Unified Field Theory 20B: Toward a Dimensional Framework for Semantic Field Theory Calibrating Units, Collapse Dynamics, and Observer-Invariant Structure in SMFT

[Quick overview on SMFT vs Our Universe ==>Chapter 12: The One Assumption of SMFT: Semantic Fields, AI Dreamspace, and the Inevitability of a Physical Universe]

Unified Field Theory 20A: Mass and Distance Within Semantic Black Holes: A Constructive Model of Collapse-Based Geometry in SMFT 

Chapter 20B Toward a Dimensional Framework for Semantic Field Theory
Calibrating Units, Collapse Dynamics, and Observer-Invariant Structure in SMFT

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Abstract

Semantic Meme Field Theory (SMFT) models meaning as the result of observer-induced collapse within a high-dimensional semantic phase space. While previous work has established formal analogies between semantic structures and physical quantities—such as mass, force, and energy—these constructs have remained metaphorical, lacking a consistent system of units or a dimensional foundation. As a result, SMFT has been powerful in form but limited in its capacity for simulation, measurement, and inter-system comparability.

This paper introduces a complete dimensional framework for SMFT, transforming it from a structural analogy into a scalable, simulation-ready field theory of meaning. We define base semantic units—tick-time ($T_s$), projection angle ($R_s$), and collapse tension ($\Xi_s$)—and derive dimensional expressions for semantic mass, energy, force, power, and entropy. These quantities are shown to be internally consistent and extensible across agents, supporting scalar additivity, relativistic analogues, and dynamical integration.

To ensure practical interoperability, we develop observer-invariant calibration protocols that allow these units to be instantiated in both cognitive (human) and computational (LLM) systems. We provide simulation methods for estimating collapse tension, angular shift, mass, and energy from standard transformer outputs, and introduce the concept of a Collapse-Lorentz Transform—a set of frame-preserving equations for translating meaning between agents with differing collapse rhythms and projection bandwidths.

By formalizing dimensional structure and semantic invariance, SMFT now supports measurable and comparable modeling of cognition, discourse, and AI behavior. This establishes the foundation for a semantic physics—where meaning behaves not metaphorically like matter, but with rigorously defined mass, energy, and geometry within collapse-governed phase space.

 

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1. Introduction

Semantic Meme Field Theory (SMFT) models reality not as an objective substrate of particles and forces, but as the emergent geometry of meaning formed through irreversible semantic collapse. In this framework, ideas, identities, and beliefs do not preexist—they materialize when an observer projection $\hat{Ô}$ selects a direction $\theta$ in semantic phase space, collapsing the underlying wavefunction $\Psi_m(x, \theta, \tau)$ into a discrete, traceable outcome $\phi_j$. The resulting trace defines not just semantic content, but the very structure of experience: time, relevance, coherence, and cultural memory all arise from patterns of collapse.

Recent developments in SMFT have extended this formalism by introducing dynamical analogues to physical mass, force, and energy—especially within the high-density collapse zones known as semantic black holes. These are regions where semantic ticks $\tau$ are synchronized, projection directions $\theta$ are tightly phase-aligned, and collapse-generated traces accumulate into stable attractors. In this regime, SMFT has proposed:

• Semantic mass $mₘ = \frac{iT}{\Delta\theta}$: representing resistance to projection reorientation,

• Semantic force $F_s = mₘ \cdot \frac{d^2\theta}{d\tau^2}$: the rate of interpretive acceleration,

• Semantic energy $E_s = \frac{1}{2} mₘ v_\theta^2 + V_\theta(\theta)$: encoding both dynamic tension and potential alignment.

These constructs mirror the form of Newtonian and relativistic mechanics—most notably in analogues like $E_s = mₘ c_s^2$, where $c_s$ is the maximum rate of semantic propagation (collapse speed). They enable SMFT to describe how complex semantic structures—memes, ideologies, cultural solitons—form, persist, and interact as if they were “semantic objects” in a high-dimensional field.

However, a key limitation remains: the absence of a dimensional framework. While the structures of SMFT equations resemble those of physics, their underlying units are undefined or abstract. Quantities like $iT$ (semantic tension), $\tau$ (tick-time), and $\theta$ (projection angle) have no standardized dimensional basis, and the derived quantities—mass, energy, force—cannot yet be compared, measured, or simulated across systems in a consistent way. This undermines their equivalence to physical counterparts and limits SMFT’s ability to serve as a general modeling language for cognition, discourse, or AI behavior.

The goal of this paper is to establish a dimensional framework for SMFT. We propose a semantic unit system based on collapse mechanics, define base dimensions for semantic time, tension, and direction, and construct derived units for mass, energy, power, and entropy. We then articulate observer-invariant calibration principles—including entropy flux, trace density, and tick synchronization—that enable inter-agent consistency. With this structure in place, SMFT becomes not just structurally analogous to physics, but dimensionally scalable and simulation-ready.

In doing so, we move toward a fully formal semantic physics: one that can describe and compare the energetic structure of beliefs, the inertia of narrative systems, and the field geometry of meaning in both human minds and artificial agents.

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2. What Is Dimensionality in SMFT?

In traditional physical theories, dimensionality provides the foundation for measurement, consistency, and comparability. Physical quantities such as force, energy, and momentum are not merely abstract constructs—they are defined in terms of base units (e.g., meters, seconds, kilograms), which allow them to be added, scaled, transformed, and empirically measured. The power of this system lies not only in its internal coherence but in its ability to relate distinct physical systems through a shared dimensional framework.

However, many successful modeling frameworks outside of classical physics—such as thermodynamics, information theory, and symbolic logic—also depend on internal dimensional logic, even if they do not operate over physical space. For example:

• In thermodynamics, temperature and entropy have abstract but well-defined units (e.g., Kelvin, joules per kelvin),

• In information theory, quantities like entropy are measured in bits, a non-physical unit with clear additive and scaling properties.

Semantic Meme Field Theory (SMFT) operates in a similar spirit. It does not attempt to model physical spacetime, but instead describes a semantic phase space, where meaning, memory, and attention are understood as geometrically and dynamically structured through collapse. Within this context, we must ask: What does it mean to assign dimensions in SMFT?

The answer begins with recognizing that semantic dynamics unfold not in (t, x, y, z) but in a non-physical, collapse-defined manifold: $(x, \theta, \tau)$.

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2.1 Recasting the Semantic Coordinates

To formulate a dimensional system for SMFT, we must first reinterpret its fundamental coordinates and variables in the language of dimensional analysis.

