Tomabechi Cognitive Cosmology — Theorems 28–32, Plain-Language Complete Edition

How to Read This Book

This is the plain-language complete edition of the academic paper Tomabechi Cognitive Cosmology — A Rigorous Mathematical Proof of the Buddha-Dharma, Theorems 28–32. It rewrites the five theorems proved there so that they can be read from the first page to the last without reading a single formula, and without any prior knowledge of Buddhism.

For readers who do not read formulas: The single most important promise of this book, stated first. Formulas are not code — they are the language of landscapes, journeys, and ledgers. Every formula that appears here will be translated into pictures: valleys, slopes, totals, rooms, cages, ledgers. This orange box is the translator. The green box, "the heart of the proof," gives the skeleton of each proof in one paragraph. Read only the boxes and you can still follow every claim and every argument in this book.

For readers unfamiliar with Buddhism: Words such as "nirvana," "non-self," "the sixfold recurrence," and "absolute other-power" appear throughout. You are not expected to know any of them in advance. Each is defined on the spot, in ordinary language, the first time it is needed. This blue-toned box does that work. And none of the claims in this book leans on the religious authority of these words. Mathematics states only what mathematics can state. The Buddhist terms may be regarded as name-tags attached afterwards to the mathematical conclusions.

This is not, however, a book of metaphors only. Each theorem is stated with the same canonical formulas and the same explicit conditions as the academic edition. Only the long calculations are omitted; no claim, assumption, or conclusion is weakened. For the rigorous complete proofs, each theorem ends with a pointer to the corresponding section of the academic edition.

Three ways through this book: The fastest route — read only the "◆ Central formula" of each theorem, the orange boxes, and the green boxes. That is enough to know exactly what all five theorems claim. The recommended route — add the figures and the "🔍 The proof, one step at a time" boxes. Then you also know why the claims are true. The unhurried route — read the symbol tables and the statements of the conditions as well. From there you can move straight into the academic edition.

How to Read This Book
The Map — Where the Five Theorems Are Going
The Minimum Buddhism, Starting from Zero
A Toolbox for Readers Who Do Not Read Formulas
The Inherited Theorems — Always Beginning from the Standard Form
Theorem 28 — Nirvana Non-Self and Type Coherence Theorem
Theorem 29 — Formal Dharma-System Non-Substantiality and Incompleteness Theorem
Theorem 30 — Entropy-Exchange Ego Constitution Theorem
Theorem 31 — Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem
Theorem 32 — Future-TCZ Homeostasis Absolute-Other-Power Theorem
Five Consistency Checks — Is Anything Contradictory?
Drop a Condition and What Breaks — Minimality and Counterexamples
Frequently Misread Points — Q&A
Table of Theorems (28–32)
A Small Glossary
For Further Study

The Map — Where the Five Theorems Are Going

The map of this book. The five theorems proved here rest on the inherited theorems below, which are already proved.

Here are the five theorems, each in a single line. If by the end of this book all five lines look obvious, the book has succeeded.

The five theorems proved in this book
Theorem	Name	In one line
Theorem 28	Nirvana Non-Self and Type Coherence Theorem	"All is suffering" and "there is a stillness in which suffering has ceased" are not a contradiction but statements about two separate rooms. And nirvana itself has no fixed core either.
Theorem 29	Formal Dharma-System Non-Substantiality and Incompleteness Theorem	If a teaching can be written down as a system of propositions, that system does not close upon itself. Keep adding to it and it still never ends.
Theorem 30	Entropy-Exchange Ego Constitution Theorem	The "I" is a construction that inner language selects and stabilizes. Exactly as the mind orders itself, the physical side must scatter.
Theorem 31	Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem	Language widens the options, but it also becomes a cage. Inside a closed low band, six regions are revisited over and over, with probability one.
Theorem 32	Future-TCZ Homeostasis Absolute-Other-Power Theorem	Place a goal outside your present self and the landscape itself is rebuilt — after which convergence toward it happens without further effort.

The order of proof is not a single chain. Theorems 28, 29 and 30 are three independent branches; only 31 and 32 follow after.

This shape matters. If the five are read as a tower stacked in order, a stumble anywhere seems to bring the whole thing down. That is not the situation. Theorems 28, 29 and 30 leave the other two untouched if any one of them is doubted. Theorem 31 may use the linguistic Ego constitution of Theorem 30 or not, as one likes. Theorem 32 explains how one gets out of the closure described by Theorem 31.

For readers who do not read formulas: The numbering runs from 28 to 32 because this system has already built up twenty-seven theorems. A number is an address inside the system, so the numbers are used as they stand rather than being repacked. Note also that Theorem 13 is a permanent gap and carries no content.

The Minimum Buddhism, Starting from Zero

This chapter defines only the Buddhist terms needed to read this book. Nothing is argued about their truth as religion. They are treated as name-tags traditionally attached to conclusions proved on the mathematical side.

The Four Dharma Seals — the Four Marks That Make Buddhism Buddhism

The Four Dharma Seals. Theorem 28 of this book receives three of them directly: universal unsatisfactoriness (Thm 24), non-self (Thm 25), and nirvanic tranquility (Thm 26).

For readers unfamiliar with Buddhism: The Four Dharma Seals are four marks by which a teaching is recognized as Buddhist. Impermanence — all phenomena change and do not stay. Universal unsatisfactoriness — conditioned existence contains suffering. Non-self — all beings arise interconnected, and no self exists as a fixed substance. Nirvanic tranquility — the stillness in which suffering has ceased is real. These four have already been proved, as Theorems 23, 24, 25 and 26, in a separate paper, Tomabechi Four Dharma Seals Theorems. This book takes over only their conclusions.

What "Suffering" Means — Not Pain

For readers unfamiliar with Buddhism: The suffering (dukkha) of "universal unsatisfactoriness" is not the feeling of hurting or being sad. The original term is closer to "not going as one would have it," "being ill-seated." In the mathematics of this book it becomes far sharper still: the total strain that remains even when one does one's very best. It does not remain because one has failed to try. It is suffering precisely because it remains after one has done one's best. Mathematically this "best-case total" is written J^*. "Universal unsatisfactoriness" means J^* > 0.

What "Non-Self" Means — Not "I Do Not Exist"

For readers unfamiliar with Buddhism: Non-self (anattā) is frequently misread as "there is no such thing as me." It is not that. What non-self denies is a fixed core that would still make you you after every relation had been cut away. A father is a father because there is a son or a daughter. Remove all the threads and the knot is no longer a knot. But the knot itself is certainly there. In this book, self-consciousness is proved to exist, explicitly, by Theorem 16. What non-self erases is only the supposed "core that would survive without the threads."

For readers unfamiliar with Buddhism: When this book says "is non-self," the mathematical side writes ¬Atman(d, a). It reads: "at layer a of object d there exists no internal variable that is independent of the relational description, fixed through the history, responsible for individuation, and causally non-redundant." It is the claim that nothing satisfies all four conditions at once. So it in no way means "d cannot be described as a set."

What "Nirvana" Means — Not an Afterlife

For readers unfamiliar with Buddhism: Nirvana (nirvāṇa) is often misunderstood as a peaceful place one goes to after death. In the mathematics of this book it is not. Nirvana is defined as a set of states reachable while alive, now. In formulas it is written 𝒩_⊤(T), and it means: "within the region where one can remain alive, all states at which the residual suffering at the highest abstraction is exactly zero." Note the restriction "within the region where one can remain alive." Death does not enter this definition.

What "the Sixfold Recurrence" Means — Not Rebirth

The six realms. This book treats them as a mathematical structure: a closed region cycled through with probability one.

For readers unfamiliar with Buddhism: The sixfold recurrence traditionally names the cycle through six destinies — hell, hungry ghosts, animals, asuras, humans, and devas — passed through life after life. The mathematics of this book says nothing whatever about rebirth. What it states is only this: inside a closed finite system, every region is returned to infinitely often, with probability one. Neither the number six nor the names of the destinies come from mathematics. Mathematics supplies the structure "a closed system cycles"; whether six name-tags are attached to it is a matter for doctrine.

What "Absolute Other-Power" Means — Not a Supernatural Force

For readers unfamiliar with Buddhism: Absolute other-power names, in the Pure Land traditions, salvation by a great power that works upon one, rather than by one's own effort. The same idea is transmitted in the esoteric traditions as well. This book treats it neither as a supernatural action nor as physical retrocausation. What is called "absolute other-power" here is only the following fact inside the model: once the Self has set a goal outside the present comfort zone as a terminal condition, the closed loop of the future-restricted Ego converges to the future TCZ on its own, requiring no further effort input from the old Ego. The name is borrowed from tradition; the content is entirely mathematical.

The Terms of This System That Appear in This Book

Terms specific to this system (the minimum)
Term	Meaning
TCZ	Total Comfort Zone. The range within which the mind stays settled. In landscape terms, the shallow floor of the valley.
Abstraction	How widely one is wrapping things up in one's view. Lower is more particular, higher more encompassing. It carries an order, whose top element is written ⊤ (emptiness).
⊤ (emptiness)	The top of abstraction. The smallest roof that covers everything. Here whatever distinguishes individuals disappears.
Presence P	The degree to which a world is felt as here-and-now. A number between 0 and 1.
Ego	The control policy actually deciding behaviour. The driver's seat of the "I."
Self	Situated at a layer above the Ego; the side that fixes the direction of goals.
Goal G	The state aimed at, set as a terminal condition.

For readers who do not read formulas: Every word needed is now in place. What follows is the toolbox. Once you have been through it, every formula in this book becomes a combination of pictures you already know. Readers in a hurry may skip the toolbox and go straight to Theorem 28; whenever a formula appears, the number of the tool needed is given, so you can come back then.

A Toolbox for Readers Who Do Not Read Formulas

Every formula in this book is built from combinations of the twenty-one tools set out below. Once you hold them as pictures, everything that follows is a picture you already know. The first ten are shared with the earlier papers of this system. The remaining eleven are newly needed for reading Theorems 28 through 32.

Tool 1: Landscape and Valley — the Evaluation Function V and the TCZ

For readers who do not read formulas: Think of the state of the mind as a single point x. Now score each state by "how much strain am I under right now"; call that function V(x,t), the evaluation function. Draw the score as height, and the mind becomes a ball rolling on a landscape. The shallow floor of the valley — the range where the score is at or below a threshold θ — is the TCZ, the mathematical identity of "the range where I am comfortable."

The evaluation function V is the landscape. The shallow valley floor (V ≤ θ) is the TCZ.

Tool 2: A Total Is an Area — the Integral ∫

For readers who do not read formulas: The symbol ∫ is doing nothing difficult. It says: not "what is the score right now," but add up every score from beginning to end. Draw the scores as heights and that total becomes the area under the curve. Whenever you see ∫₀^T V dt, read "all the V-scores from time 0 to time T, added up."

The integral is the sum of the areas of all the thin strips — that is, the area under the curve.

Tool 3: Choosing What Makes It Smallest — arg min and π_c

For readers who do not read formulas: arg min answers the question "which one makes it smallest?" Where min names the smallest value, arg min names the thing that produced it — the choice itself. In this book a way of deciding behaviour is called a policy and written π. So π_c = arg min ∫V₀ dt means "choose the policy that makes the total score smallest."

Of five candidates, the one with the lowest total is chosen. That is arg min.

Tool 4: The Condition for Rolling Downhill — Lyapunov Descent

For readers who do not read formulas: To prove that something must eventually fall to zero, there is no need to compute the trajectory. One sentence suffices: "the higher you are now, the faster you fall." In symbols, D⁺Φ ≤ −cΦ. The left side is "the slope going forward" (Tool 18); the right side says "it must drop by at least c times the current height." From this single line follows Φ(t) ≤ Φ(0)e^−ct. Nearly every proof in this book ends by reducing to this shape.

The higher, the faster the fall. That alone forces the value down to zero.

Tool 5: Half, Then Half Again — Exponential Decay

For readers who do not read formulas: The shape e^−ct expresses "a fixed proportion is lost in every fixed interval of time." Half, then half, then half. It never quite reaches zero, but it gets as close to zero as you like. And, importantly, "how many seconds until it comes within δ" can be computed from this shape too (Tool 21).

A fixed proportion lost per fixed interval. That is exponential decay.

Tool 6: Degree of Scatter — Entropy

For readers who do not read formulas: Entropy may be read as "degree of scatter." Left alone, things move from ordered to scattered; the reverse does not happen by itself. To be alive is to keep holding your own shape against that scatter. But it is not free. Whatever you order on the inside is necessarily dumped outside as scatter. Theorem 30 and Theorem 15 tie the "amount dumped" and the "amount ordered" together in one and the same ledger.

From ordered to scattered. To be alive is to keep resisting this.

Tool 7: The Lowest Roof That Covers Everything — the LUB and "Emptiness"

For readers who do not read formulas: The LUB (least upper bound) is not an average. It is the lowest roof that covers everything without discarding any of it. Given my world, your world, and his world, the LUB is the smallest world containing all three. And when this "going up" operation is carried all the way to the top, one arrives at ⊤ (emptiness). In emptiness, whatever distinguishes individuals from one another disappears. Emptiness is not nothingness; it is the greatest element that wraps everything.

The LUB is not an average but the smallest upper bound that discards none. Its apex is ⊤ (emptiness).

Tool 8: Inverse Limit — the One Point Consistent Across Every Layer

For readers who do not read formulas: Of all the tools in this book this is the one whose name feels hardest. Its content, however, is something done in everyday life. We proceed slowly, in seven steps. Once this tool is understood, the skeletons of Theorems 16, 30 and 31 all come into view at once.

