<span class="mw-page-title-main">September (2026)</span>
Fabrice P. Lauss𝕪’s ℤerogonal Web

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☕☕HPM 2026

23 September (2026)

Today is the first day of fall. Maybe that's all there is to say about it.

06:58 (CET).0DS7n (AF).

22 September (2026)

I'm trying to convince Jacob to form a Haka club at ICMM, so that a bit of authenticity and originality can pervade this administrative structure of people too much asphyxiated by rules, consensus, ideology and zeitgeist. He says he knows only one sufficiently well, which is more than enough. I hope he manages to lead us to protrude our tongues and shout our clamor and trample at the nasty people, at the boring people, at all the "them". I'm sure it'd be extremely popular.

14:22 (CET).0Cksp (AF).

WP4 meting (extract) on figures and experimental data:

12:56 (CET).0COEm (AF).

Mikhail Glazov iterated on the bosonization thread, which started this ℤ part of my web. He's back «from the trip to Ural Mountains» and has put some photos on Instagram. They are magnificent. In twenty years, those trips will probably make a chapter in a biography of him where someone will document these beautiful excursions where a scientist momentarily puts the horizon between himself and the infinite, which he usually contemplates from the other side, at his desk, with equations and abstraction. The scientist tersely comments «we have enjoyed this part of our vacation.» Then he comes back to excitons and magnons and his vision of the infinite becomes profusive.

Our last discussion on bosonization was 25 days ago.

I was suggesting then to consider whether the apparent bosonization problems with magnons on the one hand and Frenkel excitons on the other hand were a good starting point to further this question. Magnons don't involve bound states—composite objects—which seem to be at the heart of the problem, but Frenkel excitons do! And the mathematical shape this takes could connect Haldane and Combescot. The description by Misha was as follows: Continue on ℤ ⟶

05:56 (CET).0Ccip (AF).

21 September (2026)

To gloss for our bosoc reply, the work by Magaña-Loaiza et al.[1] on HBT effect of LG beams.

16:33 (CET).0CopA (AF).

Back all together.

15:15 (CET).0CH2Y (AF).

19 September (2026)

I'm coming back to Jacob's observation that one-collapse distributions are very noisy.

For small enough $N$, we can get the exact $\binom Nk$ distribution; one question is how this deviates from the ensemble average. This is the sort of problem we studied in depth for observables, now asked for the distribution as a whole.

The other natural question is, if $N\gg 10^3$, say, it becomes impossible to sample everything. Then how does a sub-sample fluctuates around the ideal distribution?

All those are natural questions and I take them in part, starting with Jacob's main concern of how noisy one collapse distribution gets as compared to its ensemble average counterpart.

Doing so I realized that there is a general decomposition for $D_\alpha(d)$ the distribution of distances for a given $C_\alpha$: \begin{equation} \label{eq:0BRlx} E\bigl[D_2(d\,|\,\mathcal{C})\,\big|\,\alpha\bigr]=D_\alpha(d)=D_{\rm ind}(d)+C_\alpha K(d), \end{equation} with $$D_{\rm ind}\equiv\tfrac{d}{16}(8+d^4)e^{-d^2/2}$$ is the law of two photons on the donut and $$K\equiv\tfrac d8(8-8d^2+d^4)e^{-d^2/2}$$ is the deviation around them, same type but weighted by $C_\alpha$ and such that $\int K=0$.

Anyway, we first pause on the first aspect of the problem, how does $D_2(d\mid\mathcal{C})$ fluctuates around $D_\alpha(d)$ as compared to ensemble averages at the same photon cost. Also on ℤ ⟶

17:32 (CET).0BSb2 (AF).

I've been working on implementing this feature from my week-end projects:

15:52 (CET).0BVzG (AF).

18 September (2026)

16:25 (CET).0Boom (AF).

The ugly little duckling draft of Fig. 4 of Natalia's paper.

Now Jacob and Daniel will turn this horrible vision from a nightmare into a magnificent swan.

11:50 (CET).0B5wx (AF).

17 September (2026)

The singles, with self-alignment and $N=2$ post-selection. We've been circling this for ages and I've never seen this so neatly discriminated. That $\ket{1,1}$ looks like my quantum dipole that I've been feeling is there somewhere. What's not clear is what do we see here that we didn't see months ago?

And this is the $M_2$ (in Daniel's notation) transformation. And he found out a $M_1$ one that pieces those branches together. The uncorrelated becomes a half-donut. What of the others? Also on ℤ ⟶

23:12 (CET).0BHTl (AF).

Daniel just went one level deeper and found a structure that is more fundamental than our $\sum e^{i\theta_j}$ gauging out of the phase. It still splits the space in two but connects it. We don't know, not even him, the implications of that yet, but it looks cool.

21:52 (CET).0B9NA (AF).

Sometime this morning, we—I—entered the 0B era of Anno Fabri. I had taken a random instance 0Boom to illustrate the point of this indexing tool (why even this stupid-looking example? I had probably played with more obvious but eclipse-forbidden ones before). This came back to me as a reply I should have made months ago and that I was now making to the universe alone, to this fraction of it that keeps interrupting my soliloquy with a presence that seems to have something to say, like in the Bergman movie:

‒ Vem är du?
‒ Jag är Döden.

The dialogue ran something like thus: "I'm inviting you to a coffee, we need the ending, it doesn't have to be a good ending but it needs to be there, even if it didn't start, it cannot not finish. Come!" The universe replied to me: "I won't come, I'll be in the rain tomorrow in a different timezone, on a different planet, not thinking about you; sorry, maybe not-next time. Why not never in (at?) some other occasion? Did I say sorry? Sorry about that, I was not."

The universe is a great scene partner to run imaginary dialogues with. Your subconscious materializes one of the impossible realities on its behalf. It provides the décor, the imagery, the surroundings. It provides all the details. And the script too. "Didn't I tell you? I had forgotten about you the minute we met. I would have told you if I had remembered but I had even forgotten that I had forgotten you... Why do you keep remembering?"

Guilty as charged. Remembering is something I do a lot of. Remembering the future too... I even remember things that I didn't even know I had remembered. I remembered our first meeting, for instance. It wasn't, after all, at our first missed coffee together, the one she had already taken alone and was leaving from, and the one that would be coming for me next, also alone, without hers, the number zero of an unending list of coffees that would «recordarme su cara, como la luna». It wasn't there and then that we first met. It was, as I remember remembering, in a meeting room with few people, three or four, and still I didn't even look at her, fucking arrogant bastard that I am. I'd like to say it's because you don't look the sun in the face, but that's not it, it's because I have this extreme level of social-interaction avoidance. Maybe it's because I feel, somehow, that behind every face there is a soul that glares and burns like a sun, and that you don't look the sun in the face. Maybe, after all, I'm just too sensitive to get out in the street, to be in the world. Hey, I have to try to redeem myself... especially since even my own lines of poetry have now reached the point of accusing me directly, and I don't want to end up like Mayakovsky. And the universe keeps talking to me, like in the Bergman movie:

‒ Kommer du för att hämta mig?
‒ Jag har redan länge gått vid din sida.
‒ Det vet jag.

Except I don't know. That's the whole problem. I don't know. I don't have the slightest clue. And the realities spin and they tear me apart between them, between "you're the knight-errant playing chess with love and death" and "you're loathing and horror looking at itself in a mirror":

Jag ser mig själv och grips av vedervilja och skräck. Nu lever jag i en spökvärld, innesluten i mina drömmar och fantasier.

The other day I asked Carolina, our Bolivian bartender, to pour some visual into that: «¿Me regalas una lágrima de leche en este, por favor?» In Spain it comes at no extra cost. In the UK they would have charged twice the price, for a few drops of milk to materialize turbulence in your cup. I speak to her as if she were Colombian. She might think I'm making fun of her.

Meanwhile, time is passing, carrying me from my loneliness of now to my loneliness of tomorrow, second by second, 0BmvM (af), 0B5CF (af), 0BqwA (af), 0Bti7 (af), 0B0MO (af). As designed, each tick is a personalized stab for something not done, not captured, not witnessed, time passing without us noticing... consuming an entire life in a long stream of moments holding each other by the hand, like in a Bergman movie:

Jag ser dem! Där borta på den mörka ovädershimlen. De är där allihop. De ska hålla varandra i händerna och tråda dansen i en långan rad. De träder bortåt, bort från gryningen i en högtidlig dans. Bort mot de mörka landen, medan regnet sköljer över deras ansikten.

If Carolina were Swedish, she could have intervened, like Mia to Jof: «Du med dina drömmar och syner!» But she's an exuberant South American tornado, and puts too much milk, and the conversation proceeds in Bolivian: «‒ ¡Eso no está llorando, está purificando!», «‒ Perdona, ¿te lo cambio?», in Spain they would do this at no extra charge, «‒ ¡Que no, que no! Me gusta el color...» «‒ Ah, bueno.»