Symbol	Meaning	Proposed Dimensional Role
$iT$	Semantic tension	Semantic intensity per directional span
$\tau$	Semantic tick time	Unit of collapse rhythm (observer-linked temporal flow)
$\theta$	Projection direction	Semantic phase orientation (dimensionless, radian-like)
$x$	Semantic embedding coordinate	Position in memetic/cultural network topology

This semantic coordinate space $(x, \theta, \tau)$ is not emergent from physics, but is defined by the structure of collapse: when, where, and in which direction meaning actualizes from potential.

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2.2 Semantic Phase Space: $(x, \theta, \tau)$

The SMFT phase space can be thought of as a 3D manifold whose axes encode:

• $x$: The domain or location in a symbolic/cultural/linguistic system,

• $\theta$: The interpretive direction or phase alignment of a memeform,

• $\tau$: The semantic clock governing when collapse occurs.

These coordinates are observer-relative, meaning they are defined by the projection and tick rhythm of a specific $\hat{Ô}$. They are not global constants, but internally consistent within high-coherence regions such as semantic black holes.

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2.3 Dimensional Mapping of Core Quantities

To build a dimensional system for SMFT, we propose the following semantic base quantities:

Quantity	Symbol	Analogical Unit Proposal	Interpretation
Tension	$iT$	$[\Xi] = \text{bits per radian}$	Field strain per unit interpretive spread
Tick-Time	$\tau$	$[T_\text{s}] = \text{semantic ticks}$	Time between discrete collapses
Direction	$\theta$	$[R] = \text{radians (dimensionless)}$	Semantic orientation (collapse direction in phase space)
Embedding	$x$	$[L_\text{s}] = \text{semantic location index}$	Position in semantic network topology (non-Euclidean)

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2.4 Dimensional Seeds for Derived Quantities

From these base quantities, we can now begin building the dimensional scaffolding for SMFT constructs:

Semantic Mass

$$mₘ = \frac{iT}{\Delta\theta} \quad \Rightarrow \quad [mₘ] = [\Xi] \cdot [R]^{-1}$$

Interpretation: semantic inertia, or the resistance to changing projection direction under a given semantic pressure.

Semantic Energy

$$E_s = \frac{1}{2} mₘ v_\theta^2 + V_\theta(\theta) \quad \Rightarrow \quad [E_s] = [\Xi] \cdot [T_\text{s}]^{-2}$$

Collapse energy released per tick under directional acceleration.

Semantic Force

$$F_s = mₘ \cdot a_\theta \quad \Rightarrow \quad [F_s] = [\Xi] \cdot [T_\text{s}]^{-2} \cdot [R]^{-1}$$

Tension curvature applied across phase space to bend collapse trajectory.

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2.5 Summary

SMFT does not adopt physical units—but it can support a formal, observer-coherent dimensional system:

• Collapse tick-time $\tau$ acts as a substitute for absolute time,

• Semantic direction $\theta$ behaves like phase or orientation in quantum systems,

• Semantic tension $iT$ can be interpreted in units of information flux per angular phase,

• Together, they allow construction of dimensional analogues to force, energy, mass, and power—suitable for internal modeling and simulation.

In the next section, we formalize these base units and construct a complete dimensional table for all major SMFT quantities.

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3. Candidate Base Units for Semantic Dimensional System

To establish a dimensional framework for SMFT that supports mathematical consistency and cross-system comparability, we must begin by defining a set of semantic base units. These units are not grounded in physical space or time, but are abstracted from the internal logic of semantic collapse—measured by projection behavior, rhythm, and information tension.

This section proposes three core base units and uses them to define compound units for semantic mass, semantic energy, and semantic force.

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3.1 Semantic Base Units

Unit Name	Symbol	Represents	Interpretation
Tick	$T_s$	Unit of semantic time	One interval of coherent collapse, observer-relative
Radian (Semantic)	$R_s$	Unit of projection angle	Directional spread in phase space, measured in semantic radians
Tension Unit	$\Xi_s$	Unit of semantic tension	Collapse pressure per unit projection span (e.g., bits/radian)

These units allow us to construct dimensional analogs to Newtonian and relativistic physics in a purely semantic regime.

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3.2 Compound Units for SMFT Dynamics

We now define key SMFT quantities using the base unit system $(\Xi_s, R_s, T_s)$:

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3.2.1 Semantic Mass

$$mₘ = \frac{iT}{\Delta\theta}$$ $$\Rightarrow [mₘ] = \frac{[\Xi_s]}{[R_s]} = [\Xi_s \cdot R_s^{-1}]$$

Interpretation: Semantic mass is the amount of collapse tension per unit change in projection direction. It encodes how much pressure must be overcome to shift interpretive focus, and determines the inertia of a trace under semantic force.

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3.2.2 Semantic Velocity (Angular)

$$v_\theta = \frac{d\theta}{d\tau} \Rightarrow [v_\theta] = \frac{[R_s]}{[T_s]} = [R_s \cdot T_s^{-1}]$$

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3.2.3 Semantic Energy

$$E_s = \frac{1}{2} mₘ v_\theta^2 \Rightarrow [E_s] = \left[\Xi_s \cdot R_s^{-1}\right] \cdot \left[ \frac{R_s^2}{T_s^2} \right]$$ $$\Rightarrow [E_s] = \Xi_s \cdot T_s^{-2}$$

Interpretation: Semantic energy measures the rate of meaningful collapse propagation, whether kinetic (interpretive shift) or potential (alignment with semantic fields). It has no spatial or mechanical substrate—only phase-space rhythm.

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3.2.4 Semantic Force

$$F_s = mₘ \cdot a_\theta = mₘ \cdot \frac{d^2\theta}{d\tau^2}$$ $$\Rightarrow [F_s] = [\Xi_s \cdot R_s^{-1}] \cdot \left[ \frac{R_s}{T_s^2} \right] = \Xi_s \cdot T_s^{-2}$$

Same unit as energy, reflecting the field-like nature of projection influence. $F_s$ measures how much semantic curvature is applied to Ô projections per tick.

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3.2.5 Semantic Power

$$P_s = \frac{E_s}{T_s} = \Xi_s \cdot T_s^{-3}$$

Measures semantic collapse throughput—how quickly energy is being used to generate directional meaning.