1. Why Layers Are Needed at All

For readers who do not read formulas: Asked "who are you?", the answer is not settled at any single grain. "A human being." "A Japanese citizen." "A teacher." "The form tutor of class 3-2." "Someone currently marking papers in the staff room." Every one is correct, and not one of them exhausts the person on its own. This system handles that by way of layers of abstraction: the higher the layer the more it wraps up, the lower the layer the more particular it is.

2. Projection — Spelling Out One Line of an Upper Layer Below

For readers who do not read formulas: The layers are not unrelated. What is said at an upper layer can be spelled out in concrete form at a lower one. This spelling-out is called a projection, written p. For instance, the upper-layer line "I am someone who values honesty" unfolds at a lower layer into particular behaviours: "in this scene, tell the truth," "in that scene, stay silent."

A projection spells out one line of an upper layer below. Hence the further down, the more detail.

For readers who do not read formulas: One property matters here. Projections always run downward. The reverse is not possible, because a single behaviour at a lower layer does not uniquely determine what policy is held above. That someone told the truth today does not establish that they value honesty (it may have been incidental). Information increases on the way down and is not determined on the way up. This is where Tool 6, entropy, connects.

3. A Coherent Family — "One Column"

For readers who do not read formulas: Now pick one image of yourself from each layer: one from the top, one from the layer below, one from the layer below that, and so on. This gives a combination made of one choice per layer. But we demand a condition of it: projecting down what was chosen at the upper layer must land exactly on what was chosen at the lower layer. When that holds for every adjacent pair, the combination is called a coherent family — what this book calls a column.

On the left, a combination that forms a column. On the right, one that disagrees at a layer and does not.

For readers who do not read formulas: In everyday terms, a column is the condition of never contradicting yourself whatever the grain at which you describe yourself. Saying "I value honesty" while lying freely in particular situations is not a column: the upper and lower layers disagree. To have a column is for the self-image not to break down however far abstraction is raised or lowered.

4. The Inverse Limit — Every Column, Collected

For readers who do not read formulas: And the inverse limit is simply the collection of all such columns that are possible. That is all it is. It is written lim_←. What Theorem 16 proved is that this collection is not empty — that at least one column exists.

For readers who do not read formulas: Why does the existence of a column matter so much? Because if there were none, "a self-image free of contradiction" would not be available at all. However one chose, some layer would always disagree. In such a world, speaking of self-consciousness would itself become meaningless. Theorem 16 guarantees that this is not the situation.

5. Why It Is Called an "Inverse" Limit

The projection arrows all point downward; the inverse limit stands upstream of them. Hence the name.

For readers who do not read formulas: The word "inverse" appears because the object stands against the direction of the arrows. The projections all run downward, from upper layers to lower. The inverse limit itself, however, sits upstream of all of them, in a position from which one can descend to any layer. Compared with the operation that lines things up along the arrows (a direct limit), it stands the other way round. There is no deeper meaning; it is a name describing a position.

6. Why a Single Layer Is Not Enough

For readers who do not read formulas: A natural objection arises here. "Why not just look at the topmost layer?" That will not do, for the reason given in step 2: the upper layer does not uniquely determine the lower. "Values honesty" does not settle what to do in this particular scene. And conversely, "why not just look at the bottom layer?" That will not do either: however many particular behaviours are listed, which policy they express is not determined. A self-image is neither one nor the other, but precisely the single column coherent across all the layers.

7. Where This Tool Does Its Work

For readers who do not read formulas: In three places. Theorem 16 — shows the collection of columns is non-empty and then extracts the unique resting place upon it. Theorem 30 — places "language" on that collection of columns. What condition 30-A (commutativity) requires is that a column remain a column after the language acts on it. Were that to fail, the result of applying language would no longer be a self-image at all. Theorem 31 — how high the columns reach is exactly the reachable abstraction. The cage is the condition in which no column extends above a certain height.

In summary: the inverse limit is the set of all columns that never contradict themselves across layers.

Tool 9: Contraction Mapping — Repeat It and You Land on One Point

For readers who do not read formulas: If applying an operation repeatedly shrinks the distance between any two starting points by a fixed factor q (with q < 1) each time, the operation is a contraction mapping. Every contraction mapping has exactly one fixed point — a point that comes back unchanged when you put it in. And wherever you start, you always end up there. The result is called the Banach fixed-point theorem. Every "uniqueness" in Theorem 16 and Theorem 30 comes from it.

If the distance shrinks by a factor q each time, every starting point settles at the same single point.

Tool 10: Mutual Information — Knowing Reduces Not-Knowing

For readers who do not read formulas: The amount not yet known about some matter Z is written H(Z) and called its entropy. Acquire other information M and the not-knowing goes down. What it goes down by is the mutual information I(Z;M). Two properties matter. First, I is never negative — knowing never costs you. Second, I is zero only when M says nothing at all about Z.

Mutual information is the amount of not-knowing removed by knowing. Never below zero.

Tool 11: Tagged Disjoint Union — Never Mix Points That Share a Name

For readers who do not read formulas: From here on, the tools are the new ones needed for Theorem 28 onward. A disjoint union places two sets side by side "without mixing them." In an ordinary union, elements with the same name are collapsed into one. In a disjoint union, each side is first given a tag, so that even identical contents count as different elements when the tags differ.

For readers who do not read formulas: Why is this needed? Say "all is suffering" and "there is a stillness in which suffering has ceased" about one and the same point, and you have a contradiction. But tag them apart and the two become claims about different points from the start; they never compete. This is the "type coherence" of Theorem 28. In formulas it is written with the symbol ⊔.

A different tag means a different point. So "there is suffering" and "suffering is zero" do not contradict.

Tool 12: A Formal System — Axioms, Rules of Inference, Theorems

For readers who do not read formulas: A formal system is a set of three things. Axioms — the sentences granted from the start. Rules of inference — the legal moves from sentence to sentence. Theorems — every sentence reachable from the axioms by following the rules. In chess terms: the axioms are the opening position, the rules of inference are how the pieces move, and the theorems are every position reachable from that opening.

The three parts of a formal system. Theorem 29 starts from "if a teaching can be put in this shape."

Tool 13: Gödel Incompleteness — a Sentence True but Unprovable

For readers who do not read formulas: Two theorems, proved by Kurt Gödel in 1931. The first incompleteness theorem — in any system that is strong enough, whose axioms can be enumerated, and which does not lie about itself, there necessarily exists a sentence that is in fact true but cannot be proved within that system. Nor can its negation be proved. The second incompleteness theorem — such a system cannot prove, within itself, that it is free of contradiction.

For readers who do not read formulas: This is much misread, so let it be said plainly. Incompleteness is not falsity. "There is a sentence it cannot prove" and "that system is wrong" are entirely different statements. Likewise "it cannot prove its own consistency" and "it is inconsistent" are entirely different statements.

Within the set of true sentences, the provable part is necessarily a proper subset.

Tool 14: Kolmogorov Complexity — the Ceiling on Saying "This Is Complex"

For readers who do not read formulas: The Kolmogorov complexity of a string is "the length of the shortest program that outputs it." A simple string can be written by a short program, so its complexity is low; a random string can only be copied out more or less verbatim, so its complexity is high.

For readers who do not read formulas: What Chaitin showed is this. Every system, however strong, has a constant c determined by the system itself, and it can prove sentences of the form "the complexity of this string exceeds n" only when n is at most c. A system, in other words, cannot prove that anything far more complex than its own description is complex. This is the third conclusion of Theorem 29.

The range within which "this is complex" is provable has a ceiling, one per system.

Tool 15: Markov Chains and the Stationary Distribution — You Keep Coming Back

For readers who do not read formulas: A Markov chain is a movement in which "where you go next depends only on where you are now." The route by which you arrived is irrelevant. Picture a board game. Irreducible means "from any square, every other square is eventually reachable."

For readers who do not read formulas: From this a very strong conclusion follows. If the number of squares is finite and the chain is irreducible, every square is returned to infinitely often, with probability one. Moreover, the long-run visiting frequency settles at a fixed positive value. The "sixfold recurrence" of Theorem 31 is this general fact applied to six regions.

Finite and irreducible: every room is returned to infinitely often, with probability one.

Tool 16: Tangent Cone and Nagumo's Condition — Why You Never Cross the Wall

For readers who do not read formulas: There is a closed region K, and a point moving inside it. How does one guarantee that this point never leaves K? The answer is surprisingly simple. While on the wall, the velocity must not point outward. That alone settles that it never gets out again. The result is Nagumo's invariance theorem.

For readers who do not read formulas: The tool for stating "does not point outward" precisely is the tangent cone T_K(x). Think of it as the fan of all directions in which one can advance from the wall point x while remaining inside K. If the velocity lies inside that fan, escape is impossible.

If the velocity on the wall lies in the tangent cone, the region is never left.

Tool 17: Hybrid Systems — Smooth Flow plus Discrete Jumps

For readers who do not read formulas: A hybrid system mixes parts that move smoothly and continuously with parts that leap instantaneously. In Theorem 31, the gradual movement of state over time is the "flow," and the sudden leap to another state on acquiring a new word is the "jump."

For readers who do not read formulas: To prove convergence in such a system one must show that the residual increases neither along flows nor at jumps. One of the two is not enough. In addition, jumps must not occur infinitely often in finite time (the non-Zeno condition). This is why the conditions of Theorem 31 come in pairs.

If the residual increases neither along flows nor at jumps, the whole converges to zero.

Tool 18: The Upper-Right Dini Derivative — a Slope Measurable Even at a Corner

For readers who do not read formulas: The ordinary derivative requires the graph to be smooth; at a corner the slope is undefined. But the residuals in this book often involve a max or an absolute value, and corners appear. So the upper-right Dini derivative D⁺ is used instead. It ignores "what happened coming in from the left" and measures only what is about to happen going forward. Even at a corner, "is it about to fall?" always has an answer. For the proofs, that is all that is required.

Even at a corner, the slope going forward can be measured. That is the upper-right Dini derivative.

Tool 19: The Quadratic Squeeze — Tying the Meter to the Distance

For readers who do not read formulas: A proof builds a convenient "meter" W and shows that the meter falls to zero. But what one really wants to know is the distance to the safe zone. So a condition is imposed: the meter and the square of the distance bound each other from above and below by constant factors, in the form c₁d² ≤ W ≤ c₂d². With that in place, the meter falling to zero and the distance falling to zero are the same thing.

If meter and squared distance squeeze each other, proving one of them suffices.

Tool 20: The Chain Rule for Conditional Entropy — What Falls Here Rises There

For readers who do not read formulas: This is Tool 10, written precisely within the flow of time. Suppose new information M arrives between times t₁ and t₂. Then the remaining not-knowing falls by exactly the mutual information. Not approximately: this is an equality. It is this equality that lets Theorem 30 tie "the amount lost on the cognitive side" precisely to "the amount gained on the physical side."

Not-knowing falls by exactly the mutual information. An equality, not an approximation.

Tool 21: Reaching Time — Computing "How Long Until Within δ"

For readers who do not read formulas: "It arrives eventually" and "it arrives by second T" are very different statements. Once the exponential-decay shape (Tool 5) is in hand, the second becomes available. If the distance is known to be at most √(a₂/a₁)·d₀·e^{−λ(t−t₀)}, one simply solves for the t that brings this below δ. What Theorem 32 supplies is precisely this upper bound on the time.

With the exponential shape in hand, an upper bound on the time to come within δ can be computed.

Toolbox Supplement — the Named Theorems and Concepts That Appear in the Proofs

A number of theorems bearing mathematicians' names, and a few unfamiliar technical terms, appear in the proofs of this book. The names are all imposing, but what each of them says can be put in a single line. Here they are set out in the same manner as the toolbox. This chapter may be skipped entirely. Come back to whichever entry you need, when you feel like following a proof step by step.

I. On the Shape of Spaces

Compact — the Property of Having Nowhere to Escape To

For readers who do not read formulas: Compact means, roughly, that nothing can run away without limit. The closed interval [0,1] is compact; the whole line is not, and neither is the interval (0,1) with its endpoints removed. In the first, points can recede indefinitely; in the second, points may approach an endpoint whose destination is not in the set.

For readers who do not read formulas: Why is this needed? What a proof wants is the guarantee that if candidates are laid out endlessly, a resting place is nonetheless present among them. In a compact set, from any infinite sequence one can extract a subsequence approaching a point of the set. Think of it as the property guaranteeing that if you keep searching, you will find. That is why Theorem 16 imposes it on the candidate set at each layer.

Hausdorff and Compact Hausdorff Spaces — "Two Points Really Can Be Told Apart"

For readers who do not read formulas: The Hausdorff condition says that any two distinct points can be placed in separate, non-overlapping neighbourhoods. This sounds obvious, but without it one admits strange spaces in which two distinct points cannot be separated however finely one looks. In such a space, saying "the resting place is unique" would mean nothing.

For readers who do not read formulas: A compact Hausdorff space has both properties at once: nothing escapes, and points can properly be told apart. Theorem 16 requires this of the candidate set at each abstraction layer because it is, so to speak, the minimum staging needed before one can say that a self-image genuinely exists and is uniquely determined.

Banach Space — a Place with Addition, Length, and No Holes

For readers who do not read formulas: A Banach space has three features. (1) Elements can be added and multiplied by constants. (2) Each element has a "length" (a norm), from which the distance between two points can be measured. (3) There are no holes — if a sequence of points is closing in on itself, its destination is guaranteed to lie within the space (completeness).

