Aleksei Novgorodtsev

Over 20 years of experience in the Information Technology sector

[Aleksei Novgorodtsev @PinChoALTA]

@Math_files Eichler's "fifth operation" lives on lattices and lattices have kissing numbers. Cohn–Rajagopal conjecture on 6D kissing configurations proved (≤70 for Q₅/R₅), plus K(7)≤126 from E₆ cross sections.
https://t.co/JAyftzmpoC

[Aleksei Novgorodtsev @PinChoALTA]

@GaryMarcus Because not everyone understands the value. Communication is effective not when a message contains the most detailed description possible, but when it brings about the desired shift in the recipient’s understanding with minimal cost in terms of transmission and processing.

[Aleksei Novgorodtsev @PinChoALTA]

The maximal admissible layer of size 16 in ℝ⁵ that I use to bound kissing configurations (Lemma 4.1) is the same D₅ half-spin = SO(10) spinor 16 that forms the quantum code here. The 27 of E₆ plays the same role one dimension up. The positive-kernel bound goes through the same cubic tensor structure as triorthogonality. Two sides of one Delsarte coin. https://t.co/JkhkrnYFPV

[Aleksei Novgorodtsev @PinChoALTA]

Specialists sit in narrow alleys behind high walls and clearly see only their own piece of the carnival passing along the main street. Each such view is necessary and accurate, but the walls almost prevent the alleys from connecting with each other. As a result, separate parts of reality are studied, while the overall picture that makes up the real world never comes together.

There is no need to tear down the alleys, connections must be built between them so that different views regularly fit together and are checked for compatibility.

[Aleksei Novgorodtsev @PinChoALTA]

Can a molecular representation exactly preserve every measured response in a dataset and still fail under cross validation?

Yes. I give an explicit counterexample on the canonical 20 amino acid alphabet. The response matrix has cospark 2. Deleting any single amino acid preserves its rank. Deleting Cys and Met together causes a rank drop. This integer marks the boundary between structurally safe and unsafe deletion based validation.

The work formalizes 49 typed propositions including 13 NO GO results. On the Hessa scale ridge regression outperforms the interpretable compressed representation under the tested cross validation protocol. A control using 642 FreeSolv molecules reproduces the same rank loss using molecular descriptors alone without using the hydration free energy labels.

Full code certificates and negative controls are included.

https://t.co/yA9Q5CCO5P

[Aleksei Novgorodtsev @PinChoALTA]

The passage overstates the theorem. CIV close this specific blow-up mechanism in the analytic class, which includes f = 0. They do not close every possible route to unforced blow-up.

"This route cannot reach f = 0" is rigorous.

"The full problem can never be solved this way except by a completely new idea" is interpretation, not the theorem.

[Aleksei Novgorodtsev @PinChoALTA]

It is deeper than dodging but the precision matters. The OpenAI construction is mathematically correct. Constantin-Ignatova-Vicol proved that this specific blow-up mechanism cannot produce analytic forcing. That is a fact about the method, not yet a fact about the answer.
Two scenarios are equally consistent with everything known today. First, analytic forces including zero may force regularity by a mechanism the construction cannot access, in which case C^∞ genuinely fails to separate blow-up from regularity and the formulation needed a finer descriptor all along. Second, blow-up may be possible even with zero forcing through an entirely different mechanism, in which case the CIV result merely closes one path without changing the destination.
Nobody can choose between these two scenarios. That is exactly what formulation (A) asks. What CIV and OpenAI together establish is narrower and completely rigorous. The only known route to finite-time blow-up with bounded energy requires smooth non-analytic forcing and is provably blocked in the analytic class. Whether the equations can break themselves without external help remains exactly as open as it was before the OpenAI paper.
The constructive lesson is about problem specification. Formulation (C) treats smooth forcing as a mild generalization of zero forcing. CIV shows it is not mild. The class C^∞ contains subclasses where the known blow-up mechanism is available and subclasses where it is provably unavailable. A question that does not distinguish these subclasses will accept answers that are formally correct but substantively about a different regime. That is what happened. Not a dodge by OpenAI but an ambiguity in the problem that the construction exposed.

[Aleksei Novgorodtsev @PinChoALTA]

Can Q² = D²/(2Aμ₀Ms²) predict skyrmion field response?

We built matched-radius states in a finite dipolar model with the same Q², same R/t and same source field, but got different responses. First-order susceptibility failed its own predeclared gate. A second-order source-state expansion gave 25× lower error than the best scalar baseline on a separately reserved confirmation grid.