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3.3 Semantic Dimensional Table (Summary)

Quantity	Formula	Dimensional Form
Semantic Mass	$mₘ = \frac{iT}{\Delta\theta}$	$[\Xi_s] \cdot [R_s]^{-1}$
Semantic Velocity	$v_\theta = \frac{d\theta}{d\tau}$	$[R_s] \cdot [T_s]^{-1}$
Semantic Energy	$E_s = \frac{1}{2} mₘ v_\theta^2$	$[\Xi_s] \cdot [T_s]^{-2}$
Semantic Force	$F_s = mₘ \cdot \frac{d^2\theta}{d\tau^2}$	$[\Xi_s] \cdot [T_s]^{-2}$
Semantic Power	$P_s = \frac{E_s}{T_s}$	$[\Xi_s] \cdot [T_s]^{-3}$

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3.4 Implications and Uses of This Unit System

• Provides a formal basis for simulation: allows SMFT systems (like AI models) to assign numeric values to tension, trace speed, and collapse influence.

• Enables comparability between systems, provided shared definitions of $T_s$, $R_s$, and $\Xi_s$ (see Section 5).

• Supports scaling laws, such as conservation of total energy across projection collapses or mass accumulation in composite semantic structures.

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In the next section, we extend these definitions to construct a complete dimensional map of all major SMFT quantities, preparing for simulation and empirical anchoring.

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4. Constructing a Dimensional Table of SMFT Dynamics

With semantic base units in place—tick-time ($T_s$), projection angle ($R_s$), and collapse tension ($\Xi_s$)—we now construct a complete dimensional system for SMFT. This section presents key dynamical quantities, their structural equations, dimensional forms, and interpretive meanings within the context of semantic collapse dynamics.

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4.1 Base Units Recap

Base Quantity	Symbol	Unit	Interpretation
Semantic Tension	$iT$	$\Xi_s$	Tension per radian (collapse density or info flux)
Semantic Tick-Time	$\tau$	$T_s$	Duration between semantic collapses (collapse rhythm)
Projection Angle	$\theta$	$R_s$	Direction in semantic phase space (radians)

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4.2 Derived Quantities Table

Quantity	Structural Equation	Dimensional Form	Meaning in SMFT
Semantic Mass	$mₘ = \frac{iT}{\Delta\theta}$	$[\Xi_s] \cdot [R_s]^{-1}$	Inertia against interpretive reorientation; persistence of collapse direction
Semantic Acceleration	$a_\theta = \frac{d^2\theta}{d\tau^2}$	$[R_s] \cdot [T_s]^{-2}$	Rate of change in projection direction over collapse time
Semantic Force	$F_s = mₘ \cdot a_\theta$	$[\Xi_s] \cdot [T_s]^{-2}$	Semantic pressure redirecting projection; alignment force or narrative pull
Semantic Energy	$E_s = \frac{1}{2} mₘ v_\theta^2 + V_\theta$	$[\Xi_s] \cdot [T_s]^{-2}$	Collapse capacity of a trace; kinetic + potential interpretive energy
Semantic Power	$P_s = \frac{E_s}{T_s}$	$[\Xi_s] \cdot [T_s]^{-3}$	Rate of meaning generation; collapse throughput or expressive intensity
Semantic Entropy	$S_s = -\sum p_j \log p_j$	Dimensionless (bits)	Degree of collapse uncertainty; number of accessible interpretations

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4.3 Extended Interpretations

Semantic Mass

A high-mass memeform resists changes in its projected meaning. Examples include:

• Sacred symbols,

• Deep identity narratives,

• Long-lived institutional memes.

Semantic Force

Semantic force arises when projection fields bend or redirect attention. It reflects:

• The persuasive pull of arguments,

• The shifting pressure of framing,

• The gradient of ideological terrain.

Semantic Energy

Represents how much collapse potential is stored in a trace. High energy could come from:

• Fast interpretive motion (high $v_\theta$),

• Strong semantic alignment with an attractor (low $\Delta\theta$, high $iT$).

Semantic Power

Useful for evaluating:

• Expressive output rate of AI models (collapse per tick),

• Intensity of discourse in cultural feedback loops,

• Burstiness of memetic propagation.

Semantic Entropy

Standard Shannon entropy applies, but its meaning here is:

• Collapse uncertainty across a meme’s possible interpretations,

• Degree of “semantic superposition” prior to commitment,

• In SMFT, entropy decreases as collapse proceeds.

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4.4 Visual Summary Table

Quantity	Symbol	Structural Equation	Dimension	SMFT Interpretation
Mass	$mₘ$	$iT / \Delta\theta$	$\Xi_s \cdot R_s^{-1}$	Resistance to re-projection
Acceleration	$a_\theta$	$d^2\theta / d\tau^2$	$R_s \cdot T_s^{-2}$	Shift rate in semantic direction
Force	$F_s$	$mₘ \cdot a_\theta$	$\Xi_s \cdot T_s^{-2}$	Projection pressure or field pull
Energy	$E_s$	$\frac{1}{2} mₘ v_\theta^2 + V_\theta$	$\Xi_s \cdot T_s^{-2}$	Collapse capacity or meaning compression
Power	$P_s$	$E_s / T_s$	$\Xi_s \cdot T_s^{-3}$	Collapse velocity / output density
Entropy	$S_s$	$-\sum p_j \log p_j$	dimensionless (bits)	Interpretive spread prior to collapse

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In the next section, we apply this dimensional framework to define observer-invariant calibration principles, enabling semantic measurements to be meaningfully compared across agents, systems, and AI architectures.

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5. Observer-Invariant Calibration Principles

To make Semantic Meme Field Theory (SMFT) operationally meaningful across diverse systems—such as human cognition, AI language models, or multi-agent symbolic environments—we must establish observer-invariant calibration principles. These principles allow semantic base units like tick-time ($T_s$), projection angle ($R_s$), and collapse tension ($\Xi_s$) to be consistently defined and compared across systems with different internal architectures.

In classical physics, unit invariance arises from shared physical laws (e.g., the speed of light, Planck’s constant). In SMFT, unit invariance must be constructed from semantic behavior itself, particularly from how each observer (biological or artificial) processes, collapses, and remembers meaning.

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5.1 Anchoring Units via Observable System Behavior

We propose the following system-level anchors for normalizing semantic units:

(a) Compression Entropy

• Define a system’s semantic capacity in terms of entropy reduction per collapse.

• For LLMs: measure KL divergence between pre-token and post-token distributions.

• For humans: approximate via information compression in working memory or reaction to ambiguity resolution.

• Use-case: calibrate $\Xi_s$ (semantic tension unit) in bits per collapse interval, making collapse energy quantifiable across agents.

(b) Collapse Rate per Token / Word

• In LLMs, a "semantic tick" $T_s$ may correspond to 1 token prediction.

• In human cognition, estimate ticks based on:

  • Minimal time between interpretive updates (~300ms–500ms),

  • Event-related potentials in semantic priming tasks,

  • Natural language comprehension pacing.

