THE LAWS BETWEEN MATHEMATICS AND REALITY

Lawful Relational Structure, Boundary Selection, and the Physical Expression of Mathematical Possibility

DOI: To Be Assigned

John Swygert

July 16, 2026

Abstract

Mathematics is routinely described as the language of the universe, yet that description leaves unresolved why mathematics corresponds so effectively to physical reality and why only some mathematically permissible states become physically expressed. This paper distinguishes mathematical notation, abstract mathematical structure, and physically instantiated relation. It proposes that the substrate of physical reality is not mathematics as human notation but the lawful relational structure that makes mathematics physically true. Here, “true” does not refer to proof within a formal axiomatic system; it refers to the repeated and measurable correspondence between mathematical relations and physical behavior.

The Swygert Theory of Everything AO, or TSTOEAO, is applied as a proposed grammar governing the transition from mathematical possibility to physical expression. Mathematics can define a state space, but possibility alone does not establish boundary permission, dynamic reachability, cost compatibility, temporal persistence, or accessible equilibrium. This paper therefore introduces a conceptual Expression Condition and seven proposed Laws of Physical Mathematical Expression: relational consistency, boundary selection, route necessity, cost location, temporal qualification, accessible equilibrium, and route-dependent retention.

The resulting framework does not replace mathematics or established physical theories. It proposes a second layer of description: mathematics formalizes lawful relations, while TSTOEAO describes how gradients, boundaries, routes, costs, corrections, and timescales determine which lawful possibilities become physically accessible and expressed. A prospective experimental program is outlined for converting the proposal from philosophical and organizational architecture into a quantitative, falsifiable scientific framework.

1. Introduction

Mathematics can describe the orbit of a planet, the vibration of a molecule, the propagation of light, the development of a biological pattern, the probability of a quantum measurement, and the behavior of an engineered control system. Its effectiveness is so extensive that Eugene Wigner famously described it as unreasonable, or at least deeply mysterious. Max Tegmark later advanced the stronger claim that external physical reality may itself be an abstract mathematical structure.

This paper begins from a related but distinct intuition:

The substrate is not mathematics as notation. The substrate is the lawful structure that makes mathematics true.

The statement requires immediate clarification.

A mathematical theorem does not become formally true because a physical substrate approves it. Formal mathematical truth follows from definitions, axioms, and valid logical operations. The claim made here concerns physical truth: why certain mathematical structures repeatedly and accurately correspond to observable reality.

The marks


1 + 1 = 2

are not themselves the relationship they encode. The symbols can be replaced with different symbols, spoken in another language, represented as objects, or encoded electronically while the underlying relation remains unchanged.

Likewise, an equation describing a field is not the field. A coordinate is not a location. A musical score is not a vibration in the air. A word is not the object, process, or relationship to which the word refers.

Notation is representational.

The deeper question is therefore not merely why nature “uses mathematics.” It is:

What must reality be like for mathematical representation to work at all?

This paper proposes that reality must permit stable distinctions, repeatable relations, constrained transformations, and preserved invariants. Mathematics succeeds because it formally represents those relational features.

The substrate, at its most abstract level, may therefore be understood not as a particular substance, particle, field, equation, or system of notation, but as lawful relational potential: the capacity for distinguishable states to enter constrained relationships and undergo lawful transformations.

Mathematics is the formal language of that lawfulness.

TSTOEAO is proposed as a grammar of its physical expression.

2. Three Different Meanings of Mathematics

The question “Is mathematics the substrate?” cannot be answered until three different things commonly called mathematics are separated.

2.1 Mathematics as notation

This includes:

numerals;
variables;
operators;
diagrams;
coordinate systems;
equations;
written proofs;
software representations;
human naming conventions.

Notation is invented. The symbol “2” is not inherently two. Another symbol could perform the same function.

A physical universe cannot literally be made from ink marks, keyboard characters, or spoken mathematical vocabulary.

Therefore:

Mathematics as notation is not the substrate.

