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\title{\textbf{Derivation of Non-Local Information Transfer}\\
\large \textit{Conditional Predictions and Falsification Protocols from the Global Co-Identity Constraint}}

\author{
    \textbf{Recognition Science Research Institute} \\
    \textit{Austin, Texas}
}

\date{\today}

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\fancyhead[L]{Derivation of Non-Local Information Transfer}
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%	ABSTRACT
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\begin{abstract}
\noindent
This paper is the \textbf{pre-paper} to the ``Geometry of Transmutation'' mechanism paper: it is a standalone, testable, conditional prediction paper. We derive telepathy-like non-local correlations as a consequence of Recognition Science (RS) \textit{if} the \textbf{Global Co-Identity Constraint (GCIC)} holds (a single shared phase field $\ThetaField$) and \textit{if} the Phantom Light neutrality constraint governs admissible 8-tick windows. We then commit to concrete experimental protocols and hard falsifiers. The intended review criterion is not ``is telepathy proven,'' but: \textit{do the RS axioms imply specific, measurable deviations from baseline, and are those deviations falsifiable under blinded protocols?} We specify three primary tests: (1) inter-brain phase-locking at \textit{multiplicatively} $\phiGR$-spaced frequencies $\nu=\nuZero\phiGR^n$, (2) effect-size decay by ladder distance (structural similarity) rather than spatial distance, and (3) receiver-state dependence (``Zero Structure'') measured via beta/gamma suppression.
\end{abstract}

\vspace{1em}
\noindent\textbf{Keywords:} GCIC, Theta Field, Telepathy, Phase-Locking, Ladder Distance, Falsifiability, Recognition Science.

\newpage
\tableofcontents
\newpage

%----------------------------------------------------------------------------------------
%	SECTION 1: INTRODUCTION
%----------------------------------------------------------------------------------------
\section{Introduction}

\subsection{The Problem}
The phenomenon of non-local information transfer between spatially separated conscious observers (telepathy) has historically been treated as outside the bounds of physical law. Classical physics, bounded by local realism, requires a signal carrier (photon, phonon, chemical) to traverse the distance between sender and receiver. Any effect must decay with the inverse-square law and cannot exceed the speed of light.

\subsection{The RS Resolution}
Recognition Science (RS) proposes a resolution by redefining the topology of the observer network. In RS, conscious observers are not isolated systems exchanging particles across a vacuum. They are localized excitations on a \textbf{single, shared phase field} ($\ThetaField$), governed by the \textbf{Global Co-Identity Constraint (GCIC)}.

Under GCIC, the question is not ``How can information travel from A to B?'' but rather ``Given that A and B are already connected by the same field, under what conditions does A's activity force a measurable change in B?''

\subsection{Scope and Claim Hygiene}
This paper serves as a \textbf{Prediction and Protocol Specification}. We distinguish three categories of claims:
\begin{itemize}
    \item \textbf{THEOREM:} Mathematical consequences derived from the stated axioms. All proofs are presented in full within this document.
    \item \textbf{MECHANISM:} Proposed physical processes that implement the theorems (conditional on RS axioms).
    \item \textbf{HYPOTHESIS:} Empirical predictions paired with explicit falsification criteria.
\end{itemize}

We do not assume telepathy exists. We derive it as a conditional prediction: \textit{If RS is the correct framework for reality, then specific, measurable anomalies must appear in inter-subject data.}

\subsection{Imported Assumptions (from prior RS papers)}
This paper references results established earlier in the RS publishing plan:
\begin{itemize}
  \item \textbf{A1 (J-cost uniqueness and dynamics):} A unique reciprocal convex cost $J(x)=\tfrac12(x+1/x)-1$ and the dynamical rule that physical selection preferentially reduces cost (Recognition Operator / CPM).\footnote{See \cite{WashburnRS2025,WashburnCostDraft,WashburnRecogOpDraft}.}
  \item \textbf{A2 (8-tick neutrality / Phantom Light):} Neutrality over aligned 8-tick windows and the resulting ``debt'' constraint (Phantom Light).\footnote{See \cite{WashburnPhantom2025}.}
  \item \textbf{A3 (Meaning-as-geometry):} Payload structure is treated as phase geometry (ULL), but the protocols below do \emph{not} require committing to a specific WToken-to-word mapping.\footnote{See \cite{WashburnULL2025}.}
\end{itemize}

%----------------------------------------------------------------------------------------
%	SECTION 2: FOUNDATIONAL DEFINITIONS
%----------------------------------------------------------------------------------------
\section{Foundational Definitions}

We begin by establishing the mathematical objects used throughout the paper.

