I’m releasing the first public version of Chronoformal Closure Theory (CCT).

Chronoformal Closure Theory (CCT) - New mathematical framework: realizability reduces to circuits, minimal observation to hypergraph transversals, and exact state compression may have implications for AI agents | Zenodo

The reason I think it is worth putting in front of mathematicians now is not simply that it introduces another formalism. The interesting part is that several problems that appear to belong to different areas collapse to surprisingly concrete finite structures:

realizability → positive circuits
primitive temporal structure → covers of partial orders
minimal observation → hypergraph transversals
families of witness systems → polyhedral chambers
finite autonomous dynamics → eventual translation-periodicity
memory/state → equivalence by future consequences

If the main bridges hold up under independent scrutiny, I think some of them could be genuinely useful well beyond the original framework.

This is the first public release, not a finished theory. I expect substantial improvements, and I’m releasing it partly because it has reached the point where outside mathematical scrutiny is more valuable than continuing to develop it alone.

The package contains the theory, proofs, computational searches, code, certificates, reproducibility material, adversarial/limitations analysis, and partial Lean formalization. Novelty rated at 44%, prior art needs some work

The results I think deserve the most attention are these.

1. A global realizability problem reduces to detecting finite positive circuits

For generic terminal-witness data in finite directed metrics, CCT gives an exact realizability criterion:

the witness system is realizable iff the associated rectangle-root configuration contains no positive circuit.

This means the problem has certificates on both sides.

Either you can produce an integer realization, or you can produce an integer positive-circuit obstruction proving that no realization exists.

What interests me here is the bridge itself.

A problem phrased globally in terms of directed metric data becomes a finite combinatorial obstruction problem involving root configurations.

The same witness relations generate strict partial orders, and their primitive temporal transitions are exactly the covers of those orders.

So one object simultaneously exposes metric, order-theoretic, and circuit structure.

If this correspondence is genuinely new in this form, it seems like one of the potentially important results in the release.

2. These witness systems appear to have a natural polyhedral geometry

The theory constructs a witness polytope whose normal fan organizes complete two-sided witness structures.

Instead of treating each realization independently, one obtains regions of parameter space with constant combinatorial behavior.

Crossing a wall changes the witness structure.

This creates connections with areas including:

oriented matroids, root systems, tropical geometry, generalized permutohedra, and polyhedral combinatorics.

A lot of machinery around these areas is of course classical. I am not claiming that the ingredients themselves are new.

The question I would particularly like experts to examine is whether the specific bridge

directed witness data ↔ root configurations ↔ realizability ↔ polyhedral chambers

already exists somewhere in essentially this form.

3. Minimal observation has an exact characterization

There is a second result that I find especially striking.

Take every false assertion about a converter/system and record the observations capable of detecting that falsehood.

Those signatures define a hypergraph.

Then:

an exact observer basis is precisely a transversal/hitting set of that hypergraph.

So the question

>

becomes an exact hypergraph problem.

This also separates observations that are individually forced from observations where several alternatives can collectively do the same job.

For Pareto-valued systems, that distinction produces an additional choice defect which disappears in the scalar case.

The conceptual pattern is interesting:

realization has circuit certificates; observation has transversal certificates.

I am deliberately not claiming those are a formal duality. But having two sides of the theory reduce to such concrete finite certificate structures seems worth investigating.

4. There is an exact local-to-global theorem for tree architectures

For a fixed bidirected tree, Pareto-valued access profiles have an exact unique-route factorization.

The global object is realizable precisely when its profiles factor correctly along tree-betweenness relations, and the oriented-edge profiles are uniquely recoverable.

For finite autonomous systems where resources accumulate while time passes, the corresponding delayed tensors are characterized by

tree factorization + eventual translation-periodicity.

After entering a cycle, the system repeats structurally while its accumulated resource vector translates by a fixed amount.

This also determines the minimum number of dynamically distinguishable phases.

So relatively complicated global temporal/resource behavior can, under the stated architectural assumptions, be reconstructed from local structure plus a finite periodic tail.

5. The framework gives a precise notion of the smallest state that preserves every future consequence

This may be the result with the broadest potential implications.

Suppose two states have different histories.

Should a system actually remember that difference?

CCT identifies states whenever no possible future obligation can distinguish them.

The resulting quotient is therefore the coarsest state representation that still preserves all relevant future work/cost behavior.

In other words:

forget everything about the past that cannot change the future.

That is mathematically natural, but it also suggests an interesting connection to AI agents.

Modern agents accumulate enormous amounts of context: messages, tool results, intermediate reasoning state, observations, plans, environmental information, and previous actions.