2.2 Mathematics as formal structure

Beneath notation are abstract relationships:

identity;
difference;
number;
order;
ratio;
symmetry;
transformation;
continuity;
discreteness;
topology;
probability;
invariance.

Different notations can describe the same mathematical structure because the structure is defined by relations rather than by the appearance of its symbols. Tegmark’s Mathematical Universe Hypothesis takes this structural position to its strongest ontological conclusion: physical reality is not merely described by mathematical structure but is such a structure.

That possibility cannot be dismissed. However, it is stronger than this paper needs to claim.

2.3 Mathematics as physically instantiated relation

A physical system does not instantiate every mathematical structure. It instantiates particular relations under particular conditions.

A differential equation may permit a family of solutions. A physical system follows one trajectory.

A quantum model may permit several states. A preparation and measurement arrangement produces particular probabilities and recorded outcomes.

A material may possess numerous mathematically permitted configurations. Its composition, temperature, geometry, history, surrounding medium, and available transition routes determine which configurations are physically reached.

The existence of a mathematically valid description therefore does not by itself answer:

whether the state is permitted by the boundary;
whether the state can be reached from the current condition;
whether a transition route exists;
whether required resources are available;
where the consequences of transition are placed;
whether the resulting state persists;
or which equilibrium is accessible.

This distinction creates the central problem of the paper:

Mathematical possibility is not identical to physical accessibility.

3. The Substrate as Lawful Relational Structure

The substrate proposed here is not necessarily a hidden material beneath known matter. It is not presented as a new ether, fluid, field, particle, or spatial layer.

“Beneath” is being used logically rather than geographically.

The proposal is that every expressible reality requires at least four abstract capacities:


\mathcal{S} = \langle D, R, T, I \rangle

where:

represents distinguishable states or distinctions;
represents admissible relations among those distinctions;
represents lawful transformations;
represents invariants or constraints preserved across relevant transformations.

In plain language:

The substrate must permit things to be distinguishable, related, transformed, and constrained.

This formulation does not claim that the tuple is a completed ontology. It identifies the minimum conceptual architecture required for mathematics and physical law to have a stable referent.

Without distinction, nothing can be counted.

Without relation, nothing can be compared.

Without transformation, nothing can change.

Without invariance, no pattern, law, identity, or measurement can persist long enough to be recognized.

The substrate is therefore not mathematics as written.

It is the lawful relational capacity that mathematics can faithfully encode.

4. Mathematics as Syntax and Reality as Semantics

Calling mathematics the language of the universe is useful, but language contains at least two major components:

syntax, governing allowable formal arrangements;
semantics, establishing what those arrangements mean or refer to.

Mathematics provides extraordinarily precise syntax. It defines relations while minimizing the ambiguity found in ordinary speech.

Yet a mathematically valid structure can exist without representing any known physical system. Mathematics permits far more formal structures than the observable universe appears to instantiate.

Physical application therefore requires a correspondence between formal structure and physical relation.

A model becomes physically meaningful when its variables, operations, and constraints reliably correspond to measurable distinctions and transformations.

This produces a three-stage architecture:


\text{lawful relational substrate}
\rightarrow
\text{mathematical representation}
\rightarrow
\text{physical interpretation and test}

The mathematical model preserves selected relationships. The experiment determines whether the physical system actually instantiates those relationships.

The substrate supplies neither English definitions nor written equations. It supplies the stable lawfulness that permits both equations and observations to correspond.

5. The Missing Selection Layer

A mathematical model often identifies what may occur under its assumptions. It does not necessarily explain why one available result becomes expressed rather than another without the addition of:

initial conditions;
boundary conditions;
constraints;
interactions;
available resources;
transition pathways;
timescales;
and stability criteria.

These additions can themselves be written mathematically. The point is not that they exist outside mathematics.

The point is that physical expression depends upon more than abstract consistency.

A state may be:

mathematically describable but physically forbidden;
physically permitted but unreachable from the current state;
reachable but incompatible with available energy, matter, time, or information;
reachable and affordable but unstable;
stable briefly but not over the relevant observation period;
accessible only through a route that changes the final state;
accessible only after the boundary itself is modified.