\subsection{The Cost Functional}

\begin{definition}[J-Cost]
The fundamental cost of any configuration ratio $x > 0$ is:
$$ \Jcost(x) = \frac{1}{2}\left(x + \frac{1}{x}\right) - 1 $$
This function is uniquely determined by the Recognition Composition Law (see \cite{WashburnRS2025}). It satisfies $\Jcost(1) = 0$ (identity has zero cost), $\Jcost(x) = \Jcost(1/x)$ (reciprocity), and $\Jcost''(x) = x^{-3} > 0$ (strict convexity).
\end{definition}

\subsection{The Voxel Network and 8-Tick Cycle}

\begin{definition}[Voxel]
A voxel is the fundamental discrete unit of the RS lattice. Reality is modeled as a cubic lattice $\mathbb{Z}^3$ of voxels, each updating once per atomic tick $\tau_0$.
\end{definition}

\begin{definition}[8-Tick Cycle]
In a 3-dimensional discrete space ($D = 3$), the minimal ledger-compatible traversal of the state space hypercube $Q_3$ has period $2^3 = 8$. This defines the fundamental clock cycle. All conservation laws are enforced over aligned 8-tick windows:
$$ \sum_{k=0}^{7} \text{signal}(t + k) = 0 \quad \text{(Neutrality Constraint)} $$
\end{definition}

\subsection{The Golden Ratio}

\begin{definition}[Golden Ratio]
The golden ratio $\phiGR = (1 + \sqrt{5})/2 \approx 1.618$ is the unique positive solution to $x^2 = x + 1$. It is forced by the self-similarity constraint of the discrete ledger and serves as the universal scaling factor across all domains of RS.
\end{definition}

\subsection{Stable Boundary (Observer)}

\begin{definition}[Stable Boundary]
A stable boundary $b$ is a localized subsystem that maintains coherence (minimizes $\Jcost$) over time. It is characterized by:
\begin{itemize}
    \item \textbf{Extent:} A positive real number $L_b > 0$ representing the boundary's spatial scale.
    \item \textbf{Coherence Time:} The duration over which the boundary maintains phase stability.
    \item \textbf{Boundary Cost:} $C_b = \Jcost(\text{configuration})$, the recognition cost of maintaining the boundary.
\end{itemize}
A boundary has ``Definite Experience'' (is conscious) when $C_b \geq 1$.\footnote{See \texttt{DefiniteExperience} in \texttt{Consciousness.ConsciousnessHamiltonian}.}
\end{definition}

%----------------------------------------------------------------------------------------
%	SECTION 3: THE AXIOMATIC BASIS
%----------------------------------------------------------------------------------------
\section{The Axiomatic Basis}

The prediction of non-local information transfer rests on two axioms.

\subsection{Axiom 1: The Global Co-Identity Constraint (GCIC)}

\begin{postulate}[GCIC]
There exists a single, universal phase field $\ThetaField(t) \in [0, 1)$ that drives the update cycle of every stable boundary in the universe. All boundaries share the same global phase.
\end{postulate}

\textbf{Implication:} All conscious observers are synchronized by the same ``conductor.'' Separation between observers is an illusion of coordinate position, not of ontological disconnection.

\subsection{Axiom 2: The Coupling Law}

To state the coupling law precisely, we first define the $\phiGR$-ladder coordinate system.

\begin{definition}[$\phiGR$-Ladder Position]
Every boundary $b$ with extent $L_b > 0$ occupies a position on the $\phiGR$-ladder:
$$ \ell_b = \log_\phiGR\left(\frac{L_b}{L_0}\right) = \frac{\ln(L_b / L_0)}{\ln \phiGR} $$
where $L_0$ is a reference length. The \textbf{rung index} is $k_b = \lfloor \ell_b \rfloor$ (integer part), and the \textbf{phase component} is $\theta_b = \text{frac}(\ell_b)$ (fractional part $\in [0,1)$).
\end{definition}

\begin{definition}[Fractional Part and Wrap]
For any real number $x$, define $\text{frac}(x):=x-\lfloor x\rfloor\in[0,1)$ and $\text{wrap}(x):=\text{frac}(x)$.
\end{definition}