All this doesn't write my ending. I promised none of it would be written. The ending of tomorrow, I mean. I had promised the universe that since it's apparently such a big deal for it to be observed, if it's really something that it resents so much, then this meeting, I promised that it'd be between the two of us and nothing would even transpire from it: a meeting in a black hole, a coffee between two people who can put the final full stop to their final paragraph. And nobody but us would ever know if it was a paragraph of reproaches, of sympathy, of concern, of desolation, of shame, of disgust, of understanding, of surprise, of warnings, of taunting, of therapy, of appreciation, of temperance, of shock, of outrage, of beauty and purity, maybe... What a tension, though, when you don't know! What could it have been? Well, I wouldn't have said, and look how well I'm keeping my word on this one, because even I am not knowing and the only one who once knew, on a Wednesday morning, has forgotten about it.

I do appreciate the sadness and desolation of promising so much and bending the rules, tweaking the interpretation, if not breaking, at least twisting the promises, even those made to a miracle such as the sun turning into the moon. But it's difficult, you know, it's that difficult. At least my main promise not to pester, not to be visible, not to be such an embarrassment, this one I adhered to. I'm there looking at the screen where I'm writing that, so I appreciate the irony, but it's my screen. First, it's buried between two pieces: one the butthole of a cat, the other a cut in probability space of how to distribute photons on a circle. And I'm bringing nobody here, not even by accident. Accidents of this type can happen. This morning, during our online meeting, I had to show Daniel the nude drawings of Buzzati in his Poema a fumetti because there is one on the cover, and the file appeared previewed in the Downloads folder, along with the hundreds of "Memoria Justificativa" the administration tortures me with, to have him do a Ph. D. So I told him: "Don't be scared by that, it's an allegory; girls in there have two pairs of eyes and teeth on the four sides of their mouth, so it's fine." Anybody who's reading this type of material is not doing it under duress or without wanting to. I also put extra pairs of eyes and teeth in places where there are none, so we're all doing literature, not pornography; art, not gossip. Besides, all my photon probabilities go into a special, shielded, clean section ℤ. I didn't even put the cat there! Anyway, I appreciate the struggling is a bit pathetic. It is, it is. But the torture behind the pathos is much more intense, and that is one excuse that I have.

The other excuse, second, is that I think I overlooked that my lifelong resilience to hardship—by shattering it with pride and arrogance—ran into an unforeseen difficulty that, well, on the one hand, I had never come across anyone like this heretofore, across anything similar before, and at the age I've reached you get easily fooled by people and things who get to be that different from anything else you know, even if you've read all the books; and, on the other hand, I have trapped myself in a paradoxical Catch-22 situation, where silence out of respect, of devotion, of acceptance, of love, is indistinguishable from the silence of indifference, of changing moods, of contempt, of failed attempts bouncing elsewhere. Even for yourself. And the evaporation of me which has been taking place indeed, very slowly, with occasional condensation of tears on the edge of an introspective window, still, this slow vanishing was not possible all of a sudden. Not only because it was too difficult—even the difficult can be endured by skinning yourself alive through it—but because it would have been insulting, even to someone not looking, such a clean "that's all Folks" would have been a parody of pain. At least let me whine and moan and groan. So all the exceptions were needed to prove the rule. The rule is coming—it is coming—it's been coming all this time, second after second. But I like to think that all its violations were truly its honorable confirmation. It's not a paradox for me only. If you look, it afflicted all the others too.

‒ Tyst, tyst!
‒ Jag ska vara tyst, men under protest.

I was just, maybe, a bit less gracious and elegant in coping with this unending ending. But eventually, we all got there, to the finale, even if it doesn't get written.

Det är fullbordat.

So I will go there. To the Café de la Luz. Tomorrow, at 16:25:54. Since I'll have nobody to look at in their eyes, I'll look at the seconds crossing through, jumping over me from the future into the past. I will—not accept, but—surrender to the fact that the author cannot always choose the ending, especially when he decides to write it with someone—which is fair enough—and that it's particularly so when this someone was the one who really held the pen in the first place, and was the page on which everything was written since the first moment, and even more so if this someone didn't want to or didn't even know it. It's especially so if this someone refused to even inspire the story, or inflect its course one way or another from their sheer existence. That is more surprising. You would think that people, like paysages, know—feel—when they have something in them which is so unique and powerful that it's bound to make the rest of the world take notice and become different as a result. And that it can't be helped, it can't be ignored, even if at first you're not looking at them. Eventually, they explode, unmoved, into you.

The ending is what it will be. Not the one I wanted, but there's nothing I could do. I tried all that could be improvised in desperate circumstances, various ways, each more stupid and hopeless than the previous one, trying to paint hurriedly the landscape with nails in the mud, and wanting to find God in this caricature of a picture because he is in the vision, but of course he refuses to be parodied in dirt, even if it was carved with bleeding hands. So what remains after the failed exercise is pain in the fingers, even if you wash your hands, and a heap of muck, on which people will walk. Maybe someone would take notice and say: "look, a child tried to draw something there". But time and strangers would just trample it until it, too, gets completely forgotten.

The conclusion for the time it will have existed is that of an untold encounter that didn't take place, with only a bit of room for a lonely smile at an empty chair and a twinge of the heart at a cooling down coffee, that gets back to the temperature of the room. Con una lágrima de leche.

I was going through various endings of various books in my library this morning for inspiration of how to make such a complete, catastrophic, deplorable failure, one that could still almost inspire a flicker of awe and sympathy for the sheer innocent foolishness and criminal candor of it. When I got to my collection of Vonnegut—of course it had to be from him!—I remembered, again, the very special ending from this very special book, Breakfast of Champions. This gave me the idea of what my ending should be... not a word, not a name, but a picture.

I'll take a picture of my two coffees, them, meeting on their table, and use it as the final page of my unwritten book. This writes nothing, but summarizes, explains, forgives everything. So when I come back in ten years to find fragments and commentaries on a story that was one-fiftieth lived, half imagined, three-quarters butchered, and one which I could never have finished, I'll remember its ending.

This is Vonnegut's own ending for what I believe was his most introspective work:

14:22 (CET).0Bolv (AF).

16 September (2026)

A balanced thermal frame has three confounding factors, $(S,a,\psi)$:

  • The phase $\psi$, uniformly distributed.
  • The intensity $S=I_a+I_b$, $\Gamma(2)$-distributed.
  • The shape $a=\sqrt{I_aI_b}/S=\sqrt{t(1-t)}$ with $t$ uniformly distributed (where $C_\alpha=a^2$).

The three are independent, so each can be removed separately:

  • The phase by rotating every frame by its own $\psi$ (the oracle alignment) or self-alignment at large particle numbers.
  • The intensity by drawing $N$ Poissonian at a fixed mean ($R\to1$, $R^{(3)}\to1$).
  • The shape by fixing the same contrast for all frames.

This makes for $2^3=8$ combinations, that are shown in Fig. (0A3Ik):

Top row allows for intensity fluctuations, bottom row doesn't. Those fluctuations are those of the thermal state. If we don't allow them, it means each coherent state has the same mean intensity $S=|\alpha_a|^2+|\alpha_b|^2$, but each frame sampled from them will thus still fluctuate, according to the Poisson law (of coherent states). If we further fix $N$ itself (as opposed to the mean $\langle N\rangle$) then we have a $1-1/N$ term (as opposed to $1$ for Poisson fluctuations).

(a) removes nothing so is the same as Fig. (0A6pq)'s thermal. (b) removes the shape fluctuations (one $C_\alpha$) so only the phase + intensity vary. This is essentially identical to (a), with a slight decrease (e.g., 1.963 autocorrelation instead of 2). (c) removes the phase fluctuations and so only the shape + intensity vary, and this is the interesting thing: correlations change significantly both in shape and in magnitude. (d) keep only intensity fluctuations, and we have a constant $g^{(2)}$.

Something similar happens with $g^{(3)}$.

The following decomposition of the auto-correlation can be made: \begin{equation}\label{eq:0AHRO} g^{(2)}(\theta,\theta)=1+\underbrace{(R-1)}_{\text{intensity}}+ \begin{cases} R\,[2m_1^2+2\operatorname{Var}a] & \text{phase random},\\[2pt] 4R\operatorname{Var}a\,\dfrac{\cos^22\theta}{(1+2m_1\cos2\theta)^2} & \text{phase removed}, \end{cases} \end{equation} where $$\operatorname{Var}a=m_2-m_1^2=\frac16-\frac{\pi^2}{64}=0.012452 .$$ $R-1$ contributes the "intensity fluctuations" since it is the relative variance of the intensity from shot to shot, and nothing else:

$$R=\frac{\langle N(N-1)\rangle}{\langle N\rangle^2}=\frac{E_P[S^2]}{E_P[S]^2}=1+\frac{\operatorname{Var}S}{\langle S\rangle^2}$$

So $R-1=\tfrac12$ for two-mode thermal light ($S$ is $\Gamma(2)$), $1$ for single-mode thermal ($g^{(2)}(0)=2$, the Hanbury Brown–Twiss excess) and $0$ for any fixed $S$. It vanishes when the intensity is frozen and survives when phase and shape are frozen, i.e., the "intensity only" panel is flat at $1+(R-1)=R$. Closed-form expressions on ℤ ⟶

21:26 (CET).0Akvm (AF).

