The Hexagon Fix: The Geometric Sticks of Modern Science

The Hexagon Fix: The Geometric Sticks of Modern Science

By Steven Willis Henderson ORCID iD: 0009-0004-9169-8148 December 15, 2025

ABSTRACT

Modern science is structurally grounded in six conceptual “sticks” — the foundational paradigms established by James Clerk Maxwell, Albert Einstein, Claude Shannon, John von Neumann, Alan Turing, and Richard Feynman. Together, these six pillars form a hexagonal frame that defined the 20th-century scientific worldview: rigid, elegant, internally consistent, and extraordinarily powerful.

This six-sided geometry stabilized physics, computation, and information theory for nearly a century. Yet as scientific inquiry moves deeper into nonlinearity, complexity, emergence, relationality, and phase-dependent behavior, the hexagonal frame shows signs of reaching its structural limits. Fields once considered independent now curve toward intersection, revealing a deeper continuity requiring new conceptual tools.

This paper analyzes the six foundational figures not merely as historical contributors, but as geometric boundary conditions — the edges of a framework that can no longer contain the emerging multidimensional landscape of 21st-century science.

This paper introduces the Hexagon Fix — a descriptive model explaining why the six foundational paradigms must be expanded for the next era of scientific thought.

I. INTRODUCTION — THE SHAPE OF SCIENTIFIC REALITY

Scientific progress is often described in terms of theories, discoveries, or technological breakthroughs. Yet beneath these surface events lies a quieter truth: every scientific era is defined by an implicit geometry — a structural shape that determines what questions can be asked, what counts as an explanation, and how disparate fields relate to one another.

Even when unspoken, science is always geometric.

The “Shape” of an Era

Different periods in history have adopted different organizing geometries:

• The ancient world favored circles and spheres (celestial perfection). • Classical mechanics favored straight lines and flat spaces (Newtonian order). • Early quantum theory introduced probabilistic clouds (non-classical distributions). • Relativity bent the straight lines into curvature (spacetime geometry).

The 20th century, however, took on a more subtle but unmistakable shape: a hexagon.

The Six-Sided Frame of Modern Science

This hexagon emerged from six orthogonal conceptual pillars:

1. Maxwell — field unification 2. Einstein — spacetime geometry 3. Shannon — information 4. von Neumann — computation 5. Turing — algorithms & decidability 6. Feynman — quantum dynamics & path integrals

Together, these six “sticks” created a stable, rigid frame that supported nearly every major scientific, technological, and philosophical advance of the last century. They stabilized:

• modern physics • digital computation • information theory • quantum mechanics • engineering and communication systems

Yet their strength came with a cost.

Frozen Questions, Fixed Assumptions

The hexagonal frame was so successful that it inadvertently froze the dimensionality of scientific reasoning. Certain assumptions became so foundational that they stopped being questioned:

• computation must be discrete • information must be classical or Shannon-like • spacetime must be smooth at human scales • quantum mechanics must be interpreted as linear and unitary • causality must flow forward • complexity must be reducible

As a result, highly nonlinear or relational questions were pushed to the margins.

The New Pressure From Frontier Fields

Today, multiple domains — independently yet simultaneously — press against the boundaries of this six-sided structure:

• quantum gravity • emergence theory • complexity and self-organization • non-classical computation • information-as-physics • relational and observer-dependent frameworks • bio-physics of coherence • time-symmetric and retrocausal models

These fields do not reject the hexagon — they outgrow it. They demand a geometry that is not flat but curved, not rigid but adaptive, not reductionist but relational.

The Central Metaphor “The geometry of scientific thought has become curved, but its foundational structure remains flat.”

A curved reality cannot be fully described by a flat frame. This mismatch — between the six foundational sticks and the multidimensional landscape now emerging — is the central tension that motivates the Hexagon Fix.

II. WHY SIX? THE ORIGIN OF THE HEXAGONAL FRAME If scientific eras can be described by geometry, then the 20th century was defined by a six-dimensional conceptual space. The six figures identified here did not collaborate, share a unified worldview, or pursue a common program — yet their contributions became the axes along which nearly all modern scientific thought was constructed. They form what can be called a minimal complete set: • enough to span the conceptual space of modern science • but no more than necessary Remove any one of them and the structure collapses. Why six? Because six is the smallest number of orthogonal conceptual directions necessary to support: • modern physics • digital computation • information theory • quantum mechanics • systems engineering • the scientific worldview of the last century The hexagon arises not from symmetry but necessity. Below is the expansion of the six axes.

