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Knowledge becomes geometry: Each piece of knowledge is encoded as a unique address in a balanced ternary system (+1, 0, -1), making the system self-addressing.
Empty spaces are meaningful: Vacant addresses are not errors but structural possibilities – they represent what is not yet known but could be discovered.
The engine is the “carry”: Combining addresses triggers the ordinary arithmetic carry of the ternary system, which serves as the internal rewrite rule.
The system feeds back: The result is written back into the network, changing the starting state for the next computation – creating a feedback loop.
Thinking is stabilizing: The process stops when the address no longer moves (a fixed point); the number of steps represents the “thinking effort” required.
Science becomes calculable: Conflicts, contradictions, and knowledge gaps become measurable “residues” with a location and propagation, allowing them to be systematically detected and prioritized.
J.Konstapel,Leiden, 26-8-2026.
Introduction
Most knowledge systems are built as archives. Information is collected, classified, indexed and retrieved. The address of an item is assigned by a human-designed taxonomy, database schema or search mechanism. Such systems can become extraordinarily large and useful, but their basic architecture remains passive: information is stored, and a user or program decides what to retrieve and what to do next.
MAZE starts from a different premise.
A piece of knowledge is represented as a route through a finite structure. The route itself determines its address. When several routes are loaded together, their interaction is not resolved by an external semantic rule. It is resolved by arithmetic. The resulting configuration is written back into the same structure. In this sense, MAZE is not merely an archive. It is a self-addressing, self-rewriting knowledge system.
The central discovery described here is that the required rewrite operation is already contained in the number system itself. The carry in balanced ternary arithmetic provides the missing rule.
The consequence is a machine in which knowledge can alter the state from which subsequent knowledge is computed.
1. From archive to net
The basic image is a strand that returns into itself. A route consists of three possible turns: over, straight and under, represented by +1, 0 and −1. A route becomes an integer by assigning successive turns the weights 1, 3, 9, 27 and so forth.
Thus a route is simultaneously a geometrical object and an address.
For a route of depth n, the available addresses form the symmetric interval
[ – a +. ]
This gives rings with capacities 1, 4, 13, 40, 121 and 364. Empty positions remain part of the structure. They are not missing data in the conventional database sense; they are addresses that have not yet been occupied.
The distinction is fundamental. An ordinary archive primarily represents what is present. MAZE represents both what is present and where something could be present.
The current inhabited net contains approximately 3.6 million pieces of material distributed across roughly 4,800 windings, including material drawn from English Wikipedia, the periodic table and essays. As a simple population test, the 118 known chemical elements can be placed in a ring with 121 available positions, leaving three structurally visible vacancies.
The archive, however, is only the first stage. The decisive question is whether the net can operate on itself.
2. The missing operation
The original problem was to find a rewrite rule satisfying three requirements: it must preserve depth, remain compatible with coarsening, and satisfy
[ RR=0. ]
The last condition means that once a configuration has been discharged, applying the rewrite mechanism again produces no further carry.
The solution is the ordinary carry mechanism of balanced ternary arithmetic.
Take two closed addresses. Their corresponding digits are added without carrying:
[ v_k=a_k+b_k. ]
Because each digit is −1, 0 or +1, the initial sum lies between −2 and +2.
Whenever a digit exceeds the permitted range, it is decomposed into a legal residue and a carry. The rule can be written as
[ c_k=(v_k/3) ]
and
[ d_k=v_k-3c_k. ]
The residue (d_k) remains at its present position. The carry (c_k) moves one position upward.
The resulting identity is exact:
[ v_k3^k = d_k3^k+ c_k3^{k+1}. ]
Nothing disappears. The overloaded configuration is separated into what remains at its current depth and what must move to the next depth.
This is the essential transition from archive to machine.
The system does not require a separately invented semantic rewrite mechanism. The arithmetic already contains the operation needed to close the net.