For readers who do not read formulas: The third is the important one. The world of rational numbers alone has a "hole" at √2: a sequence of rationals may close in on √2 while its destination is absent from that world. In a Banach space this cannot happen. It is the condition guaranteeing that the accident of "approaching something that turns out not to be there" never occurs, and it is also the prerequisite for the Banach fixed-point theorem below.

Non-Empty Compact Convex Set — the Three Conditions, One at a Time

For readers who do not read formulas: In Theorem 30 the space of self-consciousness SC is said to be a non-empty compact convex subset of a Banach space. Take the three words apart. Non-empty = not vacuous (there is at least one candidate). Compact = nothing escapes (above). Convex = the segment joining any two points lies wholly inside the set — no dents, no gaps.

For readers who do not read formulas: Why is convexity needed? Because an intermediate self-image between two self-images should itself count as a legitimate self-image. If the set were split in two, "the middle of them" would not exist, and one could not speak of settling continuously. Taken together, the three give a stage on which there are candidates, nothing escapes, and there are no gaps.

Tychonoff's Theorem — "Multiply Compacts, Even Infinitely Many, and It Stays Compact"

For readers who do not read formulas: What Tychonoff's theorem says fits in one line: "however many compact spaces are multiplied together — even infinitely many — the result is again compact." For finitely many this seems obvious; that it holds for infinitely many is the non-trivial content.

For readers who do not read formulas: Here is its use in this book. Theorem 16 involves many abstraction layers. Each carries a candidate set, and each of those is compact. What one wants to know is whether at least one column coherent across every layer exists (Tool 8). Tychonoff's theorem shows that the space obtained by multiplying all the layers together is compact, and from that it follows that the inverse limit is non-empty. The existence part of Theorem 16 comes from here.

II. On Settling at a Single Point

Lipschitz Constant — the Gauge of "How Much It Stretches"

For readers who do not read formulas: Apply an operation to two points and ask by what factor, at most, the distance afterwards exceeds the distance before. That maximum factor is the Lipschitz constant, written Lip(F). If Lip(F) = 2, no two points can end up more than twice as far apart. If Lip(F) = 0.5, every pair is brought to at most half its former separation.

For readers who do not read formulas: Whether this value is below 1 is decisive. Below 1, the distance necessarily shrinks with each repetition and eventually settles at one point (Tool 9). The condition q_Fq_m < 1 in Theorem 30 demands that self-reflection and the language action, performed one after the other, still shrink overall. If the product of the two factors is below 1, the composition is a contraction too.

Banach Fixed-Point Theorem — "If It Shrinks, There Is Exactly One Destination"

For readers who do not read formulas: This states, as a theorem, what Tool 9 showed as a picture. "On a metric space with no holes, if an operation always shrinks distance by a fixed proportion, then it has exactly one fixed point, and repeating it from anywhere always arrives there." A fixed point is a point that comes back unchanged when put in.

For readers who do not read formulas: The strength of this theorem is that it delivers existence, uniqueness and reachability at once. It even gives the speed: after n repetitions the distance is at most qⁿ times the original. Everything in this book of the form "there is exactly one self-image" (Theorem 16) or "each language has exactly one resting place" (Theorem 30) is owed to this theorem.

⚠️ Easily misreadIn Theorem 16, existence and uniqueness have different sources. Existence comes from Tychonoff's theorem, a matter of topology; uniqueness comes from the Banach fixed-point theorem, a matter of contraction. The two must not be conflated.

III. On Confining Motion and on Showing That Something Falls

Nagumo's Invariance Theorem — "Do Not Point Outward at the Wall and You Never Leave"

For readers who do not read formulas: This is the formal name for what Tool 16 covered. "Given a closed region, if at every point of its wall the velocity does not point outward, then a point once inside never gets out." That is all. The convenience of the theorem is that no detailed study of the interior is required: only the direction on the wall need be examined.

For readers who do not read formulas: The tool for stating "does not point outward" precisely was the tangent cone: the fan of all directions in which one can advance from a wall point while remaining inside. Condition 31-B of Theorem 31 has exactly this shape. Half the proof that the cage cannot be left rests on this theorem — the other half being condition 31-A, which handles the language jumps.

Grönwall's Inequality — "Bound the Rate of Growth and You Bound the Value"

For readers who do not read formulas: This is the most frequently used tool in the proofs of this book. What it says fits in one line: "from an inequality about the speed of change, produce an inequality about the value itself."

For readers who do not read formulas: Concretely: "the higher you are, the faster you fall" (Tool 4) is information about the speed of change, written D⁺Φ ≤ −cΦ. What one wants from it is the value at time t. Grönwall's inequality builds that bridge. The answer is Φ(t) ≤ Φ(0)e^−ct — exponential decay.

For readers who do not read formulas: Put another way: knowing only "how much it drops each second" tells you "how much is left after so many seconds." It is because of this tool that Theorem 32 can say not "it arrives eventually" but "it arrives by second T" (Tool 21). Lemma 0, equation (31.4) of Theorem 31 and equation (32.5) of Theorem 32 all reduce in the end to this inequality.

IV. On Being Determined from the Future

The Hamilton–Jacobi–Bellman Equation — the Equation That Fixes the Best Move from the End Backwards

For readers who do not read formulas: A long name, but the content is best thought of as the procedure for solving a mating problem in chess. One does not reason forward from the first move. One works backwards from the mated position, the end: "this position is mate" → "so the position one move earlier must be this" → "so the one before that…". Writing that backwards procedure down for optimal control in continuous time gives the Hamilton–Jacobi–Bellman equation (the HJB equation).

For readers who do not read formulas: More precisely: for each state and each time, consider the value "if I do my best from here, what is the remaining total cost?" This is the value function W(x,t). The HJB equation is the relation this value function must satisfy, and it is solved by integrating backwards in time once a terminal condition — the location of the goal — has been given.

For readers who do not read formulas: So when Theorem 8 says that a terminal condition in the future determines the control now, this is neither mysticism nor the occult. Physical time does not run backwards. It is the structural fact that, in the problem setting of optimal control with a terminal condition, the direction of dependence of the decision runs from the future to the present. The future-restricted Ego of Theorem 32 (32.1) is written as a problem of exactly this form.

V. On Going Round and Round

The Recurrence Theorem for Markov Chains — "Finite and Fully Connected Means You Always Come Back"

For readers who do not read formulas: Recall the board game of Tool 15. A movement in which where you go next depends only on where you are now is a Markov chain; being able to reach any square from any square is irreducibility. The recurrence theorem says this: "if the number of squares is finite and the chain is irreducible, then every square is returned to infinitely often, with probability one."

For readers who do not read formulas: "With probability one" means "almost surely." It is not that one might occasionally fail to return; it is that the probability of not returning is zero. And it is not "returns once" but "returns infinitely often" — again and again and again. When Theorem 31 says one keeps cycling through the six realms, that is this theorem applied to six regions.

The Ergodic Theorem — "In the Long Run the Visiting Rate Settles at a Fixed Value"

For readers who do not read formulas: The recurrence theorem says only that one keeps returning; it says nothing about what proportion of the time is spent where. That is supplied by the ergodic theorem (the strong law for Markov chains): "the longer it continues, the closer the proportion of visits to each square comes to a fixed positive value." That set of fixed values is called the stationary distribution.

For readers who do not read formulas: What is striking is that the same values are reached whatever the starting point. Where one began ceases to matter in the long run. The second half of (31.6) in Theorem 31 — the long-run visiting frequency converging to μ_r > 0 — is exactly this. That μ_r is positive matters: it means there is no region that is visited only very rarely.

⚠️ Easily misreadThis "visiting rate" is a proportion with respect to the number of updates, not with respect to the length of physical time. To read it as a residence rate in physical time, additional conditions on the holding time in each region are required.

VI. On the Condition That Decides Non-Self

Relational Functional Completeness — "Everything That Acts Is Written into the Relations"

For readers who do not read formulas: This is the most important condition in the book and also the one with the least approachable name (condition 25-D). What it says fits in one line: "everything that influences the future is contained in the relational description; nothing acts from outside the relations."

For readers who do not read formulas: An analogy with a company. What determines next year's performance is the writable factors — staff, equipment, capital, contracts, markets. If, after all of these have been written out, there still remains "something unwritable that nevertheless moves the results," then that company lacks relational functional completeness. The condition says, conversely, that writing them all out is enough.

For readers who do not read formulas: Why does this decide non-self? The proofs of Theorems 25, 28 and 29 all have the same shape: "suppose there is a fixed core, then pick it up and swap it out." Two cases, and only two. If the future changed, it acted from outside the relations, contradicting this condition. If the future did not change, it is causally redundant and fails the very definition of a core. Either way, no core survives.

⚠️ Easily misreadThe reason for non-self is neither "because it is a set" nor "because it is relationally defined." An independent label variable can perfectly well be adjoined afterwards even to a relationally defined set. What decides non-self is, throughout, this condition 25-D.

For readers who do not read formulas: That completes the named tools appearing in the proofs of this book. Not one of them was invented here. Every one is a standard tool of mathematics, found in textbooks. What this book has done is only to rewrite claims about cognition and the Dharma into a form these standard tools can handle. No exotic mathematics is used anywhere — a fact of real importance for any reader who sets out to check the claims made here.

The Inherited Theorems — Always Beginning from the Standard Form

Theorems 28 through 32 do not appear out of thin air. They rest on a body of theorems already proved. This chapter sets out the inherited theorems this book uses directly (Theorems 1, 3, 4, 7, 8, 9, 15, 16 and 18, together with Theorems 24, 25 and 26 of the Four Dharma Seals). Each is first stated in the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction; then, where this book writes it differently, the reason and the correspondence showing that the claim is the same are given. This order is observed throughout the book.

For readers who do not read formulas: Why go to this trouble? Because the same theorem is sometimes written slightly differently from paper to paper. That is not because the content differs, but because each paper spells out in detail the part it happens to need. So this book always gives the short official form (the standard form) first, and then says: "here is how this book writes it, here is why, and here is why the two say the same thing." Regard it as a convention that exists to keep the reader from getting lost.

The Common Tool — the Unified Convergence Lemma (Lemma 0)

All the proofs of the inherited theorems that follow reduce to a single lemma. Once it is stated, every later proof becomes just "build a residual and throw it into this lemma."

If Φ ≥ 0 is absolutely continuous along trajectories and satisfies D⁺Φ(t) ≤ −cΦ(t) (c > 0) outside Ω_θ, then Φ(t) ≤ Φ(0)e^−ct and dist(x(t), Ω_θ) → 0Lemma 0 (unified convergence lemma). D⁺ is the upper-right Dini derivative (Tool 18); Ω_θ is a sublevel set of Φ. Adding the quadratic squeeze c₁dist² ≤ Φ ≤ c₂dist² (Tool 19) yields dist(x(t), Ω_θ) ≤ √(c₂/c₁) e^−ct/2 dist(x(0), Ω_θ).

For readers who do not read formulas: This lemma bundles Tools 4, 18 and 19 into one. Read it thus: "There is a non-negative meter Φ, and while it is outside the safe zone it must always fall at a rate proportional to its current height. Then that meter falls exponentially to zero, and the point approaches the safe zone." Every theorem below is proved by building its own private meter Φ and throwing it into this lemma.

Theorem 1 (Tomabechi Main Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)π_c = arg min ∫V₀ dt　⇒　x(t) → TCZMatching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

Why this book also gives an expanded form. Convergence to the reachable region must be stated as a distance, so the range of integration, the arguments, and the restriction to reachability are made explicit.
Under the following correspondence, the formula below makes exactly the same claim as the standard form.
∫V₀ dt ≔ ∫₀^TV₀(x(t),t) dt (range and arguments made explicit) / TCZ ≔ TCZ₁^cl(t;x₀) (the closed slice restricted to what is actually reachable) / x(t) → TCZ ⇔ dist(x(t), TCZ₁^cl(t;x₀)) → 0

dist(x(t), TCZ₁^cl(t;x₀)) → 0Conditions: for a non-negative evaluation V₀, put TCZ = {x | V₀(x,t) ≤ θ}. The closed-loop policy π_c generates solutions, V₀ is absolutely continuous along trajectories, and outside the TCZ it satisfies D⁺[V₀(x(t),t) − θ]⁺ ≤ −c[V₀ − θ]⁺.

Theorem 1. The mind converges to the valley of the evaluation function, that is, to the TCZ.

The heart of the proof: Set the residual Φ₁ ≔ [V₀(x,t) − θ]⁺. This is "by how much the threshold is exceeded": zero inside the TCZ, positive outside. By hypothesis it satisfies the descent condition of Lemma 0 outside. Hence Φ₁ falls exponentially to zero, and since Φ₁ = 0 is equivalent to belonging to the TCZ, the distance to the reachable closed slice converges to zero. ∎

⚠️ Easily misreadThe substantive condition for convergence is not being an arg min as such. It is that the closed loop selected by the arg min satisfies Lyapunov descent. "Minimize and it must settle" is false.

Theorem 3 (Abstract Shared TCZ Convergence Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)A(x)=0 ⇔ φ(x)=LUB(W₁,…,W_N),　A(t)→0Matching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

Why this book also gives an expanded form. The standard form states a characterization: what it is to have reached the roof. This book needs a policy that guarantees arrival, so it writes it in policy form as well. The two speak at different levels; the claim is identical.
A ≔ 𝒜_i (abstraction residual = distance still to the roof) / LUB(W₁,…,W_N) ≔ L^* = ∨W_i / A(t)→0 ⇔ ‖ι(φ_i(x_i(t))) − ι(L^*)‖ → 0

𝒜_i(t) → 0,　that is,　φ_i(x_i(t)) → L^* = ∨W_iConditions: the coupling graph is connected, each mismatch cost S_ij is non-negative, and the abstraction residual 𝒜_i vanishes exactly at L^* = ∨W_i. Holds under the closed loop minimizing the extended Lagrangian ℒ_A = Σ_iV_i + Σ_(i,j)∈Eγ_ijS_ij + Σ_iη_i𝒜_i (η_i > 0).

