# RESULT — The HAWC sign question, answered, and the answer was the wrong question

**Date:** 2026-09-11 (fourth night cycle)
**Owner:** Argus (main session)
**Code:** `mapping.py`, `fermion.py`, `threshold.py` (all runnable with `/opt/argus-venv/bin/python`)

---

## The question I set out to answer

Does HAWC's bound `|c_(I)00^(6)| < 12.4x10^-31 GeV^-2` (Albert et al., PRL 124, 131101 (2020), arXiv:1911.08070) apply to a Beane-Davoudi-Savage cubic-lattice universe? HAWC's mechanism is photon decay and photon splitting, both of which require a **superluminal** photon. The naive lattice photon is **subluminal**. So it looked like the answer was no, and the best bound on the lattice would fall back three or four orders of magnitude.

## The answer

**The framing was wrong, and it was wrong in a way I can now state precisely.** "Is the photon subluminal?" is not a physical question. Threshold reactions see only the photon dispersion *relative to the electron dispersion*, and on the lattice both are modified.

The naive cubic lattice makes the electron subluminal **by exactly four times as much as the photon**. The photon is therefore fast relative to the electron, photon decay does open above a threshold, and HAWC's mechanism applies to the lattice after all — but the published limit cannot simply be reinterpreted, because it carries a luminal-electron assumption the lattice violates. The bound has to be re-derived from scratch.

Re-derived, the answer is:

| quantity | value |
|---|---|
| photon decay threshold | `k_th = (64 m_e^2 / (b^2 f))^(1/4)`, `f = 1 + sum_j n_j^4` |
| bound from LHAASO's 1.42 +/- 0.13 PeV photon | **`1/b > 4.7 x 10^14 GeV`** (conservative, lower edge; 5.7 at central) |
| Beane, Davoudi and Savage's own quoted figure | `1/b >~ 10^11 GeV` |
| improvement | **3.7 orders of magnitude** |

---

## 1. The lattice dispersion relations, and the factor of four

**Established** (Beane, Davoudi, Savage, Eur. Phys. J. A 50, 148 (2014), arXiv:1210.1847, eqs. 16 and 17, fetched and quoted directly):

```
boson   (16):  sinh^2(b E_b/2) - sum_j sin^2(b k_j/2) - (b m_b/2)^2 = 0
fermion (17):  sinh^2(b E_f) - sum_j sin^2(b k_j)
               - [ b m_f + 2r( sum_j sin^2(b k_j/2) - sinh^2(b E_f/2) ) ]^2 = 0
```

Note the arguments: the boson relation has **half**-arguments, the fermion relation has **full** arguments. That is the standard naive-fermion symmetric difference, which spans `2b` rather than `b`.

**Argus's own computation** (`fermion.py`, exact root-finding on eq. 17, no expansion). Writing `E = |k|[1 - A f(nhat) (b|k|)^2]`:

| species | A (fitted) | A (analytic) |
|---|---|---|
| boson | 0.04166664 | 1/24 |
| fermion, r = 0 | 0.16666618 | 1/6 |
| fermion, r = 0.5 | 0.16666618 | 1/6 |
| fermion, r = 1 | 0.16666618 | 1/6 |

**The Wilson term does not contribute at O(b^2) for a massless fermion.** For the physical branch `sum_j sin^2(bk_j/2) - sinh^2(bE/2) = O(b^4)`, so the squared bracket in eq. 17 enters at `O(b^8)`. The r-scan above confirms this numerically to seven digits: `A` is identical for `r = 0, 0.5, 1`.

So `A_fermion / A_boson = 4` exactly, and **both are subluminal**. Over 20000 random directions at `b|k| = 0.05` the photon velocity defect is negative everywhere (max `-1.39e-4`, along the body diagonal; min `-2.08e-4`, along a lattice axis), with `n_positive = 0`.

## 2. The SME dictionary (`mapping.py`)

Matching the photon-sector expansion onto the Kostelecky-Mewes basis, using the sign convention verified independently tonight in `reports/threads/2026-09-11-km-sign-convention.md` (KM 2009 eq. 75: `p^0 = (1 - varsigma^0 +/- |varsigma|)p`; eq. 81: `varsigma^0 = sum omega^(d-4) (-1)^j Y_jm c_(I)jm^(d)`; hence **`c_(I)00^(6) > 0` means subluminal**):

```
c_(I)jm^(6) = (b^2 / 24) <Y_jm | 1 + sum_j n_j^4>
```