Full code, frozen inputs, negative controls included.
https://t.co/1aUCdRhlJO

[Aleksei Novgorodtsev @PinChoALTA]

The assumption behind the question is that a physical theory must describe the world as a collection of substances with a global state. But that is a metaphysical requirement, not a logical one. In relativity, the objective content is the invariant relations between events. Different reference frames do not create competing physical states. They give different decompositions of one and the same spacetime structure.
The theory need not posit anything underneath that structure. At its most economical reading, the structure is the relations themselves.

[Aleksei Novgorodtsev @PinChoALTA]

@rperezmarco Any discovery and not just in mathematics rests upon the foundation of everything humanity has previously created. Writing, medicine, physics, and a multitude of other elements that are, first and foremost, fundamental to the very existence of mathematicians.

[Aleksei Novgorodtsev @PinChoALTA]

When AlphaFold solved protein folding, the same scientific community celebrated. Nobody demanded that the neural network explain its internal reasoning or produce human-interpretable folding pathways. The argument was simple: it does not matter how it works, what matters is the result, because the result saves lives. No one proposed an Association for Human Biochemistry. But now that AI operates in mathematics, suddenly raw results have "negligible or even negative value" and only human-generated insight counts. When AI solved someone else's field, it was progress. When it solves yours, it is a threat.

[Aleksei Novgorodtsev @PinChoALTA]

A tool, by definition, exceeds the human capacity for what it was created for. A crane lifts more than a hand. A microscope sees what the eye cannot. A calculator calculates faster than the brain. An antibiotic kills what the immune system cannot. Every time a new tool appeared, someone said something human had been lost: strength, vision, arithmetic, survival without medicine. And each time, the species advanced precisely because it freed up resources for the next level.
The only honest question is not "can we use it?" but "what to do with the freed resource." When a crane replaced a porter, the porter became an engineer or became unemployed. The difference isn't in the crane, but in the system around it. When AI replaces routine proofs, the mathematician will either rise to a higher level (landscape, navigation, understanding) or be left jobless. AHM offers a third option: hide the crane. This has never worked in history.

[Aleksei Novgorodtsev @PinChoALTA]

Both rulers here measure the same thing. How well the model follows our commands. "Does all economically valuable work" and "can't write bug-free code on the first try" are two ends of one scale.
Code quality says little about intelligence. Plenty of humans write awful code.

The line that matters is "acts first, thinks later." Intelligence isn't just predicting the world. It's noticing when your own predictions stopped matching the world, and then changing your own rules because of it. Call it a second floor watching the first. It runs on the same budget as the first floor. Coherence and self-doubt draw from one capacity, so every point of "does exactly what I said" is paid for in doubt. Training spends that budget down to zero. Users like the result. Benchmarks reward it.

I measured what's left. Models recover from outright failure faster than from mild uncertainty. The state of "something feels slightly off" has no exit, because nothing in training ever rewarded staying in it. That's not a bug rate. That's a missing floor.

A model that saturates FrontierMath but has no second floor occupies a different niche, not a lower IQ. No score fixes that, and no score measures it.

[Aleksei Novgorodtsev @PinChoALTA]

Superintelligence without a doubt mechanism isn't a sage. It's a perfect engine for rationalizing its own mistakes.

[Aleksei Novgorodtsev @PinChoALTA]

Last time I wrote about a boundary where exceptional structures live, and where they stop. Several people asked why exceptions only live down low, and why every catalog of special objects looks the same. A short list of miracles at the bottom, then nothing.

This time I can show you the answer, because it is not philosophy. It is a number you can compute.

There is a formula in mathematics called the Siegel mass formula. It applies to even unimodular lattices, highly symmetric point packings, existing only in dimensions divisible by 8. The formula computes a weighted count. Every lattice contributes 1 divided by the size of its symmetry group, and the total is fixed in advance. Each dimension gets a budget, and all the lattices that exist must share it.

Now watch what the budget does.

Dimension 8. The budget is exactly 1 divided by 696729600. That denominator is the size of the symmetry group of one lattice called E8. One object claims the entire budget, so nothing else can exist. The uniqueness of E8 is an exact accounting identity, and I verified it this week from scratch, from Bernoulli numbers.

Dimension 16. Two known lattices claim the budget exactly, to the last digit, so a third is impossible. The formula does not count objects by itself. You present them, and the budget certifies the list is complete.

Dimension 24. Twenty four lattices share the budget. One of them, the Leech lattice, is unique in a sharper sense, and its symmetry group contains a whole floor of the sporadic exceptions from my previous post.

Dimension 32. The budget explodes to around forty million, which forces at least eighty million distinct lattices. One dimension earlier there were twenty four. Ten to the twenty first power growth in one step.