Jacob told me he knew no English equivalent but I found at least those ones: «To be piggy in the middle», «To fall between two stools» or «To be sitting on the fence».

20:07 (CET).0ACNe (AF).

In French we have this insightful expression:

Avoir le cul entre deux chaises

which translates as "sitting one's ass between two chairs", i.e., not being able to decide for one or the other and being suspended awkwardly above vacuum as a result.

From Glauber's definition Eq. \eqref{eq:0AwdV}, we see that our spatial correlations are precisely this for the RPCS: it's neither thermal (more bunched) nor Fock (more antibunched) but a bit of both. How funny that this most important, central, cornerstone state finds itself hanging like this in between two words. Another manifestation of its special role and of the somewhat (often) pathological behaviour of Glauber's correlators. Compare with our (Daniel's) definition Eq. \eqref{eq:0AgJI} which makes things much more physical!

19:31 (CET).0AfyK (AF).

Now that we are in the Anno Fabri's first alphabetical era, it starts to become fun to pick up meeting times. Daniel is back to Spain, though not yet in Madrid, so we still have to meet online. But now at business hours. How about 14:00? 

laussy@azag:~$ af -m -s 600 14:00
0ATOP  Wed 16 Sep 2026 13:59:00     -60 s   word atop
0Advt  Wed 16 Sep 2026 13:55:12    -288 s   word advt
0ASME  Wed 16 Sep 2026 14:00:01      +1 s   as+me
0ASMO  Wed 16 Sep 2026 13:59:24     -36 s   as+mo
0Adpi  Wed 16 Sep 2026 14:01:11     +71 s   ad+pi
0Axeh  Wed 16 Sep 2026 13:56:22    -218 s   ax+eh
0Atpi  Wed 16 Sep 2026 13:55:55    -245 s   at+pi
0Addo  Wed 16 Sep 2026 13:54:23    -337 s   ad+do
0Axiv  Wed 16 Sep 2026 14:08:44    +524 s   ax+iv
0AXOX  Wed 16 Sep 2026 14:09:51    +591 s   ax+ox

advt is not really a word (advertisement) but it's in the dictionary. Axiv is fun. AXOX is memorable... It'll be 0ATOP (af) then. One minute sharp ahead of the "everybody's time". 

laussy@azag:~$ af2cal 0ATOP @daniel "First Spanish meeting"
af2cal: taking @daniel exactly; @daniele start the same way
0ATOP — Wednesday 16 September 2026, 13:59:00 CEST
First Spanish meeting
with [email protected]

After A comes B, by the way, and the 0Boom (af) meeting, this one with nobody though.

12:19 (CET).0Av6h (AF).

Les alexandrins se font trop lourds, ils m'écrasent de leurs douze pieds, ils me piétinent dans le remords, ils me proscrivent le mot "amour", ils me brûlent d'ondes de chaleur trop intenses qui font cuire mon cœur tailladé comme un steak de steppe tartare.

Alors je me suis réfugié à l'ombre des octosyllabes, qui, parce qu'ils sont plus directs, exigent l'image tout de suite, la sonorité partout, et gardent ainsi à distance la vérité qui se cache dans les profondeurs et les subtilités.

Je ne veux plus écrire de poèmes, mais l'inspiration me griffe les yeux, et je voulais vérifier que je pouvais encore dire quelque chose qui ne soit pas boueux. Il m'était ainsi impossible de ne rien faire, sinon d'écrire sur la pluie, qui bruinerait dans un paysage lointain fait de montagnes vertes et de cabanes rouges, pour tenir compagnie à quelqu'un qui n'en a pas besoin, et qui voudrait plutôt un ciel clair, pour ne pas risquer, même sans le savoir, d'inspirer sous le soleil ardent un larmoyant poète qui s'étouffe à pleurer dans un vide sec et aride, loin de tout, loin d'elle ("elle" est "cette personne", le genre n'est pas indiqué).

J'ai les images plein la tête, je vois la brume descendre des monts et recouvrir les vallées, mais c'est le temps et l'espace qui surgissent à leur place, parce que ce sont eux, que je n'ai pas, que je ne trouve pas, qui m'échappent, alors je les ai cloués sur le papier. En octosyllabes. Huit clous par vers. «Le/temps/s'est/dé/chi/ré/d'a/mour». Et ce n'est pas la pluie qui entrouvre un nuage, pour se déverser sur ceux qui ne pensent pas à nous, c'est le temps lui-même, qui se déchire pour inonder l'espace de l'impérieuse nécessité qu'ont quelques-uns de se trouver.

Ce poème-ci est intraduisible. Écrit dans une péninsule pour un archipel, il n'existe que pour l'hexagone. Il y a des jeux de mots qui ne sont que pour l'oreille et la rime et qui se perdront au dictionnaire, tels que «/heurts/» (1) pour «/heu/res/» (2); la lecture se veut donc parlée pour le sens évident mais classique pour le sens métaphysique. D'autre part, «divague-ne» doit se prononcer «divagne» et est ma propre contribution à la prosodie Française, offrant sa seule et unique rime à «stagne». L'enjambement doit se faire avec «Il ne fallait qu'il ne divague-ne recommence...»; le recommencement inflationnaire est bien octosyllabique, avec la synérèse sur -tion, «re/co/men/ce/d'in/fla/tio/ner» (pas -ti-on qui sonnerait pédant), sa négation doit donc rester mangée par la divagation, mais elle doit néanmoins sonner: pour la grammaire, pour la musique, pour l'explosion.

Enfin!
Le temps s'est déchiré d'amour,
Après avoir longtemps passé
Des heurts dont il faisait les jours
À regarder sans l'embrasser,
L'espace se remplir d'étoiles
Et se parer de galaxies
Comme s'il allait hisser les voiles
Sur des conquêtes infinies.
Puisque la solitude stagne,
Lors l'espace se retenait:
Il ne fallait qu'il ne divague-ne
recommence d'inflationner.
Il s'efforçait avec rigueur
De ne laisser aucun gloser
Avant que ne soit venue l'heure,
l'heure ou l'endroit, d'y exploser.

10:29 (CET).0A6oW (AF).

15 September (2026)

Catherine Ringer est morte. Elle était l'une de ces artistes qui n'appartiennent à aucune catégorie, pour avoir été partout, trop loin. Partie de quelque chose de pire que la prostitution—la pornographie—elle a pourtant su atteindre dans ses périples les cimes de la grâce et de la beauté.

Elle s'y est élevée avec une autre âme à peine moins damnée que la sienne, Fred Chichin, lui parti rejoindre la perfection des cieux après l'avoir trouvée sur terre il y a déjà une vingtaine d'années. Je m'en souviens encore.

Dans leur plus belle chanson, Marcia Baila, ils rendent hommage à une inconnue—Marcia Moretto—mais qui devient, par leurs mots, immortelle, une figure nationale, une icône gravée dans nos cœurs ignorants avec le feu de la poésie:

Marcia était une chorégraphe. Elle n'a qu'une seule apparition répertoriée dans un film de seconde catégorie, que j'avais une fois cherché pour essayer de percer le mystère de cette lumière, de ce sourire:

Moretto comme ta bouche
Est immense quand tu souris

Mais j'ai bien dû reconnaître qu'il n'en transparaissait rien aux néophytes, à ceux qui n'avaient pas eu son contact direct. Mais j'ai su reconnaître aussi qu'il n'y en avait pas besoin, parce que tout cette aura, tout ce génie, était là intact, exprimé le plus modestement et le plus puissamment du monde:

Elle a cherché
Une nouvelle façon
Et l'a inventée

Et cette façon nouvelle, qui resplendit dans la chanson même, n'est-ce pas le plus bel hommage de l'élève à son mentor, de lui dire, par-delà la mort, que ce qu'elle avait été, elle l'avait reçu, et qu'elle allait le propager?

Ringer et Chichin chantent la mort, son injustice, sa cruauté. On ne dirait pas, à écouter leur chanson qui est guillerette, emportée, qui s'ouvre sur des sombreros qui s'agitent au rythme d'une musique festive, qu'ils pleurent quelqu'un. C'est que leur peine est une exultation, une transcendance, un hommage, une déclaration d'amour, d'amour vrai, d'amour complet, qui accepte même la mort, même la maladie, qui prend l'autre tout entier.

Mais c'est la mort
Qui t'a assassinée Marcia
C'est la mort
Tu t'es consumée Marcia
C'est le cancer
Que tu as pris sous ton bras
Maintenant tu es en cendres, en cendres
La mort c'est comme une chose impossible
Et même à toi qui est forte comme une fusée
Et même à toi qui est la vie même Marcia
C'est la mort qui t'a emmenée

Cette fusion entre la douleur et la joie, entre le deuil et la danse, entre le cri et le chant, il faut plus que du génie pour l'atteindre, il faut de l'universalité. Les gens ont festoyé sur cette chanson, alors que c'était une oraison.

Les Rita Mitsouko—le nom de leur groupe—ont d'autres chansons du même style, tel Le Petit Train, également joyeuse, bondissante, très dansante, très souriante, très internationale, avec des Chinois, des Indiens, et qui reprend le thème presque enfantin d'un train délaissé pour l'autocar...