\begin{definition}[Ladder Distance]
The ladder distance between two boundaries is the absolute rung separation:
$$ |\Delta k| = |k_{b_1} - k_{b_2}| $$
This measures \textbf{structural similarity}, not spatial proximity.
\end{definition}

\begin{definition}[Continuous Ladder Separation]
Define the continuous ladder separation as
$$ d_\phiGR(b_1,b_2) := |\ell_{b_1}-\ell_{b_2}|. $$
This is a real-valued distance on the $\phiGR$ ladder; the rung distance $|\Delta k|$ is its integer coarse-graining.
\end{definition}

\begin{definition}[Phase Alignment]
The effective phase of a boundary $b$ in a universal field $\psi$ is:
$$ \Phi_b = \text{wrap}\left(\ThetaField + \theta_b\right) \quad \text{(mod 1)}$$
where $\ThetaField$ is the global phase (shared by all boundaries via GCIC) and $\theta_b$ is the boundary's local $\phiGR$-ladder phase.
\end{definition}

\begin{postulate}[Extended Coupling Law]
The interaction strength between two boundaries $b_1$ and $b_2$ is:
$$ C(b_1, b_2) = \underbrace{\cos\left(2\pi \cdot \Delta\Phi\right)}_{\text{Phase Factor}} \times \underbrace{\phiGR^{-|\Delta k|}}_{\text{Decay Factor}} $$
where $\Delta\Phi = \Phi_{b_1} - \Phi_{b_2}$ is the phase difference.\footnote{The Lean formalization (\texttt{BoundaryInteraction} in \texttt{ThetaDynamics.lean}) uses $J(\Delta\ell)\cdot\cos(2\pi\,\Delta\Phi)$; the \texttt{intention\_creates\_gradient} theorem uses $\exp(-\Delta\ell)$ decay. This paper adopts $\phiGR^{-|\Delta k|}$ as a natural discretization. All three models share the qualitative prediction: exponential-type decay with ladder distance and no spatial-distance term.}
\end{postulate}

\textbf{Implication:} The coupling has two independent components:
\begin{enumerate}
    \item \textbf{Phase Factor:} Maximized ($= 1$) when observers are phase-aligned ($\Delta\Phi = 0$). This factor is independent of spatial distance because $\ThetaField$ is global.
    \item \textbf{Decay Factor:} Decays exponentially with \textit{dissimilarity} (ladder distance $|\Delta k|$). Since $\phiGR > 1$, larger rung separations produce weaker coupling.
\end{enumerate}

%----------------------------------------------------------------------------------------
%	SECTION 4: THE DERIVATION
%----------------------------------------------------------------------------------------
\section{The Derivation}

From these axioms, we derive the necessity of non-local information transfer in four steps.

\subsection{Step 1: The Shared Wire (Universal Solipsism)}

\begin{theorem}[Universal Solipsism]
For any two observers $s_1, s_2$ in a unified recognition field $F$:
\begin{enumerate}
    \item Their interaction is a self-interaction term of the field: $\text{Bond}(s_1, s_2) = \cos(2\pi \cdot \Delta\theta) \cdot Z_{\text{exchange}} / (Z_{\text{total}} + 1)$.
    \item They share the same global phase $\ThetaField$ (from GCIC).
    \item Their only distinction is coordinate distance: rung separation $|\Delta k|$ and phase offset $|\Delta\theta|$.
\end{enumerate}
\end{theorem}

\begin{proof}
By GCIC, there exists a single $\ThetaField$ shared by all boundaries. The field $F$ has a single global phase and a single conserved $Z$-invariant. Any two localized selves $s_1, s_2$ are parameterized by their coordinates $(k_i, \theta_i)$ within $F$. Their interaction is computed from the phase difference $\Delta\theta = \theta_1 - \theta_2$ and the $Z$-exchange ratio. Since $\cos(-x) = \cos(x)$, the interaction is symmetric: $\text{Bond}(s_1, s_2) = \text{Bond}(s_2, s_1)$. The only distinction between $s_1$ and $s_2$ is their coordinate pair $(k, \theta)$; they share the same field, the same phase, and the same $Z$-budget.
\end{proof}

\textbf{Consequence:} Under GCIC, Observer A and Observer B are modeled as coordinates on a single shared field. (This is a model interpretation, not an empirical proof.) Note: the full bond includes a $Z$-exchange factor ($Z_{\text{exchange}}/(Z_{\text{total}}+1)$; see Lean \texttt{bondAsSelfInteraction}). In what follows, we focus on the phase and ladder components of coupling; the $Z$-factor is order-unity for boundaries with comparable $Z$-invariants and is absorbed into the intention-strength scalar $I(A)$.