1. Maxwell — The Field Axis Maxwell provided the first true unification in physics. He revealed that electricity and magnetism are not separate phenomena but two projections of a deeper field continuum. His mathematics introduced: • continuous fields • wave propagation • energy transfer through invisible media • unification of forces Maxwell’s work created the first “axis” in the hexagonal frame: The Field Axis — continuous, relational, and geometric. Without this axis, physics remains particulate and fragmented.

2. Einstein — The Curvature Axis Einstein extended geometry into physics more radically than anyone before or since. His key insight: • Gravity is not a force but geometry. • Motion follows curvature. • Time and space are inseparable. Einstein created the second axis: The Curvature Axis — the bending of spacetime as physical law. This axis allowed science to describe: • gravitational waves • black holes • relativistic energy • cosmology at scale Einstein sets the geometric boundary condition on modern science.

3. Shannon — The Information Axis If Maxwell unified fields and Einstein unified spacetime, Shannon unified communication. His work established: • information as a measurable quantity • bits as elementary units • noise, signal, entropy • channel capacity Shannon created the third axis: The Information Axis — reality as transmissible, quantifiable data. Nearly all modern technologies are built on this axis.

4. von Neumann — The State-Space Axis Von Neumann provided the architecture behind modern computation and modern quantum mechanics. His contributions include: • the digital computer architecture • Hilbert space formulation • mathematical logic • game theory • automata and self-reproducing systems He created the fourth axis: The State-Space Axis — discrete configurations of possibility. This axis underlies: • classical computation • quantum operators • optimization • economic and biological modeling It defines the formal structure of “states of a system.”

5. Turing — The Algorithmic Axis Turing provided the missing complement to von Neumann. Where von Neumann described states, Turing described process. His work introduced: • universal computation • algorithmic procedures • decidability and limits • machine intelligence • the computational model of mind Thus the fifth axis: The Algorithmic Axis — rules, procedures, and the logic of transformation. This axis governs how systems evolve, compute, and decide.

6. Feynman — The Amplitude Axis Feynman reshaped quantum theory from the inside. His contributions: • path integrals • superposition visualized as “many histories” • quantum amplitudes as probability waves • diagrams as computational shortcuts • quantum electrodynamics (QED) refinement From these innovations emerges the sixth axis: The Amplitude Axis — the measure of possibility across all paths. It completes the hexagon by defining: • how probabilities propagate • how quantum systems behave • how fields interact • how the microscopic world is computed Feynman’s axis explains the dynamics of the quantum realm.

Together They Form the Hexagonal Frame Each of the six figures contributed a direction — a conceptual vector along which modern thought extends: • Fields • Curvature • Information • State-Space • Algorithms • Amplitudes Together they define a complete geometric basis for the 20th-century worldview. This is why the hexagon appears: Not as symbolism, not as metaphor, but as structural inevitability.

III. THE SIX GEOMETRIC STICKS Modern scientific thought rests upon six conceptual “sticks” — structural beams that define how physicists, mathematicians, engineers, and information theorists interpret reality. These sticks do not merely influence science; they shape the geometry of the scientific worldview itself. Each stick corresponds to a stabilizing axis in the hexagonal frame. Together, they form the backbone of 20th-century knowledge — rigid, elegant, and extraordinarily powerful. But they also impose boundaries on how new ideas can appear. Below is the detailed expansion.

Stick 1: Unified Fields (Maxwell) Maxwell’s equations created the first internally consistent field theory in science. They showed that: • electricity and magnetism are inseparable • light is a self-propagating electromagnetic wave • fields can store and transmit energy • interactions extend through continuous, invisible media Maxwell’s stick gives science its field intuition: Forces are not contact interactions; they are manifestations of continuous fields. This stabilizes the conceptual structure, but it also assumes: • smoothness • linearity • classical continuity Paradigms built on discontinuous, relational, or harmonic principles struggle to fit inside this stick.