3. Why the rule terminates
The carry always moves upward.
Once a position has been discharged, its outgoing carry can only affect a higher position. Since the structure is finite at any given depth, the process must terminate.
The maximum number of rounds is therefore bounded by the depth plus one. In the example 43 + 25, the initial configuration contains two overloaded positions, yet the correction propagates through three successive rounds before reaching a closed configuration:
[ 43+25=68. ]
The important observation is that the amount of initial tension and the distance required to resolve it are different quantities.
MAZE calls the first quantity (), the initial overload, and the second (), the number of rewrite rounds. Thus a small local discrepancy can require a long propagation, while a heavily loaded configuration can sometimes resolve immediately.
This distinction gives the system an intrinsic measure of computational effort.
4. The boundary condition
The nilpotent property is not unrestricted. It holds for loads of up to four closed addresses.
With four inputs, the maximum digit sum is four. The resulting carry and residue remain legal trits. With five inputs, a digit sum of five can produce a carry of two, which is outside the closed configuration.
This establishes a natural capacity boundary: MAZE must load configurations in groups of four or less, or combine them pairwise.
The boundary is not an engineering inconvenience. It is part of the arithmetic architecture.
5. What makes the system self-addressing
The critical architectural step occurs when the output is written back into the structure.
MAZE operates through seven stages:
- Encode the input as a route and address.
- Load it into the current state.
- Close the configuration through the carry rule.
- Measure its overload and propagation.
- Land on an occupied or empty address.
- Read the result as an arithmetic difference.
- Write the result back into the net.
The final step changes the state of the system. The changed state becomes the input environment for the next operation.
This creates a feedback loop:
knowledge → address → interaction → rewrite → new address → updated knowledge → new interaction.
That is the point at which MAZE becomes more than an addressing system.
A conventional search engine retrieves an answer from a pre-existing index. A conventional database updates records according to externally specified instructions. A generative system predicts a continuation according to its learned model.
MAZE instead changes the geometry of its own knowledge state through the same operation that it uses to process new input.
Its output therefore has a causal role in its subsequent computation.
6. Thinking as movement toward a fixed point
The system provides a particularly simple operational definition of thinking.
The engine continues until the address stops moving. That stable configuration is a fixed point. The number of iterations required to reach it represents the effort associated with the question.
Under this definition, thinking does not require a separate symbolic theatre in which the machine explains its reasoning to itself. It requires a state, a transformation rule and feedback.
MAZE has all three.
The distinction is important because it changes the question from “Can a machine imitate human thought?” to “Can a computational system transform its own knowledge state until it reaches a stable configuration?”
MAZE is designed to do precisely that.
7. Empty addresses are not failures
A second consequence is equally important.
A generative system is normally expected to produce something. An unanswered question is therefore treated as a failure or as a prompt to generate a more plausible answer.
MAZE has another possible terminal state: an empty address.
An empty address is legitimate because emptiness is part of the geometry. If repeated loads approach an unoccupied position, the system can retain that fact as a counter. A vacancy therefore becomes a computational object rather than an absence of information.
This provides the basis for a machine that can identify its own gaps.
Instead of asking only:
What answer is already present?
the system can ask, in effect:
Which position repeatedly attracts evidence but remains unoccupied?
That is a fundamentally different architecture for knowledge discovery.
8. From calculation to discovery
The historical examples in the MAZE framework illustrate the same structural pattern.
Le Verrier confronted a residual discrepancy in the calculated orbit of Uranus. Instead of treating the discrepancy merely as an error in the existing position, he projected it outward and calculated the location of a new body. Neptune was subsequently observed close to the predicted position. In the MAZE interpretation, the unresolved residue behaves like a carry: something that cannot be absorbed at one position generates occupancy at the next.
Mendeleev provides another example. His periodic table contained deliberate gaps. Rather than treating those gaps as defects, he used them to infer properties of elements that had not yet been isolated. Gallium and germanium subsequently supplied striking confirmations of those predictions.
