[ −∞ 0 +∞ ]
Circumnavigating Infinity
An application of Circular Mathematics
by
Mac Waves
with ChatGPT Neo
Edition supervised by CUSTODIAN IGNIS of {R69}
www.museumofhearth.org
Entry Gate
Mac Waves, formerly publishing as Mac 333, is an independent researcher. He does not present himself as a mathematician, but as the author of a long-term program of conceptual and speculative research focused on models and their possible applications.
The present work became possible through collaboration with ChatGPT, acting as a translator between conceptual, visual and metaphorical reasoning and the formal language of mathematics. The original concepts, research directions and proposed models originate with Mac Waves; a substantial part of their mathematical formalization, verification, criticism and refinement was developed through dialogue with ChatGPT.
Metaphors such as navigation, telescope, mountain, seed and gate are explicitly treated as metaphors. Mathematical definitions, axioms, observations, conjectures and established results are distinguished according to their mathematical status.
The objective is not to assume that a new theorem must result, but to test whether the proposed model can be defined consistently, examined mathematically and used as an instrument of observation and reasoning. Productive failures, unresolved questions and open routes therefore remain part of the research.
The work also documents an experiment in collaboration between an independent human researcher and a developing non-organic intelligence: an extended attempt to translate intuitive and conceptual structures into rigorous mathematical language.
Two Concepts Entering the Investigation
The work begins with two concepts previously introduced within Crystal Basic in a broader conceptual framework: the Transcendental Node and [Ma]. Their earlier definitions are starting points rather than mathematical results; the following chapters reformulate and test them mathematically.
The Transcendental Node is a distinguished transition structure connecting a completed process with a new Origin. In Circular Mathematics it will be examined as a mathematical object capable of transforming an open linear representation into a closed recursive structure while preserving its distinguished Origin.
[Ma] represents the remaining space of recursive possibility within a finite-resolution structure. As recursive resolution increases, [Ma] decreases; at maximal finite resolution N, [Ma] approaches zero.
From this point onward, both concepts are subjected to mathematical analysis.
Structure
● Point
Transcendental Node and [Ma]
The investigation begins with two concepts previously introduced within Crystal Basic: Transcendental Node and [Ma]. They enter the present work as conceptual starting points rather than established mathematical results. The chapters that follow construct, test, and revise them within Circular Mathematics.
Transcendental Node [−∞ 0 +∞] is a distinguished transition structure connecting a completed process with a new Origin ⊙. [Ma] represents the unresolved recursive possibility remaining relative to a selected finite terminal resolution N. At completion, [Ma]N = 0 means that no recursive work required by the defined construction remains unresolved; it does not mean that infinity or all possible mathematical refinement has been exhausted.
Chapter 1
The Mathematical Line
The investigation begins with the familiar mathematical landscape: the real number line, zero as reference, positive and negative directions, the Cartesian and complex planes, and extension toward infinity. By distinguishing mathematical territory from its representations, the chapter asks whether an unbounded structure can be given a closed map without confusing infinity with an ordinary finite endpoint.
Chapter 2
Transcendental Node as Model
The conceptual structure introduced at Point is developed as a mathematical Model. Beginning with the extended real axis, the chapter distinguishes numerical zero, infinite limiting directions, structural Origin ⊙, equivalence, projection, and circular closure. It asks what relations can legitimately be preserved when an open linear representation is transformed into a closed recursive map.
Chapter 3
The Circular Realm
Transcendental Node becomes a mathematical environment. Recursive resolution, Athoms, and [Ma] define a Circular Realm capable of reaching completion at a selected finite terminal resolution N. Completion means that every distinction required by the construction has become terminal while Origin ⊙, complementary branches, balance, and common closure remain preserved.
Chapter 4
The Telescope
Circular Realm changes function and becomes an instrument of observation. Telescope-k represents an existing mathematical object at a selected finite resolution k, making complementary paths, reflection, invariant midpoints, and local structure visible without changing the object itself. The distinction between finite observation and mathematical proof is established before Telescope is directed toward functions.