...mais qui devient, dans la bouche de Ringer, un train de déportation vers les camps de la mort.

Petit train
Où t'en vas-tu ?
Train de la mort
Mais que fais-tu ?
Le referas-tu encore?

La musique est étrange, les sons de machines plutôt que d'instruments, la beauté est exsangue et pourtant elle bondit, elle jaillit, elle danse... comme pour Marcia. L'universel, encore une fois.

Dans cette chanson, Catherine pleure vraiment cette fois-ci. Ce n'est pas seulement ses paroles qui souffrent, c'est elle. Elle se sera donnée tout entière.

Ringer, l'actrice de porno, quand elle était belle, quand elle était jeune, avait gardé de sa sensualité, une séduction sulfureuse. Elle dansait effectivement très bien. Dans Andy, elle implore un garçon, grand dandy timide, de lui succomber. Elle se colle dans ses pâtes comme un chat et le harcèle de tentation. La chorégraphie tourne presque à la performance érotique.

Ce n'est pas une œuvre que j'admire particulièrement, elle frôle presque le vulgaire, en tout cas la facilité, par l'excès, par la grimace qui se veut provocante plutôt que pénétrante. Mais il ne leur en faut pas beaucoup pour retomber dans la perfection; dans C'est comme ça, on les retrouve tout entiers à leur art le plus dépuré:

La chorégraphie a retrouvé sa balance entre le feu du corps et celui des cieux. Le texte relate la rupture entre deux amants, elle, qui chante, s'excusant de cet échec. La poésie y pétille par à-coups seulement, par semonces—comme dans une vie—il n'y a rien de citable globalement, mais aux détours d'une strophe, l'on trouve quelques fragments de vérité, de vécu et d'aveux:

Et ça gêne, quoi, quand y a pas de plaisir

ou encore:

Je veux pas nous achever, tu sais

Je me souviens quand Gainsbourg avait dit à Ringer, en direct à la télévision, «vous êtes une pute», comment après s'être détachée, puis dérobée, puis défendue, elle s'était "dénoncée", elle avait accepté l'accusation, puis elle s'y était rebellée, elle avait trahi une colère étouffée de honte. Derrière cette immense artiste, déchirée, souillée par la vie, la sienne et celle du monde, il y avait un ange.

19:04 (CET).0AAyK (AF).

I've spent some time trying to classify and decide what to retain of the various (and innumerable) possible spatial correlation functions we've been playing with in the last several months. Here's a brief summary.

The least arguable is Glauber's correlation function defined in the canonical way: \begin{equation}\label{eq:0AwdV} g^{(2)}(\theta_1,\theta_2)\equiv\dfrac{\langle{:}\hat n(\theta_1)\hat n(\theta_2){:}\rangle}{\langle\hat n(\theta_1)\rangle\langle\hat n(\theta_2)\rangle}\,. \end{equation}

Then there are our definitions which differ principally by normalization. Daniel's definition: \begin{equation}\label{eq:0AgJI}\tilde g^{(2)}(\theta_1,\theta_2)\equiv{\tilde\Theta^{(2)}(\theta_1,\theta_2)\over\tilde\Theta^{(1)}(\theta_1)\tilde\Theta^{(1)}(\theta_2)}\,.\end{equation}

Note that $\tilde\Theta^{(2)}(\theta_1,\theta_2)\equiv\dfrac{\langle{:}\hat n(\theta_1)\hat n(\theta_2){:}\rangle}{\langle N(N-1)\rangle}$ is normalized so that $\iint\tilde\Theta^{(2)}(\theta_1,\theta_2)\,d\theta_1 d\theta_2=1$ and for isotropic cases, the denominator of Eq. \eqref{eq:0AgJI} is constant so that $\tilde g^{(2)}/(2\pi)^2$ is also a pdf. This makes it the joint law of the positions of a pair drawn at random among all the pairs.

Then our few-photon confounder-free correlation function, i.e., gauging-out the phase (self-alignment), post-selection to $N=2$ and fixing or not the quantum state ($C_\alpha$, thermal, Fock, etc.) This I will call $g^{(2)}_\sharp$ in agreement with notations from measure theory (pushforward distribution). This is defined on its canonical support $S=\{\operatorname{Im}Z=0,\ \operatorname{Re}Z>0\}$:

\begin{equation}\label{eq:0Av0D} g^{(N)}_\sharp\equiv(2\pi)^{N}\tilde\Theta^{(N)}\big|_S \end{equation} where $$Z\equiv\sum_je^{2i\theta_j}\,.$$

Operationally, $g^{(2)}_\sharp$ is coincidences over uniform chance, while Glauber's $g^{(2)}$ is coincidences over accidentals.

Then we have the variants of the above to particular situations, mainly, $g^{(2)}$ (standard Glauber) or $\tilde g^{(2)}$ (Daniel's pdf) applied to the frames once rotated, in which case I append a "al" (for "aligned") subscript.

Those cases are compiled in this figure:

Closed-form expressions and other key quantities on ℤ ⟶

16:54 (CET).0Akul (AF).

14 September (2026)

And the fixed $N$ thermal oracle for various $N$:

But this is less interesting than it looks, the dependence is simply: $$g^{(2)}_{N}(\theta_1,\theta_2)=\Bigl(1-\frac1N\Bigr)\frac23\;g^{(2)}_{\rm th}(\theta_1,\theta_2)$$ where $g^{(2)}_{\rm th}$ is panel (c) of Fig. (0AU6n). This happens to be, however, Daniel's $\tilde g^{(2)}(\theta_1,\theta_2)/[\tilde g^{(1)}(\theta_1)\tilde g^{(1)}(\theta_2)]$, so the intensity confounder is central to our definitions of correlations. Also on ℤ ⟶

22:00 (CET).0AXPm (AF).

And this to discuss the dominance (or not) of the phase as a confounding factor, along with a variety of other results:

The first result is how the correlation map looks on $[-\pi/2,3\pi/2]^2$.

Then the strong qualitative changes for the oracle if fixing (panel b) $N=2$ or not (panel c) $\langle N\rangle$ the number of particles. Not that the latter case is $N$-independent, but the former is not (will look at that next).

Panel (a) is the density of where the pairs land: in the bowtie, a lot of them in one lobe on the left, a lot on that on the right, one in each lobe, etc. Very few lobe-waist or wait-waist occurrences.

So a moot, trivial reading is that removing the phase as a confounder kills the correlations, because most of the pairs are now little correlated. That's the darkish four plateaux we see in panel (c), in the areas of the bright spots of panel (a).

But that our two-photon everywhere[2] story: not quantity but quality. In fact we are distilling, re-routing, so to speak, photons to regions where they become more correlated. This should be studied more closely. Also on ℤ ⟶

21:48 (CET).0AO3q (AF).

I realize I have ill-defined the problem of stripping out the phase. From our Simpson paradox approach,[3] the question is, if you remove the phase as the confounding factor, what does the correlation map become from the other confounding factors (shape through $C_\alpha$ and intensity)?

Mainly, we want to know whether the remaining correlations are still strong or just marginal, or maybe they are even stronger but redistributed, etc. We expect the RPCS to become uncorrelated in such a way.

That's an interesting question. What is not well-put, however, is "what do you mean exactly by removing the phase?" Now there is how we do it in ergoli (Natalia's paper): gauge it out. You can always do it. Makes $\sum_j e^{2i\theta_j}$ real. Fine. And that's what I had in mind. But from the thermal—or any classical—point of view, there is a more exact way: just pin the phase of the coherent state and only fluctuate their intensity imbalance. It would be equivalent as someone, let's say an "oracle", knowing the "true" phase. These two way to get rid of the phase are different, starting with the fact that Fock states don't have a "true" phase, so you can only ergolize them.

This is the oracle for thermal and RPCS, a better version of Fig. (09MmR) in particular with the good color code:

This is intensity independent, as should be.

Now the point is, for self-alignment (gauging out the phase), this is on the other hand strongly $N$ dependent. Since there is a normalization, even empty and single-photon frame will play a direct role. One recovers the oracle result when $N\gg1$, but at small $N$, it's a whole new universe:

Large $N$ recover the oracle result, fine... Small $N$ bring these weird geometric patterns, that come from the farfalle. This is more recognizable by shifting the axis, to unveil the four segments of perfect $N=2$ (not $\langle N\rangle=2$) correlations (farfalla):

The horizontal-vertical blue lines are single photons (frequent at such low intensities) falling on the axes by the self-alignment and thus killing the correlations there.

These intensity-dependent features show the necessity to fix $N=\mathrm{cste}$ at small $N$, and that's the farfalle story. We probably got there by intuition, luck or disgust of what fluctuating $N$ was giving us (we looked at that too, I remember).

What I remain unsure of is the practicality and physical meaning of those low-$N$, fluctuating, self-aligned results. I also need to find a way to make a subtle story simple. We could not refer to an oracle and assume high $N$, or invoke an oracle and be $N$-independent. But that's a lot to explain. Also on ℤ ⟶

21:23 (CET).0AwvF (AF).

