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\begin{tcolorbox}[defbox, title={Core Definition: Existence}] \[ \boxed{\;\ES(X_t) \;=\; \ST(X_t) \;\times\; \GR(X_t) \;>\; 0\;} \] where $\ST(X_t) > 0$ denotes the system's capacity to maintain structural integrity under perturbation at time $t$, and $\GR(X_t) > 0$ denotes the system's capacity to expand the boundar...
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\textbf{Why $\times$ and not $+$?} Four structural reasons: \begin{enumerate}[leftmargin=1.6em,itemsep=4pt] \item \textbf{No compensation (hard threshold at zero).} Under addition, $\ST = 0$ and $\GR = 10$ yields $\ES = 10$ --- the system ``exists'' despite having lost all structural integrity. This is empir...
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\item \textbf{Mutual dependency (coupling structure).} $\ST$ is the necessary condition for $\GR$ to have a substrate to act on. $\GR$ is the necessary condition for $\ST$ to persist through a changing environment (static stability is brittle). This mutual dependency is a multiplicative, not additive, r...
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\item \textbf{Consistency with TOB.} In the ontological cycle of TOB, \Dung\ (stabilization) and the \Sinh--\Dan\ arc (emergence and propagation) must co-occur for a \textit{tu niem} (cognitive segmentation) to persist in a Frame. If only \Dung\ operates without propagation, the pattern dies. If only pr...
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\item \textbf{Consistency with Box--Sphere logic.} Stability maps to \textbf{Box logic} (linear, positional, zero-sum, boundary-preserving). Growth maps to \textbf{Sphere logic} (non-linear, resonant, non-zero-sum, boundary-expanding). Box and Sphere are orthogonal operations; their joint action is a pr...
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\begin{tcolorbox}[warnbox] \small\textbf{Open question.} The case $\GR < 0$ (contraction, dormancy, retreat) requires separate treatment. Some systems exist in phases of deliberate contraction (hibernation, recession, withdrawal into \Vo). Whether contraction constitutes $\GR = 0$ or $\GR > 0$ (as a boundary-\textit{re...
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A recurring failure mode in cross-domain reasoning is \textbf{frame conflation}: carrying a term from one Frame into another as if it retained the same referent. The most frequent instance in this paper is the word ``energy''. In physical frames, energy is a conserved quantity (Two-Substrate Theory: $E = f_1 + f_2 + E...
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\begin{tcolorbox}[warnbox] \small\textbf{Category error warning.} Functional ``energy'' (activation, coupling strength) is not identical to physical energy ($E = f_1 + f_2 + E_V$). Claims that mix these referents without declaring which Frame is operative are structurally incomplete under NMF: they deploy a Name withou...
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%% ══════════════════════════════════════════════════════════════════════════ \section{Box--Sphere Logic: Stability and Growth as Orthogonal Operations} %% ══════════════════════════════════════════════════════════════════════════ Before applying the definition to specific frames, it is necessary to characterize the t...
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\textbf{Box logic} describes the mode of operation that maintains structural integrity through linear, positional, boundary-enforcing processes:
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item \textbf{Linear}: causes and effects are proportional; scaling is additive. \item \textbf{Positional}: what is inside the boundary is protected; what is outside is environment. \item \textbf{Exclusive} (zero-sum): resources allocated to one sub-function a...
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In biological terms: homeostasis, immune defense, membrane integrity, structural repair. In socio-political terms: institutional enforcement, compliance mechanisms, resource defense. In TOB terms: \Dung\ --- the anchoring of a pattern into a stable Frame. \subsection{Sphere logic: the structure of growth} \textbf{Sph...
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item \textbf{Non-linear}: small inputs can produce large structural reorganizations (resonance, phase transitions, cascade effects). \item \textbf{Relational}: the boundary expands through coupling with the environment, not by excluding it. \item \textbf{Non-...
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In biological terms: learning, allostasis, niche construction, developmental plasticity. In socio-political terms: legitimacy expansion, coalition-building, knowledge accumulation. In TOB terms: the \Sinh--\Dan\ arc --- emergence and propagation of a new cognitive segmentation before it encounters an anchor. \subsecti...
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Box logic operates on the current boundary; Sphere logic operates on what the boundary \emph{could} become. They do not trade off: increasing Box operations does not decrease Sphere capacity (within normal ranges), and vice versa. Their combination is therefore captured by a product, not a sum.
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\begin{center} \begin{tikzpicture}[ node distance=1.6cm and 2.6cm, box/.style={rectangle,rounded corners=3pt,draw=primary,fill=subtle, line width=0.8pt,minimum width=2.6cm,minimum height=0.85cm,align=center, font=\small}, sph/.style={ellipse,draw=accent,fill=accent!8, line width=0.8pt,mini...