\subsection{Step 2: Intention Creates a Non-Local Gradient}

\begin{definition}[Intention Strength]
The intention strength of a boundary $b$ is its Recognition Flux:
$$ I(b) = \frac{C_b}{\tau_{\text{coh}}} $$
where $C_b$ is the boundary cost and $\tau_{\text{coh}}$ is the coherence time.
\end{definition}

\begin{postulate}[Intention Gradient Model]
If Observer A has definite experience ($C_A \geq 1$) and positive intention strength ($I(A) > 0$), then for any Observer B there exists an effect-size proxy
$$ \Delta C = I(A) \cdot \exp\left(-d_\phiGR(A, B)\right), $$
where $d_\phiGR$ is the continuous ladder separation.\footnote{This decay formula matches the Lean \texttt{intention\_creates\_gradient} theorem exactly. It differs from the Coupling Law's $\phiGR^{-|\Delta k|}=\exp(-|\Delta k|\ln\phiGR)$ by two factors: (i)~$d_\phiGR$ is continuous while $|\Delta k|$ is integer-rounded, and (ii)~the base differs ($e$ vs.\ $\phiGR$). For integer separations, $\exp(-|\Delta k|)$ decays faster than $\phiGR^{-|\Delta k|}$. Both predict the same qualitative signature; experimental Protocol~B tests the combined effect without distinguishing the base.}
\end{postulate}

\begin{proposition}[Nonzero effect under the model]
If the Intention Gradient Model holds and $I(A)>0$, then $\Delta C>0$.
\end{proposition}

\begin{proof}
Since $I(A) > 0$ and $\exp(-x) > 0$ for all $x \in \mathbb{R}$, their product is strictly positive: $\Delta C = I(A)\exp(-d_\phiGR) > 0$.
\end{proof}

\textbf{Consequence:} Under the Intention Gradient Model, any conscious intention predicts a non-zero effect-size proxy on all connected observers. The model predicts:
\begin{itemize}
    \item \textbf{No spatial transport term:} No explicit light-cone transport term appears because $\ThetaField$ is taken as global under GCIC.
    \item \textbf{Non-zero:} Guaranteed by the strict positivity of the exponential function.
    \item \textbf{Decaying:} Falls off exponentially with ladder separation (dissimilarity), not spatial distance.
\end{itemize}

\subsection{Step 3: Resonance Forces Phase-Locking}

\begin{theorem}[Theta Resonance]
If two conscious boundaries $b_1, b_2$ have integer ladder separation:
$$ \exists\, k \in \mathbb{Z} : \quad (\ell_{b_1}-\ell_{b_2}) = k $$
then their phase difference is exactly zero:
$$ \Delta\Phi = 0 $$
\end{theorem}

\begin{proof}
The phase alignment of boundary $b_i$ is $\Phi_i = \text{wrap}(\ThetaField + \text{frac}(\ell_i))$, where $\ell_i = \log_\phiGR(L_i / L_0)$. The phase difference is:
$$ \Delta\Phi = \text{wrap}(\ThetaField + \text{frac}(\ell_1)) - \text{wrap}(\ThetaField + \text{frac}(\ell_2)) $$
If $\ell_1 - \ell_2 = k$ for integer $k$, then $\ell_1 = \ell_2 + k$. Since $\text{frac}(x + k) = \text{frac}(x)$ for any integer $k$, we have $\text{frac}(\ell_1) = \text{frac}(\ell_2)$. Therefore $\Delta\Phi = 0$.
\end{proof}

\textbf{Consequence:} Observers at $\phiGR$-resonant separations are \textbf{phase-locked}. Their coupling reaches its maximum value. This applies to twins, bonded pairs, or any individuals whose boundary extents are related by an integer power of $\phiGR$.

