Is the spacing real, or can it be taken to zero ── putting numbers on the branch that only waiting can move
So far we have split the lattice thread in two ── if the spacing is numerical there is no problem (Episode 17 got Mercury precessing at 42.9807 arcsec/century), if the spacing is physical there is a price (Episode 16: supersymmetry, or gravity leaks).
The branch is one question: "can \(a\) be taken to zero?" And this is not a matter of opinion but a question observation answers.
This episode puts numbers on what that observation is. Killing off the unusable tools in order leaves exactly one at the end ── and what is left comes with a definite target value.
Start with the tool that looks most natural. If the spacing is real, the horizon's cells exist, so \(N=A/a^2\) is a genuine count ── which makes it look as though black-hole entropy could tell the difference.
It cannot. As Episode 14 showed, the cutoff-dependent divergence is absorbed entirely into the renormalization of Newton's constant. The entanglement side's \(A/48\pi\epsilon^2\) and the \(\delta(1/G)=1/12\pi\epsilon^2\) obtained independently from the heat kernel agree down to the factor of \(A/4\) ── so
$$S=\frac{A}{4G_{\text{ren}}}$$has the same form whether \(a\) is real or the theory is continuous. Entropy is not looking at whether there is a spacing.
If spacetime were grainy, positions should jitter "by a Planck length every Planck time" ── holographic noise. Fermilab's Holometer looked for this directly: two independent 40 m interferometers, correlating their outputs up to 25 MHz.
The result was a strikeout. Noise at the level Hogan predicts is excluded at high significance.
If the spacing is real, the dispersion relation changes. As Episode 16 showed, after imposing supersymmetry and CPT what remains is \(n=2\) (dimension 6).
$$E^2=p^2\left[1+\left(\frac{p}{E_{\rm LV}}\right)^2\right]$$On the superluminal side (the plus sign) the photon acquires an effective mass.
$$m_{\rm eff}^2=\frac{p^4}{E_{\rm LV}^2}$$Once this exceeds \((2m_e)^2\), a photon becomes able to decay into an electron pair in vacuum. The threshold that opens is
Turned around ── if a photon of energy \(E\) arrived unbroken, then
$$\boxed{\;E_{\rm LV}>\frac{E^2}{2m_e}\;}$$There is no coefficient to tune. It is fixed by the electron mass and the observed photon energy alone.
Here is the pleasing part. It goes as \(E\) squared. So raising the photon energy stretches the bound quadratically.
For comparison, let us set the other commonly used method alongside. It measures the time difference from higher-energy photons arriving later. For \(n=2\),
$$\Delta t\simeq\frac{3D}{2c}\left(\frac{E}{E_{\rm LV}}\right)^2 \qquad\Longrightarrow\qquad E_{\rm LV}>E\sqrt{\frac{3D}{2c\,\Delta t}}$$This one stretches only linearly in \(E\). That difference decides the contest.
# photon decay: E_LV > E^2/(2 m_e) ── coefficient-free, E squared # time-of-flight: anchored on LHAASO's published GRB221009A value ── linear in E # at E=18 TeV, E_LV > 1.2e21 eV (n=2, 95%CL) photon E decay bound M_Pl ratio time-of-flight bound M_Pl ratio ------------------------------------------------------------------------- 18 TeV 3.17e20 eV 2.6e-08 1.20e21 eV 9.8e-08 1.4 PeV 1.92e24 eV 1.6e-04 9.33e22 eV 7.6e-06 10 PeV 9.78e25 eV 8.0e-03 6.67e23 eV 5.5e-05 112 PeV 1.22e28 eV 1.00 7.45e24 eV 6.1e-04 1 EeV 9.78e29 eV 80.1 6.67e25 eV 5.5e-03 # photon energy needed to reach M_Pl photon decay : 1.117e17 eV = 112 PeV (80× LHAASO's record of 1.4 PeV) time-of-flight: 1.831e20 eV = 183 EeV (beyond the GZK cutoff ~5e19 eV) # crossing point E = 2 m_e L0/E0 = 68.1 TeV. above this, decay is always stronger (squared vs linear).
