Hi Bob — I’ve been trying to use AI to understand all this and the AI is saying there seems to be a discrepancy?
When checking the two scaling laws against each other. Yue gets B×R = const (field falls as 1/R, ring radius, tube thickness fixed). Vishnevskii’s r·V = const with Ω = B gives B×r² = const (field falls as 1/r²). Different exponents — so as written they’re not the same statement. Is there an identification between the radii I’m missing, or do you mean they’re both frozen-in-flux laws rather than literally the same formula? Keen to get clarity on this one.
You're right, and your AI is right. Thank you - that sentence was wrong and it's now corrected in the repo, with you credited by name. I have also amended the prose here in the blog article. The good news is it has no effect on the code and everything that has been shared from it thus far, detail below.
To answer your question directly: there's no identification between the radii that you're missing. And "both frozen-in-flux laws" is closer, but still not quite it - they conserve different things. The sentence collapsed two laws that only look alike because both get written with the letters B and R. In fact B means two different quantities and R means two different radii.
In Yue, B is B_θ, the toroidal field inside the tube, and R is the ring's major radius - the size of the smoke ring. Crucially the tube's minor radius stays roughly fixed while R shrinks. Wrap Ampère's law around the ring and B_θ·2πR = μ₀I, so B_θ·R = const is really the statement that the current threading the ring is conserved. Constant ampere-turns, arrived at via frozen-in flux, but a current-conservation law underneath.
Vishnevskii's eq 4.5 is m₀Vr = ħ: the angular momentum of the circulating flow is one ħ. Here r is the radius of the circulation itself and V is the tangential speed of the medium, not a field. Apply B = Ω (which is only a unit rescaling - in that system 1 T ≡ (e/m₀) Hz, so a field in tesla is numerically the cyclotron rate it induces) and you get exactly your B·r² = const.
And that exponent isn't an artifact of the substitution - it's the interesting bit. B·r² is a flux, and πħ/m₀ = h/2m₀ = h/2e, since e = m₀ in that system. So Vishnevskii's law says one flux quantum per circulation. Which is what you'd hope for, given the paper is about the superconductivity of a Dirac monopole. Your 1/r² landed on the physics rather than stumbled over it.
So: two radii, two B's, two conserved quantities. Not one law.
What survives, and is what I should have written in the first place: both have the form amplitude × length = invariant, so both make amplitude rise as 1/σ under the similarity transform the app's Collapse control performs. Same scaling class, not the same equation. (There's a second reason to keep it at class level, which you didn't even need to raise: Yue's law is measured for a ring shrinking at fixed tube thickness, whereas the hierarchy shrinks self-similarly. It's being carried to a different geometry regardless.)
Two things I'd add, because they make your catch more useful than a typo fix.
First, no code was affected, and I checked rather than assumed. B = Ω is confined totest; it never touches the level scaling. The hierarchy has always used Yue's branch, and the currents module has always stated the consequence out loud in its docstring - that the flux per level shrinks as 4⁻ᵏ rather than staying quantised. The defect was entirely in the text joining the two.
Second, this is a genuine fork rather than a wording slip. Constant ampere-turns perone implemented) and constant flux quantum per level (Vishnevskii's, which would give B ∝ 4²ᵏ instead of 4ᵏ) are different models, differing by a factor of 4ᵏ in field strength at depth k. Which one a nested hierarchy should obey is an open question, and it's now labelled as open - Tier 3 in the project's provenance ledger - rather than papered over by a claim thst. That's a better state than before you asked.
The full algebra is in PHYSICS.md under "CORRECTION (2026-07-27)" if you want to check my correction as carefully as you checked the original I will email it to you. Please do..
I will look at the implications of adding the scaling law as per Vishnevskii - the thing is, neither Yue or Vishnevskii's models consider factuality, additionally, Yue, as stated above has the flux tube at constant thickness.
Any chance of adding dark matter flows to the O-Animator simulation? This might be helpful in seeing how spherical ball lightning shapes transform into toroids and fractal bagels. Thanks.
It is intended, but the maths is not readily accepted for it. It is challenging enough to establish if Yue or Vishneveskii are right in the ratio of scaling with size.
Right, I was reading your exchange here with Martin so it makes sense to be sure the math is correct. Perhaps relic neutrino characteristics could be a place to start at some point building off the work of Parkhomov which you translated.
Hi Bob — I’ve been trying to use AI to understand all this and the AI is saying there seems to be a discrepancy?