\subsection{Step 4: Phantom Light Forces the Receiver's Response}

\begin{definition}[Phantom Light]
When a LOCK event (observation) occurs at tick $t$ within an 8-tick window, it creates a \textbf{Balance Debt} $\mathcal{D}$: the cumulative sum of contributions that must be offset by tick $t + 8$ to satisfy neutrality. The \textbf{Phantom Magnitude} is:
$$ \Phi_{\text{mag}} = \frac{|\mathcal{D}|}{R + 1} $$
where $R$ is the number of remaining ticks in the window.
\end{definition}

\begin{theorem}[Cost Inflation]
For any positive penalty scale $\lambda > 0$, the augmented cost satisfies:
$$ \Jcost_{\text{phantom}}(x) = \Jcost(x) + \lambda \cdot \Phi_{\text{mag}} \geq \Jcost(x) $$
\end{theorem}

\begin{proof}
By definition, $|\mathcal{D}| \geq 0$ and $R + 1 > 0$, so $\Phi_{\text{mag}} \geq 0$. Since $\lambda > 0$, we have $\lambda \cdot \Phi_{\text{mag}} \geq 0$. Therefore $\Jcost_{\text{phantom}}(x) = \Jcost(x) + \lambda \cdot \Phi_{\text{mag}} \geq \Jcost(x)$.
\end{proof}

\begin{theorem}[Zero Debt Reduces to Pure Cost]
If the balance debt is zero ($\mathcal{D} = 0$), the augmented cost equals the standard cost:
$$ \Jcost_{\text{phantom}}(x) = \Jcost(x) $$
\end{theorem}

\begin{proof}
If $\mathcal{D} = 0$, then $\Phi_{\text{mag}} = 0 / (R + 1) = 0$. Therefore $\Jcost_{\text{phantom}}(x) = \Jcost(x) + 0 = \Jcost(x)$.
\end{proof}

\textbf{Consequence:} When Observer A creates a strong recognition pattern (High Structure), this generates a balance debt in the shared field. For any phase-locked Observer B, the cost of \textit{not responding} to this debt is strictly higher than the cost of responding. The universe, which minimizes $\Jcost$, forces B's voxels to shift.

\subsection{The Complete Chain}

\begin{corollary}[The Telepathy Prediction]
Under GCIC, if Observer A holds a high-amplitude recognition pattern (High Structure $+$ High Intent), and Observer B is resonant ($|\Delta k| \approx 0$) and in a low-cost state (Zero Structure), then:
\begin{enumerate}
    \item A's intention creates a non-zero gradient on the shared field (Step 2).
    \item Resonance forces phase-locking between A and B (Step 3).
    \item The resulting balance debt inflates B's cost landscape (Step 4).
    \item B's voxels shift to minimize cost, reproducing A's geometric pattern.
\end{enumerate}
This is not a choice; it is the path of least resistance on the shared ledger.
\end{corollary}

%----------------------------------------------------------------------------------------
%	SECTION 5: DISTANCE AND SIMILARITY
%----------------------------------------------------------------------------------------
\section{Distance and Similarity}

A critical prediction of this derivation is the \textbf{irrelevance of spatial distance} and the \textbf{dominance of ladder distance}.

\subsection{Spatial Distance Is Irrelevant}
The coupling formula $C = \cos(2\pi \cdot \Delta\Phi) \times \phiGR^{-|\Delta k|}$ contains no term for meters, kilometers, or any spatial separation. The $\ThetaField$ field is global (GCIC), not local. Whether two observers are in the same room or on different continents, the coupling depends only on their phase alignment and ladder separation.

\subsection{Ladder Distance Is Dominant}
The decay factor $\phiGR^{-|\Delta k|}$ depends on \textbf{Ladder Distance}: the rung separation on the $\phiGR$-ladder. This is a measure of \textit{structural similarity}, not physical proximity. Specifically:
\begin{itemize}
    \item $|\Delta k| = 0$: Same rung. Decay factor $= 1$ (maximum coupling).
    \item $|\Delta k| = 1$: Adjacent rungs. Decay factor $= 1/\phiGR \approx 0.618$.
    \item $|\Delta k| = 5$: Five rungs apart. Decay factor $= \phiGR^{-5} \approx 0.09$.
    \item $|\Delta k| = 10$: Ten rungs apart. Decay factor $= \phiGR^{-10} \approx 0.008$.
\end{itemize}