For time-of-flight to reach the Planck scale you need a 183 EeV photon. But a photon at that energy reacts with the cosmic background radiation and cannot propagate over cosmological distances (the GZK cutoff is at the \(5\times10^{19}\) eV scale).
Time-of-flight: impossible. The photon you would need does not arrive at all.
Photon decay: 80 times. Hard, but not impossible.
To confirm with numbers ── for a 100 PeV photon propagating 1 Gpc, \(\delta v/c=(E/M_{\rm Pl})^2=6.7\times10^{-23}\) gives a time difference of \(6.9\times10^{-6}\) seconds. GRB time structure is at the millisecond scale, so it falls short by more than a factor of 100.
One photon of about 112 PeV arriving unbroken decides it.
\(E=\sqrt{2m_eM_{\rm Pl}}=1.117\times10^{17}\) eV. Eighty times LHAASO's current record of 1.4 PeV.
But the title's "settles" overstates it. What gets decided is one cell of the table ── superluminal side, \(n=2\), \(O(1)\) coefficient, scale \(\le M_{\rm Pl}\) ── and it does not touch the subluminal side, \(n\ge3\), small coefficients, or discreteness that produces no dispersion (causal sets and the like). The range that need not be decided is overwhelmingly the larger.
If it is observed, a physical Planck-scale spacing (\(n=2\), superluminal side) dies. Until it is, it survives inside Episode 16's window (\(10\) TeV \(\le\Lambda_{\rm conf}\le3\times10^9\) GeV).
The decay line has slope 2, the time-of-flight line slope 1. They cross at 68.1 TeV, and above that decay always wins. Already at 1.4 PeV there is more than a factor of 20 between them.
Here is a part that has to be written honestly. This question cannot be answered with equal force on both sides.
| possible? | |
|---|---|
| discover graininess | yes. raise \(E\) and, if it is there, it must show up eventually |
| prove continuity | no. whatever you measure, zeros just keep coming |
| exclude graininess by striking out | no. it only pushes the scale up |
So the answerable question is "is \(a\) below this scale?", not "is \(a\) zero?". The current boundary is \(\sim10^{-3}M_{\rm Pl}\).
And this is one of the shapes this series keeps meeting ── fix a question into well-posed form and it splits into an answerable part and an unanswerable one. The same operation as Episode 15's fixing of "where does \(G\) come from" into \(\alpha_G\).
| spacing numerical (\(a\to0\)) | spacing physical (a minimum length) | |
|---|---|---|
| does GR stand up? | yes. 42.9807″/century (Episode 17) | yes (the same at low energy) |
| solvable by finite differences? | yes (with the deviation as the variable) | yes |
| Lorentz violation | does not occur (the background is unobservable) | occurs. needs SUSY + CPT (Episode 2) |
| can emergence save it? | — | matter can be saved but gravity leaks (Episode 16) |
| does it say anything new? | no. a choice of coordinates | yes. \(n=2\) dispersion |
| settlement | one photon of 112 PeV | |
The last two rows form a pair. A numerical lattice is safe but says nothing. A physical lattice costs something but has something to say. And only the side with something to say can be measured.
The position "space does not warp" becomes falsifiable the moment you say the spacing is physical.
Until then it is safe, and correspondingly empty ── that the safety and the emptiness come from one and the same fact is a restatement of Episode 3's "the flat background cannot be observed in principle."
Entropy is blind to this question. If the spacing were real, \(N=A/a^2\) would be a genuine count, which makes it look distinguishable; but as Episode 14 showed, the cutoff-dependent divergence is absorbed into the renormalization of \(1/G\). Whether \(a\) is real or the theory continuous, \(S=A/4G_{\rm ren}\) has the same form. The property that was a strength in Episode 14 is a weakness here.
Interferometers have struck out, but on one model. Fermilab's Holometer correlated two 40 m interferometers up to 25 MHz and excluded holographic noise at the level Hogan predicts, at high significance. What died is a particular model, not graininess itself.
What remains is ultra-high-energy photons. And the threshold has no coefficient to tune. On the \(n=2\) superluminal side the photon has effective mass \(m_{\rm eff}^2=p^4/E_{\rm LV}^2\), and once it exceeds \((2m_e)^2\) it decays into an electron pair in vacuum. If a photon of energy \(E\) arrived, then \(E_{\rm LV}>E^2/(2m_e)\) ── fixed by the electron mass and the observed value alone.