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%% ══════════════════════════════════════════════════════════════════════════ \section{Biological Frame: Survival and Development} %% ══════════════════════════════════════════════════════════════════════════ \subsection{Existence as survival $\times$ development}
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In the biological frame: \[ \ES_{\text{bio}} \;=\; \underbrace{\ST_{\text{bio}}}_{\text{survival}} \;\times\; \underbrace{\GR_{\text{bio}}}_{\text{development}} \;>\; 0 \] with the constraint: \textit{development is the expansion of survival boundaries so that survival can continue under broader conditions.}
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This makes development instrumentally coupled to survival, not merely optional growth. Development is what prevents the system from being trapped in a narrow niche that eventually becomes untenable. \subsection{Survival as absorption and assimilation (Box logic)} Survival is the Box-logic component: maintaining organ...
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item \textbf{Energy flow}: metabolic intake and conversion. \item \textbf{Material flow}: nutrients, building blocks, waste regulation. \item \textbf{Information flow}: sensing and regulatory feedback. \end{itemize}
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The key is not the substance absorbed but the maintenance of organizational constraints through managed throughput. This is broadly consonant with autopoietic accounts of living systems as self-producing networks of processes (Maturana and Varela, 1980). \subsection{Development as boundary expansion (Sphere logic)} D...
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item improving regulatory range (allostasis, not merely homeostasis), \item widening accessible resource regimes (new niches, new behaviors), \item increasing predictive and adaptive capacity (learning and plasticity). \end{itemize} \subsection{Hoai and the necessit...
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A component absent from the prior formulation is \Hoai: the phase in which patterns that do not survive anchoring are displaced, compressed, or transferred to other levels of organization.
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\begin{principle}[Hoai as constitutive of biological existence] No biological system grows without discarding. \Hoai\ is not a failure mode; it is a constitutive mechanism. Cell apoptosis, synaptic pruning, niche abandonment, and immune culling are all instances of structured displacement that enable the $\GR > 0$ cond...
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Without \Hoai, the system accumulates structural debt and eventually loses the capacity to anchor new patterns (\Dung). The absence of displacement is itself a failure of growth. %% ══════════════════════════════════════════════════════════════════════════ \section{Socio-Political Frame: Stability and Expansion} %% ══...
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In socio-economic and political systems, ``survival'' is less biological but remains meaningful as system viability:
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item \textbf{Stability} ($\ST_{\text{soc}}$): continuity of institutional operation, coordination mechanisms, and conflict management. Box logic: boundary-enforcing, positional, zero-sum within the governed territory. \item \textbf{Growth} ($\GR_{\text{soc}}$):...
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\subsection{NMF analysis of frame-specific meanings} The NMF framework makes explicit why the same Name generates different Meanings in different Frames:
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\begin{center} \small \begin{tabular}{@{}p{2.2cm}p{3.4cm}p{5.5cm}@{}} \toprule \textbf{Name} & \textbf{Frame} & \textbf{Meaning} \\ \midrule \textit{stability} & RF\textsubscript{biology} & homeostatic maintenance of metabolic constraints \\[3pt] \textit{stability} & RF\textsubscript{political} & institutional ...
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Same Names. Different Frames. Structurally analogous Meanings. \subsection{Failure modes of the coupling} The multiplicative structure predicts two symmetric failure modes:
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\begin{itemize}[leftmargin=1.4em,itemsep=3pt] \item \textbf{$\ST \to 0$, $\GR > 0$}: overextension without structural coherence --- expansion outpaces the system's capacity to maintain integrity. Result: collapse of coordination, fragmentation. \item \textbf{$\ST > 0$, $\GR \to 0$}: brittle stagnation --- t...
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Under addition, the first failure mode would still yield $\ES > 0$ (the expanding system would appear to ``exist'' despite losing coherence). Multiplication correctly returns $\ES \to 0$ as either factor approaches zero. %% ══════════════════════════════════════════════════════════════════════════ \section{Consciousne...
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In systems whose substrate is primarily dynamical (oscillatory, process-based, activity-dependent), stability means a \emph{maintained pattern}, not a fixed structure. The Box metaphor of a wall holding steady gives way to the Sphere metaphor of a standing wave: structurally coherent, dynamically maintained. \subsecti...
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\begin{definition}[Resonant coherence] In process-based systems, stability is operationalized as \textbf{resonant coherence}: the system's capacity to maintain the alignment of its internal relational structure despite perturbation. In neural terms, this corresponds to maintained phase alignment enabling inter-areal co...