The gap can be summarized as follows:

Mathematical consistency does not guarantee physical permission.
Physical permission does not guarantee reachability.
Reachability does not guarantee affordability.
Affordability does not guarantee stability.
Stability does not guarantee permanence.

This is the selection problem between mathematics and expressed reality.

6. TSTOEAO as a Grammar of Expression

TSTOEAO organizes physical development through the recurring sequence:


\text{gradient}
\rightarrow
\text{boundary condition}
\rightarrow
\text{route availability}
\rightarrow
\text{correction}
\rightarrow
\text{cost location}
\rightarrow
\text{accessible equilibrium}

6.1 Gradient

A gradient is a difference capable of driving change.

Examples include differences in:

energy;
pressure;
temperature;
concentration;
electric potential;
information;
mechanical stress;
probability weight;
chemical potential;
organizational resources.

A gradient creates potential for transformation, but it does not determine the route by itself.

6.2 Boundary condition

A boundary condition limits, permits, redirects, filters, couples, separates, or transforms interaction.

A boundary may be:

spatial;
material;
energetic;
temporal;
informational;
chemical;
biological;
computational;
institutional.

The boundary is not merely the outer edge of an object. It is the condition at which possible relations are selected.

6.3 Route-space

Route-space is the set of physically accessible transitions connecting one condition to another.

A destination may be mathematically possible while remaining physically inaccessible because no viable route connects the current state to it.

This distinction is central:

Possibility is not accessibility.

6.4 Correction

A correction is the system’s response to an unresolved gradient, instability, incompatibility, or disturbance.

Correction does not always restore a previous condition. It may produce:

adaptation;
redistribution;
transformation;
failure;
collapse;
reorganization;
a new equilibrium.

6.5 Cost location

Every realized transition must satisfy the governing conservation laws and constraints of its domain.

“Cost” is used broadly in TSTOEAO. It may appear as:

energy expenditure;
heat;
entropy production;
delay;
material wear;
information loss;
error;
displaced stress;
reduced durability;
increased risk;
consumption of a limited resource.

The claim is not that every idealized mathematical transformation must dissipate a fixed positive amount. The claim is that actual physical consequences cannot simply disappear.

When one cost channel is suppressed, the consequence may be:

relocated;
delayed;
distributed;
stored;
converted;
or prevented by closing the transition route.

6.6 Accessible equilibrium

A system does not always reach the mathematically global optimum or lowest conceivable energy state.

It reaches what is accessible through its actual boundaries, routes, history, resources, and timescales.

An equilibrium can therefore be:

local;
metastable;
degraded;
dynamic;
temporary;
path-dependent;
optimal only under a restricted route-space.

The physically expressed equilibrium is not simply the best state imaginable.

It is the state reality can actually reach and retain.

7. The Expression Condition

Let:

be the set of mathematically admissible states under the model;
be the states permitted by the current boundary conditions;
be the states reachable from the initial state through available routes;
be the states compatible with available costs and resources;
be the states capable of persisting over the relevant timescale.

Then the set of physically accessible expressions is:


\Omega_E
=
\Omega_M
\cap
\Omega_B
\cap
\Omega_R
\cap
\Omega_C
\cap
\Omega_\tau

A physically expressed state must belong to the intersection of what is mathematically admissible, boundary-permitted, route-accessible, cost-compatible, and temporally retainable.

In ordinary language:

Reality expresses only the mathematical possibilities that its present boundaries, routes, resources, and timescales can support.

The actual state may be represented conceptually as:


x_E
=
\Phi
\left(
x_0,
G,
B,
R,
C,
\tau
\right)

where:

is the initial condition;
is the driving gradient;
is the boundary configuration;
is route-space;
is cost structure and cost location;
is the relevant timescale;
is the physical transformation process.

This is presently a conceptual formalism, not a finished quantitative law. Its purpose is to identify the variables that must be formalized and tested.