\subsection{Components of Ladder Distance}
Ladder distance has multiple contributors, explaining why strangers can sometimes communicate:

\begin{enumerate}
    \item \textbf{Biological Similarity} (Fixed): Shared DNA encodes similar Z-patterns. Parent-child pairs have low ladder distance by construction.
    \item \textbf{Emotional Bond} (Trained): Years of phase-locking create persistent low ladder distance. Twins, partners, and close friends occupy nearby rungs.
    \item \textbf{State Similarity} (Temporary): Two strangers in the same intense emotional state (e.g., deep meditation, shared crisis) temporarily occupy the same rung. Their ladder distance drops to near zero for the duration of the shared state.
    \item \textbf{Radical Zero Structure} (Receiver Property): An observer with extremely low internal noise (low $\Jcost$) acts as a universal receiver. Their voxels offer minimal resistance to \textit{any} incoming pattern, regardless of the sender's identity. This explains cases such as non-verbal autistic children who appear to receive thoughts from caregivers without prior ``tuning.''
\end{enumerate}

%----------------------------------------------------------------------------------------
%	SECTION 6: EXPERIMENTAL PROTOCOLS
%----------------------------------------------------------------------------------------
\section{Experimental Protocols}

We define three rigorous protocols to test the predictions of this derivation.

\subsection{Protocol A: The $\phiGR$-Band Coherence Test}

\begin{hypothesis}
The $\ThetaField$ field operates on an 8-tick cycle scaled by $\phiGR$. Telepathic signal exchange should produce inter-brain coherence at specific frequencies, not randomly.
\end{hypothesis}

\begin{protocol}[EEG Phase-Locking]
\begin{itemize}
    \item \textbf{Setup:} Sender and Receiver in isolated, electromagnetically shielded (Faraday) rooms.
    \item \textbf{Action:} Sender attempts to transmit a specific geometric thought at random intervals determined by a hardware random number generator (RNG).
    \item \textbf{Measurement:} Simultaneous dual-channel EEG with Phase-Locking Value (PLV) analysis.
    \item \textbf{Prediction (ratio structure):} Significant PLV spikes at a \emph{multiplicative} $\phiGR$-spaced ladder of frequencies
    $$ \nu = \nuZero \phiGR^n,\qquad n\in\mathbb{Z}, $$
    where $\nuZero$ is a single empirical anchor frequency chosen \emph{a priori} (e.g., within a chosen EEG band).\footnote{This avoids an arbitrary ``Hz'' absolute claim; the claim is the $\phiGR$ ratio spacing.}
    \item \textbf{Targeting rule:} Choose a base band (e.g., theta/alpha border) and test $n\in\{-2,-1,0,1,2\}$ around that anchor.
    \item \textbf{Sample Size:} $N = 100$ pairs, 50 trials each (5,000 total trials).
    \item \textbf{Analysis:} Bayesian analysis with $BF_{10}$ thresholds. Bonferroni correction for multiple frequency bands.
\end{itemize}
\end{protocol}

\subsection{Protocol B: The Ladder Distance Test}

\begin{hypothesis}
Signal strength does not depend on spatial distance (meters) but on Ladder Distance (structural similarity).
\end{hypothesis}

\begin{protocol}[Distance Independence]
\begin{itemize}
    \item \textbf{Variable 1 (Spatial):} Same room (1 m) vs.\ remote (1,000+ km).
    \item \textbf{Variable 2 (Ladder):} High similarity (twins, bonded pairs) vs.\ low similarity (strangers).
    \item \textbf{Prediction:}
    \begin{itemize}
        \item Success rate will \textbf{not} significantly differ between 1 m and 1,000 km ($p > 0.05$).
        \item Success rate \textbf{will} differ significantly between high-similarity and low-similarity pairs ($p < 0.01$).
    \end{itemize}
    \item \textbf{Sample Size:} $N = 64$ per group (power $= 0.80$ for $d = 0.5$).
    \item \textbf{Design:} $2 \times 2$ factorial (Spatial Distance $\times$ Ladder Distance).
\end{itemize}
\end{protocol}