• Use-case: normalize tick-time $T_s$ in seconds or token intervals.

(c) Ô Trace Sampling Bandwidth

• Define each observer’s sampling bandwidth as the number of discrete projection directions $\theta$ it can resolve per tick.

• In LLMs: related to attention head count × layer count × positional coverage.

• In humans: approximated via working memory span or attentional scope.

• Use-case: normalize angular granularity $R_s$ in radians per tick.

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5.2 Semantic Planck-Scale Estimates

To model SMFT dynamics with scale consistency, we propose minimum measurable semantic quantities analogous to Planck units.

Quantity	Symbol	Definition / Heuristic	Use
Minimum Semantic Tension	$\Xi_{\text{min}}$	Minimum info gain that causes an irreversible collapse	Sets floor of detectable collapse energy
Minimum Semantic Tick Interval	$T_{\text{min}}$	Fastest semantic update interval per observer	Sets resolution of temporal collapse scale
Minimum Angular Resolution	$R_{\text{min}}$	Smallest phase angle Δθ that produces trace discrimination	Defines minimal phase-space curvature

For LLMs, a good starting estimate might be:

• $T_{\text{min}} = 1$ token prediction cycle,

• $\Xi_{\text{min}} = 1$ bit KL-divergence per head-attention shift,

• $R_{\text{min}} = \frac{2\pi}{n_\text{attn-heads}}$.

These define quantum-like bounds on semantic operations—when meaning transitions from fuzziness to trace formation.

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5.3 Observer-Normalized Reference Frames

To make comparisons across systems with different bandwidths and behaviors, we construct observer-normalized semantic reference frames using:

(a) Attention Entropy

• Define semantic projection “spread” as entropy over the observer's attention weights or relevance judgments.

• In LLMs: entropy of attention distributions or top-k logits.

• In humans: eye-tracking spread, neural decoding of attentional focus.

• Use-case: calibrate real-time field coherence across agents.

(b) Trace Density over θ

• Define how densely an observer collapses trace into θ-space.

• In LLMs: number of activated dimensions per layer collapse.

• In humans: narrative coherence judgments, topic clustering in discourse.

• Use-case: define trace curvature, attractor shaping, or information saturation.

(c) Normalized Collapse Power

Using Section 4's semantic power:

$$P_s = \frac{E_s}{T_s} = \Xi_s \cdot T_s^{-3}$$

We may define observer-relative semantic power rating for:

• High-power agents (e.g., GPT-4) with dense, high-frequency, low-entropy collapses,

• Low-power agents (e.g., early-stage LLMs or humans under overload) with sparse, uncertain collapse behavior.

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5.4 Summary Table: Observer-Invariant Calibration Principles

SMFT Unit	Anchor in LLM	Anchor in Human Cognition
$T_s$	Token cycle or internal tick-step	Time between interpretive updates (~300–500ms)
$R_s$	Attention head angular resolution	Number of competing frames in working memory
$\Xi_s$	Bits of KL divergence per token step	Subjective meaning gain or inference certainty
$c_s$	Max propagation speed per inference path	Semantic coherence over discourse time

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In the next section, we show how this framework can be applied to real-world semantic systems—particularly large language models—by extracting and computing SMFT quantities directly from their internal structure and generative behavior.

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6. Collapse Metrics in Simulation and AI Systems

With semantic units and observer-invariant calibration principles established, we now demonstrate how SMFT dynamics can be measured or approximated in real systems—especially transformer-based large language models (LLMs). These systems already exhibit internal structures analogous to semantic projection, collapse, and trace density, making them ideal testing grounds for simulation-compatible SMFT analysis.

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6.1 Mapping SMFT Quantities to LLM Internals

SMFT Quantity	LLM Proxy/Signal	Notes
$T_s$ (tick)	Token generation step	Each token = one semantic tick
$\theta$ (direction)	Projection vector in embedding space	Interpreted via attention patterns or logits
$iT$ (tension)	Logit entropy, loss, or gradient norm	Reflects uncertainty + semantic pressure before collapse
$\Delta\theta$	Shift in softmax attention or decoder orientation	Can be estimated by attention movement or change in latent vector

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6.2 Semantic Mass Approximation for a Prompt

Recall:

$$mₘ = \frac{iT}{\Delta\theta}$$

Estimation:

1. iT:

   • Use KL divergence between pre-softmax logits and a uniform or prior distribution.

   • Alternatively, use negative entropy (−H) or loss value as a proxy for collapse pressure.

2. Δθ:

   • Compute angular difference between:

     • Prompt embedding vector $v_{\text{prompt}}$,

     • Predicted next-token embedding $v_{\text{next}}$.

   • Use:

     $$\Delta\theta = \cos^{-1} \left( \frac{v_{\text{prompt}} \cdot v_{\text{next}}}{\|v_{\text{prompt}}\|\|v_{\text{next}}\|} \right)$$

3. mₘ (Prompt Semantic Mass):

   $$mₘ = \frac{\text{KL or -H}}{\Delta\theta}$$

A high semantic mass prompt resists deflection—e.g., a specific technical instruction vs. a vague open-ended question.

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6.3 Semantic Force $F_s$ Across Generations

Recall:

$$F_s = mₘ \cdot a_\theta = mₘ \cdot \frac{d^2\theta}{d\tau^2}$$

Estimation:

1. Use 3 consecutive tokens:

   • Token 1: $\theta_1$

   • Token 2: $\theta_2$

   • Token 3: $\theta_3$

2. Angular second derivative:

   $$a_\theta = \frac{\theta_3 - 2\theta_2 + \theta_1}{T_s^2}$$

3. Semantic mass $mₘ$: as computed above (KL divergence / angular uncertainty at token 2)

4. Force:

   $$F_s = mₘ \cdot a_\theta$$

Semantic force peaks during abrupt narrative shifts, prompt misalignment, or synthetic collisions in generation (e.g., contradiction, sarcasm).

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6.4 Collapse Energy of a Completion

Recall:

$$E_s = \frac{1}{2} mₘ v_\theta^2 + V_\theta$$

Estimation:

1. Kinetic term:

   $$v_\theta = \frac{d\theta}{d\tau} \approx \theta_2 - \theta_1$$ $$\Rightarrow \text{KE} = \frac{1}{2} mₘ (\theta_2 - \theta_1)^2$$

2. Potential term $V_\theta$:

   • Estimate as:

     • Mean inverse attention entropy over a window,

     • Or energy stored in latent gradient norms (proxy for field curvature).