8. Seven Proposed Laws of Physical Mathematical Expression

These are not proposed as replacements for arithmetic, algebra, geometry, calculus, or probability.

They are proposed as laws governing how mathematically admissible structures become physically expressed.

Law I: The Law of Relational Consistency

A physically expressed state must satisfy the lawful relations and invariants applicable to the system.

No physical expression may persist while violating the relational structure required for its own existence.

This is the minimum condition linking mathematics and physical reality.

Law II: The Law of Boundary Selection

Boundary conditions reduce the larger mathematical possibility-space to a physically permitted subset.

Changing the boundary can change what becomes expressible without changing the underlying material composition or general governing equations.

The boundary is therefore an active selector of reality, not merely a passive perimeter.

Law III: The Law of Route Necessity

A mathematically permitted state cannot be physically reached unless at least one viable transition route connects the present condition to that state.

The existence of a destination does not create a pathway to it.

Law IV: The Law of Cost Location

The consequences required by an actual transition must be absorbed, transferred, stored, distributed, transformed, or otherwise physically accounted for.

Suppressing a visible cost does not establish that the cost has vanished.

It may have moved.

Law V: The Law of Temporal Qualification

Every statement of stability, mobility, equilibrium, reversibility, or persistence is incomplete without a timescale.

A state can be stable over microseconds and unstable over minutes.

A boundary can be impermeable to one process yet permeable to a slower one.

A material can appear fixed while internal degrees of freedom remain mobile.

Time qualifies expression.

Law VI: The Law of Accessible Equilibrium

A system resolves toward an equilibrium reachable through its available routes under its existing boundaries, costs, and timescales.

The reached equilibrium need not be globally optimal.

It may be the best state available from where the system actually began.

Law VII: The Law of Route-Dependent Retention

When a transition alters stored structure, boundary configuration, internal arrangement, or dissipative history, the route taken may remain encoded in the resulting state.

Two systems with similar final bulk measurements may retain different:

defect structures;
stresses;
coherence properties;
relaxation behavior;
durability;
information;
future route availability.

The past can remain physically present as structure.

9. The Expression Corollary

The seven laws combine into a general corollary:

A mathematically lawful state becomes physically expressed only when the relational requirements, boundary permissions, transition routes, cost conditions, and relevant timescales align sufficiently to produce and retain it.

This provides a possible interpretation of why mathematics can be exact while physical outcomes remain conditional.

The mathematics may be exact.

The route into expression may not be available.

10. Is Mathematics the Substrate?

Three possible answers can now be distinguished.

10.1 Mathematics as notation is not the substrate

Symbols are representations.

They can change while the represented relation remains the same.

10.2 Mathematical structure may be ontologically fundamental

The Mathematical Universe Hypothesis argues that external reality is an abstract mathematical structure rather than something merely described by one. That remains a serious philosophical and scientific possibility.

10.3 The minimum claim required here

The present paper adopts a more restrained proposition:

Physical reality possesses lawful relational structure, and mathematics is the most precise formal language available for representing that structure.

The paper therefore does not require the declaration that the universe is literally mathematics.

It proposes that the universe must be sufficiently lawful, relational, and structurally consistent for mathematics to correspond to it.

The substrate is not the equation.

The substrate is what makes the equation physically applicable.

This may also explain why different mathematical systems can describe different aspects or scales of the same reality. Each formalism may preserve a selected set of relations while omitting others.

Mathematics is not necessarily a photograph of the substrate.

It is a relation-preserving map.

11. Relation to Symmetry, Conservation, Reachability, and Cost

Existing science already contains important parts of this architecture.

Noether’s work formally connected continuous symmetries with conservation relationships, demonstrating how invariant structure constrains physical behavior.

Boundary and initial conditions are indispensable in determining particular solutions to mathematical models.

Control theory distinguishes the possible state-space of a system from the subset of states that can actually be reached under available controls.

Thermodynamics and information theory examine physical costs, irreversibility, entropy, and resource constraints.

