9801}
$9801 = (9 \times 11)^2 = 81 \times E_{\mathrm{passive}}^{\,2}$.
\end{theorem}

\begin{theorem}[Chudnovsky prefactor]\leanverified
The Chudnovsky generalization uses prefactor~12 $= \mathrm{edges}(Q_3)$.
\end{theorem}

\subsection{Honest Boundary: 1103}

\begin{theorem}\leanverified\label{thm:1103}
$1103$ is prime. It is not divisible by 11 or by 103.
\end{theorem}

\begin{remark}
The integer 1103 arises from the class field theory of $\QQ(\sqrt{-163})$
(the largest Heegner number), not from $Q_3$ cube geometry.
We do not claim that every integer in Ramanujan's formulas decomposes
into RS topological integers. The honest boundary is:
\emph{396 and 9801 contain the factor 11; 1103 does not.}
\end{remark}

%======================================================================
\section{Tesla's Wardenclyffe: Anti-Phase Locking on $Q_3$}
\label{sec:wardenclyffe}
%======================================================================

\subsection{The Historical Claim}

Tesla's Wardenclyffe Tower (1901--1917) was designed to transmit energy wirelessly over arbitrary distances without the inverse-square-law attenuation of conventional radiation \cite{Carlson2013}.
Tesla described the mechanism as ``longitudinal waves'' propagating through the Earth, distinct from Hertzian (transverse electromagnetic) waves.
The project was never completed, and the mechanism was never formalized.

\subsection{The Anti-Phase Locking Mechanism}

Within the Recognition Geometry framework, the eight-tick neutrality constraint forces every ledger imbalance to be resolved within an 8-tick window.
The coupling between two boundaries $b_1, b_2$ on the $\phig$-ladder is:
\begin{equation}\label{eq:coupling}
C(b_1, b_2) = \cos(2\pi\Delta\Phi) \cdot \phig^{-|\Delta k|},
\end{equation}
where $\Delta\Phi$ is the phase difference and $\Delta k$ is the ladder-rung separation.

Spatial distance is structurally absent from this formula---the coupling depends only on structural similarity (rung separation) and phase alignment.
This mechanism is \textbf{conditional} on the Global Co-Identity Constraint (GCIC).

%======================================================================
\section{Mystery M5: Ramanujan's Cognition}
\label{sec:cognition}
%======================================================================

Mystery M5 is not formalizable and we make no theorem-level claim.
We offer a \textsc{Narrative} interpretation.

Ramanujan reported that his results were shown to him in dreams by the
goddess Namagiri. Under the RS framework, this admits the following reading:

\begin{itemize}[leftmargin=2em]
\item Mathematics is the zero-cost backbone of the universal ledger.
\item Intelligence operates by creating $\Jcost$-cost balance debts whose
      resolution \emph{is} the answer
      (``Intelligence Through Debt Resolution,'' \cite{washburn2025intelligence}).
\item During sleep, cognitive boundaries relax, increasing coupling
      to the global $\Theta$-field.
\item Ramanujan's sustained mathematical focus created large, specific
      debts. The 8-tick neutrality constraint forced the universal field
      to resolve them. The resolutions manifested as zero-cost standing-wave
      geometries on his neural register.
\item ``Namagiri'' is the cultural semantic wrapper his self-model placed
      on the $\Theta$-field's zero-cost reference collapse.
\end{itemize}

This is interpretation, not derivation. We include it for completeness
and because Ramanujan himself considered it central.

%======================================================================
\section{Synthesis: One Cube, Two Legacies}
\label{sec:synthesis}
%======================================================================