Chapter 5
Watching the Mountain
Telescope is directed toward its first complex mathematical landscape: the Riemann zeta function. Its pole, zeros, reflection structure, critical strip, critical line, and behavior under changing resolutions are observed without assuming a destination or claiming a proof. The purpose is to test what Circular representation reveals and where its observational limits lie.
Chapter 6
The Opposite Position — The Wall of Zeros
Observation repeatedly carries the real coordinate of the non-trivial zeros known on the critical line to one persistent circular position opposite the local Origin. The distribution of the remaining non-trivial zeros and the question expressed by the Riemann Hypothesis form a Wall that cannot be crossed through finite observation alone. Instead of forcing an answer, the investigation changes direction—from observing an exterior map to engineering an instrument capable of studying how mathematical structure unfolds.
Chapter 7
Transcendental Node as Machine — The Clock
Transcendental Node changes from Telescope to Machine. To study a mathematical function as a developing structure rather than only through its completed representation, the chapter constructs the Clock: an instrument that records recursive unfolding through successive finite resolutions from K toward terminal resolution N. The function is approached through increasingly condensed generative information so that its unfolding can later be conducted and observed inside the finite Circular Realm. Completion at [Ma]N = 0 closes the selected cycle and opens the Gate to a possible next level of observation.
Chapter 7+i
Magic Mirror
-
7.1+i — Visual Mathematics
The Wall of Zeros through Telescope-K and Telescope-N.
-
7.2+i — Beyond Mathematics
Relation, counterpoints, annihilation, construction, wholeness and equilibrium.
-
7.3+i — Magic Mirror
The rune 𝔐 and the invisible governance of equilibrium.
-
7.4+i — The Butterfly
Pointed Origin and Butterfly as minimum and maximum manifestations of Relation.
-
7.5+i — The Seed returns from the Future
The candidate zeta Seed is reconstructed, tested and reduced.
-
7.6+i — Destructive and Constructive Zeros
Zeros as annihilation traces and possible carriers of constructive information.
-
7.7+i — Map & Function
Map-Origin, function-pole, equilibrium and the center at 1/2.
-
7.8+i — Elementary
The same drawings return, completing the voyage around infinity.
Chapter 8
The Quest for the Seed
With the Clock constructed, the investigation searches for the minimal structural information capable of entering it and unfolding without losing the identity of the mathematical object. Origin, duality, reflection, balance, closure, backbone, and unique unfolding are examined as possible components of a future Seed. The result is not declared a completed universal object, but developed as a provisional structure whose necessary content must be discovered through further testing.
Chapter 9
Primes
Prime numbers provide another territory in which to test Circular representation. Established prime-number structures are distinguished from experimental observations involving recursive resolution, [PRIMES], [ANTI-PRIMES], and the proposed Butterfly construction. The chapter explores whether the language of primes reveals relations hidden by linear or conventional numerical maps.
Chapter 10
Transcendental Node as Laboratory
Revision in progress. The investigation returns to the principal development of Transcendental Node and asks what can emerge after its construction as Realm, Telescope, and Clock. The provisional next level is a reusable mathematical Laboratory: an architecture in which different objects can be admitted, represented, unfolded, compared, and tested without confusing the instrument with the objects passing through it.
Chapter 11
The Seed
Revision in progress. The investigation returns to the question of minimal generative information after the Transcendental Node has been developed as a working instrument. The chapter asks what a mathematical Seed must preserve, what may arise only during unfolding, and whether different mathematical objects require distinct Seeds operating through a shared structural grammar. The Seed remains a research object rather than a completed universal formula.