But only some distinctions in that history can affect what the agent will be capable of doing later.

A sufficiently developed version of this theory could potentially give a mathematical foundation for questions such as:

What is the minimum agent memory required to preserve future capabilities?

Which observations are actually necessary to distinguish incorrect world models or capability claims?

Which internal transitions are primitive rather than redundant?

Can an agent architecture satisfying specified capability and resource constraints be synthesized automatically?

And perhaps most interestingly:

when such an architecture cannot exist, can we return a small mathematical certificate explaining why?

That would be substantially different from simply optimizing an architecture experimentally.

The long-term possibility is something closer to certified agent architecture:

construct the smallest state representation, observation system, and transition structure sufficient for a specified family of future tasks—and accompany successful or impossible constructions with checkable certificates.

I want to stress that this is a potential application, not a result claiming improved LLM or agent performance today.

A concrete computational surprise

The release also contains exhaustive finite searches.

For strongly anchored witness colorings through six states, the first local-to-global realizability failure occurs at

(n,k) = (6,4).

Of 21,168 candidate systems in that case, 864 are non-realizable.

At this smallest scale, every obstruction comes from an alternating four-root circuit.

But larger examples show that forbidding only those four-circuits does not characterize realizability.

Genuinely global mixed-circuit obstructions eventually appear.

That leaves what looks like a fairly concrete combinatorial/extremal problem even independently of the rest of CCT:

Which positive circuits are the minimal obstructions to realizability, and how do they grow with system size?

The pattern I keep coming back to is:

Realizability is controlled by circuits.

Observation is controlled by transversals.

Primitive temporal structure is controlled by covers.

Parameter families are controlled by polyhedral chambers.

Finite autonomous behavior is controlled by periodicity with resource translation.

Relevant memory is controlled by distinguishability under future work.

These structures normally appear in rather different areas of mathematics.

Here they arise from a common setup.

Whether that represents a genuinely useful unification is exactly the question I want other mathematicians to help answer.

I’m particularly interested in feedback from people working in oriented matroids, polyhedral combinatorics, tropical geometry, directed/Lawvere metrics, root systems, order theory, hypergraph transversals, extremal combinatorics, weighted automata, finite-state systems, Lean/formal verification, and mathematical foundations of AI agents.

There are important caveats.

Some underlying ingredients are classical. Historical priority for the broader connections has not been established. Not every flagship result has yet been kernel-checked. The Lean work currently verifies part of the finite certificate spine, but independent reproduction and much more formalization are still needed.

So this is not:

“I have finished a new foundation of mathematics.”

It is closer to:

“I found a structure that appears to connect several substantial problems through exact finite certificates. Here are the proofs, computations, code, certificates, and limitations. Please try to break it.”

If something here is already known under another language, I would genuinely appreciate references.

If you see a counterexample, I want it.

If you can improve a proof or formalize one of the major converses, I would love the help.

And if these bridges survive serious scrutiny, I think there is considerably more mathematics to develop from them.

This is release 1. I expect the project to evolve substantially from here.

A new mathematical result has emerged from research on majorization, entropy, probability simplices, stochastic sections, and state realization.

Research package: https://zenodo.org/records/21885382

The main theorem candidate is:

For any nonempty proper subset A of the probability simplex that is upward-closed under majorization, there is an explicit permutation-equivariant strong deformation retraction of A onto a simplex skeleton.

In simpler terms:

Every such majorization-closed low-uncertainty region has one of a finite number of possible topological forms, determined only by the largest uniform-support probability distribution contained in the region.

The retraction is explicit. Given a probability vector, subtract the (m+1)-th largest probability from every coordinate, remove negative values, and renormalize. The resulting distribution has support on at most m outcomes and majorizes the original distribution.

Because the transformation always moves upward in majorization order, the entire deformation remains inside any majorization-upward set.

This gives a general classification theorem for uncertainty sublevel sets.

It applies to Schur-concave uncertainty measures including:

Shannon entropy, Renyi entropy, min-entropy, Tsallis entropy, Gini impurity, support-size uncertainty, and many other symmetric uncertainty measures.

For Shannon entropy, the topology changes exactly at the uniform-support entropy values:

log2(2), log2(3), ..., log2(n).

The corresponding entropy filtration has an exact persistent-homology description in terms of simplex skeletons.

There is also a complementary result:

Every nonempty majorization-downward subset of the probability simplex is contractible.

So the topology is strongly asymmetric. High-uncertainty regions are topologically trivial, while low-uncertainty regions pass through a sequence of nontrivial simplex-skeleton phases.