\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{p{2.8cm}p{3.2cm}p{4cm}p{2cm}}
\toprule
\textbf{Observation} & \textbf{$Q_3$ source} & \textbf{Formula} & \textbf{Status} \\
\midrule
Tesla's ``3'' & Dimension & $D = 3$ & THEOREM \\
Tesla's ``6'' & Faces & $f_2 = 2D = 6$ & THEOREM \\
Tesla's ``9'' & Parities & $D^2 = 9$ & THEOREM \\
Tesla's Wardenclyffe & 8-tick neutrality & $C = \cos(2\pi\Delta\Phi)\cdot\phig^{-|\Delta k|}$ & CONDITIONAL \\
\midrule
Ramanujan's 24 & Directed edges & $2 \times 12 = 24$ & THEOREM \\
Ramanujan's 11 & Passive edges & $12 - 1 = 11$ & THEOREM \\
$\tau(2) = -24$ & Directed flux & $-\vec{f}_1(Q_3)$ & THEOREM \\
$396 = 4 \times 9 \times 11$ & Passive edge factor & $E_{\mathrm{passive}} = 11$ & THEOREM \\
$9801 = (9 \times 11)^2$ & Passive + parity & $D^2 \times E_{\mathrm{passive}}$ & THEOREM \\
$|\tau(p)| \le 2p^{11/2}$ & Weight $12 = f_1$ & $(f_1 - 1)/2$ & HYPOTHESIS \\
Rogers--Ramanujan & $\phig$-ladder stability & $\phig^n + \phig^{n+1} = \phig^{n+2}$ & THEOREM \\
Zeckendorf & $\phig$-ladder stability & Non-consecutive $=$ stable & THEOREM \\
Continued fracs $\to \phig$ & Fixed point & $\phig = 1 + 1/\phig$ & THEOREM \\
Mock theta orders & 8-tick coprimality & $\gcd(\{3,5,7\}, 8) = 1$ & THEOREM \\
Mock $\to$ Maass & Phantom Light & Window + shadow $=$ balanced & HYPOTHESIS \\
Primes $\{5,7,11\}$ & $Q_3$ structural triple & Disc., modes, passive & HYPOTHESIS \\
\bottomrule
\end{tabular}
\end{center}

\begin{remark}
The ``THEOREM'' entries are arithmetic consequences of $D = 3$ and the double-entry axiom; they are machine-verified with zero \texttt{sorry} instances.
The ``CONDITIONAL'' entry depends on the GCIC hypothesis.
The ``HYPOTHESIS'' entries are structural correspondences that require additional work to elevate to theorems.
\end{remark}

%======================================================================
\section{Discussion}
\label{sec:discussion}
%======================================================================

The central finding of this paper is that Tesla's and Ramanujan's apparently unrelated numerical intuitions share a common origin: the combinatorial structure of the three-dimensional hypercube $Q_3$ under a double-entry ledger constraint.
This is not a claim that Tesla or Ramanujan were ``right'' in any scientific sense---neither provided derivations---but rather that their specific numerical observations are \emph{natural outputs} of a structure that arises independently from the dimensional selection $D = 3$.

\textbf{1. No free parameters.}
The integers $3, 6, 8, 9, 11, 12, 24$ are all arithmetic functions of $D = 3$ and the double-entry axiom.
The golden ratio $\phig$ is the unique positive root of $x^2 = x + 1$, forced by self-similarity in the discrete ledger.
No numerical input beyond $D = 3$ is used.

\textbf{2. Falsifiability.}
Each interpretive claim is equipped with explicit falsification criteria.
The most consequential are: (i) the mock theta correspondence (Section~\ref{sec:mock}) would be falsified by exhibiting a shadow structure that cannot be expressed in 8-tick neutrality terms, and (ii) the Wardenclyffe mechanism (Section~\ref{sec:wardenclyffe}) would be falsified by the absence of $\phig$-spaced frequency coherence in controlled EEG experiments.

\textbf{3. Machine verification.}
All arithmetic claims and the core $\phig$-ladder stability theorems are compiled and verified in Lean~4 using Mathlib.
The verification modules compile with zero \texttt{sorry} instances and zero axioms beyond Mathlib's standard foundation.

\textbf{4. What remains open.}
The deepest question is whether the directed flux count $\vec{f}_1(Q_3) = 24$ is \emph{structurally} (not just numerically) related to the exponent in $\Delta(\tau) = \eta(\tau)^{24}$.
A positive answer would require constructing a $Q_3$ partition function whose modular properties reproduce $\Delta(\tau)$.

%======================================================================
\section{Conclusion}
\label{sec:conclusion}
%======================================================================

Tesla's triple $(3, 6, 9)$ and Ramanujan's recurrent integers $(11, 24, \phig)$ are not numerology.
They are the topological coordinates, edge combinatorics, and algebraic fixed point of the three-dimensional hypercube $Q_3$---the unit cell of any discrete three-dimensional recognition lattice.
The dimensional selection $D = 3$, established independently by topological, dynamical, and geometric constraints \cite{PardoGuerraThapa2026}, generates the entire cascade.