When checking the two scaling laws against each other. Yue gets B×R = const (field falls as 1/R, ring radius, tube thickness fixed). Vishnevskii’s r·V = const with Ω = B gives B×r² = const (field falls as 1/r²). Different exponents — so as written they’re not the same statement. Is there an identification between the radii I’m missing, or do you mean they’re both frozen-in-flux laws rather than literally the same formula? Keen to get clarity on this one.
Martin,
You're right, and your AI is right. Thank you - that sentence was wrong and it's now corrected in the repo, with you credited by name. I have also amended the prose here in the blog article. The good news is it has no effect on the code and everything that has been shared from it thus far, detail below.
To answer your question directly: there's no identification between the radii that you're missing. And "both frozen-in-flux laws" is closer, but still not quite it - they conserve different things. The sentence collapsed two laws that only look alike because both get written with the letters B and R. In fact B means two different quantities and R means two different radii.
In Yue, B is B_θ, the toroidal field inside the tube, and R is the ring's major radius - the size of the smoke ring. Crucially the tube's minor radius stays roughly fixed while R shrinks. Wrap Ampère's law around the ring and B_θ·2πR = μ₀I, so B_θ·R = const is really the statement that the current threading the ring is conserved. Constant ampere-turns, arrived at via frozen-in flux, but a current-conservation law underneath.
Vishnevskii's eq 4.5 is m₀Vr = ħ: the angular momentum of the circulating flow is one ħ. Here r is the radius of the circulation itself and V is the tangential speed of the medium, not a field. Apply B = Ω (which is only a unit rescaling - in that system 1 T ≡ (e/m₀) Hz, so a field in tesla is numerically the cyclotron rate it induces) and you get exactly your B·r² = const.
And that exponent isn't an artifact of the substitution - it's the interesting bit. B·r² is a flux, and πħ/m₀ = h/2m₀ = h/2e, since e = m₀ in that system. So Vishnevskii's law says one flux quantum per circulation. Which is what you'd hope for, given the paper is about the superconductivity of a Dirac monopole. Your 1/r² landed on the physics rather than stumbled over it.
So: two radii, two B's, two conserved quantities. Not one law.
What survives, and is what I should have written in the first place: both have the form amplitude × length = invariant, so both make amplitude rise as 1/σ under the similarity transform the app's Collapse control performs. Same scaling class, not the same equation. (There's a second reason to keep it at class level, which you didn't even need to raise: Yue's law is measured for a ring shrinking at fixed tube thickness, whereas the hierarchy shrinks self-similarly. It's being carried to a different geometry regardless.)
Two things I'd add, because they make your catch more useful than a typo fix.
First, no code was affected, and I checked rather than assumed. B = Ω is confined totest; it never touches the level scaling. The hierarchy has always used Yue's branch, and the currents module has always stated the consequence out loud in its docstring - that the flux per level shrinks as 4⁻ᵏ rather than staying quantised. The defect was entirely in the text joining the two.
Second, this is a genuine fork rather than a wording slip. Constant ampere-turns perone implemented) and constant flux quantum per level (Vishnevskii's, which would give B ∝ 4²ᵏ instead of 4ᵏ) are different models, differing by a factor of 4ᵏ in field strength at depth k. Which one a nested hierarchy should obey is an open question, and it's now labelled as open - Tier 3 in the project's provenance ledger - rather than papered over by a claim thst. That's a better state than before you asked.
The full algebra is in PHYSICS.md under "CORRECTION (2026-07-27)" if you want to check my correction as carefully as you checked the original I will email it to you. Please do..
I will look at the implications of adding the scaling law as per Vishnevskii - the thing is, neither Yue or Vishnevskii's models consider factuality, additionally, Yue, as stated above has the flux tube at constant thickness.
Bob
Bob,
I would love to take a look at the physics markdown document my email is discombobulationion [at] proton [dot] me
Amazing work!
Cheers,
Martin
I will send this to you when the beta keys have gone out to the annual subscribers.
This is deep beta until sometime in 4th Quarter.
Any chance of adding dark matter flows to the O-Animator simulation? This might be helpful in seeing how spherical ball lightning shapes transform into toroids and fractal bagels. Thanks.
It is intended, but the maths is not readily accepted for it. It is challenging enough to establish if Yue or Vishneveskii are right in the ratio of scaling with size.
Right, I was reading your exchange here with Martin so it makes sense to be sure the math is correct. Perhaps relic neutrino characteristics could be a place to start at some point building off the work of Parkhomov which you translated.
https://starfire.tech/O-Animator/index.html
games are here