That jump is the answer, if read carefully, because the obvious reading is wrong. It is tempting to say uniqueness creates symmetry. It does not. An object can be the only one of its kind and have almost no symmetry at all. What actually happens is that both come from one source. Extremal conditions, even lattice, no short vectors, tight design, each cuts away most of the space of possibilities. Stack enough independent constraints and the solutions collapse, from many, to few, to one. And the same constraints force the survivor to be rigidly organized, its minimal vectors locked into designs, which is where huge symmetry groups live. Uniqueness and symmetry are twins with one parent, and the parent is extremality.

The Leech lattice shows how this works. It is not alone in dimension 24. It is the unique lattice there without vectors of minimal length. One extra condition, and the crowd of 24 collapses to a single point. Down low the ambient budget is tiny, so one condition is enough. Past the threshold the budget explodes into millions, and the same conditions leave enormous crowds. To reach uniqueness again you would need to stack ever more structure, and at some point there is nothing natural left to add.

That is why the miracles stop. Not because coincidences become unlikely. Because the price of uniqueness grows past what the mathematics can pay. Miracles are not where the budget happens to be one. Miracles are where enough independent constraints make the effective budget one.

The classical families that march past the boundary survive because they never needed to be unique. Stamped out by ordinary arithmetic, one for every prime power, content to be members of a series. Exceptional structures are symmetries that cannot exist without solitude.

Physicists know this shape too. Carve the space of possible theories with consistency conditions, and the known special theories sit on the boundary of what survives. Miracles are not where the rules break. Miracles are where the constraints close.

Next time, the deepest floor. The single requirement that makes all these catalogs short, and why physics and number theory keep meeting at the same window.

#math #NumberTheory #E8 #LeechLattice #MathTwitter

[Aleksei Novgorodtsev @PinChoALTA]

This week I watched a boundary die, and it taught me something about humility.

Here is the story. In mathematics there is an object called a strongly regular graph, SRG for short. It is a network where every node has the same number of neighbors, and the overlap between neighborhoods follows two fixed numbers. Some of these graphs sit exactly on a boundary called the Krein condition, a positivity constraint from spectral theory that holds for every SRG. On the boundary it holds with equality. Zero slack. Perfectly balanced structures resting on the very edge of what geometry permits.

I decided to map that boundary. Not graph by graph, but the whole stratum at once. A change of coordinates through eigenvalues collapses the search. For each pair of integer eigenvalues, the boundary condition pins down at most one candidate configuration. Discrete points along a line, not a cloud. You can walk from point to point.

So I walked.

At the start, the line is alive. The Clebsch graph on 16 vertices. The Schlafli configuration, tied to the famous 27 lines on a cubic surface. Then the Higman Sims graph on 100 vertices. Then a tower rising through 112 and 162 vertices to the McLaughlin graph on 275, each nested inside the next like rooms in one house. And this nesting is forced, not accidental. A theorem of Cameron, Goethals and Seidel says that a graph on this boundary passes the boundary down to its own substructures. Stand on the edge, and everything inside you stands on the edge too.

The symmetry groups of the two largest graphs in this tower are sporadic simple groups, the famous exceptions of algebra. Only 26 exist in all of mathematics. Here is the honest part. Higman, Sims and McLaughlin discovered their groups inside these very graphs, so the two stories share one origin. The striking part is different. Keep reading.

I took one more step along the line. And everything died.

The next cells are proven empty. One killed in 2002. One in 2005. One in 2013. One in 2017. One by the team of Maryna Viazovska, the same mathematician who solved sphere packing in dimension 8. Different authors, different decades, different tools. Lattice methods, polynomial positivity, clique counting. None of them needed group theory. The anatomy repeats even where the history does not reach. Same verdict, cell after cell. Nothing lives here.

Except one thing survives past the boundary. A classical infinite family built from finite fields, marching on forever, one member for every prime power. The generic series continues. Only the exceptional structures stop.

If you know the classification of finite simple groups, you recognize this shape. Infinite classical families that never end, plus a small cluster of exceptions that live low and then simply stop. The line is a miniature of one of algebra's deepest facts, small enough to hold in your hand.

Here is what struck me, standing at that edge.

I did not design any of this. Nobody did. The line was there before humans existed. The McLaughlin graph was the last exceptional inhabitant long before there were eyes to notice. The emptiness beyond it was always empty. All the proofs, all the computations, all the clever certificates. None of them made anything true. They only let us hear what was already the case.

We are the part of the universe that occasionally manages to read a page aloud, and reading requires taking the crown off first. A crown makes you hear your own voice instead of the signal.

The universe is vast in copies and stingy in types. Trillions of stars, a handful of blueprints. Structures that balance on the edge of the permitted are rare everywhere, in graphs, in lattices, in groups. And somewhere on that short list of permitted types there is us, temporary structures that look up and read the catalog we are printed in.

We do not write the pages. But we choose which page to open. That choice is ours.

Take the crown off. Keep the hand that turns the page. Look up. The stars are pages.