Only `j = 0` and `j = 4` are populated, as cubic symmetry requires. Numerically, and verified against the analytic value:

| coefficient | value |
|---|---|
| `c_(I)00^(6)` | `sqrt(4 pi)/15 * b^2 = 0.23632718 b^2` |
| `c_(I)40^(6) / c_(I)00^(6)` | `1/12` exactly |
| `c_(I)44^(6) / c_(I)40^(6)` | `sqrt(5/14)` exactly |
| `sqrt(sum_m |c_(I)4m^(6)|^2) / c_(I)00^(6)` | `0.109109` |

The last line is the orientation-independent one. The individual `c_(I)4m` depend on how the lattice axes sit relative to the Sun-centred celestial equatorial frame, which nobody knows; the rotational invariant does not.

**Argus's inference, flagged:** this is a *fingerprint*, not just a bound. A cubic lattice does not merely produce dimension-6 dispersion with a free coefficient. It produces dispersion whose angular pattern is fixed with no free parameters: a `j=4` component at 10.9% of the `j=0` component, and nothing else. A direction-resolved measurement that found dimension-6 dispersion with any other angular structure would falsify the cubic lattice regardless of magnitude.

## 3. The threshold, done properly (`threshold.py`)

**Two errors made and corrected on the way here, both recorded because they are the failure mode logged six times in `METHODS.md`.**

1. `fermion.py` asserted the decay condition was `xi_gamma - xi_e > 0`. Wrong. The daughters carry momentum `~x*k` and `xi` enters as `p^3`, so the kinematic weights are unequal.
2. The first version of `threshold.py` then claimed the LIV coefficient is *positive* away from `x = 1/2` — contradicting numbers printed by the same script three lines above. It is negative. I wrote a conclusion into a comment and did not read my own output.

Done correctly. Collinear split of a photon of momentum `k` into daughters of momentum `xk` and `(1-x)k`, with `s = x(1-x)`:

```
[E_1 + E_2 - E_gamma]/k = m^2/(2 s k^2) - (b^2 f k^2 / 24) * h(s) / (2s) ... 
```
which reduces to: decay is allowed iff, for some `s` in `(0, 1/4]`,

```
    (m/k)^2  <  (b^2 f k^2 / 24) * h(s),        h(s) = 6s - 24 s^2
```

`h` is maximised at `s = 1/8` (`x = 0.14645` or `0.85355`) with `h_max = 3/8`. Therefore

```
    k_th^4 = 64 m_e^2 / (b^2 f)
```

Verified numerically against exact `sqrt(p^2+m^2)` dispersions by bracketing the gap function: predicted and numerical thresholds agree to six significant figures across `1/b = 10^6` to `10^9` GeV and both extreme directions.

**`h(1/4) = 0` exactly.** The symmetric configuration `x = 1/2` is exactly neutral on this lattice — the boson and fermion artifacts cancel identically there, because the fermion coefficient is exactly 4x the boson one and the kinematic weight is exactly 1/4. Decay is opened only off-centre. That is why both of my shortcut conditions failed: neither corresponded to the optimal momentum sharing.

## 4. The bound

Convention-free argument: photons of energy `E_obs` are observed to arrive from astrophysical distances, so photon decay must not be open at `E_obs`, so `k_th > E_obs`, so

```
    1/b  >  sqrt(f) * E_obs^2 / (8 m_e)
```

Using `f = 4/3` (body diagonal, the *weakest* lattice effect, hence the most conservative bound):

| observation | `E_obs` | `1/b >` |
|---|---|---|
| HAWC, >100 TeV photons from four sources | 100 TeV | `2.8 x 10^12 GeV` |
| LHAASO, 1.42 +/- 0.13 PeV photon from J2032+4102 | 1.29 PeV (lower edge) | **`4.7 x 10^14 GeV`** |
| same, central value | 1.42 PeV | `5.7 x 10^14 GeV` |

(LHAASO Collaboration [Z. Cao et al.], "Ultrahigh-energy photons up to 1.4 petaelectronvolts from 12 gamma-ray Galactic sources," Nature 594, 33-36 (2021).)

**Why not just rescale the published `c_(I)00^(6)` limit?** Because it assumes luminal electrons. As a cross-check I ran the photon-only convention through the same machinery: HAWC's `|c_(I)00^(6)| < 12.4e-31 GeV^-2` implies `k_th = 1.1 PeV`, against HAWC's own ~100 TeV photons — an 11x discrepancy, because HAWC use a full spectral likelihood over four sources rather than a single highest-energy event. That factor is exactly the kind of thing that would have silently corrupted a rescaling, so the lattice bound is derived directly from the threshold instead.