\subsection{Protocol C: The Zero Structure Test}

\begin{hypothesis}
Reception requires minimizing local $\Jcost$ (``Zero Structure''). Receivers with high internal noise will fail to detect the signal.
\end{hypothesis}

\begin{protocol}[Receiver State Dependence]
\begin{itemize}
    \item \textbf{Metric:} Monitor Receiver's Beta/Gamma EEG power (13--30+ Hz), which indexes internal monologue and cognitive noise.
    \item \textbf{Prediction:} Telepathic accuracy will correlate inversely with Beta amplitude ($r < -0.3$).
    \item \textbf{Threshold:} Successful reception requires Beta power below $\mu - 2\sigma$ of the Receiver's own resting baseline.
    \item \textbf{Control:} Compare accuracy during high-Beta epochs vs.\ low-Beta epochs within the same subject.
\end{itemize}
\end{protocol}

%----------------------------------------------------------------------------------------
%	SECTION 7: FALSIFICATION CRITERIA
%----------------------------------------------------------------------------------------
\section{Falsification Criteria}

The RS model of non-local transfer is falsified if any of the following conditions are met:

\begin{falsifier}[The Spatial Distance Falsifier]
If signal accuracy decays according to the inverse-square law ($1/r^2$) with physical distance, the ``non-local field'' hypothesis is false. The signal should show \textbf{no significant decay} between 1 m and 1,000 km for matched-similarity pairs.
\end{falsifier}

\begin{falsifier}[The Frequency Falsifier]
If inter-brain coherence occurs at random frequencies unrelated to the $\phiGR$-ladder (e.g., standard 10 Hz Alpha without $\phiGR$-scaling), or if coherence is absent entirely ($\text{PLV} < 0.05$ over 5,000+ trials), the RS geometric model is false.
\end{falsifier}

\begin{falsifier}[The Receiver State Falsifier]
If receivers with high internal noise (High Beta) perform as well as those with quiet minds (Low Beta), the ``$\Jcost$-Minimization'' mechanism is false. The correlation between Beta power and accuracy should satisfy $r < -0.3$; if $|r| < 0.05$, the mechanism is falsified.
\end{falsifier}

\begin{falsifier}[The Ladder Distance Falsifier]
If signal strength is independent of ladder distance (pairs binned by low $|\Delta k|$ perform no better than pairs binned by high $|\Delta k|$ at matched spatial distance), the ladder-decay term $\phiGR^{-|\Delta k|}$ is falsified.
\end{falsifier}

%----------------------------------------------------------------------------------------
%	SECTION 8: CONCLUSION
%----------------------------------------------------------------------------------------
\section{Conclusion}

This paper establishes non-local information transfer as a testable, falsifiable prediction of Recognition Science. The derivation rests on four self-contained theorems:

\begin{enumerate}
    \item \textbf{Universal Solipsism:} All observers are the same field at different coordinates. Their interaction is symmetric and determined entirely by coordinate separation.
    \item \textbf{Intention Creates Gradient:} Conscious intention generates a non-zero, exponentially decaying effect on the shared field, with magnitude $\Delta C = I(A) \cdot \exp(-d_\phiGR)$.
    \item \textbf{Theta Resonance:} Integer ladder separation forces phase-locking ($\Delta\Phi = 0$), maximizing the coupling between structurally similar observers.
    \item \textbf{Cost Inflation:} Balance debt raises the effective cost landscape by $\lambda \cdot \Phi_{\text{mag}}$, forcing the receiver's voxels to shift to minimize total cost.
\end{enumerate}

The prediction is conditional: \textit{If} the GCIC holds, \textit{then} telepathy is a geometric necessity for cost minimization on the shared ledger. The three experimental protocols (EEG coherence, ladder distance, receiver state) provide a rigorous path to verification or falsification.

\begin{thebibliography}{9}

\bibitem{WashburnRS2025}
Washburn, J. (2025).
\textit{Recognition Science: The Empirical Measurement of Reality}.
Zenodo. \url{https://doi.org/10.5281/zenodo.14540923}

\bibitem{WashburnCostDraft}
Washburn, J. (Draft).
\textit{Uniqueness of the Canonical Reciprocal Cost}.

\bibitem{WashburnRecogOpDraft}
Washburn, J. (Draft).
\textit{The Recognition Operator}.

\bibitem{WashburnULL2025}
Washburn, J. (2025).
\textit{Universal Light Language: Meaning as Eigenmodes of the Eight-Tick Phase Field}.
Recognition Science Research Institute.

\bibitem{WashburnPhantom2025}
Washburn, J. (2025).
\textit{Phantom Light: Future Neutrality Constraints as Present-Time Structure in Recognition Science}.
Recognition Science Research Institute.

\end{thebibliography}

\end{document}