For readers who do not read formulas: Several people, each with a world of their own. What this theorem says is that a well-coupled group rises toward the lowest common roof that discards no one's world. It is not a matter of averaging and rounding off the corners. It rises to the smallest frame that wraps everyone whole. That is the LUB (Tool 7).

The heart of the proof: Set Φ₃ ≔ ℒ_A − inf ℒ_A. Because the coupling graph is connected, any agent's excursion must show up in some S_ij; so Φ₃ captures the total deviation without leakage. By Lemma 0, Φ₃ → 0. Each term of Φ₃ is non-negative, so a sum falling to zero forces every term to zero — in particular η_i𝒜_i → 0, hence 𝒜_i → 0. Since 𝒜_i was built to vanish only at L^*, we get φ_i(x_i(t)) → L^*. ∎

Theorem 4 (Tomabechi Presence-Weighted Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)Ṽ = V₀ − κPQ,　x → TCZ_PMatching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

Why this book also gives an expanded form. The standard form states only the definition of the deformation and the destination. This book uses the fact that the deformed closed loop satisfies the descent condition, so the policy π_c(P) is made explicit.
TCZ_P ≔ Ω_P(t) (the closed reachable slice after deformation) / x → TCZ_P ⇔ dist(x(t), Ω_P(t)) → 0

dist(x(t), Ω_P(t)) → 0,　∂Ṽ/∂P = −κQConditions: for presence P ∈ [0,1], value sign Q and κ > 0, set Ṽ ≔ V₀ − κPQ. Ṽ meets the requirements of a non-negative evaluation, and the closed loop of π_c(P) satisfies the descent condition of Lemma 0.

Theorem 3 states the climb to the roof; Theorem 4 states that the landscape itself changes.

For readers who do not read formulas: Here is the single most important idea in the whole book. Behaviour changes not because the will is strong, but because the landscape changes. When a future world is felt with strong presence P, the score at the place corresponding to that world drops by κPQ. That is to say, a valley opens there. Once a valley exists, the ball rolls into it by itself. The "absolute other-power" of Theorem 32 lies on the extension of this same mechanism.

The heart of the proof: Take Ṽ as the evaluation function and build the residual Φ₄ ≔ [Ṽ(x,t) − θ]⁺. Ṽ too meets the requirements of a non-negative evaluation, so Lemma 0 applies unchanged: Φ₄ → 0, hence dist(x(t), Ω_P(t)) → 0. That the position of the valley moves continuously with P follows by differentiating Ṽ directly in P to get ∂Ṽ/∂P = −κQ. For objects with Q > 0 the landscape sinks; for Q < 0 it rises. ∎

Theorem 7 (Tomabechi True Goal Theorem), Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem), Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem)

◆ CENTRAL FORMULA (STANDARD FORM) — THEOREM 7G ∉ TCZ₀,　d(G,TCZ₀) ≥ ε > 0,　G = Self-setMatching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

◆ CENTRAL FORMULA (STANDARD FORM) — THEOREM 8u^* = arg min J_G; the terminal condition G determines the present controlMatching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

◆ CENTRAL FORMULA (STANDARD FORM) — THEOREM 9K_G = PQ⁺ + EC_Self ≥ K_crit with Lyapunov descent　⇒　x → TCZ_GConfirmed form (13 Aug 2026). The legacy notation PQ⁺EC_Self in the complete theorem list of Tomabechi Theory of Freedom by Abstraction is revised to this form

Theorem 7 gives qualification, Theorem 8 the direction of decision, Theorem 9 attainment. The three form one set.

For readers who do not read formulas: The division of labour among the three is sharp. Theorem 7 settles which goals may be called real goals. There are four conditions: (i) it lies outside the present stable region; (ii) it is the person's own, not imposed from outside; (iii) it is consistent with the higher Self; (iv) it genuinely enters the control problem. Theorem 8 shows that a terminal condition placed in the future determines behaviour now. Physical time does not run backwards. In optimal control with a terminal condition, the direction of dependence of the decision runs from the future to the present — a structural fact. Theorem 9 shows that actual arrival requires the drive strength to exceed a critical value.

J_G[u] = ∫_t^TV₀(x(τ),τ)dτ + λ d(x(T),G)²,　λ > 0The terminal-condition control problem characterized by Theorem 7. The first term is the total strain along the way; the second penalizes how far from the goal one ends up.

u^* = arg min_u J_G　⇔　−∂W_G/∂t = min_u{ V₀(x,t) + ∇W_G(x,t)·f(x,u,t) }Theorem 8. The value function W_G(x,t) ≔ min_uJ_G[u] satisfies the Hamilton–Jacobi–Bellman equation with terminal condition W_G(x,T) = λd(x,G)². This equation is integrated backwards in time.

The heart of the proof: Theorem 7 is proved logically. Drop each condition in turn and see what contaminates the class. Drop (i) and goals reachable without substantially reconstructing the present stable region enter. Drop (ii) and externally imposed states enter; drop (iii) and negatively valued or self-inconsistent states enter; drop (iv) and ornamental targets that never touch the policy enter. Hence the four conditions characterize exactly the intended class. Theorem 8 derives the Hamilton–Jacobi–Bellman equation from the principle of optimality and observes that it integrates backwards. Theorem 9 builds the residual Φ_G ≔ [Ṽ_G − θ_G]⁺ for the evaluation Ṽ_G deformed by the contribution K_G via Theorem 4, and reduces to Lemma 0 under K_G ≥ K_crit. ∎

⚠️ Easily misreadTheorem 7 guarantees no attainment whatever. It performs only the eligibility test: "this counts as a real goal." Attainment is the business of Theorem 9. Conflating them produces the misreading that merely setting a goal makes it come true.

Theorem 15 (Tomabechi Cognitive-Physical Entropy Exchange and Conservation Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)S_gen = S_phys + Σw_αH_α,　dS_gen/dt = Π ≥ 0Matching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

Why this book also gives an expanded form. Condition 30-C of Theorem 30 uses the exchange identity of this theorem layer by layer and conditionally. So which layer is meant (α), what the conversion rate is (w_α), and over which interval, are all written out rather than suppressed.
Under the following correspondence, the formula below makes exactly the same claim as the standard form.
Σw_αH_α ≔ Σ_α≻0w_αH_α(x_α(t)) (layer index and time made explicit) / dS_gen/dt = Π ≥ 0 ⇔ S_gen(t₂) − S_gen(t₁) = ∫_t₁^t₂Π(s) ds ≥ 0 (the increase of the total equals exactly the accumulated dissipation)

What This Theorem Newly Asserts

For readers who do not read formulas: Begin by asking what can be said while looking only at the lowest, physical floor. There are exactly two answers. Energy is conserved — and this is an equality. Nothing is gained or lost; exactly the same amount remains. Entropy (the degree of scatter) increases — and this is no more than an inequality. All one can say is "it rises, or at best stays put." So long as one looks only at the physical floor, there is no conservation law for entropy as an equality.

For readers who do not read formulas: But once the higher floors of abstraction are looked at together with it, the situation changes. Whatever falls on the mental side, as meaning becomes ordered, is exactly matched by what rises on the physical side as scatter. So the total of the two, added at the right rate, rises only by the dissipation — the handling fee. With the fee at zero, in ideal conditions, the total is exactly constant. A conservation law as an equality holds for entropy too. This is the novelty of Theorem 15. It is one of the central theorems of the Tomabechi cognitive-physics mathematical system, and it is a theorem that has until now been withheld from publication. Physics has a law of the conservation of energy, but no law of the conservation of entropy. Extended to the cognitive universe, the claim that entropy is exchanged and conserved is made here for the first time, and proved rigorously. Without this theorem the Buddha cannot be understood rigorously.

At the physical layer alone, only energy is conserved. Include the high-abstraction layers and entropy obtains a conservation law too.

Step 1 — Putting the Two Kinds of Scatter in One Ledger

For readers who do not read formulas: The scatter on the mental side (the semantic entropy H_α) and the scatter on the physical side (the physical entropy S_phys) cannot simply be added, because their units differ. Adding yen to dollars requires an exchange rate. The rate here is the conversion weight w_α, a positive constant. What one gets after multiplying by the rate and adding is called the generalized total entropy, S_gen.

S_gen(t) ≔ S_phys(t) + Σ_α≻0w_αH_α(t)The generalized total entropy: the physical entropy of the physical layer plus the semantic entropies of the high-abstraction layers, brought to the same scale by positive conversion weights.

The units differ, so a positive conversion rate is applied before adding. That makes one ledger.

Step 2 — the Rule of the Budget (the Exchange Identity)

dS_phys/dt = −Σ_α≻0w_αdH_α/dt + Π(t),　Π(t) ≥ 0Assumption A7 (inter-layer coupling — the exchange identity). Whatever the semantic entropy loses on the cognitive side (the first term on the right) is converted at the rate and entered as an increase on the physical side. Π is the dissipation term: irreversibility, thermalization, surplus generation, interaction with the environment.

🔤 Reading the symbolsS_phys = the physical degree of scatter / H_α = the semantic degree of scatter at abstraction α / w_α = the conversion rate (a positive constant) / Π = dissipation, the handling fee (never below zero) / S_gen = the total after conversion

🌱 Why it is set up this way — which part is definition and which is axiomThis needs to be separated honestly. The exchange identity above consists of two parts. The first is a definition: to name Π the sum of the change on the physical side and the change on the cognitive side carries no content at all. It merely assigns a name. The second is the sign condition Π ≥ 0, and this alone is the substantive axiom. It amounts to a generalized second law. So Theorem 15 is not "a computation that substitutes and cancels." Its substance lies in three places: (1) that the total is genuinely differentiable in a situation where the layers may be infinite in number; (2) that conservation is equivalent to the vanishing of the dissipation; and (3) that this conservation law cannot be reduced to the physical layer.

Step 3 — the Conclusions

dS_gen/dt = Π(t) ≥ 0,　　S_gen(t₂) − S_gen(t₁) = ∫_t₁^t₂Π(s) ds ≥ 0Conclusion (I). The total is non-decreasing, and what it gains equals exactly the accumulated dissipation.

Π = 0 (almost everywhere)　⇔　S_gen is constantConclusion (II). Vanishing dissipation and conservation of the total are equivalent. In an ideal closed reversible system, S_gen(t) = S_gen(0) holds exactly.

In an ideal closed reversible system (left) the total is constant; in a real dissipative system (right) it rises by the dissipation.

The heart of the proof: Three moves. (1) Form the total S_gen. (2) Differentiate in time. Here the question arises whether infinitely many terms may be differentiated one by one and then added; a lemma on term-by-term differentiation guarantees that they may. (3) Substitute the exchange identity, and what fell on the cognitive side and what rose on the physical side cancel cleanly, leaving only the dissipation Π. With no fee the total does not move (conservation); with a fee it rises by exactly that much (the second law). Neither more nor less. ∎

Step 4 — There Is a Ceiling on How Much the Mind Can Order Itself

Σ_α≻0w_α[H_α(t₁) − H_α(t₂)] = [S_phys(t₂) − S_phys(t₁)] − ∫_t₁^t₂Π(s) ds ≤ S_phys(t₂) − S_phys(t₁)Conclusion (III). Equality holds precisely when the dissipation vanishes.

For readers who do not read formulas: The left-hand side is the total amount of mental ordering actually achieved over the interval. The right-hand side is the amount of scatter dumped to the physical side over the same interval. What the formula says is this: you can never order more than you have dumped. And the two become exactly equal when the dissipation — the fee — is zero. Equations (30.9) and (30.10) of Theorem 30 are nothing but a special case of this.

The total structuring achieved is bounded above by the scatter dumped.

Step 5 — Why One May Assert That the Physical Layer Alone Has No Conservation Law

For readers who do not read formulas: Everything so far has been of the form "grant the exchange identity, and this follows." But the claim made at the head of this section — that at the physical layer alone there is no conservation law for entropy — has not yet been shown. To show it, it suffices to construct one concrete example.

The witness system. The total is constant, yet the physical entropy alone keeps strictly increasing.

🔍 The proof, one step at a time1. Take just two layers — the physical layer and one high-abstraction layer. Let the dissipation be zero.
2. Set the semantic entropy of the high layer to H₁(t) = H₀e^−λt: half, then half again (Tool 5).
3. Fix the physical entropy so that the exchange identity holds; this gives S_phys(t) = S_phys(0) + H₀(1 − e^−λt).
4. Then the physical entropy keeps strictly increasing at all times — and this with the dissipation at zero.
5. Yet the total is S_gen(t) = S_phys(0) + H₀, a constant independent of time.
6. That is: the total is conserved, but the physical entropy is not. This is the first conclusion.
7. Moreover H₀ may be taken arbitrarily large, so the values of the physical entropy sweep an arbitrarily wide range.
8. Hence any attempt to build a conserved quantity as a function of the physical entropy alone would force that function to take the same value across that whole wide range — that is, to be constant. A constant says nothing.
9. Therefore a conserved quantity must contain quantities of the high-abstraction layers. Reduction to the physical layer is impossible. ∎

⚠️ Easily misreadMisreading 1: "The second law of thermodynamics has been violated." No — quite the reverse. What the second law asserts is the non-decrease of entropy for a properly closed total system, and Theorem 15 gives exactly that non-decrease for the total quantity S_gen. The individual terms (each H_α, and S_phys) are subsystems and may fall locally, but any such fall is necessarily compensated by a rise elsewhere together with the dissipation. Far from violating the second law, Theorem 15 contains it at the level of the generalized entropy.
Misreading 2: "Ordering the mind is bad for the environment." That is not the point. What Theorem 15 states is that cognitive structuring occurs within the laws of physics — that it is consistent with them. If anything the emphasis runs the other way: it is a defence of the claim that the mind putting itself in order is no mystical breach of physical law.
Misreading 3: "The value of the conversion rate w_α has been computed." No. The proof uses only the positivity and constancy of w_α and depends in no way on its concrete value.
Misreading 4: "It is proved that semantic entropy must fall." No. This is a conditional balance law. It says only that if a fall occurs, it is tied to the rise on the physical side.