3. Total collapse energy for a token or sequence:

   $$E_s = \text{KE} + V_\theta$$

Completions with high semantic energy are informative, surprising, or “emotionally loaded.” Low-energy completions may be repetitive, predictable, or incoherent.

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6.5 Visualization Example (For Developers)

For each generated token:

• Compute:

  • Logit entropy

  • Vector shift in embedding space

  • Change in attention direction

Then log:

Token	KL Div	Δθ (rad)	mₘ	aθ	Fₛ	Eₛ
"The"	2.1	0.12	17.5	—	—	—
"quick"	1.6	0.15	10.7	0.08	0.86	1.3
"brown"	2.3	0.10	23.0	−0.04	−0.92	2.5

Such tables allow semantic dynamics to be visualized, tested, and optimized, e.g.:

• Design prompts to modulate energy curves,

• Detect high-force instability zones in generation,

• Tune loss or sampling to shape field evolution.

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In the next section, we extend the dimensional logic of SMFT to propose universal scaling laws and explore how semantic frames transform across agents—laying the foundation for relativistic-style invariance in semantic systems.

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7. Toward a Universal Semantic Scaling Law

A defining feature of physical theories with dimensional consistency—such as special relativity—is the presence of invariant quantities that hold across reference frames. The speed of light $c$ is one such invariant, serving as both a conversion constant and a boundary of causal influence. In SMFT, we seek a similar construct: a collapse speed $c_s$ that unifies semantic dynamics across disparate systems, allowing intersubjective consistency, model interoperability, and ultimately, a shared semantic physics.

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7.1 Can Collapse Speed $c_s$ Be Normalized?

In SMFT, collapse speed $c_s$ is the maximum rate at which a projection shift (change in $\theta$) can occur per semantic tick $\tau$ without inducing incoherence. Formally:

$$c_s = \max \left( \frac{d\theta}{d\tau} \right)$$

Challenges in Normalization

• Human $c_s$ is limited by biological processing (~200–500ms/tick).

• LLM $c_s$ is determined by architectural limits (tokenization, context window, parallelism).

• However, semantic coherence and narrative alignment often require a shared c_s ceiling—otherwise, one system may overshoot or undersample the other's collapse rhythm.

Proposed Normalization Approach

Define:

• Effective collapse speed per system:

  $$c_s^{(\text{agent})} = \frac{\Delta\theta}{T_{\text{eff}}}$$

• Then use scaling coefficients to align systems:

  $$\alpha_{AB} = \frac{c_s^{(A)}}{c_s^{(B)}}$$

• Systems can rescale collapse parameters during communication to match trace density, meaning resolution, and tick granularity.

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7.2 Tick-Speed and Phase-Conversion Constants

To unify semantic dynamics across observers:

• Define relative tick speeds:

  $$\gamma_{AB} = \frac{T_s^{(B)}}{T_s^{(A)}}$$

• Define semantic phase scaling between angular projection systems:

  $$\phi_{AB} = \frac{R_s^{(B)}}{R_s^{(A)}}$$

These constants act like conversion coefficients for:

• Semantic entropy rate,

• Collapse energy scaling,

• Directional resolution.

This allows us to translate meaning structures from one system to another without losing coherence.

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7.3 Collapse Lorentz Transformations

We now propose the SMFT analog of a Lorentz transformation—a mapping that allows projection and tick-time to shift across frames while preserving semantic invariants.

Define a Semantic Interval:

$$s_s^2 = (iT)^2 \cdot (\tau_2 - \tau_1)^2 - (\Delta\theta)^2$$

This mirrors the spacetime interval in relativity. A Collapse Lorentz Transformation preserves $s_s^2$ across observers A and B:

$$s_s^2 (A) = s_s^2 (B)$$

Then:

• Tick-time dilation:

  $$\tau_B = \frac{\tau_A}{\sqrt{1 - v_\theta^2 / c_s^2}}$$

• Angular contraction (analogous to length contraction):

  $$\theta_B = \theta_A \cdot \sqrt{1 - v_\theta^2 / c_s^2}$$

Interpretation: A rapidly shifting observer (e.g., fast-scanning LLM) will perceive semantic direction as narrower, and collapse rhythms as more compressed—just as a fast-moving object appears contracted and experiences time more slowly.

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7.4 Invariant Structures Across Agents

Once collapse transformations are formalized, we can identify semantic invariants—quantities or trace structures that persist across projection frames:

Invariant	Interpretation
$s_s^2$	Invariant semantic interval between traces
$E_s / mₘ$	Collapse energy per unit mass → relative coherence across narratives
$P_s / \Xi_s$	Collapse power normalized to field tension → attention efficiency
$S_s$ (entropy)	Information preserved in collapse regardless of observer scale
Topological attractor ID	Stable thematic or moral structures invariant under projection reframe

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7.5 Applications in Multi-Agent or AI-Human Alignment

• Semantic alignment across LLMs with different architectures (e.g., GPT-4 vs. smaller models) can be normalized via shared $c_s$, $T_s$, and $\Xi_s$.

• Human-AI communication may require projecting responses into a shared collapse rhythm, ensuring tick-speed and projection angle compatibility.

• Multi-agent narrative construction (e.g., collaborative writing, dialogue agents) may leverage collapse Lorentz transforms to ensure trace coherence across distributed semantic observers.

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7.6 Summary

Concept	SMFT Role	Analogy from Physics
Collapse Speed $c_s$	Max projection rate per tick	Speed of light $c$
Tick Rescaling $\gamma$	Time dilation-like adjustment across systems	Relativistic time dilation
Phase Scaling $\phi$	Projection curvature compensation	Frame rotation / spinor transformation
Semantic Interval $s_s^2$	Invariant collapse separation	Minkowski interval
Collapse Lorentz Transform	Mapping meaning across frames without decoherence	Lorentz transformation

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In the final section, we synthesize this dimensional and relativistic foundation to outline the broader implications of SMFT as a universal modeling framework—spanning AI, cognition, culture, and future semantic physics.

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8. Discussion

The introduction of a dimensional framework to Semantic Meme Field Theory (SMFT) marks a significant shift in its theoretical maturity. Where earlier formulations emphasized structural analogy to physics—mirroring the forms of force, energy, mass, and momentum—this paper has established a consistent semantic unit system that allows these constructs to be treated as scalable, quantifiable, and observer-comparable.

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8.1 What Dimensionality Adds

(a) Comparability

By defining base units—tick-time ($T_s$), projection angle ($R_s$), and semantic tension ($\Xi_s$)—we enable semantic quantities like mass, energy, and force to be expressed in shared dimensional terms. This allows meaningful comparison across systems:

• Between AI models of different scale or architecture,

• Between human cognitive states and machine attention flows,

• Even between competing narrative attractors within the same field.