Two of the most creative minds in the history of science and mathematics intuited fragments of this structure without possessing the framework to derive it.
That their observations converge on the same combinatorial object---separated by decades, disciplines, and continents---is, at minimum, a remarkable coincidence.
Within the Recognition Geometry framework, it is an expected consequence of the fact that $D = 3$ is forced, and its combinatorial consequences are determinate.

%======================================================================
\appendix
\section{Lean Verification Map}
\label{app:lean}
%======================================================================

Each claim is mapped to an exact Lean symbol.
Referee verification: \texttt{lake build IndisputableMonolith.Mathematics.RamanujanBridge}.

\begin{center}
\small
\begin{tabular}{@{}lll@{}}
\toprule
\textbf{Paper Ref} & \textbf{Lean Symbol} & \textbf{Status} \\
\midrule
Thm~\ref{thm:tesla}
  & \texttt{Tesla369.tesla\_triple\_from\_D3}
  & \textsc{Theorem} \\
Nine parities
  & \texttt{Foundation.NineParities.nine\_parities\_master}
  & \textsc{Theorem} \\
\midrule
Thm~\ref{thm:collapse}
  & \texttt{PhiLadderStability.adjacent\_collapses}
  & \textsc{Theorem} \\
Thm~\ref{thm:jcost-positive}
  & \texttt{PhiLadderStability.adjacent\_Jcost\_positive}
  & \textsc{Theorem} \\
Thm~\ref{thm:coherence-cost}
  & \texttt{PhiLadderStability.coherence\_cost\_aperiodicity}
  & \textsc{Theorem} \\
RR Stability
  & \texttt{PhiLadderStability.rogers\_ramanujan\_stability}
  & \textsc{Theorem} \\
Zeckendorf
  & \texttt{ZeckendorfJCost.fibonacci\_lattice\_is\_complete}
  & \textsc{Theorem} \\
\midrule
Thm~\ref{thm:24}
  & \texttt{DirectedFlux24.directed\_edges\_Q3}
  & \textsc{Theorem} \\
Cor~\ref{cor:deligne}
  & \texttt{DirectedFlux24.ramanujan\_deligne\_exponent}
  & \textsc{Theorem} \\
\midrule
Thm~\ref{thm:coprime}
  & \texttt{MockThetaPhantom.mock\_orders\_coprime\_to\_8}
  & \textsc{Theorem} \\
Thm~\ref{thm:mock-defect}
  & \texttt{MockThetaPhantom.coprime\_order\_forces\_mock\_defect}
  & \textsc{Theorem} \\
Hyp~\ref{hyp:phantom}
  & \texttt{MockThetaPhantom.MockThetaPhantomCorrespondence}
  & \textsc{Hypothesis} \\
Thm~\ref{thm:two-time}
  & \texttt{MockThetaPhantom.two\_time\_unique}
  & \textsc{Theorem} \\
\midrule
Thm~\ref{thm:396}
  & \texttt{RamanujanPiFactors.factor\_11\_in\_396}
  & \textsc{Theorem} \\
Thm~\ref{thm:9801}
  & \texttt{RamanujanPiFactors.nine801\_eq\_9\_times\_11\_sq}
  & \textsc{Theorem} \\
Thm~\ref{thm:1103}
  & \texttt{RamanujanPiFactors.one103\_is\_prime}
  & \textsc{Theorem} \\
\midrule
$\phig = 1 + 1/\phig$
  & \texttt{ContinuedFractionPhi.phi\_continued\_fraction\_eq}
  & \textsc{Theorem} \\
Fixed point
  & \texttt{ContinuedFractionPhi.phi\_is\_cfrac\_fixed\_point}
  & \textsc{Theorem} \\
Uniqueness
  & \texttt{ContinuedFractionPhi.sequential\_optimization\_forces\_phi\_strong}
  & \textsc{Theorem} \\
\bottomrule
\end{tabular}
\end{center}

%======================================================================
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%======================================================================

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\end{document}