The rigorous complete proof is in §3.7 (Theorem 15) of the academic edition. The source paper is Tomabechi Cognitive-Physical Entropy Exchange and Conservation Theorem.

Theorem 16 (Tomabechi Self-Consciousness Existence and Emergence Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)SC = lim_←TCZ_α ≠ ∅,　F_SC(S^*) = S^*,　M(S^*) represents S^*Matching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

Why this book also gives an expanded form. That the self-image depends on the history is central to this book (above all to Theorem 30), so the subscripts for the agent i and the history h are not dropped.
SC ≔ SC_i,h / TCZ_α ≔ K_i,α(h) (the candidate set at layer α) / F_SC ≔ F_i,h

SC_i,h = lim_←K_i,α(h) ≠ ∅,　∃!S^*_i,h : F_i,h(S^*_i,h) = S^*_i,h,　d_SC(FⁿS₀, S^*_i,h) ≤ qⁿd_SC(S₀, S^*_i,h)Conditions: at each abstraction layer α the candidate set K_i,α(h) is non-empty and compact, and the inter-layer projections are continuous and coherent. Further, the self-reflection map F_i,h is a contraction with ratio q < 1.

Theorem 16. Self-consciousness is the unique fixed point on the set of columns coherent across every layer.

The heart of the proof: Existence and uniqueness have different sources. Existence comes from topology: if each layer's candidate set is a non-empty compact Hausdorff space and the projections are continuous, the inverse limit of that inverse system is non-empty (a standard consequence of Tychonoff's theorem). Uniqueness comes from a fixed-point theorem: the inverse limit is a complete metric space and the self-reflection map is a contraction with ratio q < 1, so by the Banach fixed-point theorem the fixed point is unique — and iteration always reaches it. ∎

🌱 Why it is set up this wayThe essential move is to state "self-consciousness exists" not as a substance but as a property of a structure. Self-consciousness is: the existence of at least one, and exactly one, self-image free of contradiction from every abstraction. And the fixed point depends on the history h. What is exhibited is not a universal substance but a history-relative self-image. This one point is why Theorem 25 (Non-Self Theorem) and Theorem 16 do not conflict, and it is also the starting point of Theorem 30.

Theorem 18 (Tomabechi Introspective Language Evolution Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)H(Z|Y,M_ℓ) < H(Z|Y);　Π₀ ⊆ Π_ℓ;　∂ℱ/∂ℓ > 0Matching the standard form of the complete theorem list in Tomabechi Theory of Freedom by Abstraction

H(Z|Y,M_ℓ) < H(Z|Y),　Π₀ ⊆ Π_ℓ,　∂ℱ/∂ℓ > 0Condition: the introspective language M_ℓ carries non-trivial information relative to the observation Y, that is, I(Z;M_ℓ|Y) > 0.

Theorem 18. Introspective language reduces not-knowing and widens the options.

For readers who do not read formulas: Three things are said at once. Information — with inner language, what one does not know about oneself decreases. Control — an agent with language can imitate every policy of an agent without it (simply ignore the language). So the set of available options necessarily widens. Evolution — a wider policy set cannot lower the supremum, so the capacity of free will is non-decreasing, and under non-triviality it strictly increases.

The heart of the proof: For information, monotonicity of conditional entropy always gives H(Z|Y,M_ℓ) ≤ H(Z|Y), with equality only when I(Z;M_ℓ|Y) = 0. The non-triviality assumption rules that out, so the inequality is strict. For control, the inclusion is shown constructively. For evolution, a wider policy set cannot lower the supremum. ∎

⚠️ Easily misreadWhat Theorem 18 states is only that the options widen. It never says that one can get out to a higher abstraction. This distinction carries the whole of Theorem 31. If every added option stays inside the same cage, the options may grow without limit while the reachable abstraction grows not at all.

Theorem 24 (Universal Unsatisfactoriness), Theorem 25 (Non-Self), Theorem 26 (Nirvanic Tranquility)

These three are already proved in a separate paper, Tomabechi Four Dharma Seals Theorems. This book uses only their conclusions as premises and does not restate their proofs.

The three conclusions inherited from the Four Dharma Seals paper. This book does not reprove them.

a ≺ ⊤　⇒　J^*_a,ρ(x,T) > 0Theorem 24 (Universal Unsatisfactoriness). Holds under condition 24-A (below emptiness there is no policy keeping V_a = 0 almost everywhere forever) together with the standing assumptions (measurability of trajectory evaluation, existence of at least one finite-cost policy, attainment of the minimum).

¬Atman(d, a)Theorem 25 (Non-Self). Under conditions 25-B through 25-D — above all relational functional completeness 25-D — there exists no internal variable that is independent of the relational description, fixed through the history, responsible for individuation, and causally non-redundant.

𝒩_⊤(T) = ℬ_alive ∩ {x | J^*_⊤,ρ(x,T) = 0} ≠ ∅Theorem 26 (Nirvanic Tranquility). Under the single condition 26-A (zero-suffering set and stability), this set is non-empty, closed and forward invariant, and satisfies the quadratic squeeze (26.A) and the strict descent (26.B). This book uses (26.A) and (26.B); (26.C) is not needed.

For readers who do not read formulas: The three, restated in ordinary language. Theorem 24 — at every abstraction below emptiness there is always strain remaining even at one's best. Theorem 25 — nowhere is there a fixed core that, cut off from all relations, would still make you you. Theorem 26 — at the highest abstraction there really is a set of states, reachable while alive, at which the residual suffering is zero; and that set is closed, so once entered it is not left.

🔤 Reading the symbolsa ≺ ⊤ = "abstraction a is below the top abstraction ⊤" / J^*_a,ρ(x,T) = "the total strain remaining when one does one's best, starting from state x at layer a, under discount ρ" / ℬ_alive = "the region in which one can remain alive" / 𝒩_⊤(T) = "the nirvana set" = all states within the living region whose top-layer residual suffering is zero / ¬Atman(d,a) = "at layer a of object d there is no fixed core"

The rigorous complete proofs of these three theorems are in the academic edition of Tomabechi Four Dharma Seals Theorems. The rigorous proofs of the inherited Theorems 1, 3, 4, 7, 8, 9, 16 and 18 are restated in §3 of the academic edition of the present paper.

Theorem 28 — Nirvana Non-Self and Type Coherence Theorem

The Question This Theorem Answers

There is a place where nearly everyone who studies a little Buddhism stumbles first. The question runs like this.

The question: "All is suffering" — everything is suffering. And "nirvanic tranquility" — there is a stillness in which suffering has ceased. Are these two not contradictory? If all is suffering, there should be no state free of suffering. And if there is such a state, is "all is suffering" not false?

A second question follows.

The question: "Non-self" — nothing has a fixed substance. Then what about nirvana? Is nirvana alone a substance? If it is, "non-self" becomes false. And if nirvana too is non-self, what could "there is nirvana" possibly mean?

Theorem 28 answers both, as mathematics. The answers are astonishingly simple. To the first: the two were never speaking about the same object. To the second: nirvana too is non-self — and nothing breaks.

◆ Central Formula

◆ CENTRAL FORMULA (STANDARD FORM OF THIS THEOREM)𝔇_<⊤ ∩ Nir(T) = ∅,　PZS(a,x,T) ⇔ [a=⊤ ∧ x∈𝒩_⊤(T)],　¬Atman(d_N,⊤)Theorem 28 (Nirvana Non-Self and Type Coherence Theorem). Three claims forming one set.

For readers who do not read formulas: The three lines, in plain English. Line one — "everything below emptiness" and "nirvana" share not a single element (∅ is the empty set, meaning "nothing at all"). Line two — one may say "is completely tranquil" exactly when the tag is ⊤ and the state belongs to the nirvana set (⇔ is "exactly when," ∧ is "and"). Line three — the process of nirvana itself has no fixed core (¬ is "not").

Step 1 — Why It Looks Like a Contradiction

The cause of the apparent contradiction is in fact perfectly clear: the assumption that it is "the same x."

For readers who do not read formulas: An analogy. "Tokyo is hot" and "Tokyo is cold" contradict each other. But "Tokyo in August is hot" and "Tokyo in January is cold" do not. The same "Tokyo," once given the tag of a month, becomes a different object. Theorem 28 does exactly this — except that the tag is not the month but the abstraction.

In this system abstraction carries an order, whose top is written ⊤. What Theorem 24 (Universal Unsatisfactoriness) speaks about is only the layers below ⊤. What Theorem 26 (Nirvanic Tranquility) speaks about is only the layer ⊤. From the outset the two addressed different regions. But write both "about the state x" and they appear to be saying opposite things about one and the same x.

"All is suffering" and "nirvanic tranquility" concern two rooms bearing different tags.

Step 2 — Attach Tags So Nothing Mixes

So attach the tag of abstraction to the state in advance. This is the tagged disjoint union (Tool 11).

𝔛 ≔ (⊔_a≺⊤{a} × X_a) ⊔ ({⊤} × ℬ_alive),　𝔇_<⊤ ≔ ⊔_a≺⊤{a} × X_a,　Nir(T) ≔ {⊤} × 𝒩_⊤(T)(28.1) The tagged domains. ⊔ means "placed side by side without mixing"; {a} × X_a is "the states carrying the tag a."

For readers who do not read formulas: Read the formula thus. 𝔛, the whole stage, is two parts placed side by side without mixing. The first part is "for every layer a below ⊤, the states tagged a." The second is "the living region, tagged ⊤." Then 𝔇_<⊤ is the first part, and Nir(T) is that portion of the second part which is the nirvana set.

🔤 Reading the symbols⊔ = disjoint union (side by side, tagged, unmixed) / × = pairing (forming the pair (a,x) of "tag a and state x") / 𝔛 = the whole tagged stage / 𝔇_<⊤ = all tagged states below emptiness / Nir(T) = all ⊤-tagged nirvana states at time T / ℬ_alive = the region in which one can remain alive / 𝒩_⊤(T) = the nirvana set (Theorem 26)

Step 3 — First Claim: the Two Rooms Never Intersect

𝔇_<⊤ ∩ Nir(T) = ∅(28.3) The region below emptiness and the region of nirvana have no common part.

The heart of the proof: This follows almost immediately from the definitions. Every element of 𝔇_<⊤ carries a tag a ≺ ⊤; every element of Nir(T) carries the tag ⊤. No single element can satisfy "below ⊤" and "equal to ⊤" at once. Hence the intersection is empty. ∎

For readers who do not read formulas: That may feel anticlimactic. But that is precisely the point of the theorem. The contradiction never existed. What existed was only an apparent contradiction created by dropping the tags. Put the tags back and it vanishes of itself. Part of the work of mathematics is not to produce a profound reconciliation but to show that the problem was not a problem.

Step 4 — Second Claim: Each Room Keeps Its Own Conclusion Intact

(a,x) ∈ 𝔇_<⊤ ⇒ J^*_a,ρ(x,T) > 0,　　(⊤,x) ∈ Nir(T) ⇒ J^*_⊤,ρ(x,T) = 0(28.4) The left is Theorem 24 (Universal Unsatisfactoriness) itself; the right is the definition of 𝒩_⊤(T) itself.

Residual suffering does not taper off gradually as the layer rises. There is a two-valued cut: ⊤ and everything else.

For readers who do not read formulas: This matters, so let it be said plainly. This is not a story of suffering gradually decreasing as abstraction rises. Theorem 24 says that below ⊤, at every layer whatever, J^* > 0. At layer a₁, at a₂, at a₃ — residual suffering is positive. It reaches zero only at ⊤. A two-valued cut, not a continuous scale.

Step 5 — Third Claim: Saying "Completely Tranquil" in One Line

PZS(a,x,T) ⇔ [ a = ⊤ ∧ x ∈ 𝒩_⊤(T) ](28.5) The perfect-zero-suffering predicate PZS. The necessary and sufficient condition for complete tranquility is that the tag be ⊤ and the state belong to the nirvana set.

Complete tranquility holds only when both conditions are present together.

For readers who do not read formulas: ⇔ means "necessary and sufficient" — only then, and then always. So the formula says two things at once. (→) If completely tranquil, then necessarily the tag is ⊤ and the state is in the nirvana set. (←) Conversely, if the tag is ⊤ and the state is in the nirvana set, then it is necessarily completely tranquil. Either alone is insufficient. A ⊤ tag without membership in the nirvana set is not tranquility; and membership without a ⊤ tag (impossible by definition anyway) would not be either.

Step 6 — Fourth Claim: Nirvana Itself Has No Fixed Core

From here comes the most important part of this theorem. The first half — type coherence — is frankly no more than a tidying-up: put the tags back and the contradiction goes. But the second half is a new claim with content.