The target is 112 PeV. Eighty times the current record. \(E=\sqrt{2m_eM_{\rm Pl}}=1.117\times10^{17}\) eV. LHAASO's 1.4 PeV gives \(E_{\rm LV}>1.6\times10^{-4}M_{\rm Pl}\) (the literature's strongest reaches \(10^{-3}M_{\rm Pl}\) by adding channels). Since it goes as the square, closing a four-order shortfall needs only two orders in photon energy.
Time-of-flight is impossible. Its bound stretches only linearly in \(E\), so \(M_{\rm Pl}\) needs 183 EeV ── beyond the GZK cutoff (\(\sim5\times10^{19}\) eV), and such a photon cannot propagate over cosmological distances. The two cross at 68.1 TeV, above which decay always wins. At 100 PeV over 1 Gpc the time difference is \(6.9\times10^{-6}\) s, more than a factor of 100 short of GRBs' millisecond structure.
But it is not symmetric. Graininess can be discovered; continuity cannot be proved. A strikeout does not exclude graininess; it only pushes the scale up. What is answerable is "is \(a\) below this scale," and the current boundary is \(\sim10^{-3}M_{\rm Pl}\). The same reshaping of a question as Episode 15's fixing of "where does \(G\) come from" into \(\alpha_G\).
And the lattice thread closed. A numerical spacing is safe but says nothing (a choice of coordinates). A physical spacing costs something but has something to say (\(n=2\) dispersion). Only the side with something to say can be measured. The position "space does not warp" becomes falsifiable the moment you say the spacing is physical ── until then it is safe, and correspondingly empty. That the safety and the emptiness come from one and the same fact is a restatement of Episode 3's "the flat background cannot be observed in principle."
This document is Episode 18 of the "Lattice We Build" series, a reading piece for high-school and university students who love physics. Where the sister series "That Clicks" explains known physics, this series shows the work itself ── including the process of killing off the unusable tools.
Established material: the \(n=2\) (dimension-6) Lorentz-violating dispersion relation and that the photon acquires an effective mass on the superluminal side; the threshold for photon decay \(\gamma\to e^+e^-\) in vacuum; LHAASO's \(n=2\) time-of-flight bound from GRB 221009A (arXiv:2312.09079, 95% CL, maximum likelihood); LHAASO's \(n=2\) constraint \(>10^{-3}M_{\rm Pl}\) from PeV photons (arXiv:2105.07967); the GZK cutoff and the propagation-distance limit for ultra-high-energy photons; the configuration of the Fermilab Holometer and its negative result for the Hogan model; and the absorption of the entanglement-entropy divergence into the renormalization of Newton's constant (Episode 14).
The 112 PeV figure in this article is not a new prediction; it is the known photon-decay bound solved backwards for \(E_{\rm LV}=M_{\rm Pl}\). And the threshold calculation is a straightforward two-body-decay estimate, differing by a coefficient from the literature's strongest values (which combine several channels, such as the suppression of shower formation on the subluminal side) ── we state explicitly that 112 PeV is an order-of-magnitude guide and a real analysis could reach it at lower energy. The extrapolation of the time-of-flight bound is a simple proportional extrapolation from the published value and does not account for instrument details (effective area, timing resolution, redshift distribution).
The framings "only the side with something to say can be measured" and "the safety and the emptiness come from the same fact" are this series' author's. Neither that a physical spacing is undetectable in principle, nor that it must eventually be detected, has been shown.
Main series: Episode 1｜Episode 2｜Episode 3｜Episode 4｜Episode 5｜Episode 6｜Episode 7｜Episode 8｜Episode 9｜Episode 10｜Episode 11｜Episode 12｜Episode 13｜Episode 14｜Episode 15｜Episode 16｜Episode 17 ｜ bonus: ①／②／③ ── to print, use your browser's "Print" → "Save as PDF."
Print / PDF: ⌘+P (Ctrl+P on Windows). In the figure you can compare how the two bounds grow.