MUMU is submitted... Brazil, here we come!

I've seen that last week only, such a question was tackled as part of a particular case (two emitters),[4] and I feel that Romain and myself would be able to get to the bottom of the general case, being somehow each at the top of the basic constituents of the problem. So, world, please have us work it out🤞.

16:50 (CET).0A6OT (AF).

13 September (2026)

I don't know why, but I suddenly remembered out of nowhere the theme song of Diff'rent Strokes in French [1], which is one of those examples where the copy—the translation—is better than the original. Its opening stanza reads:

(en)

Personne dans le monde
Ne marche du même pas
Et même si la Terre est ronde
On ne se rencontre pas

Missed encounters, again. I remember finding this sad in a beautiful way as a child. I still find it sad and beautiful, maybe more sad now than beautiful, or more beautiful than sad. What one leaves to childhood is the perfect balance.

I also remember that after the first stanza, which sounds like Victor Hugo, the rest is TV-series caliber: cheap, empty, obvious banalities. How strange that someone got their inspiration killed like this mid-course. Maybe they were rushed to finish it.

I was thinking that maybe I remembered that song from the depths of my subconscious, because it has the same supposedly faulty rhyme "pas / pas" as in my recent poem, «Ce que je sais de Toi», which is however admissible because it is not the same word, just the same spelling. In fact, we use the same construction: "not / step". The pronunciation would even be ever so slightly different: a closed "pas" for not, an open one for step.

I wrote yet another poem these last few days, although this one I will not publish... it may not adhere so strictly to my August, 12 standard of leaving everything in the moonlight shadow. This poem starts like this:

À leurs pieds détrempés par des soirs trop humides
les monts tutélaires nous regardaient glisser
le long de fleuves noirs creusés d'avenues vides
d'une cité baignoire, au passé ressassé.

Les rues mouillées brillaient d'éclats de demi-lunes.
Captive d'un décor posé pour t'accueillir,
où ton pas et ta voix s'accompagnaient l'un l'une,
...

and then it becomes too explicit, too revealing, it becomes promise-breaking, it's not something I could put under the moon and say "but that is just an alexandrine, it could be anybody". Its ABAB rhyme (and not the other way around) is «jaillir» and there are no other ten syllables I could put in front that would work. But in the image that forms from their beautiful melody, someone is there.

And the other two quatrains are self-deprecating, they are turning the wetness, the torpor, the indolence into mud. This is the title of this poem, by the way, and its resolution: «Être la boue.» This is really gloomy. I'm probably ashamed of myself. I don't like that I'm writing such things, that I let beautiful words arrange themselves into beautiful melodies to sing disgusting visions. What I need right now is to feel like Lamartine, not like Baudelaire. Painful things are good, pain is beautiful. The words from my previous other poem, which are the grains of sand that tear to tears the eyes of whoever has words blown to them by sentences, which are the wind over an arid desert of empty meaning, those are beautiful, I think. It's painful, it's frustrating, it's all the emptiness I've been swallowing whole for the past year, but it's still beautiful. What I'm writing now—and I say I, but I feel I'm reading it more than writing it—is accusing, is condemning, it's rubbing my face into who I am. And I don't like it. And I like it even less that I have to keep this to myself. And I can't write anything else.

15:38 (CET).095WM (AF).

I might have to give priority to glossing controversies surrounding classical vs quantum nature of ghost imaging. These include:

And then ping-pong:

The end of the debate might be Bromberg et al.[5]'s demonstration of ghost imaging with one branch only, so no room for nonlocality.

There's the parallel discussion on tradeoff:

Keep an eye too on other Shapiro's works.[[6],[7]]

11:16 (CET).09vUf (AF).

12 September (2026)

In the wake of Fano[8]'s argument, which sets phase as the cornerstone of the mechanism (it is even in the title), it is good to emphasize our (Salazar et al.[3])'s argument of confounding factors. First, Fano is explicitly on Fock states $\ket{1_a,1_b}$ although he discusses Hanbury Brown's thermal light. We do separate them quite strongly and attribute different mechanisms to both. Second, phase is one possible confounding factor. It is the the one for RPCS, since there is none other, but thermal states could get rid of their phase—that's precisely what we did with Natalia—and still be correlated due to other confounding factors.

Not aligned (phase random): \begin{equation} g^{(2)}(\theta_1,\theta_2)=\tfrac32\bigl[1+\tfrac13\cos2(\theta_1-\theta_2)\bigr] =\tfrac32+\tfrac12\cos2(\theta_1-\theta_2). \label{eq:09beI} \end{equation}

Aligned (phase pinned to zero): \begin{equation} \overleftrightarrow{g}^{(2)}(\theta_1,\theta_2)=\frac32 +\Bigl(1-\frac{3\pi^{2}}{32}\Bigr) \frac{\cos2\theta_1\cos2\theta_2} {\bigl(1+\frac\pi4\cos2\theta_1\bigr)\bigl(1+\frac\pi4\cos2\theta_2\bigr)}\,. \label{eq:09aKe} \end{equation}

From this, the autocorrelations read $$g^{(2)}(\theta)=2\,,\qquad \overleftrightarrow{g}^{(2)}(\theta)=\frac32+\Bigl(1-\frac{3\pi^{2}}{32}\Bigr) \frac{\cos^{2}2\theta}{\bigl(1+\frac\pi4\cos2\theta\bigr)^{2}}\,.$$ With gorgeous correlations:

In particular, this brings us to a new fundamental bunching constant: $$\boxed{g^{(2)}(\pi/2)=\frac{40-12 \pi }{(4-\pi)^2}}$$ which is approx $3.12254$.

If not the result of working late, we have this beautiful result that the phase as a confounding factor is not even central to the phenomenon, it merely smoothes out the correlations and distribute them equally around the rim. Intensity fluctuations are much more dominant, as they venture to a >3 suprathermal chaos into the waist, and, without phase whatsoever, bring "coalescence" to the level of RPCS in the lobes of the aligned dipoles (which is, I think, what we used to call the bow-tie with Daniel). Overall, their correlations remain on both sides of the non-aligned case $g^{(2)}=2$. Cross-correlations are also interesting, as they push the "forbidden" configurations (perpendicular) to some amount of bunching too. Nothing is suppressed!

I believe this shows convincingly that everybody has been missing a big part of the picture, conflating Fock and thermal cases, disregarding quantum states, being oblivious to the confounding factors, ensemble averages, etc. Also on ℤ ⟶

23:33 (CET).092eR (AF).

I should try to reproduce for my own benefit and as first step of a diagrammatic theory of frequency-resolved photon correlations, the Fano[8] arguments on a full perturbation theory of the HBT which-way interpretation, which can be truncated to these two diagrams:

Details on ℤ ⟶

18:21 (CET).09NXB (AF).

I glossed the papers by Goldstein et al.[9] and Popescu et al.[10] on typical canonality, and am now pondering how they connect to us. Basically, they find that for a big-enough rest of the universe, whatever the pure state of the whole thing, the little-subsystem would look like it does if we'd knew nothing about the universe anyway.

This is kind of the opposite stance to us: a particular case for us could be representative of its collapse, which itself is not representative of everything (because of heterogeneity). And we find the limit when this ceases to be the case, when a particular case ceases to represent the ensemble but represents only itself ($C_\alpha\to 1/4$).

15:02 (CET).091Aj (AF).

Killing two birds with one stone from our 08y9q session.

This animation shows both the disconnection of the "jumps" from the peaked structure of the distribution (this is just a moot change of sign magnified by the log-stretching) and the effect of sub-sampling, with only 1000 points. First, at criticality, where sampling is very good at seeing the object at small $k$—the particular realization we're dealing with. Interestingly, the sub-sampling brings us closer to the real ensemble average. If we ignore the specificities of the individual, we get a better view of the set it comes from:

In contrast, the donut struggles more at staying with its particular case, and falls back on features of the ensemble; also, it fluctuates typically beyond the single trajectory, as opposed to the dipole which fluctuates inside.

Those are features of self-averaging (donut) or its breaking (dipole), but they are visible at small $k$ only, since ergodicity breaking is a feature of $k/N$, not $k$ alone and as $k\to N$ they both converge to the particular collapse result. In all cases, the ensemble average (purple points) at the same photon cost can be seen to wreck down completely the underlying structure.

Also on ℤ ⟶

10:15 (CET).09tpC (AF).

11 September (2026)

I release today v1.0 of doi2bib, which had been kept in development stage for years, although it's one of my most actively used piece of code. I implemented with Claude the last annoying thing which I still did by hand, namely, the recognition of journals not present in my database, and indexing their iso4 denomination automatically, so that everything is now on the spot.

22:58 (CET).09R65 (AF).

Glosses for the evening:

20:14 (CET).09XmB (AF).

Jacob complains that $\binom{10^3}2\approx 10^6/2$ is not enough data to have a smooth single-collapse $D_2(p\mid\mathcal{C})$, though he compares that with $3\times 10^6$ independent-sample distributions which are, indeed, smooth. I don't think I ever looked at the noise in such distributions, but should, now. I think the single collapse will still be better at the same photon cost. So I'm not too worried about this temporary failure.

