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11 September (2026)
J 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.
Adam and Eve's law in statistics are actually fringe terminology, from Blitzstein et al.[1] 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.
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.
8 September (2026)
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...
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.
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.
2 September (2026)
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 $N>2$).
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).
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:
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.
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.
Not very enlightening but that settles it.
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 ($C_\alpha=1/4$):
- ↑ 🕮Introduction to Probability. J. K. Blitzstein and J. Hwang. CRC Press, 2nd ed., 2019. [ISBN: 9781138369917]