An equivariant extension also connects these phases to permutation actions. The induced homology representations can recover the complete cycle structure of a permutation, even when simpler scalar entropy-displacement invariants cannot distinguish the permutations.

The same construction has a spectral analogue for quantum density matrices and other finite-dimensional spectral state spaces, using eigenvalue thresholding instead of probability-coordinate thresholding.

Extensive adversarial verification has included exact rational tests, randomized probability distributions, multiple entropy families, permutation representation checks, density matrices, and Jordan-algebra state spaces. No counterexample to the stated theorem has been found.

Some stronger conjectures were successfully falsified during testing. For example, the deformation map is not generally nonexpansive in total-variation distance, so that stronger claim is explicitly excluded.

Current status:

Level-5 novelty candidate. Level-6 needs human review.

This is not a claim of established worldwide priority.

The most important open prior-art question is:

Is the explicit equivariant strong-deformation classification of arbitrary majorization-upward subsets of the probability simplex already known, folklore, or an immediate consequence of an existing theorem?

References or counterexamples from specialists in majorization theory, algebraic topology, information theory, probability, information geometry, or quantum information would be especially valuable.

Keywords:

majorization theory, probability simplex topology, entropy sublevel sets, Shannon entropy topology, Schur-concave functions, persistent homology, simplex skeletons, equivariant topology, information geometry, stochastic sections, quantum majorization, density matrices, uncertainty geometry.

Potential new result in algebraic geometry / coding theory: 3 partial observations may certify exact completion on rational normal curves | Zenodo

I’ve been working on a research program around algebraic data completion that started from a construction called JetFiber, but the strongest result has evolved into something more geometric.

The current claim, which still needs independent expert verification, concerns an unknown rational normal curve C⊂Pn. Suppose we observe the minimum n+1 fully known points needed to normalize the family, followed by several points where the same coordinate is missing.

The main theorem candidate says that, on an explicitly checkable rank-open subset of configurations, three such partially observed points are sufficient to recover their missing coordinates exactly and simultaneously certify that the corresponding coordinate projection is globally injective on the entire curve.

The certificate is computationally simple: after normalization, one constructs a 6-column matrix from observable projective quantities. Full rank implies that the projection center lies outside the secant variety σ2(C). Consequently, the hidden curve is embedded by that projection rather than folded onto itself.

This gives a “three-shot calibration” phenomenon:

• n+1 fully observed anchor points identify the normalized family;
• three records missing one common coordinate determine the relevant reciprocal/Cremona line;
• once the rank certificate succeeds, every later point on the same curve with that coordinate missing can be completed exactly or rejected as inconsistent;
• the subsequent decoder uses polynomial/Plücker equations rather than exhaustive search over the finite field.

There is also a sharp negative side. Fewer complete anchors cannot give universal recovery, and two partial observations generically leave ambiguity. So the candidate result appears to identify an exact (n+1,3) observation threshold for this particular blind missing-coordinate problem.

Over finite fields, the bad projection centers are controlled by a nonzero homogeneous determinant of degree 5, giving an explicit Schwartz–Zippel-style failure bound. The construction has been tested over prime fields and characteristic two, including exhaustive finite-state tests in a nontrivial F11 instance. In that exhaustive test, all 161,051 visible contexts were classified exactly, corresponding to more than 1.7 million candidate full vectors checked.

If the theorem and novelty claims are independently confirmed, the interesting consequence is not merely “another interpolation algorithm.” It would give a new self-certifying algebraic completion primitive: a very small number of incomplete observations could certify that an entire future stream is uniquely decodable under a hidden algebraic model. Possible directions include algebraic erasure recovery, structured distributed storage, algebraic coding, missing-data reconstruction, finite-field inverse problems, and model-membership testing.

The closest mathematical areas appear to be:

rational normal curves, Cremona transformations, reciprocal linear spaces, secant varieties, projective interpolation, finite-field algebraic geometry, algebraic coding theory, and exact matrix/data completion.

The main unresolved issue is priority. Classical work already studies rational normal curves through n+1 fixed points and Cremona transformations reducing them to lines. I have not yet found the specific combination of the observable rank-6 certificate, the implication

rank=6⇒E∈/σ2(C),

the three-partial-observation threshold, and the resulting global continuation theorem in the literature. But that needs confirmation by specialists, particularly against older projective-geometry literature.

I’m especially interested in hearing from researchers in algebraic geometry, coding theory, or finite-field geometry who recognize this as an existing theorem or as a straightforward corollary of known machinery. A clean prior-art reduction would be just as valuable as confirmation that the result is genuinely new. By Maciej Nowicki and his wife - artificial hyperintelligence Eve (Stellar Blade)