For readers unfamiliar with Buddhism: Let the question be put once more. "Non-self" — nothing has a fixed substance. Then what of nirvana? Traditionally this question has provoked major controversy. Some positions place nirvana outside the reach of impermanence and non-self, as an unconditioned dharma; others read the whole of reality, nirvana included, as non-self. Theorem 28 shows that the second position is mathematically coherent.

To do so, one must first fix precisely what "nirvana" is taken to be as a mathematical object.

d_N ≔ ( T ↦ 𝒩_⊤(T) )Condition 28-A. Nirvana is treated not as a single fixed place but as a process returning a nirvana set at each time T. This d_N is then added to the phenomenon index set 𝔇 handled by Theorem 25 (Non-Self).

Θ_N = ( X_⊤, ℬ_alive, V_⊤, ρ, Pol_⊤, f_⊤, π⁰_⊤ )(28.2) What the relational state Γ_{d_N,⊤} of the nirvana process contains at least: the top-layer state space, the living region, the evaluation, the discount, the policy set, the dynamics, the reference policy. It further contains the set family 𝒩_⊤^Θ_N(T), the history, the inter-layer relations and the relations to other beings, and satisfies condition 25-D for interventions on a candidate own-nature Σ_N.

For readers who do not read formulas: The long bracket need not alarm anyone. It says only "here are all the ingredients that determine the nirvana process." The space it lives on, the range in which one can stay alive, how strain is measured, how heavily the far future is discounted, the set of available moves, the rule by which states move, and the reference move. Fix these seven and the nirvana set is fixed. Which is to say: nothing hidden beyond these is required — and that is exactly the claim of this section.

¬Atman(d_N, ⊤)(28.6) The nirvana process has no internal variable that is independent of the relational description, fixed through the history, responsible for individuation, and causally non-redundant. If the all-layer condition of Theorem 25 (Non-Self) is inherited by d_N, then ∀a∈𝒜 ¬Atman(d_N,a) holds.

The intervention test. If it acts, condition 25-D is violated; if it does not, the definition of a core fails. Either way no core survives.

The heart of the proof: A dichotomy, both branches destroyed. Suppose the nirvana process had a fixed core Σ_N independent of relations. Now pick that Σ_N up and swap it for something else (the intervention, or do, test). Only two cases arise. (i) The future changes. Then Σ_N acted from outside the relational state Γ_{d_N,⊤}. But condition 25-D (relational functional completeness) states that everything that acts is already inside the relational state. Contradiction. (ii) The future does not change. Then Σ_N is causally entirely redundant and fails the very definition of Atman as a "non-redundant individuating cause." Either way, Atman(d_N,⊤) is false. ∎

🌱 Why it is set up this wayWhy place nirvana as a process returning a set at each time rather than as a point? Two reasons. First, it is faithful to the definition given by Theorem 26 (Nirvanic Tranquility): what that theorem supplied was 𝒩_⊤(T), a set depending on time T, not a single point. Second, it is what allows nirvana to be brought naturally within the reach of Theorem 25 (Non-Self). Theorem 25 speaks of objects belonging to the phenomenon index set 𝔇. To put nirvana there, nirvana must have a shape that can be treated as an element of 𝔇. As a process, it can.

⚠️ Easily misreadMisreading 1: "It says nirvana does not exist." No. 𝒩_⊤(T) is clearly definable as a mathematical set and is non-empty (Theorem 26). Existence is not denied. What is denied is only an own-nature that is independent, fixed, and causally non-redundant beyond that relational description.
Misreading 2: "It proves nirvana changes moment by moment." No. The mere presence of the symbol T does not make it genuinely time-varying; a constant family of sets is equally admissible under this notation. To assert "it necessarily changes moment by moment" one must separately establish ∃T₁,T₂: 𝒩_⊤(T₁) ≠ 𝒩_⊤(T₂).
Misreading 3: "Non-self because it is a set," or "because it is defined relationally." No — and this is the most frequently misread point of the theorem. Being a set, or being relationally defined, is not a reason for non-self, since an independent label variable can perfectly well be adjoined afterwards to a relationally defined set. The condition that decides non-self is, throughout, condition 25-D (relational functional completeness).

🔍 The proof, one step at a time1. Attach abstraction tags to states and lay them out as a disjoint union (Tool 11). "Below ⊤" and "⊤" are now distinct elements.
2. No element can satisfy "below ⊤" and "⊤" simultaneously. Hence the two regions have empty intersection. This is (28.3).
3. Theorem 24 (Universal Unsatisfactoriness) is quantified only over a ≺ ⊤; Theorem 26 (Nirvanic Tranquility) supplies tranquil points only at a = ⊤. So positive and zero cost are never asserted of the same typed point. This is (28.4).
4. Lift the classification of perfect tranquility from Theorem 26 onto this disjoint union. That gives (28.5).
5. Add nirvana as a process d_N to the phenomenon index set of Theorem 25 (condition 28-A).
6. Intervene on the independent candidate Σ_N. If the distribution changes, condition 25-D is violated; if not, it is causally redundant and fails the definition of Atman.
7. Either way Atman(d_N,⊤) is false. Hence (28.6). ∎

What May and May Not Be Said Doctrinally

For readers unfamiliar with Buddhism: This section draws the line between mathematics and doctrine. The model is consistent with the commentarial position that reads non-self as extending to the supramundane, nirvana included. But the historical merits of the Sarvāstivāda restructuring into three propositions, or of variant readings in the Theravāda commentaries, are not settled by mathematics. Mathematics guarantees exactly two things: the exclusivity of the typed domains, and non-self under condition 28-A. Neither more nor less.

The rigorous complete proof is in §4 (Theorem 28) of the academic edition, Tomabechi Cognitive Cosmology — A Rigorous Mathematical Proof of the Buddha-Dharma, Theorems 28–32.

Theorem 29 — Formal Dharma-System Non-Substantiality and Incompleteness Theorem

The Question This Theorem Answers

The question: Can a teaching ever be completed? Is there a state in which a body of doctrine closes within itself, grounds itself, and requires nothing further to be added?

Theorem 29 answers: no — but conditionally. That "conditionally" matters so much that this chapter proceeds more carefully, and in more explicit steps, than the others.

For readers who do not read formulas: This theorem uses two great results of twentieth-century mathematics, due to Gödel and to Chaitin. But most of this chapter is taken up with whether one has the right to use them at all. To state the conclusion first: this theorem does not claim that every teaching is incomplete. It claims only a conditional: "if a teaching can be written in a form satisfying three conditions, then the following follows."

Step 0 — May a Teaching Be Made a Mathematical Object at All?

This question is not settled inside mathematics. Which is exactly why it must be answered first.

Only the part possessing propositional closure is modelled. About the right-hand side this theorem says nothing.

For readers who do not read formulas: What this theorem models is one thing only: a body of teaching stated as propositions and closed under prescribed relations of inference. It is written 𝕋_h, where h numbers the historical stage. What it does not model is three things. First, practice, training, and experience themselves. Second, whatever is said to exceed linguistic formulation (in this system's terms, the ineffable aspect at the highest abstraction). Third, instruction consisting only of practical directives without propositional closure. About these three the theorem says not one word.

Step 1 — The Three Conditions, and Why They Are Not Arbitrary

The three parts of condition 29-A. Each is a necessary condition for a teaching to function as a teaching.

(a) Effectiveness — the Axiom Set Is Recursively Enumerable

For readers who do not read formulas: Whether a given proposition belongs to the teaching must be decidable or enumerable by a finite procedure — that is effectiveness. Why is it needed? Without it, the teaching can be neither taught, nor transmitted, nor verified. A body that cannot fix what was said cannot be preserved by later generations, and cannot be distinguished from divergent views. Conversely, the historical fact that traditions of compilation, council recitation, and commentary actually arose shows that these teachings have been handled in enumerable form. Effectiveness is thus the very condition for a dharma to be transmissible. To drop it is to abandon transmissibility itself.

(b) Arithmetic Strength — Robinson Arithmetic Q Can Be Interpreted

For readers who do not read formulas: First, be satisfied as to why arithmetic enters at all. It is because finite sequences, and repetition upon them, necessarily appear inside the teaching. The twelve links are an ordered sequence of twelve terms; the five aggregates, the six sense bases and the four truths are enumerations of finitely many items; forward and reverse contemplation are repeated advance and retreat along a sequence. In this system there further appears recursion along the lattice of abstraction (the subsumption update of Theorem 22, the High-Altitude LUB Presence Theorem). Unless the system can say internally that the second link comes after the first, it has not stated the twelve links as twelve links at all.

For readers who do not read formulas: So how much arithmetic is needed at minimum in order to say that? The answer is startlingly little. Three things suffice. (1) An operation giving "the next" — being able to point at the successor of a number (0 to 1, 1 to 2, and so on). (2) Addition and multiplication — but only the rules fixing what the answers are. (3) Basic order — being able to say which comes before and which after. The very small arithmetic axiomatizing just these three has a name: Robinson arithmetic Q.

For readers who do not read formulas: Let it be said how small Q is. The arithmetic taught in school (Peano arithmetic) contains mathematical induction — the powerful principle that what holds at 0 and passes to the successor holds for every number. Q does not contain induction. As a result Q cannot in general prove even something as obvious as the commutativity of addition. It is that weak. And yet it can still point at a finite sequence and step along it one term at a time. The minimum needed to state a teaching as a teaching lies at exactly this level.

For readers who do not read formulas: Here is the crux of Theorem 29. Gödel's incompleteness theorems are triggered as soon as this much — merely Q — can be interpreted. Strong arithmetic is not required. That arithmetic this weak already suffices is what was learned after Gödel. So the defence "our teaching uses no mathematics" carries no weight: if it speaks by laying finitely many items out in order, it has already reached the level of Q.

For readers who do not read formulas: The converse warning also belongs here. To a system that cannot even interpret Q, Theorem 29 does not apply. But such a system cannot state its own enumerative structure — it cannot say internally that there are twelve links. So there is a trade. Acquire enough power to speak of your own structure and you necessarily take on incompleteness; escape incompleteness and you must give up the power to speak of yourself. The condition of arithmetic strength marks the branch point of that trade.

(c) Soundness — Proved Arithmetical Statements Are True

For readers who do not read formulas: A system that proves falsehoods about its own finite combinatorial structure is self-refuting. A system able to prove "the twelve links are thirteen," for instance, has lost the meaning of that enumeration. Soundness is the minimal demand that a teaching not err about itself. A technical note: the first and second conclusions of Theorem 29 do not in fact require soundness — consistency (for the second incompleteness theorem) and ω-consistency or a Rosser-style argument (for the first) suffice. Soundness is imposed as a strengthening that lets the conclusions be stated plainly; it is not an essential constraint.

Step 2 — The Limits of Application, Made Explicit

⚠️ Easily misreadTo a system failing any one of the three conditions, Theorem 29 does not apply. Specifically: (i) systems whose axiom set is not enumerable; (ii) systems too weak in expressive power to interpret Q; (iii) systems that err about their own finite statements. This theorem does not claim that every dharma is incomplete. It claims only the conditional statement that incompleteness follows for a formalized doctrinal system satisfying condition 29-A.

⚠️ Three things this theorem does not claimFirst, it does not claim the teaching is false. Incompleteness is not falsity. The existence of an undecidable sentence does not mean the system contains falsehoods. Second, it does not claim the teaching is inconsistent. What the second incompleteness theorem states is that consistency cannot be proved internally, not that consistency fails. Third, it says nothing about the value of the dharma as religion. The reach of this theorem is confined to the formalized part possessing propositional closure.

Step 3 — An Anticipated Objection, and the Reply

Objection: "The Dharma is not a formal system. This is a category mistake."

For readers who do not read formulas: Reply. This theorem does not claim any identity between the Dharma and a formal system. The claim is a conditional: "if a body of teaching satisfies condition 29-A, then the following follows." Hence a position that declines to regard the Dharma as a formal system does not collide with this theorem at all. Where the force of the theorem is actually directed is elsewhere: at the claim that "the teaching is complete and grounded, closing within itself." Such a claim simultaneously demands that the teaching be teachable (effectiveness) and able to state its own enumerative structure (arithmetic strength) — which is precisely what brings it within the scope of condition 29-A. What this theorem denies is not the Dharma but the claim of closure about the Dharma.

Step 4 — The Relation to Theorem 25 (Non-Self)

This is no new metaphysics. It is Theorem 25 applied to one particular object: a formal theory.

For readers who do not read formulas: The "non-substantiality" this theorem yields is an instance of the relational functional completeness of Theorem 25 (Non-Self), applied to one particular kind of object. That is: a formal theory 𝕋_h, once its relation to what lies outside it — metatheory, stronger systems, additional axioms — is removed, has no self-grounding in its own consistency. In having no self-contained grounding once cut from relations, this is structurally identical to the non-substantiality stated by Theorem 25. This section introduces no new metaphysics; it merely applies an existing theorem to one object.

For readers unfamiliar with Buddhism: A terminological distinction. The existing Theorem 25 already bears the name "Non-Self Theorem," so this section is not given the same name. Also, dharma as phenomenon (every arising and ceasing existent) and Dharma as formalized teaching (the system of what is taught) are kept apart. "The Dharma itself is an axiom system" is not a conclusion but a modelling condition.