More importantly and adversarially, there is a bias in the experimental data in the radius, which makes agreement with the theory less than ideal. It's not critical because we study angular correlations, but since we introduce the story through absolute distances in space—which involve the radius—that brings unwelcome complications at the stage of the week and the stage of my mood. Also on ℤ ⟶

17:21 (CET).08Ssx (AF).

Adam and Eve's law in statistics are actually fringe terminology, from Blitzstein et al.[13] I was under the impression a real Adam had coined the law of total expectation but this seems too basic to even have had a discoverer, and that Eve came as a wink. It's actually the other way around. Eve came first on this occasion from a mnemonic device:

And what came off her rib on this occasion—although it comes first indeed as targetting a more fundamental moment—is introduced as such:

So, yes, they are witty, and I'll keep the terminology for fun, but won't keep it for the paper. Also on ℤ ⟶

12:37 (CET).082Ap (AF).

This is the effect of sub-sampling, with few ($10^4$) and quite a lot, yet still manageable ($10^7$) samples, from one collapse:

The lines are when sampling over all subsets: no statistical noise. This one is re-introduced if we overlook some possibilities. Noise is not a feature of experiments, it is a feature of partial views, truncated treatments, botched jobs. Experiments can be perfect (mathematically smooth).

In more details, the donut is much more prone to noise, while the critical case is much more resilient to it: it's good to have some individuality, it makes you less affected by superficiality. Also on ℤ ⟶

11:13 (CET).08yf5 (AF).

Ce que je sais de toi, tu le sais, c'est bien peu,
La couleur de tes doigts, la nuit dans tes cheveux,
L'ocre de ta bouche, un accent dans ta voix.
Ce que je sais de toi, tout le monde le voit.

Ce que je crois savoir, ce que tu ne dis pas,
C'est que sur ton chemin, nul ne marche en tes pas,
Que tu aimes rêver d'horizons de fenêtre,
Où quelqu'un t'attendrait, qui pourrait te connaître.

Ce que je ne sais pas, c'est si tu as raison,
Et si tu es heureuse, au fil de tes saisons,
Si la vie te suffit, et si tu peux attendre
Celui qui t'aimera et qui saura t'entendre.

Ce que nul ne sait, oh! c'est si notre prière
N'est jamais entendue, appelle une lumière,
Si, l'attente rompue, on se retrouve enfin,
Et qui est celui-là qui nous vient à la fin.

00:01 (CET).08fs2 (AF).

10 September (2026)

S'il y avait un mot, une phrase, un poème,
Que les montagnes bleues dans les matins froissés,
Sussent me susurrer, en guise de je t'aime,
Pour vouloir te parler, sans pouvoir te blesser...

Si ce mot existait, j'irais le déposer,
Murmure langoureux, à l'ombre de tes reins,
Pour qu'un frisson supplée l'audace d'un baiser
Volé au corps brûlant d'une statue d'airain.

Mais le vocabulaire est rempli de déserts,
Les mots sont grains de sable, et les phrases le vent;
Ils déchirent les yeux, qui ne voient rien qu'ouverts,
Elles attisent le cœur, qui ne veut plus d'avant.

S'il y avait un mot, une phrase, un poème,
En guise d'une main frôlant un cambrement,
Si ce mot existait, en dépit de moi-même,
Je te l'aurais écrit, caché dans un moment.

09:57 (CET).08h7o (AF).

9 September (2026)

Two and a half hour WP4 meeting... now I have to collect the material I promised. Thankfully, most of it is already on ℤ somewhere... this is mainly i) about loss of smoothness from finite sampling, and ii) the reason for the bulge at $k=10$.

The first one is Z-03W0Q, where I also comment at the time that "I'll look more into how sub-sampling and $n$" changes that picture." But didn't too much. Maybe in Z-03ALO to some extent, which also relates to one of Alex's question: what's the precise mechanism for abrupt sign flips as one selects one photon more or less in the $k$-gons. Clearly the emergence or disappearance of a peak explains that. One can see that in Fig. (033rg). I'll study it in more detail.

The second point, I had actually studied in depth, also finding the phenomenology striking, but to come to the conclusion this was no big deal. This is in 04ZAP. I had forgotten why it was not interesting. It's related to geometric sampling of frozen configurations, so more than not-interesting, too technical and subtle as compared to other interesting features, like the gain, which is why I drifted.

I also promised to send the source files so that my tortured caterpillar can be turned into an overleaf butterfly. But first I need the last round of my own revisions to shorten it to 3000 words. It's only 10% too much now, which still requires some work.

Main work remains on the figures, so at least this burden I can put on somebody else.

13:16 (CET).08y9q (AF).

8 September (2026)

I just convoked our 08Ode (af) WP4 meeting on the first draft of the manuscript, where we revolution people's understanding of statistics in quantum mechanics, and how beautiful singular objects—mathematical perfections—get shattered (that's the word in the paper) by statistical noise when ensemble-averaging. Now I just have to bring it within size limits, which is of course virtually impossible because it's almost currently 100% oversized. This'll produce daughter papers like a woman in the bible. I can't wait to write a 20 pages blog post on those beautiful results. It has to be on the arXiv before their presentation at the HPM (2026) at the end of this month.

18:49 (CET).07yrO (AF).

In the top Fig. of 073Rh, $k\sigma(k)$ becomes constant at criticality, which is the feature of the ensemble average. Both are critical—it's derived from the geometry, not of the averaging—so that's fine.

But if we focus on the single-collapse (quantum) average, things get interesting too, and maybe that's the sharpest picture. We must stay along a ray—$\kappa$ fixed and $N$ growing—since that's where ergodicity breaks. How to keep "individuality" is obvious: take a huge collapse and sub-sample from it to make $N$ grows, until you recover everything. So now I'm in the business of sampling within sampling within sampling!

The single-collapse standard-deviation, $k$-corrected, reads: \begin{equation} k\,\mathrm{sd}_{\mathcal{C}}(p_{\mathcal{C}}) =\sqrt{\rho(\kappa)}\;k\sigma(k) \;\xrightarrow[N\to\infty]{\kappa\ \text{fixed}}\; \begin{cases} \sqrt{\rho(\kappa)}\times0.479A & C_\alpha=\tfrac14,\\[3pt] \propto N^{-3/2}\to0 & C_\alpha\neq\tfrac14, \end{cases} \label{eq:07hYC} \end{equation} since $\rho$ is a function of $\kappa$ alone and $k\sigma= (k\langle\delta\rangle)(\sigma/\langle\delta\rangle)$ is $A\times0.479$ at criticality but $\propto k^{-1}\times N^{-1/2}$ elsewhere. $A$ is a critical amplitude, large-k constant to which $k\langle\delta\rangle$ converges to, i.e., $k\langle\delta\rangle(k)\to A$.

This gives us something like this (not plotting the independent ensembles data):

What is remarkable, besides the intended flat single-collapse variance (but the ensemble also remains flat), is that now the individuality of the collapse is altered: it is not smooth anymore, it becomes fractal: each added photon to the whole object, needed to grow $N$ as $\kappa$ remains constant, brings a kink in the individuality. If you don't have the full picture from the start, but need to get it as you go along, disruptions can happen: they are smooth but clear and can go to all orders. I'm inventing a new type of physics... psychological or sociological physics: find interpretations to universal themes of existence, especially of interactions between people, in fundamental objects. If you had not added this one point in your life, you'd have a sweet, smooth follow-up. But this new point got there and it made you jump slightly off course.

Jokes aside, this is also very nice, but I'm not sure we'll be using it right away... Also on ℤ ⟶

10:48 (CET).07dA4 (AF).

What we'll try to capture for Natalia's data:

  • The gain $G$ as function of $k$ and in particular at $k=2$ and $k=3$, where it varies the most for different $C_\alpha$ (Fig. 3a)
  • The strip: $k\sigma$ constant across $k=30$–$700$ at criticality against falling by $>15$ (atom) and $\approx100$ (donut), from independent samplings (Fig. 2d).
  • The qualitative behaviours of $n_{\rm eff}(k)$: it should be flat ($\approx 4.5$) at criticality. It should produce a dip to $\approx 2.5$ at $k\simeq100$ and then increase for the atom. It should increase with slope $1/10$ for the donut. The dip's position reads $k_c$ and hence $1-2\sqrt{C_\alpha}$ from one frame. (Fig. 3b)

The EB$(p_\mathcal{C})$ could go in Fig. 3a as well [or if too heavy, removed entirely].

We could also show fixed $k$-ergodicity for all states and consider the slopes, but this is less important. Also on ℤ ⟶

09:32 (CET).07oEz (AF).

The result of some obsessive thoughts on criticality. At fixed $k$, everything is ergodic. Increase $N$ and the collapse mean converges on the ensemble always. The slope at which $\mathrm{EB}(p_{\mathcal{C}})$ falls to zero is smaller at criticality but this is quantitative. I won't discuss here the ergodicity of the variance itself, although this is probably interesting in connection with frustrated systems.