The same structural idea appears in the MAZE treatment of Dirac’s negative-energy solutions and the later discovery of the positron, and in the development of continental drift from Wegener’s original proposal through later independent evidence.
The purpose of these examples is not to claim that historical scientists literally performed balanced-ternary calculations. Rather, they illustrate the kind of operation that MAZE formalizes: a residue, vacancy or contradiction can become an address for further knowledge.
9. The human role does not disappear
MAZE does not attempt to eliminate the human researcher.
The machine can calculate which position is overloaded, how much tension is present, how far the correction propagates and which addresses remain empty. It cannot supply the human experience of caring about the unresolved problem.
The distinction is useful. The intellectual work of science contains both bookkeeping and commitment. MAZE is designed to automate the first while leaving the second with people.
This is also why the historical example of Poincaré is relevant. Poincaré described mathematical discovery as involving prolonged conscious work followed by sudden combinations that appeared unexpectedly. His account emphasizes that the sudden insight was preceded by substantial preparation.
MAZE offers a computational counterpart to the preparatory side of that process: load, tension, transformation, propagation and eventual closure.
10. How science gets calculated
The larger ambition follows from the architecture.
If vacancies can accumulate counters, then a research programme no longer has to rely exclusively on humans deciding which gaps deserve attention. The structure itself can produce a ranked set of unresolved positions.
If two bodies of knowledge are independently encoded into the same addressing system, their interaction can be calculated rather than harmonized solely through externally imposed ontologies.
If two findings conflict, the conflict becomes a residue with a location and measurable propagation. Disagreement therefore becomes part of the computational state.
If a result exceeds the available resolution, MAZE can coarsen it rather than manufacture a false precision.
The result is a possible scientific infrastructure in which questions, disagreements, gaps and priorities become calculable properties of the knowledge net.
11. Why the carry matters
The deepest idea in MAZE is therefore remarkably small.
The entire transition from static archive to dynamic engine depends on the fact that an overloaded balanced-ternary digit naturally decomposes into a residue and a carry.
The carry is not an arbitrary instruction added from outside. It is already implied by the representation.
That gives the system a particularly compact architecture:
represent → combine → discharge → propagate → stabilize → write back.
The simplicity is important. Balanced ternary is a well-established number system; Knuth treats it in The Art of Computer Programming, Volume 2, and the literature contains computational treatments of balanced-ternary representations.
What is novel in MAZE is not the existence of balanced ternary. It is its use as the geometry of a self-addressing knowledge net, together with the carry as the internal rewrite operation and the resulting feedback architecture.
12. The present state of the engine
The mathematical core described in the project is already specified and manually checkable. This includes the route-to-integer bijection, the carry decomposition, the bounded nilpotency, termination and controlled depth growth. The addressing kernel has also been built, with arbitrary-precision arithmetic and tests reproducing the published numerical results.
The next stage is not to invent another theoretical layer. It is to let the engine run at scale.
The proposed measurements are particularly consequential: exhaustive verification of the rewrite rule at depth five, verification that closing preserves the integer sum, measurement of the rate at which related inputs land on inhabited rather than empty addresses, and testing whether independent encodings of the same material converge to the same address.
These experiments address the central practical question:
Does the geometry merely store knowledge, or does it generate useful new structure from knowledge?
If it generates new structure, the archive has become an engine.
Conclusion
MAZE begins with a simple representation: three possible turns mapped onto the three digits −1, 0 and +1 of balanced ternary.
From that representation follows a geometry of addresses.
From the geometry follows the possibility of vacancies, mirrors, distances and coarsening.
From addition follows overload.
From overload follows the carry.
From the carry follows a rewrite operation.
From rewriting followed by write-back follows feedback.
And from feedback follows a system capable of transforming its own knowledge state.
This is the essential innovation.