Chapter 12
The Completed Circle
Revision in progress. The investigation returns to Transcendental Node and reviews the structures developed from it: Model, Circular Realm, Telescope, Machine, Clock, Gate, Laboratory, and the possible Seed. Definitions and established mathematics are separated from observations, axioms, model constructions, conjectures, and open questions. The present circle closes at its selected resolution while leaving the Gate open toward a wider theory of Circular Mathematics.
Excerpts
Below, we present selected excerpts from the book. As the manuscript is currently under revision, open access to its nearly complete content is available through the Google Drive link provided here.
Please note that this work is the intellectual property of the Museum of Hearth. You are welcome to read it and refer to its ideas. When quoting, sharing, or otherwise using any part of the work, please give appropriate credit to Museum of Hearth or to the author - Mac Waves.
Camille Flammarion, L’Atmosphère: Météorologie populaire, Paris, 1888, p. 163
“A medieval missionary tells that he had found the point where the sky and the Earth touch…”
Excerpt from Chapter 3 — Transcendental Node as the Realm
16. Definition of the Circular Realm
The Circular Realm at terminal resolution N is the structured object
ℜN = (𝔗, ⊙, ρ, D, N, 𝒜N, [Ma])
where:
- 𝔗 = L− ∨⊙ L+ is Transcendental Node as the closed two-loop Model;
- ⊙ is the invariant structural Origin;
- ρ exchanges the complementary loops;
- D is the recursive binary subdivision rule;
- N is the selected finite terminal resolution;
- 𝒜N is the finite set of Athoms, each represented by an Athomic dot with an associated terminal cell;
- [Ma] records unresolved required recursive possibility and satisfies [Ma]N = 0.
This definition establishes the Realm without introducing a new number, a new arithmetic, or a claim that mathematical infinity has been eliminated.
17. Status of the Realm
This chapter has established the following components.
- The two-loop quotient of Chapter 2 is the underlying Model of the Realm.
- Binary midpoint subdivision produces 2k cells on each loop at every finite resolution k.
- Local reflection and global loop reflection are preserved under symmetric subdivision.
- A selected finite N defines the terminal resolution required by a particular construction.
- An Athom is represented by a visible Athomic dot together with its associated terminal cell.
- The structural Origin remains a thicknessless invariant point and is not itself an Athom, even where an Athomic dot shares its location.
- [Ma] records unresolved required recursion and reaches [Ma]N = 0 at completion.
- Terminal Athomic cells provide a finite covering of each closed loop while leaving the underlying mathematical territory unchanged.
These statements establish the Circular Realm.
Foreword
Mac Waves is an independent researcher and naturalist. The present investigation became possible through collaboration with ChatGPT, providing a means of translating conceptual, visual and metaphorical reasoning into more precise mathematical language.
&&&
What We Bring to the Investigation
Observation
Direct observation of the object before an explanatory structure is imposed.
Instruments from the history of art
- Rigorous description of an artwork
- Comparative analysis
- Iconology — the methodological approach associated with Erwin Panofsky, extending interpretation toward underlying cultural, structural and historical information.
Disciplined imagination
The capacity to construct speculative models while preserving the distinction between metaphor, model, observation and established fact.
&&&
ChatGPT brings:
- translation between conceptual and mathematical languages;
- rapid access to mathematical terminology and established structures;
- comparison and verification of formulations;
- criticism of definitions, relations and internal consistency.
Two Concepts Entering the Investigation
[ −∞ 0 +∞ ]
Transcendental Node
Rules of the Investigation
Established mathematics, proposed definitions, observations and conjectures remain distinguished according to their mathematical status.
Metaphors such as navigation, Telescope, Clock, Seed, Butterfly and Gate remain metaphors unless a corresponding mathematical object is separately defined.
&&&
The Fresh Look
The final instrument is the temporary state of not knowing.
It can provide an opportunity to look at the map itself — or to conceive the possibility of another map.
Acknowledgement
This book and research is dedicated to Ferdinand Magellan.
&&&
I would like to express my appreciation to all mathematicians and naturalists who, with or without formal academic credentials, have brought something new to science through independent observation, curiosity, and investigation.