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This is \emph{not} a claim about exotic physics. ``Resonance'' is used here in the functional sense of maintained relational alignment---not as a statement about literal wave interference in a physical medium. \subsection{Growth as widening resonance bandwidth} Growth in this frame is the expansion of the \emph{domai...
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item handling greater complexity without loss of coherence, \item maintaining pattern stability under higher noise, \item integrating broader ranges of input without pattern collapse. \end{itemize} This is compatible with predictive-processing accounts (Friston, 201...
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\subsection{TOB integration: consciousness through the cycle} The TOB ontological cycle maps onto the consciousness frame as follows:
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\begin{center} \small \begin{tabular}{@{}p{2.6cm}p{8.2cm}@{}} \toprule \textbf{TOB phase} & \textbf{Consciousness frame interpretation} \\ \midrule \Sinh & Emergence of a distinguishable experiential segment \\[3pt] \Dan & Propagation of the pattern across neural or cognitive substrates \\[3pt] \Chuyen & Encounter wit...
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\Dung\ corresponds to the $\ST > 0$ condition: a pattern has been anchored and stabilized. The \Sinh--\Dan\ arc corresponds to the $\GR > 0$ condition: new patterns are emerging and propagating. Existence of a conscious formation requires both. \Hoai\ is the necessary complement: patterns that do not achieve \Dung\ are...
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%% ══════════════════════════════════════════════════════════════════════════ \section{Integration with Two-Substrate Theory} %% ══════════════════════════════════════════════════════════════════════════ \subsection{The two substrates} Two-Substrate Theory proposes that the physical basis of existence involves two or...
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\begin{itemize}[leftmargin=1.4em,itemsep=3pt] \item \textbf{Space--Matter substrate}: positional, boundary-defining, structural. Maps directly onto Box logic and $\ST$. \item \textbf{Time--Energy substrate}: processual, relational, dynamical. Maps onto Sphere logic and $\GR$. \end{itemize}
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The total energy of a system in this framework is: \[ E \;=\; f_1 + f_2 + E_V \] where $f_1$ is substrate-bound energy (structural, conservative), $f_2$ is interference or activation energy (dynamic, relational), and $E_V$ is void-sector energy (the ledger of mismatch, dissipation, and unresolved structural tension)....
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\begin{itemize}[leftmargin=1.4em,itemsep=2pt] \item patterns that entered \Hoai\ and were not re-integrated, \item structural mismatches between $f_1$ and $f_2$ that were not resolved, \item boundary conditions that were abandoned rather than expanded. \end{itemize}
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\begin{principle}[Void as structural necessity] No system exists without a Void sector. The Void is not failure; it is the accumulation of \Hoai\ outputs that makes space for new \Sinh. A system that produces no Void is a system that never displaces anything---which means it never truly grows ($\GR = 0$). The Void is t...
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\begin{center} \small \begin{tabular}{@{}p{3.2cm}p{2.8cm}p{4.8cm}@{}} \toprule \textbf{Two-Substrate} & \textbf{$\ES$ formula} & \textbf{TOB phase} \\ \midrule Space--Matter ($f_1$) & $\ST > 0$ & \Dung \\ Time--Energy ($f_2$) & $\GR > 0$ & \Sinh--\Dan \\ Void ($E_V$) & Complement of both & \Hoai--\Vo \\ \bott...
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%% ══════════════════════════════════════════════════════════════════════════ \section{Integration with The Ontology of Being (TOB)} %% ══════════════════════════════════════════════════════════════════════════ \subsection{Existence as the output of cognitive segmentation}
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Under TOB, what we call ``existence'' in any frame is the result of a \textit{tu niem} (cognitive segmentation act) that has successfully passed through \Chuyen\ and achieved \Dung. What appears to exist is what has been stabilized within a Frame. What does not appear to exist has entered \Hoai\ or \Vo.
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This reframes the definition: \[ \ES(X_t) = \ST(X_t) \times \GR(X_t) > 0 \] not as a claim about absolute reality, but as a claim about what is \emph{legible} within a given Frame at time $t$. Existence is always frame-indexed. \subsection{Why the observation-after-Dung principle matters here} The Observation-After...
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\begin{itemize}[leftmargin=1.4em,itemsep=3pt] \item When we observe that something ``exists,'' we are observing a pattern that has already achieved \Dung\ within some Frame. \item The \Sinh--\Dan\ arc that preceded stabilization is usually invisible. \item \Hoai\ outputs (the patterns that failed anchoring) a...
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This explains why existence so often \emph{appears} to be a static primitive rather than a dynamic achievement: observers encounter the product of the cycle, not the cycle itself. \subsection{NMF analysis of ``existence'' across frames} Applying the Name--Meaning--Frame engine: \[ \textit{Meaning}(\text{``existence...
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