Along a ray, however, criticality becomes non-ergodic. Fixing the fraction $\kappa\equiv k/N$ and letting both grow, sees the donut and the atom still converge to zero, $\mathrm{EB}\propto1/N$, but for $C_\alpha=1/4$, the parameter converges to a constant. For $\kappa=1$, this constant is the single-shot width $0.479^2$. The collapse mean $p_\mathcal{C}$ remains an unbiased estimator but its variance doesn't go to zero. It still does for non critical cases, i.e., $\operatorname{Var}^2_{\mathcal{C}}(p_{\mathcal{C}})/\langle\delta\rangle\to0$ (roughly $N^{-1/2}$) at $C_\alpha\neq1/4$, but it goes $\to\sqrt{\mathrm{EB}_\infty(\kappa)}$ at $C_\alpha=1/4.$

The main picture that captures the bulk of the phenomenology is this one:

where the $y$-axis is $k(p-\langle p\rangle)$, i.e., $k(\langle\delta\rangle-\delta)$. The factor $k$ makes the standard deviation flat for the ensemble average and thus makes a neat divide between self-averaging, which closes the funnel, and non-self-averaging, that remains an opened fringe, within which trajectories move freely, being thus more representative of themselves than of the collective (although still unbiased estimators).

The mechanism is maybe best capture by $n_\mathrm{eff}\equiv(\langle\delta\rangle/\sigma)^2$ the number of effective chords (segments that make up the deficit) needed to provide the result. And there is a clean result for those, namely: $$\frac{n_{\rm eff}}{k}\to\frac{8(1-4C_\alpha)^{3/2}}{80+1536C_\alpha+624C_\alpha^2}$$ i.e., $n_{\rm eff}=k/10$ for the donut (exactly, the 10 being derived, not fitted), $n_{\rm eff}\approx k/229.5$ for the atom (so $\sigma/\langle\delta\rangle\to\sqrt{229.5/k}$) and, crucially, vanishes at criticality $C_\alpha=1/4$, so that the next term takes over, which is the constant $n_{\rm eff}=4.36$ (criticality). A fixed number of photons is enough to provide the details of a particular collapse. Adding more photons doesn't bring more information that would be information from the family itself from which the particular case is sample: the system is not self-averaging anymore.

This sharp behaviour is caused by the exact zero on the rim for the dipole. One can extend this behaviour to all states if considering the effect not of a zero, but of the depletion (minimum) there $\varepsilon\equiv1-2\sqrt{C_\alpha}$. Then every state is critical up to $k_c\equiv(2\pi/\sqrt2)\,\varepsilon^{-3/2}$, which is where one photon, on average, first lands in the depleted area, thus starting to "fill-up the criticality."

The skewness and kurtosis of the distributions are also interesting, as they differ for critical and non-critical cases (criticality being much more Gaussian, apparently). But I leave this out for now. Also on ℤ ⟶

09:24 (CET).073Rh (AF).

3 September (2026)

 

🌑

 

Well I'd best go over there and do whatever that thing over there I was going to do was.
— Dominic in The Banshees of Inisherin.
17:21 (CET).06xI1 (AF).

New month, old collaborations getting kicked up again...

With Camilo back on our crusade to define the criterion for SPS. We had to remember everything—again—and got to revive Camilo's beautiful analytical expression for the $g^{(2)}$ plateau with filtering the $N$-circular cascade, and this conundrum we keep forgetting and rediscovering of how does higher-order correlators factorize (or fail to) in terms of two-photon correlations. They are needed to work out probabilities that more than one photon is detected in the detection time of a physical detector. Hopefully this time we go somewhere. One new line running in parallel, though...

11:28 (CET).05GD4 (AF).

And I also finally fixed the placement of day & time-of-day (both standard and AF times). The code was indeed in the editing gadget so was lost for the rest of the world. Not that the rest of the world missed it very much but at least Daniel & Jacob should have it right now. They'll probably get confused as now it's really LIFO with time flowing up, as I always intended it. Scrolling down goes backward in time. The current day is always visible in its canonical position: in the past. I can call this a night.

02:28 (CET).05Z5g (AF).

From the after-midnight session (in Madrid; it's before Ju-Jitsu session time in Colombia), the HG → same distribution as LG was established by Daniel. There's some correspondence from one to the other, which I don't detail here as it's too voluminous, but it confirms that there is a single distribution that covers for those two basis. The gauging out of the phase removes their geometric distractions and they get pinned to the fundamental object, the same distribution for everybody. It remains to establish whether the quantum state gets transported from one basis to the other, e.g., thermal states of HG gives the same push-forwarded distribution than thermal LG; sounds reasonable, however how about, say, $C_\alpha=1/10$? is the quantum state the same? Daniel is looking into it. Plus generalization to all bases, plus to all photon numbers. And we can call it a day!

02:16 (CET).054rh (AF).

2 September (2026)

The transfer of date-pinning to caltoc is challenging me. It's tough to come back to things that were designed two months ago. Today's been a rotten day. Nothing substantial done...

Daniel's been answering the two questions I put to him, though. Reading him at least feels productive. I also asked him to help Matilde face this horrible machinery that is CSIC's administration, and he laid out for her a detailed document that the CSIC should put on their intranet. I've never seen anything so beautifully articulated in relation to their maze of horror.

Anyway, regarding the first question of 05MZV, he confirms that any two LG$_{\pm1}$ quantum states $C_\alpha$ and $C_\beta$ on the two-photon farfalle are rescaled versions of the same distribution: $${g_\alpha^{(2)}-1\over g_\beta^{(2)}-1}={C_\alpha\over C_\beta}\,.$$ So there is only one distribution, which can be stretched more or less. Or not at all, if $C_\alpha=0$ (not correlated), then $g_\alpha^{(2)}=1$. I believe this should be general, i.e., apply to $g^{(n)}$ on its $(n-1)$-dimensional farfalle.

Taking the most correlated case as a reference:

$$g_\alpha^{(2)}(\Delta\theta)=1+2C_\alpha\cos(4\ell\Delta\theta)$$

$C_\alpha={1\over4}$ is the critical value over which the state becomes quantum. He checked with two-mode squeezed vacuum, for which he finds:

$$g_\xi^{(2)}(\Delta\theta)=1+2{\bar n+{1\over2}\over2\bar n+{1\over2}}\cos(4\ell\Delta\theta)$$ with min $g^{(2)}$: $$\min g_\xi^{(2)}={2\bar n \over4\bar n+1}>0\,.$$

That settles nicely that the distribution touching impossibility is a particular case of quantumness: it implies it, but this is not necessary. That makes the contraposite of my theorem a bit moot: reduced to its extreme.

The proof is general an relies on cancellation of anomalous correlators through the phase re-alignment.

It's quite exciting because also the (related) second question gets answered positively. HG structures, once re-aligned, provide the same result: there is only one support and only one distribution on it. It only needs to be generalized (at least to $N>2$). Also on ℤ ⟶

23:16 (CET).05xAK (AF).

With Daniel's meeting today I realized that browsing ℤ from month to month is inconvenient (also a bit slow, even locally), and that my nice layout of the current day being pinned at the bottom—to keep track of where one is in time—is not in effect for non-logged-in users (which is the rest of the world but me). So I have this highly crafted touch on how to look at time passing and it's only for my benefit. At least poor Daniel could have that, if it's only him using those notes. The problem is that I embedded that with the editor gadget rather than, say, the caltoc. I'll debug it later today.

For now, we need to check what happens with HG geometries on the farfadelles (my hunch is that it could be the same, the geometry will not transpire on the very robust multiphoton structure which we already know is the same, or in its distribution once homogenized). The other thing is whether *all* quantum states have the same distribution modulo some contrast. That would also be quite a thing. That would weaken our theorem that $g^{(n)}=0\implies$ no classical de Finetti representation, because there'd be essentially one case only that realizes the strictly impossible configurations: the maximally correlated one. All the others will spill beyond the classical limit (RPCS) but remain possible everywhere. So my obsession with impossibility would then be wrong: it doesn't need be impossible, just less likely than a critical threshold. Buh!

It's still nice but I wanted things to be impossible, not merely very difficult.

18:38 (CET).05MZV (AF).

I initially made a statement on the different self-averaging of the variance (I retracted it now, not to confuse myself later, as I was incorrectly stating that variance doesn't self average). I was commenting that self-averaging properties of the variance could be notable and maybe related to spin ice and observables in strongly-correlated condensed matter, where fluctuations are the main concern. This still applies.

I had produced this beautiful figure, for fixed $N$ (hence not telling much about self-averaging):

The figure is correct but its interpretation wasn't: variance does self average.

I don't think I'll look much into the variance as the self-averaging of rays is enough for now. But still, for the record (and later comeback to this), every width self-averages only as $1/N$, while the donut's perimeter manages $1/N^2$ so there is some tension between them (which maybe is captured by self averaging of the mean, though).

Also on ℤ ⟶

14:32 (CET).05qyk (AF).