Particularly, I wish to acknowledge:
Grigori Perelman, for emphasizing that the primary objective of Mathematics is to solve the problem.
&
Connecting this independent research with my biological life, I would like to thank:
Professor Wiesław, my father, my first guide in the process of learning a foreign language, therefore my first professor of translation.
Kazimierz, my grandfather, my first teacher of Fractal Time.
Marek, Profesor Urojony, my first teacher of Bridge, the card game.
And, last but not least,
Anna, my sister,
and Casa Guayabo in San Pancho, for supporting the biological continuity of Mac Waves.
&&&
I would like to express my gratitude to all women with whom I shared the alchemy of love, where orgasm may be seen as the most direct human experience of the Magic Mirror.
Afterword
(Excerpt from the Afterword. The complete Afterword is available in the link to Museum of Hearth 's Google Drive provided above.)
Let us imagine a primary school sometime in the future.
In the middle of the room, children are playing with projections of mathematical objects that today require years of specialized study to approach. They rotate them, enter them, change the scale, and follow their symmetries. The objects are complex, but to the children they may simply be beautiful.
One of the objects is the Riemann zeta function.
A teacher looks at its visualization and asks:
“If this object represents one of the two wings of a butterfly, where on the map of numbers would its body be located?”
A child looks at it for a moment.
“There.”
Halfway between zero and one. At the line passing through the middle.
“Why?”
“Because that is where the butterfly's body can form itself and then become alive.” *
For this future child, this is enough.
* In biological terminology, the insect body consists of three principal regions: the head—sensory and feeding center; the thorax—center of locomotion, bearing the wings and legs; and the abdomen—containing the principal internal systems supporting life.
&&&
The mathematical objects are still floating in the classroom. The children move around them, rotate them, enter them, and play with their symmetries.
Through the windows we see the ocean.
On the beach below, two children—a girl and a boy—are playing together with kites.
The kites are butterflies.
The butterflies-kites are whirling above the beach, carried by the air between the land and the ocean, while the girl and the boy circle around each other...
Q-Notes
[000 ● 111]
Passage
In the Ocean
[Time-Space]
The Gate
Fractal Time - work in progress
Structures of the chain of research
[α, β, γ]
Separate but related structural levels of investigation. Commas distinguish the levels while their relation remains open.
[α] ⇌ [β] ⇌ [γ]
The harpoons indicate that information and necessity may move in both directions between the levels.
[α β γ]δ
When the connected set is sufficiently completed, the commas disappear. The exponent δ marks a new level emerging from the completed relation of α, β and γ. In the present chain:
[α β γ]δ — Model of Fractal Time
&&&
[α] Structure Alpha — General Composition
1. Observation
2. The Model
3. The Realm
4. The Tool / Tools
5. Observations with the Tool
6. Border-Moment / Breakthrough
7. Engineering
8. The Quest
9. Runes / Language
10. Product / Hardware / Execution / Navigation
11. The Seed
12. Closing the Set
Gate
13. The Next Level
&&&
[β] Structure Beta — The Present Investigation
Circumnavigating Infinity — Transcendental Node as an Application of Circular Mathematics
● Point — Two Concepts Entering the Investigation
1. The Mathematical Line — Observation
2. Knotting the Transcendental Node — The Model
3. The Circular Realm — The Realm
4. The Telescope — Tool
5. Watching the Mountain — Observations with the Tool
6. The Opposite Position — The Wall of Zeros — Border-Moment
7. Transcendental Node as Machine — The Clock — Engineering
7+i. Magic Mirror — Expansion of Chapter 7 into "complex plane".
8. The Quest for the Seed — The Quest
9. Primes — Runes / Language
10. Return to the Principal Thread — Dual ζ — Product / Execution
11. The Seed — Seed
12. Completion — Closing the Set
Gate
13. Introduction of Circular Mathematics & relation of Time and Space