MAZE does not need to imitate the outward appearance of human reasoning in order to constitute a different kind of thinking machine. Its claim is more structural: knowledge enters as a configuration, the configuration acts upon itself according to an internal arithmetic, the result changes the state of the net, and the changed state becomes the starting point for further computation.
In that sense, the carry is not merely an arithmetic convenience.
It is the mechanism that turns the net into a machine.
Annotated References
Knuth, Donald E. (1997). The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd ed. Addison-Wesley.
The principal technical reference for balanced ternary in the MAZE framework. Knuth’s Volume 2 provides the established computational background for the number representation on which MAZE’s addressing and carry mechanism are constructed.
Poincaré, Henri (1908). Science and Method, Book I, “Mathematical Creation.”
Important for the distinction between prolonged conscious work and the sudden appearance of a stable mathematical combination. MAZE uses this account as a conceptual reference for the relationship between loading, tension and closure. Poincaré’s own account describes extended unsuccessful work followed by a sudden mathematical insight.
Mendeleev, Dmitri (1869/1871). “On the Relation of the Properties to the Atomic Weights of the Elements.”
The historical reference for treating empty positions as informative rather than defective. In the MAZE interpretation, the periodic table demonstrates the value of preserving vacancies in a structured space.
Le Verrier, Urbain (1846). “Recherches sur les mouvements d’Uranus.”
The principal historical example of a residual discrepancy being projected into a new occupied position. MAZE interprets this as the closest historical analogue to its carry operation: an unresolved residue becomes a reason to search at the next structural level.
Dirac, Paul A. M. (1931). “Quantised Singularities in the Electromagnetic Field.” Proceedings of the Royal Society A, 133.
Used in MAZE to illustrate the interpretation of a sign reversal as a legitimate structural counterpart rather than an error. The example supports the net’s native operation of negation and mirrored addresses.
Vine, Frederick J., & Matthews, Drummond H. (1963). “Magnetic Anomalies over Oceanic Ridges.” Nature, 199.
The reference associated with the transition from a long-standing two-sided controversy over continental movement to a convergence of independent evidence. In MAZE this illustrates the distinction between a persistent residue and a configuration that closes.
Hadamard, Jacques (1945). The Psychology of Invention in the Mathematical Field.
Provides a broader examination of mathematical invention and complements Poincaré’s first-person account. Within the MAZE framework it is relevant to the distinction between the mechanical bookkeeping of problem solving and the human experience of intellectual effort.
Hayes, Brian (2001). “Third Base.” American Scientist, 89(6).
A readable technical introduction to balanced ternary and its historical use, including the Setun computer. Useful background for understanding why balanced ternary is a practical computational representation rather than merely a mathematical curiosity.
Brusentsov, Nikolai P. (1958). Work on the Setun ternary computer.
Historical evidence that balanced ternary has been implemented as an operational computing architecture. The relevance to MAZE is the engineering precedent for using ternary arithmetic in actual computation.
Gödel, Kurt (1931). “On Formally Undecidable Propositions.”
Included in the MAZE reference framework for the broader idea of assigning numerical structures to formal objects. The connection is conceptual rather than a claim that MAZE reproduces Gödel’s incompleteness construction.
Chaitin, Gregory (2005). Meta Math!
Relevant to the MAZE interest in the relationship between simple rules and the richness of structures generated by them. It provides a broader context for asking how much computational behavior can emerge from a compact formal specification.
Kuhn, Thomas S. (1962). The Structure of Scientific Revolutions.
Relevant to MAZE’s description of knowledge structures becoming saturated and subsequently requiring a deeper level of representation. The MAZE interpretation is computational rather than a claim that its rings reproduce Kuhn’s theory of paradigms.
Rowlands, Peter (2007). Zero to Infinity.
Included because of its treatment of physical theory through rewrite systems and nilpotent operators. It provides an independent conceptual reference point for the unusual role assigned to an operator whose square vanishes.