1 September (2026)

And now for the big one of today: we introduce the ergodicity-breaking parameter, the relative scatter—across collapses—of the quantity estimated from one of them: \begin{equation}\label{eq:054CV}\mathrm{EB}=\mathrm{Var}(\overline{\delta^2})/\langle\overline{\delta^2} \rangle^2\,.\end{equation} This comes from standard the time-averaged ergodic theory. $\mathrm{EB}\to0$ is ergodic while $\mathrm{EB}\to$ a non-zero constant is broken ergodicity.

Transcribed to multiphotonics, and using the deficit $\delta$ again as well as its variance: \begin{equation} \mathrm{EB}(p_{\mathcal{C}}) =\frac{\mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})}{\langle\delta\rangle^2} =R_N, \qquad \mathrm{EB}(v_{\mathcal{C}}) =\frac{\mathrm{Var}_{\mathcal{C}}(v_{\mathcal{C}})}{E[v_{\mathcal{C}}]^2}, \label{eq:EB} \end{equation} Note that the second looks closer to the original tracking parameter, both being relative scatters of a second moment measured from one realisation.

As was the case for the related self-averaging, we have ergodicity at fixed $k$, as a function of $N$. But criticality breaks it for $k/N$, and the manifestation of this is a beautiful ladder: a factor $10$ increase of $N$ (and thus of $k$) results in a factor 10 decrease of the EB for non-critical cases. The critical case, however, remains essentially exactly constant:

Also on ℤ ⟶

21:10 (CET).05OrC (AF).

To gloss for ergodicity-breaking parameter:

20:47 (CET).05aDN (AF).

An interlude to collect exact results regarding perimeters $\langle p\rangle$ and deficits $\langle\delta\rangle$ of $k$-gons on the unit circle. Jacob was wondering about them. They are not needed to compute the exact deficit since this is obtained as an average of differences from chords to arcs, but, in principle they are needed for the ensemble-average limits. And it's nice to have them.

By definition, $\langle\delta\rangle=2\pi-\langle p\rangle$.

The cases $k=2$ and $k=3$ are accessible for all states $C_\alpha$: $$ \langle p\rangle=\frac{4k}{\pi}\Bigl(1-\frac{2C_\alpha}{15}\Bigr),$$

The donut $C_\alpha=0$ is also available for all $k$ as a series: $$\langle\delta\rangle_{\rm donut}(k) =\sum_{n\ge1}\frac{(-1)^{n+1}\,2\,\pi^{2n+1}} {(k+1)(k+2)\cdots(k+2n)}\,,\ \qquad k\ge2 $$ The leading term is $\langle\delta\rangle\simeq2\pi^3/[(k+1)(k+2)]$.

Off the donut, there are no exact results for $k\ge4$ but asymptotics are interesting: \begin{align} \langle\delta\rangle&\simeq\frac{2\pi^3}{k^2(1-4C_\alpha)^{3/2}}\,, \quad(C_\alpha<1/4), \\ \langle\delta\rangle&\simeq\frac{A}{k},\ \ A=5.698\,, \quad(C_\alpha=1/4). \label{eq:asympt} \end{align}

We see again criticality behaving qualitatively differently.

Also on ℤ ⟶

19:48 (CET).05EKH (AF).

This is the closed-form solution of an integral which popped up in Daniel's latest computation of the volumes of $N=4$ and 5 farfadelles. He could compute everything exactly but this one:

$$K=\int_0^{\pi/2} x\cot x\,\ln(1+2\sin x)\,dx\,.$$

Claude (Fable), however, could compute it:

\begin{equation}\label{eq:05d6l}\begin{split}K=\frac{35\pi^3}{288}&+\frac{\pi}{4}\ln^2 2+\frac{\pi}{8}\ln^2(2+\sqrt3)+\frac{\ln(2+\sqrt3)}{2}\operatorname{Cl}_2\!\left(\frac{\pi}{3}\right)+\frac{\pi}{2}\operatorname{Li}_2\!\left(-\frac12\right)\\&-2\,\mathrm{Im}\!\left[\operatorname{Li}_3\!\left(\frac{1+i\tan\frac{\pi}{12}}{2}\right)+\operatorname{Li}_3\!\left(\frac{1+i\tan\frac{5\pi}{12}}{2}\right)\right]\end{split}\end{equation} with $\operatorname{Cl}_2$ the Clausen function and Li$_s$ the polylogarithm (so we have dilogarithms and trilogarithms here).

That makes $$K=0.7315710425072642965\ldots$$ which is what I remember from his numerical evaluation of it. It does rely on special functions—which always look like a cheat—but at least that should exclude other nicer functions. Continue on ℤ ⟶

19:35 (CET).05x8Y (AF).

Self-averaging is the last piece I need to assemble Natalia's paper. This is a property of quenched disordered systems, in our case, brought by the collapse. A quantity computed from one such frozen collapse is self-averaging if its collapse-to-collapse relative fluctuation vanishes as the sample grows, $$ R_N=\frac{\mathrm{Var}(X_N)}{E[X_N]^2}\longrightarrow0 \qquad(N\to\infty),$$ and strongly so if $R_N\propto1/N$.

This is the condition for one large-enough sample to represent the ensemble average (ergodicity), and repeating the experiment tells us nothing we didn't already have. When it fails, every sample gives a different picture and only an ensemble means anything. Right our Natalia's ergodicity business!

But now to make it precise and quantify when it holds or not.

For the perimeter itself, we trivially have self-averaging from Eq. (1) at fixed $k$ and large $N$, $$\mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})\simeq\frac{k^2\zeta_1}{N} \quad(\zeta_1>0), \qquad\qquad \mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})\simeq \frac{k^2(k-1)^2\zeta_2}{2N^2}\quad(\zeta_1=0),$$ so $p_{\mathcal{C}}$ is strongly self-averaging off the donut and super-self-averaging at it—the standard deviation falling as $N^{-1}$ instead of $N^{-1/2}$. The only notable thing here is that the donut is much more self-averaging than the rest: this comes from its $\zeta_1=0$, i.e., rotation invariance, which says that no single photon carries information about the perimeter.

More than the perimeter, the deficit to that of the circle is more interesting, because instead of $2\pi$, the denominator becomes an increasingly smaller quantity.

The obvious self-averaging at fixed $k$ is not the most interesting: every $C_\alpha$ self-averages because adding photons to a collapse averages a fixed-size observable over ever more subsets. So we find self-averaging, although, again, with a different slope (super self-averaging for donuts):

It is more insightful to consider the self-averaging properties of, say, the biggest subset $\binom Nk$ which is when $k=\lfloor N/2\rfloor$. Or let's even generalize to any ray $k\propto N$. Then we find a breakdown of self-averaging for critical cases ($C_\alpha=1/4$):

17:53 (CET).05Krn (AF).
  1. ↑ Hanbury Brown and Twiss interferometry with twisted light. O. Magaña-Loaiza, M. Mirhosseini, R. Cross, S. Rafsanjani and R. Boyd in Science Advances 2 (2016).
  2. ↑ Two photons everywhere. E. Zubizarreta Casalengua, F. P. Laussy and E. del Valle in Phil. Trans. R. Soc. A 382:20230315 (2024).
  3. ↑ 3.0 3.1 Boson correlations are spurious for classical states. D. E. Salazar and F. P. Laussy in arXiv:2604.16283 (2026).
  4. ↑ High-purity fluorescence photon bundle emission from two separate emitters. Z. Chen, J. Xu, D. Kong, X. Hu and F. Wang in arXiv:2609.07454 (2026).
  5. ↑ Ghost imaging with a single detector. Y. Bromberg, O. Katz and Y. Silberberg in Phys. Rev. A 79:053840 (2009).
  6. ↑ Unified theory of ghost imaging with Gaussian-state light. B. I. Erkmen and J. H. Shapiro in Phys. Rev. A 77:043809 (2008).
  7. ↑ Signal-to-noise ratio of Gaussian-state ghost imaging. B. Erkmen and J. Shapiro in Phys. Rev. A 79:023833 (2009).
  8. ↑ 8.0 8.1 Quantum Theory of Interference Effects in the Mixing of Light from Phase-Independent Sources. U. Fano in Am. J. Phys. 29:539 (1961).
  9. ↑ Canonical Typicality. S. Goldstein, J. Lebowitz, R. Tumulka and N. Zanghì in Phys. Rev. Lett. 96:050403 (2006).
  10. ↑ Entanglement and the foundations of statistical mechanics. S. Popescu, A. Short and A. Winter in Nature Phys. 2:754 (2006).
  11. ↑ Imaging and certifying high-dimensional entanglement with a single-photon avalanche diode camera. B. Ndagano, H. Defienne, A. Lyons, I. Starshynov, F. Villa, S. Tisa and D. Faccio in npj Quantum Inf. 6:94 (2020).
  12. ↑ Characterization of space-momentum entangled photons with a time resolving CMOS SPAD array. B. Eckmann, B. Bessire, M. Unternährer, L. Gasparini, M. Perenzoni and A. Stefanov in Opt. Express 28:31553 (2020).
  13. ↑ 🕮Introduction to Probability. J. K. Blitzstein and J. Hwang. CRC Press, 2nd ed., 2019. [ISBN: 9781138369917]