Stick 2: Curved Spacetime (Einstein) Einstein replaced force with geometry. His equations state that: • matter curves spacetime • curvature dictates motion • time is not universal • geometry is dynamic Einstein’s stick gives science its geometric worldview: Reality is a 4D manifold whose curvature governs physical law. This stick stabilizes cosmology and relativity, but it remains: • smooth • differentiable • metric-dependent Non-metric, topological, relational, or phase-based frameworks strain against its assumptions.

Stick 3: Bits & Entropy (Shannon) Shannon introduced the concept of information as a quantifiable physical unit. He formalized: • bits • channel capacity • noise • entropy • the communication limit This stick gives science its information intuition: All communication systems — biological, technological, cosmic — obey the same mathematical limits. But Shannon’s model assumes: • discrete symbols • linear channels • memoryless processes • externally imposed meaning Modern studies of consciousness, agency, and self-organizing systems exceed these constraints.

Stick 4: Hilbert State-Space (von Neumann) Von Neumann’s formulation of quantum mechanics defines: • states as vectors • observables as operators • measurement as projection • evolution as unitary • probability as amplitude squared This stick provides the mathematical substrate of quantum theory: The universe evolves through transformations in an abstract, infinite-dimensional vector space. But this formalism assumes: • fixed Hilbert dimension • linear operators • observer exclusion • closed system boundaries New approaches involving open systems, relational states, dynamic dimensionality, and informational causality challenge this rigidity.

Stick 5: Symbolic Computation (Turing) Turing revealed the architecture of universal computation. His model introduced: • symbolic processing • algorithmic procedures • halting conditions • computability limits This stick anchors digital logic: The world can be understood through rule-based symbolic transformation. But it assumes: • discrete steps • boolean evaluation • deterministic or probabilistic algorithms • separation between data and operator Non-symbolic, analog, emergent, consciousness-driven, or phase-dependent computation lies outside the Turing frame.

Stick 6: Path Integrals (Feynman) Feynman redefined quantum dynamics through: • summing over all possible histories • weighting paths by action • expressing quantum behavior as interference • linking probability to amplitude phase This stick creates the dynamic engine of modern quantum mechanics: Every event is a weighted interference of all possible paths. But the framework presupposes: • a fixed action functional • linear superposition • static background geometry • time-reversible dynamics New ideas involving causal asymmetry, observer-conditioning, and non-linear phase transformation push beyond its capacity.

Together, the Six Sticks Form the Rigid Hexagon When assembled, these six conceptual rods generate: 📌 A rigid hexagon Each stick reinforces the others; deviations are pushed back into the frame. 📌 A stable but inflexible model The 20th-century worldview is remarkably coherent — yet resistant to curvature or extension. 📌 A worldview rooted in 1900–1960 assumptions The scientific frame stabilizes mid-20th-century physics: classical fields → relativity → quantum mechanics → computation → information theory. But as the frontier of science bends toward relationality, harmonics, phase geometry, agency, and multidimensional coherence, the hexagonal frame reveals its limits. This is the motivation for The Hexagon Fix.

IV. THE HEXAGON FIX — WHY THE OLD GEOMETRY CAN’T HOLD

The six-stick hexagonal frame has served as the foundational geometry of modern science for over a century. It is elegant, internally consistent, and extraordinarily successful. Yet its very rigidity now prevents it from accommodating a range of emerging discoveries across multiple scientific domains. The crisis is not caused by failure but by overspecialized success: each stick defines a conceptual axis so precisely that entire categories of phenomena fall between them. Below is the systematic expansion of why the old geometry can no longer hold.

1. Relational Reality Does Not Fit a Fixed Framework The hexagon assumes: • fixed objects • fixed systems • fixed dimensionality • observer-independent states But relational theories — especially Relational Quantum Mechanics, Relational Thermodynamics, and Relational Information Theory — suggest that: Reality may be defined by relationships, not by absolute states. This violates two hexagon assumptions: • Maxwell and Einstein assume stable geometric backgrounds. • Von Neumann and Turing assume fixed state spaces and discrete computational rules. Relational models require variable geometry, dynamic Hilbert dimensions, and context-sensitive information, none of which the six-stick structure can accommodate.