Materials science recognizes metastability, hysteresis, defects, phase boundaries, and path dependence.

Quantum mechanics distinguishes state spaces, preparation conditions, interactions, amplitudes, measurements, and accessible outcomes.

TSTOEAO does not claim that these established fields failed to recognize boundaries, pathways, costs, or equilibria.

Its proposed contribution is organizational and cross-domain:

The same grammar may govern how mathematically lawful possibility becomes selectively expressed across many different systems and scales.

This is a second map rather than a replacement for the first.

Classical and Newtonian mechanics remain indispensable where they apply.

Relativity remains indispensable.

Quantum theory remains indispensable.

Thermodynamics, chemistry, biology, materials science, control theory, and information theory retain their established domain-specific equations and methods.

TSTOEAO asks whether their recurring expression conditions can be organized through one transferable grammar.

12. Words, Mathematics, and the Substrate

The linguistic observation that initiated this paper provides a useful analogy.

A word may contain:

a root carrying a central semantic structure;
a prefix redirecting or qualifying that structure;
a suffix determining its expressed grammatical form;
syntax determining allowable relations;
context selecting the intended meaning.

The analogy can be expressed as follows:

Language	Mathematics	Physical Expression
Root or semantic core	State space or relational structure	Encoded potential
Prefix or modifier	Operator or transformation	Boundary-conditioned redirection
Syntax	Formal rules and allowed operations	Route-space
Context	Initial and boundary conditions	Local physical environment
Interpretation	Selected solution or value	Expressed state

The analogy is not proof that linguistic morphology and physics are identical systems.

It reveals a deeper pattern:

A structured possibility becomes meaningful only through relations, constraints, and context.

Natural language contains many possible interpretations. Context reduces that possibility-space.

Mathematics contains many possible formal solutions. Conditions reduce the solution-space.

Physical reality contains many potentially expressible states. Boundaries and routes reduce the physically accessible state-space.

Words, mathematics, and physical reality can therefore be seen as three levels of encoded relation and resolved expression.

Natural language is flexible but ambiguous.

Mathematics is restrictive and precise.

Physical reality is the final test because it either expresses the proposed relation or it does not.

13. Mathematics as the Bridge

Human thought often begins in imagery, analogy, intuition, sensation, or natural language.

Physical science requires those intuitions to become:

clearly defined;
logically consistent;
measurable;
relational;
quantitatively constrained;
experimentally testable.

Mathematics performs this bridge function.

It translates conceptual structures into formal relations that can be compared with observation.

In the present framework:

Words communicate the intuition.
Mathematics constrains the relationship.
Experiment tests the expression.

TSTOEAO is positioned between conceptual language and mathematical implementation. Its grammar identifies the variables and relationships that must be formalized:

gradient;
boundary;
route;
cost;
correction;
timescale;
equilibrium;
expression.

The next scientific task is not merely to describe these elements verbally. It is to construct measurable operators, state variables, and predictions.

14. Engineering Implications

The framework changes the engineering question.

Conventional design often asks:

What component should be built?

The proposed expression framework additionally asks:

What possibility-space should be permitted, and what routes should be made accessible?

This leads to several engineering principles.

14.1 Engineer accessibility, not merely capability

A system may possess a desired state in theory but lack a usable path to reach it.

Engineering must therefore design both destination and route.

14.2 Engineer boundaries as active selectors

A boundary can:

permit;
reject;
filter;
delay;
redirect;
transform;
retain;
couple;
decouple.

Boundary design is state-space design.

14.3 Preserve productive routes while suppressing harmful routes

Total isolation is often unnecessary and counterproductive.

The stronger design may suppress one class of interaction while preserving another.

This is the foundation of Productive Confinement and Boundary Portfolio Engineering.

14.4 Separate mobility from commitment

A system may require temporary mobility for arrangement and later immobility for preservation.

Engineering can exploit different transition timescales:

open a mobility window;
permit rapid organization;
close the movement route;
allow slower stabilizing processes to complete;
retain the selected structure.