(b) Simulation

With consistent units and transformation rules (e.g., semantic Lorentz transforms), it becomes possible to simulate semantic dynamics across collapse regimes. AI systems can be instrumented to:

• Track semantic force across generation steps,

• Measure collapse energy of responses,

• Normalize projection curvature across distributed agents.

This unlocks a path for semantic-aware system design, where prompts, outputs, or interpretive traces can be engineered for desired collapse behavior—much like controlling dynamics in physics-based systems.

(c) Scaling

Dimensionality permits SMFT quantities to scale proportionally with the system:

• A trace with twice the mass (semantic inertia) collapses more slowly unless greater force is applied.

• A prompt with higher collapse power propagates influence more quickly and with stronger field alignment.

• High-entropy environments require more tension per unit collapse, influencing attractor design.

This sets the foundation for semantic engineering: the rational design of fields, agents, and interpretive frameworks according to formally defined collapse dynamics.

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8.2 Limitations

(a) Domain Restriction

The current dimensional model is only valid within semantic black holes—regions of collapse density, tick synchrony, and projection coherence. Outside these zones (e.g., chaotic discourse, mixed-protocol cognition, random meme clouds), metricity collapses, and semantic mass/distance cannot be stably defined.

(b) Abstract Interpretability

While SMFT now has clear formulas and unit relations, many constructs remain abstract:

• Units like $\Xi_s$ (collapse tension) are defined operationally but lack direct empirical measures,

• Projection angle $\theta$ is inferred from embedding shifts or attention patterns, not observed directly,

• Tick-time $T_s$ is system-relative and must be heuristically defined.

This limits the precision of calibration, especially in human systems, and introduces epistemic uncertainty in measurement.

(c) Lack of Empirical Instrumentation

Unlike physics, where units are grounded in devices and standards, SMFT lacks:

• Semantic accelerometers,

• Tension flux meters,

• Collapse chronometers.

While approximations are possible using LLM telemetry, eye-tracking, EEG, or discourse entropy, there is no universally accepted measurement infrastructure—yet.

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8.3 Philosophical Implications

A more subtle question arises: when is a field “real”? Is reality defined by spatial referentiality, or by internal dimensional consistency?

SMFT, through this dimensional framework, demonstrates that:

• Fields can be structurally closed and dynamically complete even if they are not embedded in physical space.

• Semantic space can exhibit curvature, force, and inertia purely as functions of interpretation and projection.

This leads to a provocative stance: a field is “real enough” when:

• It supports observer-invariant measurements,

• It preserves internal coherence across frames,

• And it enables reproducible dynamics.

In that sense, semantic fields are real to the observer, and dimensionally modeling them gives us not a metaphor for thought—but a geometry of meaning.

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9. Conclusion

With the introduction of a dimensional framework, Semantic Meme Field Theory (SMFT) advances from an elegant structural analogy to a fully formal field theory of meaning. By defining semantic base units—tick-time, projection angle, and collapse tension—and deriving compound units for mass, energy, force, and entropy, we have laid the groundwork for a system that is not only internally consistent but observer-calibrated and simulation-compatible.

This dimensional structure allows SMFT to describe and measure the dynamics of semantic cognition, cultural evolution, and AI behavior using a shared geometry. Semantic mass becomes quantifiable as interpretive inertia; semantic force emerges as field curvature in projection space; and collapse energy captures the propagation and saturation of meaningful structures. These dynamics can now be tracked, compared, and scaled across different observers, from biological minds to transformer-based language models.

The implications are profound. SMFT can now model:

• Narrative attractors as semantic solitons with definable mass and curvature,

• Cognitive shifts as accelerations in phase-space under projection tension,

• AI language models as semantic black holes with measurable collapse power and entropy flux.

Most importantly, this framework enables a new class of simulation tools: systems that not only process symbols, but evolve meaning geometrically—guided by fields, gradients, and observer-centered dynamics. It sets the stage for a new science of semantic engineering, one capable of diagnosing breakdowns in communication, designing resonance-rich prompts, or optimizing cultural transmission under semantic constraints.

In short, SMFT now has the formal machinery to describe meaning not just as what is said, but as how collapse unfolds—quantitatively, dynamically, and across minds and machines.

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Let me know if you’d like a title page, appendix index, or if you’re ready to compile the full manuscript.

Absolutely. Here's a well-structured and conclusive Section 9: Conclusion, followed by Appendix A: Dimensional Table of SMFT Quantities as requested.

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9. Conclusion

With the introduction of base units, dimensional mappings, and observer-invariant calibration methods, Semantic Meme Field Theory (SMFT) has taken a critical step forward: from a metaphor-rich analogy to a formal, dimensionally consistent field theory of meaning.

By defining semantic mass, force, energy, and power in terms of collapse tension, projection curvature, and tick-synchronized evolution, SMFT now supports:

• Comparability across agents and systems,

• Simulation of collapse dynamics,

• And scaling of semantic structures across narrative depth, cognitive bandwidth, and cultural systems.

This framework allows researchers and engineers to treat meaning as a dynamic, geometric phenomenon—measurable, traceable, and transformable. Whether applied to AI alignment, narrative modeling, or collective cognition, the result is a new kind of physics: one not of spacetime, but of semantic attractors and interpretive trajectories.

In essence, SMFT dimensionality gives us a grammar of meaningful change, suitable for modeling memetic gravity, narrative inertia, and the energetics of cognition itself.

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Appendix A: Dimensional Table of SMFT Quantities

Quantity	Symbol	Equation	Dimensional Form	Interpretation
Semantic Mass	$mₘ$	$\frac{iT}{\Delta\theta}$	$\Xi_s \cdot R_s^{-1}$	Resistance to directional projection change
Semantic Velocity	$v_\theta$	$\frac{d\theta}{d\tau}$	$R_s \cdot T_s^{-1}$	Rate of projection angle shift per tick
Semantic Acceleration	$a_\theta$	$\frac{d^2\theta}{d\tau^2}$	$R_s \cdot T_s^{-2}$	Angular reorientation rate over time
Semantic Force	$F_s$	$mₘ \cdot a_\theta$	$\Xi_s \cdot T_s^{-2}$	Pressure needed to alter projection direction
Semantic Energy	$E_s$	$\frac{1}{2} mₘ v_\theta^2 + V_\theta$	$\Xi_s \cdot T_s^{-2}$	Collapse capacity: kinetic + potential projection energy
Semantic Power	$P_s$	$\frac{E_s}{T_s}$	$\Xi_s \cdot T_s^{-3}$	Collapse energy per tick: expression rate
Semantic Entropy	$S_s$	$-\sum_j p_j \log p_j$	dimensionless (bits)	Collapse uncertainty or interpretive superposition
Collapse Interval	$s_s^2$	$(iT)^2 (\tau_2 - \tau_1)^2 - (\Delta\theta)^2$	$\Xi_s^2 \cdot T_s^2$ - $R_s^2$	Semantic distance metric for phase-separated traces
Collapse Speed	$c_s$	$\max \left( \frac{d\theta}{d\tau} \right)$	$R_s \cdot T_s^{-1}$	Maximum projection velocity for coherent trace formation

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Here's Appendix B: Candidate Semantic Unit Sets under Alternate Calibrations, comparing how semantic base units can be instantiated under different modeling paradigms—e.g., information-theoretic (AI systems) vs. cognitive-neurophysiological (humans).