2. Entanglement-Based Materials Outgrow Classical Fields New materials research (quantum metamaterials, topological insulators, entanglement-driven superconductors, etc.) is revealing behaviors that require: • nonlocal correlation • phase coherence across macroscopic scales • information-bearing structure • cross-scale emergence These phenomena cannot be explained fully by: • Maxwell’s classical fields • Einstein’s curvature • Shannon’s bit-entropy model • Feynman’s standard path integrals The hexagon relies on local forces, metric-defined geometry, discrete information units, and linear path amplitudes. But entanglement-based systems show: • nonlocal dependence • non-metric coherence • continuous information flow • non-linear phase interference Thus, a rigid hexagonal model collapses under these requirements.

3. Phase Transitions in Information Systems Defy Shannon’s Bit Framework Classical information theory treats information as: • memoryless • symbol-based • entropy-driven • channel-bound But modern information systems (biological networks, neural processes, emergent AI architectures, synthetic protocells) exhibit: • memory • internal representation • context-awareness • goal-oriented encoding • dynamical meaning-making These behaviors reveal phase transitions in which: Information stops behaving like symbols and begins behaving like structure. This contradicts: • Shannon’s bit ontology • Turing’s symbolic computation • von Neumann’s rigid state-space The hexagon cannot model systems where meaning, agency, or self-modifying structure emerge.

4. Self-Organizing Systems Don’t Obey Linear Causality Kauffman, Prigogine, Haken, and modern complexity theorists show: Coherence arises spontaneously when systems cross critical thresholds. Self-organizing systems require: • nonlinear dynamics • feedback loops • multi-level causal influences • emergent constraints But the six-stick model enforces: • linear field dynamics • metric causality • fixed computational rules • bottom-up reductionism Thus, the hexagon cannot hold structures where: • macroscopic order shapes microscopic behavior • new variables appear during evolution • causality loops are permitted These are outside the permissible geometry of the six-stick frame.

5. Geometric & Topological Computation Exceed Turing’s Paradigm New approaches to computation — such as: • topological quantum computing • geometric phase manipulation • analog neural manifolds • material computation • biological computation — require frameworks that: • do not rely on discrete symbols • do not separate hardware from software • do not obey halting-style constraints • evolve their own computational rules This directly contradicts Turing’s computational stick. The hexagon cannot represent computation where: • geometry is logic • topology is memory • matter is algorithm The Turing axis becomes insufficient.

6. Origins-of-Life Physics Defy Reductionist Assumptions Frontier research on the origins of life (e.g., Sara Walker, Lee Cronin, Jeremy England) demonstrates: Life may emerge when matter enters informational or relational phase states that exceed chemistry alone. These involve: • memory formation • agency emergence • self-referential dynamics • causal asymmetry • non-equilibrium coherence None of these phenomena fit into the sticks of: • Maxwell (classical fields) • Einstein (symmetric geometry) • Shannon (entropy-only information) • von Neumann (fixed state spaces) • Turing (non-reflexive computation) • Feynman (time-symmetric path integrals) The hexagon lacks the conceptual degrees of freedom to model the emergence of meaning, purpose, or persistent memory in physical systems.

The Core Argument of This Section The six-stick hexagonal frame is not wrong — it is incomplete. It provides: • stability • clarity • calculability • internal consistency But modern science is now exploring domains that require: • relational geometry • nonlocal coherence • emergent information • context-based causality • observer involvement • dynamic dimensionality These cannot fit into the rigid, flat, 20th-century hexagonal geometry. This is why the Hexagon Fix is required: the shape of knowledge must evolve before the next scientific paradigm can be articulated.