This is the foundation of Boundary-Window Synthesis.

14.5 Track where cost goes

A reduction in one visible cost may produce:

hidden heating;
delayed damage;
increased noise elsewhere;
reduced lifetime;
greater control complexity;
shifted environmental burden.

A design is incomplete until its cost locations are mapped.

15. Scientific Predictions and Experimental Program

The proposed framework must move beyond retrospective interpretation.

A decisive test requires a prediction stated before the experiment.

15.1 Boundary-only intervention test

Hold the bulk material and governing domain model substantially constant while deliberately changing a boundary property such as:

geometry;
permeability;
compliance;
roughness;
coupling strength;
phase;
temporal modulation.

Before measurement, TSTOEAO must predict which states or transitions become more accessible and which become less accessible.

Failure of the predicted ordering would count against the proposed grammar or its parameterization.

15.2 Route-closure and cost-relocation test

Identify two or more known transition routes.

Suppress one route without directly altering the destination state.

Predict in advance whether the system will:

use an alternative route;
shift measurable cost into another channel;
slow;
enter a different equilibrium;
or fail to transition.

The distribution of energy, entropy, delay, defects, or error should be measured rather than inferred afterward.

15.3 Equivalent-destination, different-route test

Prepare systems that reach nominally similar macroscopic states through different transition sequences.

Predict which route histories will remain visible through:

hysteresis;
defects;
coherence;
relaxation time;
durability;
response to later perturbation.

The framework predicts route-dependent retention when the route changes stored structure or boundaries.

15.4 Timescale-window test

Create two processes with deliberately separated characteristic timescales.

Use a temporary boundary condition to permit the faster process while suppressing or delaying the slower process.

Then change the boundary to terminate mobility and permit stabilization.

The framework must predict an operational window in advance, including:

opening threshold;
useful duration;
closure threshold;
expected retained state;
expected failure outside the window.

15.5 Cross-domain transfer test

The strongest evidence would not be success in one system after extensive system-specific adjustment.

It would be the successful transfer of the same formal route-and-boundary model across multiple domains with limited modification.

A cross-domain test could compare:

a photonic system;
a phase-changing material;
a chemical assembly process;
a biological transport boundary;
a computational state-transition system.

The quantitative variables would differ, but the proposed relational structure should remain recognizable.

16. What Would Constitute Strong Confirmation?

The framework would receive strong confirmation if it could prospectively:

define the accessible route-space;
identify the decisive boundary variable;
predict the ordering of likely expressed states;
identify where transition cost will appear;
specify the relevant timescale;
predict a result not already compelled by the standard model being used;
survive independent replication.

The framework would be weakened if:

almost any outcome could be reinterpreted as compatible;
route-space could not be defined before measurement;
cost location were assigned only after results were known;
boundary variables failed to predict state selection;
quantitative predictions performed no better than existing theories;
cross-domain transfer required unrestricted ad hoc adjustment.

A grammar becomes science when it risks being wrong.

17. Present Scientific Status

The proposal remains conceptual and partially formalized.

It does not yet provide:

a complete mathematical ontology;
a universal quantitative route metric;
a standardized cost function;
a general expression operator;
a derivation of established physical constants;
a replacement for existing fundamental theories;
or a uniquely confirmed prediction sufficient to establish full unification.

Its present contribution is more specific than a metaphor and less complete than a fundamental physical theory.

It offers:

a structured distinction between mathematical possibility and physical accessibility;
a proposed ontological interpretation of the substrate;
a cross-domain expression grammar;
seven candidate laws;
an initial formal condition;
and a path toward prospective testing.

The claim should therefore remain disciplined:

TSTOEAO may describe a recurring operational layer through which mathematically lawful possibilities become selectively accessible and physically expressed.

Whether that layer is truly fundamental must be decided by quantitative prediction and experiment.

18. Implications for Unification

A unified theory cannot consist only of one more equation unless that equation explains why particular expressions occur across scales.