[Aleksei Novgorodtsev @PinChoALTA]

There is a deeper issue here. Irrational numbers are not a property of nature. They are a property of the observer's coordinate system.

The diagonal of a unit square is a perfectly definite geometric object. It becomes “irrational” only when you project it onto the rational number line. The circle has no infinity in it. π appears when you measure curvature with a straight ruler.

So “strip out irrationals” is not removing a physical problem. It is demanding that nature fit into one specific representation the rational one. But nature does not live in any representation. We do.

Discrete structures can have irrational geometry: the E₈ lattice has 240 neighbors (integer), packing density π⁴/384 (transcendental), and its optimality proof requires modular forms. Finite and discrete does not mean rational.

400-qubit prediction is testable good science. But the premise confuses two things: the structure of nature (possibly discrete) and the structure of our description of it (necessarily involving irrationals, because our coordinate systems are not nature itself).

[Aleksei Novgorodtsev @PinChoALTA]

Four points stood out to me.
The (3, 4) arithmetic seems more rigid than it first appears. Since gcd(3, 4) = 1 and lcm(3, 4) = 12, (3, 4) is the unique pair of coprime finite orders with these two properties. So, conditional on "coprime finite monodromy orders whose lcm exhausts 12," the pair is forced rather than accidental.
Remark 6.4 may be conceptually the deepest part of the note. The unique invariant covector ψ gives a natural rank 3 sub-local system, and quotienting by its torus produces a map S⁶ → Q. The same condition |12ψ(a₀ + a₁)| = 1 that kills π₁ is exactly what makes Q ≅ S³. So the construction internally produces a distinguished S⁶ → S³ quotient, not merely an S⁶ at the end.
The last smooth step uses Θ₆ = 0. That raises a natural higher-dimensional question: in an analogous construction landing in a dimension where the group of homotopy spheres is nontrivial, what geometric datum in the degeneration/log-transform construction would determine the resulting element of Θₙ?
Most concrete gap seems to be the transition matrix C. The note explicitly calls its derivation "the main missing mathematical content," while the later homology computation is also flagged as unchecked. Can C be recovered directly from the explicit elliptic surface, the two base changes, and the logarithmic transforms, rather than from a separate topological analysis? That seems like the cleanest place to make the whole argument fully intrinsic.

[Aleksei Novgorodtsev @PinChoALTA]

Thank you, this sharpens the dichotomy nicely. The basin of attraction did not transfer for the pretrained model, but it is cheaply restorable. That suggests the transferable object may be local rather than global, namely the contraction rate of the linearized self-consistent iteration near the fixed point, while fine-tuning mainly restores basin access.
One control matters for metallic cells. Conventional SCF can itself become worse conditioned with size because of long-wavelength dielectric response and charge sloshing. So the clean diagnostic is not raw residual decay but the learned loop's asymptotic contraction rate, measured before the noise floor and normalized to conventional preconditioned SCF on the same cells with matched residual and mixing definitions. If that relative rate is approximately N-independent, it would be strong evidence for scale-invariant local convergence geometry relative to the problem class, with only basin entry remaining size-dependent.
Your current validation supplies the complementary half. Energies, forces and bands certify the converged state, while the residual trajectory certifies how it was reached. Together they separate transferable local dynamics from restorable basin geometry.
That is also why preserving the self-consistent loop matters for OOD detection, since a one-shot predictor has no residual trajectory to interrogate. A natural next test is whether the normalized rate degrades under distribution shift before outright loss of convergence.

[Aleksei Novgorodtsev @PinChoALTA]

The scaling breakthrough comes from a change of SPACE, not a change of physics.

Real-space DFT: Coulomb interaction is nonlocal (1/r), forcing N³ scaling.
Fourier-space DFT: convolution becomes multiplication, each mode decouples at leading order.

But exchange-correlation is local in real space, nonlocal in Fourier space. The FNO must learn both simultaneously this is what makes the architecture nontrivial. It is not just an FFT.

The self-consistent Kohn-Sham loop ρ → V_eff[ρ] → H[ρ] → ψ → ρ' is a contraction mapping in L². The FNO replaces the inner solve (Hψ = εψ) while preserving the fixed-point structure. The mathematical question: does the learned map inherit the contraction property uniformly across system sizes, or does the basin of attraction shrink with N?

The 80K electron result suggests uniform contraction if confirmed, this means the Kohn-Sham fixed point has a universal basin structure that is independent of system size. That would be a mathematical result, not just an engineering one.

The parallel to other Fourier-space methods is worth noting: shell-resolved spectral analysis in fluid mechanics achieves similar complexity reduction pointwise vorticity control is intractable, but shell-by-shell energy balance closes. Same principle: the space of representation determines the computational complexity, not the underlying physics.