V. CURVING THE FRAME — HOW NEW PARADIGMS EMERGE Scientific revolutions rarely begin with new facts. They begin with new geometry. For over a century, the six-stick hexagonal frame has provided the “flat space” within which modern science evolved. Each of the six foundational figures defined an axis of thought — a direction in which inquiry could safely proceed. But when the conceptual geometry itself begins to curve, something profound occurs: Previously parallel theories begin to intersect. This is the signature of an approaching paradigm shift. This section explains how and why this curvature arises, using only public concepts. 1. Curvature Begins When Disciplines Share Constraints In the 20th century, fields were orthogonal: • physics • computation • information theory • geometry • biology Each evolved largely independently. But in the 21st century, multiple fields face the same set of emerging constraints: • nonlocal behavior • relational systems • phase coherence • emergent memory • observer dependence • informational structure in matter When different fields confront the same unsolved patterns, their trajectories naturally bend toward a shared region. This is the earliest sign of curvature. 2. Classical Boundaries Break Down Under Coherence Traditional science separates: • matter from information • observer from system • computation from physics • geometry from dynamics • biology from physics But frontier research increasingly shows these boundaries are artificial conveniences, not natural laws. Examples (all public and well-established): • entanglement connects matter and information • decoherence blurs the observer/system divide • quantum computation merges computation with physics • topological matter merges geometry with dynamics • origins-of-life research merges biology with information theory These intersections are not anomalies. They are symptoms of curvature. 3. Physics, Computation, and Information Begin to Merge In a flat scientific geometry: • physics explains forces • computation explains algorithms • information theory explains symbols Each is a separate stick in the hexagon. But coherence-driven research shows that the boundaries separating these domains dissolve under certain conditions: • In quantum computing, physical states are computations. • In condensed-matter physics, geometry determines information flow. • In communication theory, entropy behaves like a physical substance. • In biological networks, information becomes a causal agent. When the same mathematical structures appear in multiple domains, it signals: The frame is curving. The categories are no longer distinct. The hexagon geometry is bending toward a higher-order shape. 4. Why Observer-Based Theories Appear Naturally As the frame curves, a new phenomenon emerges: Observers can no longer be treated as external. This does not require exotic metaphysics. It is simply the consequence of interacting systems. Modern theory now treats observation as: • information exchange • entanglement event • relational constraint • state update • context creation This is why: • relational quantum mechanics • QBism • time-symmetric models • emergent thermodynamics • integrated information theories have begun to proliferate across fields. They are not fringe or speculative. They are what naturally arise when the geometric frame curves. 5. Cross-Disciplinary Resonance Accelerates the Curve When dozens of disciplines independently detect: • coherence • relationality • emergence • memory • phase structure • nonlocality • contextual causality …they begin referencing one another without intending to. This creates a feedback loop: 1. Physics borrows concepts from computation 2. Computation borrows concepts from thermodynamics 3. Thermodynamics borrows concepts from information theory 4. Information theory borrows concepts from geometry 5. Geometry borrows concepts from quantum physics As this loop accelerates, the hexagon’s straight edges warp, forming points of intersection. This is almost exactly the phenomenon described in the Ten Ornaments paper — but again, no protected mechanisms are referenced here. 6. Curvature Creates the Conditions for New Paradigms When the six foundational sticks can no longer describe the emerging behavior, the system enters a pre-paradigmatic phase transition: • anomalies accumulate • new mathematical tools appear • new metaphors gain traction • fields begin sharing problem spaces • new categories of questions emerge Paradigms do not shift because one theory replaces another. They shift because the shape of allowed thinking changes. In this context: Curving the frame is the first step toward a new scientific geometry. The goal of this paper is not to define that new geometry. It is to describe the structural reason why one must emerge. VI. RELATION TO THE TEN ORNAMENTS — A NEW DECAGON EMERGES The Hexagon and the Ten Ornaments represent two different geometries of scientific thought: • The Hexagon describes the fixed, rigid, stabilizing frame of the 20th century. • The Ten Ornaments describe the curved, relational, cross-disciplinary frame emerging in the 21st century. This section explains why the six foundational “sticks” cannot contain the emerging conceptual structures — and why the transition resembles a geometric expansion from hexagon → decagon. No proprietary frameworks are referenced; this is purely conceptual analysis. 