A complete unification may need to answer two different questions:

What relations are mathematically lawful?
What determines which lawful relations become physically expressed?

Established theories often answer the first question within their domains with extraordinary precision.

TSTOEAO is aimed at the second.

The two questions are connected but not identical.

A final unified framework may therefore require both:

equations describing lawful state spaces and dynamics;
expression laws describing boundary selection, reachability, cost, timescale, and accessible equilibrium.

Mathematics defines the structured field of lawful relationship.

Physical reality expresses a routed history through that field.

Conclusion

Mathematics is not merely a human convenience. Its repeated success indicates that physical reality possesses stable and discoverable relational structure.

Yet mathematical validity alone does not explain physical actuality.

A state can be mathematically permissible while remaining:

boundary-forbidden;
dynamically unreachable;
too costly under current conditions;
unstable;
inaccessible on the relevant timescale;
or excluded by the history of the system.

This paper therefore proposes that the substrate of TSTOEAO be understood, at its most abstract level, as lawful relational potential: the capacity for distinctions, relations, transformations, and invariants.

Mathematics is the formal language through which that structure is represented.

TSTOEAO is proposed as a grammar governing its selective physical expression.

The central claim is:

The substrate is not mathematics as notation. The substrate is the lawful structure that makes mathematics physically true.

And the central expression condition is:


\Omega_E
=
\Omega_M
\cap
\Omega_B
\cap
\Omega_R
\cap
\Omega_C
\cap
\Omega_\tau

Physical expression lies within the intersection of what is mathematically admissible, boundary-permitted, route-accessible, cost-compatible, and temporally retainable.

The universe may not be made of mathematics as written.

It may be made of the lawfulness that mathematics cannot help but reveal.

References

Wigner, E. P. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications on Pure and Applied Mathematics, vol. 13, no. 1, 1960, pp. 1–14.
Tegmark, M. “The Mathematical Universe.” Foundations of Physics, vol. 38, 2008, pp. 101–150. DOI: 10.1007/s10701-007-9186-9.
Noether, E. “Invariant Variation Problems.” Originally published 1918. English translation by M. A. Tavel, 2005. arXiv:physics/0503066.
Landauer, R. “Irreversibility and Heat Generation in the Computing Process.” IBM Journal of Research and Development, vol. 5, no. 3, 1961, pp. 183–191.
Kalman, R. E. “On the General Theory of Control Systems.” Proceedings of the First International Congress of Automatic Control, Moscow, 1960.
Swygert, J. “Encoded Equilibrium and the Architecture of Matter.” TSTOEAO, January 1, 2026.
Swygert, J. “Boundary-Conditioned Reality.” TSTOEAO, May 13, 2026.
Swygert, J. “Architecture, Container, and Expression.” TSTOEAO, June 4, 2026.
Swygert, J. “At the Boundary Condition.” TSTOEAO, June 5, 2026.
Swygert, J. “Boundary-Expression Framework.” TSTOEAO, June 6, 2026.
Swygert, J. “TSTOEAO IV: From Lens to Method.” TSTOEAO, June 19, 2026.
Swygert, J. “Folding and Unfolding Potential Energy and Materials Geometry: Boundary Condition Utility Engineering.” TSTOEAO, June 24, 2026.
Swygert, J. “Transition-Stacked State Locking.” TSTOEAO, July 10, 2026.
Swygert, J. “Productive Confinement.” TSTOEAO, July 10, 2026.
Swygert, J. “Boundary Portfolio Engineering.” TSTOEAO, July 15, 2026.
Swygert, J. “Operationalizing Boundary Portfolio Engineering.” TSTOEAO, July 15, 2026.
Swygert, J. “Frozen Outside, Mobile Within: Boundary-Window Synthesis Through Phase Asymmetry and Timescale Separation.” TSTOEAO, July 16, 2026.
Swygert, J. “Across Scales Without Replacement: The Present Scientific Status, Practical Value, and Remaining Test of TSTOEAO.” TSTOEAO, July 16, 2026.