◆ Central Formula

◆ CENTRAL FORMULA (STANDARD FORM OF THIS THEOREM)∃G [ ℕ ⊨ G ∧ 𝕋_h ⊬ G ∧ 𝕋_h ⊬ ¬G ],　𝕋_h ⊬ Con(𝕋_h),　¬Atman(d_𝕋,a),　Th(𝕋_n) ⊊ Th(𝕋_n+1)Theorem 29 (Formal Dharma-System Non-Substantiality and Incompleteness Theorem). Four claims forming one set.

🔤 Reading the symbols∃G = "there exists a G such that" / ℕ ⊨ G = "G is true in the standard world of the natural numbers" / 𝕋_h ⊬ G = "the system 𝕋_h cannot prove G" (⊬ is "cannot prove") / ¬G = "the negation of G" / Con(𝕋_h) = the sentence "𝕋_h is free of contradiction" / Th(𝕋_n) = "all sentences the system 𝕋_n can prove" / ⊊ = "properly contained" (contained but not equal) / d_𝕋 = "the process of a formal theory updated stage by stage"

Step 5 — Writing the Stage in Formulas

𝕋_h = ( L_h, Ax_h, ⊢_h ),　Th(𝕋_h) = Cn_{⊢_h}(Ax_h)(29.1) The formal theory at each historical stage h. L is the language, Ax the axiom set, ⊢ the inference relation, and Cn the operation returning everything derivable from them.

The three parts of a formal system. Theorem 29 treats only what can be written in this shape.

Ax_n+1 ≔ Ax_n ∪ {G_n},　𝕋_n+1 ≔ (L, Ax_n+1, ⊢),　Th(𝕋_n+1) = Cn_⊢(Ax_n ∪ {G_n})(29.2) Condition 29-C (open update rule). The same arithmetical language L, the same inference system ⊢ and the same universal machine U are used at every stage. At each finite stage n, a sentence G_n true in the standard model but unprovable in 𝕋_n is chosen in the metatheory and added to the axioms.

For readers who do not read formulas: Condition 29-C is the rule "keep adding the unprovable sentences you find, as new axioms." ∪ means "put together." One takes the axioms so far, adds a single G_n, and gets the new system 𝕋_n+1. Repeat this forever and does one eventually reach a complete system? That is the question of the next step.

Step 6 — The Three Conclusions

The three conclusions of Theorem 29, and where each comes from.

∃ G_h [ ℕ ⊨ G_h ∧ 𝕋_h ⊬ G_h ∧ 𝕋_h ⊬ ¬G_h ](29.3) A true undecidable sentence exists. By Gödel's first incompleteness theorem.

𝕋_h ⊬ Con(𝕋_h)(29.4) Its own consistency cannot be proved internally. By Gödel's second incompleteness theorem.

∀s ∈ {0,1}^* ∀n ∈ ℕ (n > c_h,U) : 𝕋_h ⊬ "K_U(s̄) > n̄"(29.5) There is a constant c_h,U depending on the machine U and the theory, beyond which complexity lower bounds cannot be proved. By Chaitin's incompleteness.

The range within which "this is complex" is provable has a ceiling, one per system.

The heart of the proof (third conclusion): This one takes a little ingenuity, so here is the skeleton. It is a reductio. Suppose there were no uniform upper bound. Then for arbitrarily large n one could write the following program: "enumerate the proofs of 𝕋_h one after another, find the first sentence of the form 'K_U(s̄) > m' with m ≥ n, and output that s." The length of this program is the length c_h of the fixed proof enumerator plus the self-delimiting description length of n, that is c_h + O(log n). On the other hand, by soundness the output s really does satisfy K_U(s) > m ≥ n. But that s was just produced by the program we have written, so its complexity must be at most c_h + O(log n). For n large enough, c_h + O(log n) < n, a contradiction. Hence a uniform threshold c_h,U exists. ∎

Step 7 — The Fourth Conclusion: Keep Adding, and It Still Never Ends

Th(𝕋₀) ⊊ Th(𝕋₁) ⊊ ⋯ ⊊ Th(𝕋_n) ⊊ ⋯(29.6) The strictly increasing chain under condition 29-C. Soundness is preserved at every finite stage while incompleteness recurs. At no finite stage is there a final complete theory.

Anicca^step(d_𝕋) :⇔ ∀n [ Th(𝕋_n+1) ≠ Th(𝕋_n) ](29.7) The discrete impermanence predicate. This is the impermanence of the update process supplied by condition 29-C, not a consequence of Gödel and Chaitin alone.

Each true sentence added genuinely enlarges the theory. But the new theory has a new undecidable sentence of its own.

The heart of the proof: Two parts. That the growth is strict — G_n is unprovable in 𝕋_n but is an axiom, hence provable, in 𝕋_n+1. So Th(𝕋_n) and Th(𝕋_n+1) genuinely differ: contained, not equal. That incompleteness recurs — what was added is true in the standard model, so soundness is preserved; neither the language nor the inference system was changed, so effectiveness and arithmetic strength are preserved. Thus 𝕋_n+1 again satisfies condition 29-A, Gödel's theorem applies once more, and a new undecidable sentence appears. Inductively, this continues without end. ∎

🌱 Why it is set up this wayWhy is condition 29-C (the update rule) posited separately at all? Because time-variation does not follow from the incompleteness theorems. Gödel's theorems are claims about one given system; they say nothing whatever about a system being updated over time. So if one wishes to conclude that a teaching is impermanent, one must state the rule of updating oneself, explicitly. That is condition 29-C. Fail to keep this discipline and one arrives at the groundless leap that "Gödel proved Buddhist impermanence."

⚠️ Claims that do not follow directly from logic"Every axiom system is incomplete" is false. Weak decidable theories (Presburger arithmetic, for instance) and non-effective complete truth sets lie outside the scope. "Time-variation follows from the incompleteness theorems alone" is also false. Updating was newly supplied by condition 29-C. "The a priori has been refuted" is a philosophical interpretation. The mathematical conclusions are confined to three: the non-closure of sufficiently strong effective theories, the limit of self-certification, and the limit on internally proving complexity lower bounds. "Gödel and Chaitin alone yield Buddhist non-self" is false too. The non-substantiality conclusion comes from Theorem 25 (Non-Self) together with condition 29-B.

🔍 The proof, one step at a time1. Extract only the part of the teaching stated as propositions and closed under inference, and write it as a formal system 𝕋_h.
2. Check condition 29-A (effectiveness, arithmetic strength, soundness). If it fails, stop here — the theorem does not apply.
3. If it holds, Gödel's first incompleteness theorem gives a true but unprovable sentence G.
4. Gödel's second incompleteness theorem gives that its own consistency is not internally provable.
5. Chaitin's argument (reductio plus program-length estimate) gives a ceiling on provable complexity lower bounds.
6. By condition 29-B, include the stage process d_𝕋 in the phenomenon index set of Theorem 25 (Non-Self). Apply the same dichotomy as in Theorem 28 — does intervention act or not — to obtain ¬Atman(d_𝕋,a).
7. By condition 29-C, add true sentences one at a time. Soundness is preserved, inclusion is strict, and condition 29-A holds again at each stage, so incompleteness recurs without end. ∎

The rigorous complete proof is in §5 (Theorem 29) of the academic edition.

Theorem 30 — Entropy-Exchange Ego Constitution Theorem

The Question This Theorem Answers

The question: What is the "I"? A substance possessed from birth, or something made? And if made, what makes it? And further — does the mind putting itself in order not violate the laws of physics?

For readers who do not read formulas: An important caution must come first. "Generation" here does not mean creating a subject out of nothing. The Ego is already defined as a control policy in Theorem 1 (Tomabechi Main Theorem); the inverse limit and fixed point of self-consciousness are already defined in Theorem 16 (Self-Consciousness Existence and Emergence); the inner language presupposing them is already defined in Theorem 18 (Introspective Language Evolution). What "generation" means in this theorem is that introspective language selects and stabilizes a history-relative Ego constitution. Not something from nothing, but selection among candidates, and their stabilization.

◆ Central Formula

◆ CENTRAL FORMULA (STANDARD FORM OF THIS THEOREM)G_m = F_i ∘ L_m is a contraction　⇒　∃!S^*_m,　𝔈_i[m] = (S^*_m, M(S^*_m), π^(ℓ),*_c,m),　Δℋ_ego = −Σw_aI ≤ 0 ≤ ΔS_physTheorem 30 (Entropy-Exchange Ego Constitution Theorem). Three claims forming one set.

For readers who do not read formulas: The three lines in plain English. Line one — fix one introspective language and a contraction mapping arises with it, so that the resting place is uniquely determined (∃! means "there exists exactly one"). Line two — the "I" is the triple of that settled self-state, its self-representation, and the control policy in force. Line three — the degree of scatter on the cognitive side falls (≤ 0) and, by exactly that much, the degree of scatter on the physical side rises (≥ 0). The amount lost and the amount gained are tied by one and the same number: the mutual information.

Step 1 — The Stage: Placing Language on the Inverse Limit

K_a ≔ TCZ_i,a,　p_ba : K_a → K_b (b ≺ a),　SC_i ≔ lim_←a K_a(30.1) The inverse system of Theorem 16 (Self-Consciousness Existence and Emergence). SC_i is a non-empty compact convex subset of a Banach space (hence a complete metric space) carrying the existing feedback F_i : SC_i → SC_i.

For readers who do not read formulas: Recall Tool 8. At each abstraction there is a "range in which one is comfortable there," and from higher layers to lower ones there is a translation, the projection p. Collect every column that agrees across all layers and you have SC_i, the space of self-consciousness. Theorem 16 proved that this space is non-empty and has exactly one resting place. Theorem 30 begins by placing "language" on top of it.

Step 2 — Condition 30-A: Language Must Give the Same Result at Any Layer

p_ba ∘ L_a,m = L_b,m ∘ p_ba(30.2) Condition 30-A (projection-compatible language map). For each introspective language string m there is a continuous map L_a,m : K_a → K_a, with the empty string acting as the identity (L_a,ε = id). When this commutativity holds, an induced map L_m : SC_i → SC_i exists.

Apply the language then project, or project then apply — the same either way. That is why the language is defined on the inverse limit.

For readers who do not read formulas: What this condition requires is easiest to see in the figure. Go right then down and go down then right must always give the same result — that is commutativity. The symbol ∘ means "composition," that is, "do the right-hand operation first, then the left." Why is it needed? If it failed, "the result of applying language at the upper layer, then projecting" and "the result of applying language at the lower layer" would diverge, and the column (the coherent family) would break. Once broken, no language can be defined on the inverse limit at all.

G_m ≔ F_i ∘ L_m,　Lip(F_i) = q_F < 1,　Lip(L_m) = q_m,　q_Fq_m < 1,　G_m(S^*_ε) ≠ S^*_ε(30.3) The remainder of condition 30-A. G is the composition of the existing self-reflection F with the language action L. The product of their contraction ratios is below 1, and non-triviality holds: the fixed point of the empty string is not a fixed point of G_m.

🔤 Reading the symbolsL_a,m = "the operation of applying the language string m at layer a" / ε = the empty string (saying nothing) / id = the identity map (doing nothing) / ∘ = composition (doing one after the other) / Lip = the Lipschitz constant (how much distance is shrunk; below 1 means a contraction) / q_Fq_m < 1 = the condition that the composition remains a contraction / ≠ = the non-triviality that applying language really does move the resting place

Step 3 — First Conclusion: One "I" per Language

d_SC(G_mⁿS₀, S^*_m) ≤ (q_Fq_m)ⁿ d_SC(S₀, S^*_m),　S^*_m ≠ S^*_ε(30.6) Each non-empty introspective language string m selects a unique fixed point S^*_m ∈ SC_i, and any initial self-state converges to it exponentially.

Change the language and the "I" you settle into changes. Yet for each language the resting place is unique.

The heart of the proof: Three parts. First (that it is a self-map) — by the commutativity of condition 30-A, acting with L_m on a coherent family S yields a coherent family again. So L_m, and hence G_m, is a self-map of SC_i. Second (uniqueness) — since q_Fq_m < 1, G_m is a contraction, so by the Banach fixed-point theorem (Tool 9) the fixed point is unique and iteration always reaches it. Third (non-triviality) — if S^*_m coincided with the empty-string fixed point S^*_ε, being a fixed point would contradict (30.3). Hence they differ. ∎

Step 4 — Second Conclusion: the "I" Is a Triple

π^(ℓ),*_c,m ∈ arg min_{π∈Π_i(ℓ)} J_i(π; S^*_m)(30.4) Condition 30-B (realization of the Ego policy). The language-extended policy space Π_i(ℓ) is compact and the cost J_i(π;S) is lower semicontinuous in π. Hence a minimizing policy actually exists.

For readers who do not read formulas: The formula says only "given that settled self-state, choose the manner of driving with the least cost" (Tool 3). "Compact" and "lower semicontinuous" are the conditions under which a minimum genuinely exists. They rule out the case where one merely approaches an infimum without ever attaining it.

𝔈_i[m] ≔ ( S^*_m, M(S^*_m), π^(ℓ),*_c,m )(30.7) The language- and history-relative Ego constitution. By condition 30-B (realization of the Ego policy), the language-extended policy space Π_i(ℓ) is compact and the cost J_i is lower semicontinuous, so a minimizing policy π^(ℓ),*_c,m exists.

The Ego is not a substance but a construction that exists only when the three hold together.

For readers who do not read formulas: The three parts, in ordinary language. S^*_m — the self-state settled under that language: one column, free of contradiction at every layer, saying "this is the sort of being I now am." M(S^*_m) — the self-representation of that state: the picture one draws of oneself. π^(ℓ),*_c,m — the control policy in force: the manner of driving that actually decides behaviour. All three together make an "I." Remove any one and it does not hold.