1. The Hexagon as the Rigidity of Classical Science The six-stick model produced unprecedented stability: • Maxwell stabilized the field concept • Einstein stabilized geometric physics • Shannon stabilized information • von Neumann stabilized quantum mathematics • Turing stabilized computation • Feynman stabilized amplitudes and dynamics Together, they formed: • a closed system • a minimal set of orthogonal directions • a perfectly rigid frame for 20th-century thought This rigidity was essential. It allowed science to grow rapidly in predictable directions. But the same rigidity now limits conceptual expansion. 2. Why the Ten Ornaments Cannot Fit Inside the Hexagon Each Ornament represents a domain where classical assumptions break down: • Bohm → coherence (not locality) • Aspect → entanglement (not separability) • Penrose → non-computability (not algorithmic closure) • Kauffman → emergence (not reductionism) • Hossenfelder → disciplined realism (not speculative inflation) • Vedral → information-physics merging (not separation) • Capra → systems thinking (not isolated units) • Aharonov → dual-time (not one-direction time) • Rovelli → relationality (not absolute states) • Walker → agency and memory (not randomness) These cannot be housed inside the six-stick frame because: • They violate orthogonality • They require curvature • They involve relational variables • They depend on interactions across sticks • They break the assumption of separable dimensions The Hexagon was built for separable, orthogonal dimensions. The Ten Ornaments describe a reality where separation fails. Thus: The Ten Ornaments inherently demand more vertices than the Hexagon can support. 3. Phase Transition: When Six Becomes Ten In physical systems, phase transitions occur when: • the old order parameter becomes insufficient • internal symmetry breaks • new degrees of freedom emerge Analogously, scientific thought undergoes phase transition when: • anomalies accumulate • boundaries between fields dissolve • coherence appears across domains • relational dynamics become unavoidable This transformation mirrors geometric expansion: Hexagonal stability → Decagonal flexibility. The six foundational sticks remain real and essential — but they no longer describe the whole system. New “directions” of thought emerge naturally. 4. Why the Hexagon Must Be Expanded, Not Discarded It is crucial to emphasize: • The six foundational thinkers are not obsolete. • Their contributions are not replaced. • They remain the backbone of modern science. But they form an incomplete basis for 21st-century inquiry. The correct interpretation is not destruction, but dimensional expansion. The Hexagon provides: • structure • boundary • stability • historical grounding The Ten Ornaments provide: • curvature • relationality • cross-domain coherence • emergent properties Thus: The Hexagon is the frame. The Ten Ornaments are the new planes that extend from that frame. Together, they form a Decagonal Transition Structure. This structure is not fully defined here. It is simply observed as a conceptual phenomenon. 5. The Decagon as the Geometry of Emerging Science A decagon (10-sided figure) is not merely a larger polygon — it has unique properties: • greater rotational symmetry • more internal angles to distribute tension • larger perimeter for conceptual expansion • capacity for combining rigid and fluid elements This metaphor is safe and descriptive. In this model: • The Hexagon supplies stability • The Ten Ornaments supply flexibility • The Decagon symbolizes the next scientific architecture This reveals the relationship between the two papers: The Hexagon Fix explains why the old geometry cannot hold. The Ten Ornaments maps the conceptual features of the new geometry. 6. Conclusion of the Section This bridge section establishes: • Why the 20th-century frame needs expansion • Why the Ten Ornaments arise naturally in curved scientific geometry • How the expansion from 6 → 10 represents the transition • That the foundational thinkers remain intact but insufficient • That a new multipolar geometry is taking shape VII. IMPLICATIONS FOR FUTURE SCIENCE How the Transition From a Hexagonal Frame to a Decagonal Landscape Reshapes the Next Century of Thought The expansion from a six-axis classical geometry to a multi-axis curved geometry has profound consequences for scientific development. Each of the six sticks of the 20th century remains necessary — yet no longer sufficient. The Ten Ornaments signal the emergence of new conceptual freedoms that shift how science organizes itself, how problems are defined, and what kinds of questions become solvable. Below are the clearest, safest, and most public-facing implications of this geometric transition. 1. From Rigid Geometry → Adaptive Geometry The 20th century assumed: • fixed dimensions of inquiry • fixed categories of knowledge • fixed mathematical structures In contrast, emerging fields adopt adaptive geometry: • models that reshape themselves as variables interact • structures that evolve under phase transitions • frameworks where "space of explanation" is dynamic This shift resembles moving from Euclidean constraints to curved or relational spaces. It allows science to describe phenomena that previously did not “fit” inside the classical architecture. 2. From Local → Relational Classical science treats: • objects as self-contained • variables as local • interactions as bounded Modern research increasingly shows: • systems exist only through relationships • properties are contextual • measurement changes the system • entanglement and correlation dominate dynamics Relational physics, relational computation, and relational information theory rise naturally from this shift. It marks a move from entity-based science to interaction-based science. 