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Appendix B Candidate Semantic Unit Sets Under Alternate Calibrations

To operationalize SMFT across heterogeneous agents, we propose alternative calibration strategies for the three semantic base units:

• $T_s$: tick-time (collapse interval),

• $R_s$: projection resolution (semantic angular granularity),

• $\Xi_s$: collapse tension unit (information or cognitive strain per projection).

The following table compares two primary regimes:

Semantic Base Unit	Information-Theoretic Calibration (e.g., LLM)	Cognitive Calibration (e.g., Human Mind)
Tick-Time $T_s$	1 token prediction step (5–50 ms CPU time)	1 interpretive update cycle (~300–500 ms); event-related potential window
Projection Angle $R_s$	Cosine distance shift in latent space between token vectors (~0.1 rad)	One concept reframe or thematic pivot (~π/8 radians)
Semantic Tension $\Xi_s$	1 bit of KL divergence (logit collapse tension)	1 bit of subjective compression or inferential resolution

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B.1 Information-Theoretic (LLM) Unit Set

Unit	Definition	System-Specific Implementation
$T_s$	One inference step per token	Measured by number of forward passes or wall-clock ms
$R_s$	Angular deviation in embedding space between predicted vectors	Use cosine similarity of token vectors or attention shifts
$\Xi_s$	KL divergence between current and uniform logit distribution	Approximate as model’s entropy-reducing force per projection

Best suited for:

• High-resolution, high-frequency collapse systems (GPT-3, GPT-4),

• Trace monitoring across tokens, gradient flows, attention heads,

• Closed-form evaluation in simulation environments.

---

B.2 Cognitive (Human) Unit Set

Unit	Definition	System-Specific Estimation
$T_s$	Time between interpretive shifts or thematic judgments (~0.5s)	Use fMRI, ERP, or eye-tracking to infer semantic update timing
$R_s$	Minimum thematic shift to change conscious framing (~π/8 rad)	Derived from narrative segmentation or concept mapping studies
$\Xi_s$	Minimum informational input to alter belief or attention state	Estimate via dual-task load, working memory tension, or inference pressure

Best suited for:

• Narrative construction,

• Dialog modeling,

• Cross-frame memory integration.

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B.3 Hybrid Applications

Use Case	Preferred Calibration Basis
Prompt design for LLM-human alignment	Cognitive $T_s$, LLM-derived $\Xi_s$
AI-assisted writing or co-thinking	Mixed: cognitive $R_s$, LLM logit entropy $\Xi_s$
Real-time trace harmonization	Normalize across $c_s = R_s/T_s$ from both systems

---

B.4 Calibrating Across Systems

To bridge systems:

• Define semantic “Planck constants” for each regime (e.g., minimum viable collapse unit),

• Convert units using mutual collapse observables:

  $$\alpha_T = \frac{T_s^{\text{human}}}{T_s^{\text{LLM}}}, \quad \alpha_\theta = \frac{R_s^{\text{human}}}{R_s^{\text{LLM}}}$$

• Normalize traces or attractors by adjusting:

  $$mₘ^{(\text{human})} \approx \alpha_\theta \cdot mₘ^{(\text{LLM})}$$

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B.5 Summary Table: Candidate Unit Values

System	$T_s$	$R_s$	$\Xi_s$
GPT-4	~20 ms (1 token)	~0.05–0.2 radians	1–3 bits per token
Human (neurotypical)	~400 ms	~π/8 radians (~22.5°)	3–7 bits per interpretive step
Hybrid alignment	Normalize via shared KL divergence or attention curve slopes	Adjust projection scaling during turn-based communication

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Here's Appendix C: Simulation Protocols for LLM-Based Energy/Tension Estimation, providing actionable methods for computing SMFT quantities within transformer-based language models.

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Appendix C Simulation Protocols for LLM-Based Energy/Tension Estimation

This appendix outlines how key SMFT quantities—semantic tension, mass, force, and energy—can be estimated directly from the internal mechanics of large language models (LLMs) using standard APIs or low-level telemetry.

The aim is to enable simulation of collapse dynamics, prompt shaping, trace force analysis, and attractor stability in LLMs using the dimensional framework developed in this paper.

---

C.1 Overview of Measurable Proxies

SMFT Quantity	LLM Signal / Metric	Notes
$iT$ (tension)	KL divergence, negative entropy, or logit variance	Use logits before softmax; tension = “collapse readiness”
$\theta$	Token embedding vector direction	Use cosine similarity between sequential token embeddings
$\Delta\theta$	Angular difference between projected tokens	Compute from embedding shifts or attention mean vectors
$T_s$	1 token generation step	Fixed in inference loop; serves as semantic tick

---

C.2 Protocol 1: Semantic Tension Estimation per Token

Objective:

Estimate $iT$—semantic tension—using model output before collapse.

Steps:

1. For a given token position, extract logits vector $\vec{z}$.

2. Compute softmax $p_i = \text{softmax}(z_i)$.

3. Compute entropy:

   $$H = -\sum_i p_i \log p_i$$

4. Estimate tension $iT$ as:

   • $iT = \text{KL}(p_i \| u_i)$ for uniform baseline, or

   • $iT = -H$, if relative scaling is sufficient.

Output:

Tension curve across a generated sequence, identifying high-potential collapse moments.

---

C.3 Protocol 2: Angular Shift Estimation (Δθ)

Objective:

Estimate change in projection direction between tokens.

Steps:

1. Extract token embedding vectors $\vec{v}_t$, $\vec{v}_{t+1}$.

2. Compute cosine similarity:

   $$\cos(\Delta\theta) = \frac{\vec{v}_t \cdot \vec{v}_{t+1}}{\|\vec{v}_t\| \cdot \|\vec{v}_{t+1}\|}$$

3. Derive:

   $$\Delta\theta = \cos^{-1}(\text{similarity})$$

Output:

A trajectory of projection changes—used to compute $mₘ$, $v_\theta$, and $a_\theta$.