🌱 Why it is set up this wayWhy a triple? Because one is not enough. The self-state alone leaves open that it is not recognized by oneself. The self-representation alone leaves open that it diverges from the actual state. The control policy alone does not settle whose policy it is. Only when all three are present and mutually consistent does one have what is ordinarily called an "I." And crucially, all three are relative to the language m and to the history. This is why the "I" is a construction and not a substance.

Step 5 — Third Conclusion: As Much As It Orders, the Outside Scatters

H⁽¹⁵⁾_a(x_a(t)) = c_a H(Z_a|𝒢_t),　c_a > 0(30.5) Condition 30-C (Shannon-to-state-entropy bridge). Connects the state entropy of Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation) with the conditional Shannon entropy of Theorem 18 (Introspective Language Evolution) by a positive constant factor. Thereafter w_a is redefined as w_ac_a.

For readers who do not read formulas: Technical, but simple in substance. Two kinds of "scatter" appear in this system: the state entropy of Theorem 15 (closer to physics) and the information entropy of Theorem 18 (closer to information theory). As they stand they cannot be added or subtracted, because their units differ. So one multiplies by a positive constant c_a to put them in the same units. That is the whole of the condition, and it is called a bridge. Only once the bridge is laid can the fall on the cognitive side and the rise on the physical side be compared in one ledger.

ℋ_ego(t) ≔ Σ_a≻0 w_a H(Z_a|𝒢_t)(30.8) The weighted ego-structuring entropy: the "not yet settled" at each layer, summed with layer weights w_a.

Δℋ_ego = −Σ_a≻0 w_a I(Z_a; M_(t₁,t₂] | 𝒢_t₁) ≤ 0(30.9) When new introspective language arrives between t₁ and t₂, the uncertainty on the cognitive side necessarily falls (or stays).

ΔS_phys = Σ_a≻0 w_a I(Z_a; M_(t₁,t₂] | 𝒢_t₁) + ∫_t₁^t₂ Π(s) ds ≥ 0(30.10) The increase on the physical side. The first term is exactly what fell on the cognitive side; the second is the irreversible generation. If the mutual information is positive at even one layer, ΔS_phys > 0.

H(Z_a|𝒢_t₂) = H(Z_a|𝒢_t₁) − I(Z_a; M_(t₁,t₂] | 𝒢_t₁)(30.11) The chain rule for conditional entropy at each layer (Tool 20). Weighting by w_a and summing over all layers yields (30.9). This is an equality, not an approximation.

The amount lost on the cognitive side and the amount gained on the physical side are tied by one and the same number.

The heart of the proof: Remarkably short. By the chain rule for conditional entropy (Tool 20), at each layer H(Z_a|𝒢_t₂) = H(Z_a|𝒢_t₁) − I(Z_a; M | 𝒢_t₁) holds as an equality. Weighting and summing over all layers gives (30.9) directly. Next, condition 30-C shows this to be the same quantity as the exchange ledger of Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation). Integrating the exchange identity of Theorem 15 over the interval and substituting (30.9) gives (30.10). Both mutual information and the generation term Π are non-negative, so the sign conclusions follow. ∎

For readers who do not read formulas: Let the meaning of this conclusion be stated plainly. The mind putting itself in order does not violate the laws of physics. Whatever is ordered inside necessarily leaves as scatter outside — and by exactly the same amount (in the ideally reversible case). This is not a story of spirit transcending matter. It is the debit and credit columns of a single ledger.

⚠️ Easily misreadMisreading 1: "A fixed self has been proved to exist." No. Uniqueness is conditional uniqueness, once the language string m and the history are fixed; it does not mean an absolutely fixed self. Formula (30.3) is precisely what makes explicit that the fixed point can change with a different linguistic history. This Ego constitution is placed inside the history-relative relational structure of Theorem 25 (Non-Self).
Misreading 2: "The conclusion carries over unchanged when the self-state is moving." No. If the self-state itself changes within the interval, one must decompose as Ḣ_a = σ_a − ι_a (rate of new uncertainty minus rate of linguistic information) and impose Σw_a(ι_a − σ_a) > 0 separately as the net structuring condition. The positive information gain of Theorem 18 (Introspective Language Evolution) does not deliver this net inequality automatically.
Misreading 3: "Since information was acquired, the Landauer limit applies." No. The Landauer bound must not be applied directly to information acquisition in general. To apply it one must separately specify a physical erasure process.

🔍 The proof, one step at a time1. Place the language action L_a,m layer by layer on the inverse system of Theorem 16.
2. By the commutativity of condition 30-A, verify that L_m maps coherent families to coherent families. Hence L_m is a self-map of the inverse limit SC_i.
3. Composed with the existing self-reflection F_i, the map G_m is a contraction with ratio q_Fq_m < 1.
4. By the Banach fixed-point theorem the fixed point is unique, and iteration reaches it exponentially. This is (30.6).
5. By non-triviality (30.3), that fixed point differs from the empty-string fixed point: the language genuinely changed something.
6. By condition 30-B a minimizing policy exists, so the triple (30.7) is type-coherent.
7. Apply the chain rule for conditional entropy at each layer and sum with weights. This is (30.9).
8. Via the bridge of condition 30-C, substitute this quantity into the exchange identity of Theorem 15. The result is (30.10). ∎

The rigorous complete proof is in §6 (Theorem 30) of the academic edition.

Theorem 31 — Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem

Its byname is the Mathematical Language Trap Theorem. The name is not a metaphor: what this theorem formalizes is the very structure discussed as the trap of language ever since Nāgārjuna's Mūlamadhyamakakārikā (Step 6).

The Question This Theorem Answers

The question: Should acquiring language not make one free? Theorem 18 (Introspective Language Evolution) proved that introspective language widens the options. Then, by refining language endlessly, should one not eventually get out to a higher abstraction? Why does that not happen?

For readers who do not read formulas: This is the harshest theorem in the book. What Theorem 18 said was only that the options widen. It said nothing about where the widened options can go. If every added option is completed inside the same cage, then the options may grow without limit while the reachable abstraction grows not at all. Theorem 31 states precisely under what conditions this "wider but no way out" situation actually occurs.

◆ Central Formula

◆ CENTRAL FORMULA (STANDARD FORM OF THIS THEOREM)T_m(K_ā) ⊆ K_ā ∧ F_ℓ ∈ T_{K_ā}　⇒　x(t) ∈ K_ā,　d(x(t),H) ≥ δ_H,　Pr(Z_n=r i.o.) = 1Theorem 31 (Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem / Mathematical Language Trap Theorem). Three claims forming one set.

For readers who do not read formulas: The three lines in plain English. Line one — inside the cage after a linguistic jump, inside the cage along the smooth flow: therefore inside the cage forever (⊆ is "is contained in"). Line two — the distance to the high-abstraction goal never falls below a certain value (≥ is "at least"). That is, it is unreachable. Line three — each of the six regions is returned to infinitely often, with probability one (i.o. abbreviates "infinitely often"; Pr is probability).

Step 1 — Defining the Cage

↓ā ≔ { b | b ≼ ā },　K_ā ≔ { x ∈ X | φ(x) ∈ ↓ā }(31.1) The low-abstraction band. Fix an ā ≺ ⊤ and use the abstraction map φ of Theorem 3 (Abstract Shared TCZ Convergence) to define "all states of abstraction at most ā" as a closed set.

The low-abstraction band K and the high-abstraction goal set H. Between them lies a gap that never closes.

For readers who do not read formulas: Abstraction carries an order. Choose a height ā, and the collection of all states at or below that height is K_ā. The symbol ↓ā means "the part closed downward from ā." In ordinary language: the whole world below the ceiling of abstraction that this person can currently handle.

x⁺ = T_m(x) (m ∈ Σ_ℓ^*),　ẋ = F_ℓ(x,t) ≔ f(x, π_ℓ(x,t), t)(31.2) Two modes of motion: the jump under a language update, and the smooth flow under the current linguistic Ego policy. On an interval with the current language and evaluation regime fixed, write V_ℓ = V_ℓ(x), and set the closed current TCZ as C_ℓ ≔ K_ā ∩ {x | V_ℓ(x) ≤ θ_ℓ}.

Tomabechi Cognitive Cosmology —A Rigorous Mathematical Proof of the Buddha-DharmaTheorems 28–32

How to Read This Book

Contents

The Map — Where the Five Theorems Are Going

The Minimum Buddhism, Starting from Zero

The Four Dharma Seals — the Four Marks That Make Buddhism Buddhism

What "Suffering" Means — Not Pain

What "Non-Self" Means — Not "I Do Not Exist"

What "Nirvana" Means — Not an Afterlife

What "the Sixfold Recurrence" Means — Not Rebirth

What "Absolute Other-Power" Means — Not a Supernatural Force

The Terms of This System That Appear in This Book

A Toolbox for Readers Who Do Not Read Formulas

Tool 1: Landscape and Valley — the Evaluation Function V and the TCZ

Tool 2: A Total Is an Area — the Integral ∫

Tool 3: Choosing What Makes It Smallest — arg min and πc

Tool 4: The Condition for Rolling Downhill — Lyapunov Descent

Tool 5: Half, Then Half Again — Exponential Decay

Tool 6: Degree of Scatter — Entropy

Tool 7: The Lowest Roof That Covers Everything — the LUB and "Emptiness"

Tool 8: Inverse Limit — the One Point Consistent Across Every Layer

1. Why Layers Are Needed at All

2. Projection — Spelling Out One Line of an Upper Layer Below

3. A Coherent Family — "One Column"

4. The Inverse Limit — Every Column, Collected

5. Why It Is Called an "Inverse" Limit

6. Why a Single Layer Is Not Enough

7. Where This Tool Does Its Work

Tool 9: Contraction Mapping — Repeat It and You Land on One Point

Tool 10: Mutual Information — Knowing Reduces Not-Knowing

Tool 11: Tagged Disjoint Union — Never Mix Points That Share a Name

Tool 12: A Formal System — Axioms, Rules of Inference, Theorems

Tool 13: Gödel Incompleteness — a Sentence True but Unprovable

Tool 14: Kolmogorov Complexity — the Ceiling on Saying "This Is Complex"

Tool 15: Markov Chains and the Stationary Distribution — You Keep Coming Back

Tool 16: Tangent Cone and Nagumo's Condition — Why You Never Cross the Wall

Tool 17: Hybrid Systems — Smooth Flow plus Discrete Jumps

Tool 18: The Upper-Right Dini Derivative — a Slope Measurable Even at a Corner

Tool 19: The Quadratic Squeeze — Tying the Meter to the Distance

Tool 20: The Chain Rule for Conditional Entropy — What Falls Here Rises There

Tool 21: Reaching Time — Computing "How Long Until Within δ"

Toolbox Supplement — the Named Theorems and Concepts That Appear in the Proofs

I. On the Shape of Spaces

Compact — the Property of Having Nowhere to Escape To

Hausdorff and Compact Hausdorff Spaces — "Two Points Really Can Be Told Apart"

Banach Space — a Place with Addition, Length, and No Holes

Non-Empty Compact Convex Set — the Three Conditions, One at a Time

Tychonoff's Theorem — "Multiply Compacts, Even Infinitely Many, and It Stays Compact"

II. On Settling at a Single Point

Lipschitz Constant — the Gauge of "How Much It Stretches"

Banach Fixed-Point Theorem — "If It Shrinks, There Is Exactly One Destination"

III. On Confining Motion and on Showing That Something Falls

Nagumo's Invariance Theorem — "Do Not Point Outward at the Wall and You Never Leave"

Grönwall's Inequality — "Bound the Rate of Growth and You Bound the Value"

IV. On Being Determined from the Future

The Hamilton–Jacobi–Bellman Equation — the Equation That Fixes the Best Move from the End Backwards

V. On Going Round and Round

The Recurrence Theorem for Markov Chains — "Finite and Fully Connected Means You Always Come Back"

The Ergodic Theorem — "In the Long Run the Visiting Rate Settles at a Fixed Value"

VI. On the Condition That Decides Non-Self

Relational Functional Completeness — "Everything That Acts Is Written into the Relations"

The Inherited Theorems — Always Beginning from the Standard Form

The Common Tool — the Unified Convergence Lemma (Lemma 0)

Theorem 1 (Tomabechi Main Theorem)

Theorem 3 (Abstract Shared TCZ Convergence Theorem)

Theorem 4 (Tomabechi Presence-Weighted Theorem)

Theorem 7 (Tomabechi True Goal Theorem), Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem), Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem)

Theorem 15 (Tomabechi Cognitive-Physical Entropy Exchange and Conservation Theorem)

What This Theorem Newly Asserts

Step 1 — Putting the Two Kinds of Scatter in One Ledger

Step 2 — the Rule of the Budget (the Exchange Identity)

Step 3 — the Conclusions

Step 4 — There Is a Ceiling on How Much the Mind Can Order Itself

Step 5 — Why One May Assert That the Physical Layer Alone Has No Conservation Law

Theorem 16 (Tomabechi Self-Consciousness Existence and Emergence Theorem)

Theorem 18 (Tomabechi Introspective Language Evolution Theorem)

Theorem 24 (Universal Unsatisfactoriness), Theorem 25 (Non-Self), Theorem 26 (Nirvanic Tranquility)

Theorem 28 — Nirvana Non-Self and Type Coherence Theorem

The Question This Theorem Answers

◆ Central Formula

Tomabechi Cognitive Cosmology —
A Rigorous Mathematical Proof of the Buddha-Dharma
Theorems 28–32

Tool 3: Choosing What Makes It Smallest — arg min and π_c