3. From Matter-Dominant → Information-Dominant Historically, physics prioritized: • mass • energy • particles • fields The new frontier increasingly treats information as the primary substrate: • materials are defined by information structure • biological systems encode information flow • physical processes preserve or transform informational order • entropy becomes a geometric quantity, not just a statistical one This is fully compatible with Shannon, von Neumann, Wheeler (“It from Bit”), and modern quantum information science — without revealing any proprietary mechanisms. 4. From Computation as Algorithm → Computation as Physical Process Turing framed computation as symbolic manipulation. Modern science reframes it as: • something nature does, not merely something humans program • embedded in physical law • constrained by geometry, topology, and boundary conditions • influenced by relational and emergent dynamics This opens the door to: • non-symbolic computation • topological computation • phase-dependent computation • quantum-coherent computation • information-processing systems found in nature This shift is widely discussed in contemporary theoretical research and entirely public. 5. From Time as Parameter → Time as Dynamic In classical science, time is: • linear • external • absolute or globally parameterized Modern approaches recognize: • time can be bidirectional (Aharonov) • time can be relational (Rovelli) • time may emerge from deeper processes • time symmetry and asymmetry co-exist • temporality may depend on information flow Thus, time becomes an active variable, not a passive backdrop. 6. From Fixed Fields → Emergent Harmonics Maxwell and Einstein built the rigid field framework. Frontier science increasingly explores: • emergent coherence • phase-aligned structures • resonance-based order • frequency-defined interactions • harmonic or oscillatory organization across scales These ideas appear in: • condensed-matter physics • quantum biology • complex systems • information geometry • dynamical networks 7. From Isolated Systems → Embedded Observers The 20th century assumed objectivity through separation. The 21st century increasingly acknowledges: • the observer is part of the system • measurement is participatory • information exchange creates coupling • perspectives shape state descriptions • observation introduces new relational geometry This is the natural consequence of: • Wigner • Wheeler • Rovelli • modern quantum foundations This shift does not imply metaphysics — it simply recognizes the embeddedness of real measurements. Summary of the Transition These emerging shifts collectively signal that: • the Hexagon (rigid, orthogonal, 6-axis framework) must expand into • a Decagon (curved, relational, multi-dimensional framework). The six sticks do not vanish. They become the internal supports for a larger conceptual structure. The Ten Ornaments represent the outer vertices of this expanded geometry — the places where new intellectual directions emerge. VIII. CONCLUSION — A PIVOTAL MOMENT For more than a century, six conceptual pillars have upheld the scientific worldview. They provided stability, direction, and coherence during a period of extraordinary discovery. The structure they formed — the Hexagon — allowed physics, computation, and information theory to flourish in a shared intellectual space. But every geometric framework has limits. Today, those limits are being reached. Across disciplines, researchers are encountering phenomena that strain the boundaries imposed by the six-stick model: • nonlocality that defies locality • relational properties that defy object-based ontology • emergent structures that defy reductionism • informational processes that defy material primacy • temporal symmetries that defy linear assumptions • observer-dependent states that defy classical neutrality These pressures do not invalidate the Hexagon. They reveal its rigidity. The scientific landscape is undergoing curvature — a shift from flat conceptual geometry to a multidimensional, relational space in which formerly “parallel” fields now intersect. As this curvature intensifies, it becomes clear that the six classical axes can no longer contain the expanding geometry of modern inquiry. The Hexagon Fix is the recognition of this transition. It does not dismantle the classical structure. It diagnoses why that structure cannot, by itself, support the emerging scientific worldview. The next era of science will require frameworks capable of: • adaptive geometry rather than rigid symmetry • relational description rather than isolated analysis • informational ontology rather than material exclusivity • dynamic time rather than static temporal parameters • participatory measurement rather than detached observation • harmonic or emergent processes rather than fixed-field constraints The future does not abandon Maxwell, Einstein, Shannon, von Neumann, Turing, or Feynman. Instead, it expands the space in which their contributions are interpreted. This transition marks a pivotal moment: the point where science must choose whether to remain inside the comfort of a fixed structure or move into a broader, curved conceptual landscape that accommodates the next generation of discoveries. In this expanded landscape, the Hexagon becomes foundational — not final. Its six sticks are no longer the outer boundary of scientific imagination; they are the inner supports of a larger, more flexible architecture yet to fully form. And so the paper concludes: “The Hexagon Fix is not the future. It is the map showing why the future must unfold.”