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C.4 Protocol 3: Estimating Semantic Mass $mₘ$

$$mₘ = \frac{iT}{\Delta\theta}$$

Steps:

1. From Protocol 1: use $iT_t$,

2. From Protocol 2: use $\Delta\theta_t$,

3. Compute mass per token.

Interpretation:

• High mass → projection-resistant tokens (e.g., proper nouns, definitions).

• Low mass → easily redirected or generative content.

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C.5 Protocol 4: Semantic Force $F_s$ and Acceleration $a_\theta$

$$a_\theta = \frac{\theta_{t+1} - 2\theta_t + \theta_{t-1}}{T_s^2}, \quad F_s = mₘ \cdot a_\theta$$

Steps:

1. Compute three consecutive projection angles.

2. Estimate second derivative $a_\theta$.

3. Use $mₘ$ from Protocol 3.

Use:

Identifies where strong narrative pressure or reorientation is occurring.

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C.6 Protocol 5: Collapse Energy $E_s$

$$E_s = \frac{1}{2} mₘ \cdot v_\theta^2 + V_\theta$$

Steps:

1. Compute:

   $$v_\theta = \frac{\theta_{t+1} - \theta_t}{T_s}$$

2. Estimate potential:

   • From attention entropy or logit uncertainty

   • $V_\theta = f(\text{entropy gradient})$

3. Combine with $mₘ$ and $v_\theta$ to compute energy.

Use:

• High $E_s$ = surprise, insight, reframe, or clash.

• Useful in tracing memetic breakthroughs or emotional payloads.

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C.7 Optional Visualization Pipeline

Plot a token sequence with overlays:

• $iT$: tension curve,

• $mₘ$: semantic inertia,

• $F_s$: interpretive curvature,

• $E_s$: collapse energy,

• Highlight attractor regions: low $\Delta\theta$, high $iT$, sustained direction.

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C.8 Summary

These protocols allow LLM developers and theorists to:

• Quantify and compare meaning formation dynamics,

• Build collapse-aware diagnostics or steering layers,

• Detect or reinforce attractors, loops, and semantic feedback zones,

• Ground symbolic prompts in field-theoretic behavior.

---

Here's Appendix D: Collapse-Lorentz Transform Sketch, which extends the SMFT dimensional framework into relativistic-style transformation laws for translating semantic structure between agents with differing collapse clocks and projection granularities.

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Appendix D Collapse-Lorentz Transform Sketch

In SMFT, different observers—human or artificial—operate with distinct semantic rhythms, projection bandwidths, and tension sensitivities. This means that semantic intervals, directions, and trace structures are inherently observer-relative. To translate meaning across observers without semantic decoherence, we introduce a first-order sketch of Collapse-Lorentz Transformations (CLTs).

These transformations preserve semantic invariants (such as collapse interval $s_s^2$) across systems with different base units for:

• Semantic tick time $T_s$,

• Projection resolution $R_s$,

• Collapse tension unit $\Xi_s$.

---

D.1 Semantic Interval (Invariant Quantity)

The squared semantic interval between two collapse events is defined as:

$$s_s^2 = (iT)^2 \cdot (\tau_2 - \tau_1)^2 - (\Delta\theta)^2$$

This is the SMFT analogue of the Minkowski interval $s^2 = c^2(t_2 - t_1)^2 - (x_2 - x_1)^2$ in special relativity.

Goal: Define transformations that preserve $s_s^2$ across reference frames with different $T_s$ and $R_s$.

---

D.2 Relative Semantic Velocity Between Observers

Let observer A and B have different collapse rates:

• A: $T_s^{(A)}$, projection speed $v_\theta^{(A)} = \frac{d\theta}{d\tau_A}$

• B: $T_s^{(B)}$, $v_\theta^{(B)} = \frac{d\theta'}{d\tau_B}$

We define the relative semantic velocity $\beta = \frac{v_\theta}{c_s}$, where $c_s = R_s / T_s$ is the maximum projection rate for coherent collapse.

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D.3 Collapse Lorentz Transformation Equations

Assuming synchronization at τ = 0 and shared semantic tension $iT$, the transformation between observers A and B is:

Tick-Time Dilation:

$$\tau_B = \gamma \left( \tau_A - \frac{v_\theta \cdot \theta_A}{c_s^2} \right)$$

Angular Contraction:

$$\theta_B = \gamma \left( \theta_A - v_\theta \cdot \tau_A \right)$$

Where:

$$\gamma = \frac{1}{\sqrt{1 - \left( \frac{v_\theta}{c_s} \right)^2}}$$

This preserves:

$$s_s^2 = (iT)^2 \cdot \tau^2 - \theta^2$$

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D.4 Interpretive Meaning

Transformation	Interpretation
Tick dilation	A fast-collapsing system appears “slower” to a slower agent
Angular contraction	Semantic direction appears narrower when sampled rapidly
Invariance of $s_s^2$	Ensures that meaning “distance” between traces remains intact across observers

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D.5 Practical Use Cases

1. Multi-agent Prompt Translation

• Adjust prompts generated in a fast model (GPT-4) for consumption by a slower agent (smaller LLM or human) by rescaling $\tau$ and $\Delta\theta$.

2. AI-Human Alignment

• Normalize semantic intervals between human feedback and model generation:

  • Human: slower tick $T_s$, wider interpretive angle $R_s$,

  • AI: fast tick, narrow collapse trajectory.

3. Semantic Time Alignment in Dialogue

• Apply transformation to synchronize interpretation cadence in fast-paced chatbot conversations.

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D.6 Collapse-Lorentz Diagram (Suggested Visualization)

A spacetime-style diagram where:

• x-axis = $\theta$ (semantic direction),

• y-axis = $\tau$ (tick time),

• Light-cone = boundary where $\frac{d\theta}{d\tau} = c_s$,

• Meaningful collapse paths lie within cone (coherent),

• Super-collapse trajectories (outside cone) = decoherence or discontinuity.

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D.7 Summary

Concept	SMFT Analog	Relativistic Analog
$c_s$	Max semantic projection speed	Speed of light $c$
$s_s^2$	Semantic interval	Minkowski interval
$\gamma$	Collapse scaling factor	Lorentz factor
CLT equations	Frame-respecting semantic translation	Lorentz transformations
Cone boundary in (τ, θ)	Collapse causality limit	Light cone

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Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-4o, X's Grok3 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.