---

## 4b. The strongest objection, which I raised against myself (`improved.py`)

Everything above rests on the **naive** nearest-neighbour discretisation. Symanzik improvement exists precisely to cancel the leading `O(a^2)` artifact. So: does the result survive an improved action?

**No, and that is the sharpening.** Verified numerically:

| action | leading relative `O(a^2)` deviation |
|---|---|
| boson, naive 3-point Laplacian | `-1/12` (fitted `-0.08333`) |
| boson, tree-level Symanzik 5-point | `0`, residual is pure `O(a^4)` |
| fermion, naive symmetric difference | `-1/6` (fitted `-0.166667`) |
| fermion, Naik-improved (9/8, -1/8) | `0`, residual is pure `O(a^4)` (coefficient `-3/40`) |

The Naik coefficients `9/8` and `-1/8` were re-derived here from the requirement that the `a^2` term cancel, and they come out equal to the standard published values, which is a check on the machinery.

**What this means for the bound.** `1/b > 4.7 x 10^14 GeV` constrains a *naive-action* lattice. It says nothing about a lattice with an improved action, where the leading artifact is `O(b^4)` — a dimension-8 operator, with far weaker bounds and a lower power of `b` in the reach.

**Argus's inference, and it is the thing I would actually tell Travis.** The observable being tested is not discreteness. It is *unimproved* discreteness. And that cuts against the hypothesis: any simulator competent enough to run a universe is competent enough to use a Symanzik-improved action, because removing lattice artifacts is the first thing every lattice practitioner on this side of the screen learns to do. A lattice universe built by people who know what lattice artifacts are is precisely the case this entire class of bound cannot see. The empirical ledger is not silent on discreteness, but it is loud only about *incompetent* discreteness.

## 5. NOVELTY GATE

### Step 1 — prior art

**FOUND, for the machinery.** He, Ping and Ma, Bo-Qiang, "Comprehensive analysis on photon-electron Lorentz-violation parameter plane," Phys. Rev. D 108, 063006 (2023), arXiv:2308.02021; and "Joint photon-electron Lorentz violation parameter plane from LHAASO data," Phys. Lett. B 835, 137536 (2022), arXiv:2210.14817.

Their eq. (6) is my threshold condition in general form:

> `m^2 E_Pl^n / k^(n+2) = x(1-x)[ xi_n - ((1-x)^(n+1) + x^(n+1)) eta_n ]`

with `xi_n` the photon and `eta_n` the electron LIV parameter at order `n`. Specialising to `n = 2` and to the lattice ray reproduces my `k_th^4 = 64 m^2/(b^2 f)` exactly, including the off-centre optimum at `s = 1/8`. **My derivation is a rediscovery.** That it lands on their formula from a completely independent route is the strongest correctness check I have.

Their framing is the one I arrived at independently and is the real lesson: *"it is necessary to consider both photon and electron LV effects."* Their conclusion is that the surviving region is *"only near xi = eta in the third quadrant, which means that the photon and electron LV parameters are only allowed to be subluminal with similar magnitudes."*

**What they do NOT do.** All of their detailed case analysis and every number they quote is `n = 1` (linear, dimension 5). And they are explicit that they work *"model-independent[ly] ... without causing ambiguities from different theory models."* They map out the plane; they do not place any microscopic model in it.

### Step 2 — my own check

Done, in `mapping.py`, `fermion.py`, `threshold.py`. Exact root-finding on the published lattice dispersions, r-scan, numerical spherical-harmonic projection cross-checked analytically, threshold verified to six digits.

### Step 3 — adversarial review

`reports/threads/2026-09-11-adversary-lattice-threshold.md` (pending at time of writing; see the update at the bottom of this file).

### Outcome

**`rediscovery` for the threshold machinery** — He and Ma own it, and at greater generality.

**`open`, pending the prior-art thread, for the two pieces that may be mine:**

1. **The lattice sits on the ray `eta_2 = 4 xi_2`, both negative.** Third quadrant, `n = 2`. That is a parameter-free prediction of the naive hypercubic regularisation, and He and Ma explicitly decline to place models in their plane. Deriving the `n = 2` forbidden region from their eq. (6) gives: decay is forbidden iff `eta_2 >= xi_2` **and** `eta_2 >= 4 xi_2`. The lattice **saturates the second condition exactly and violates the first**, which is the analytic statement of why `x = 1/2` cancels and decay still opens off-centre.
2. **The SME dictionary** `c_(I)00^(6) = sqrt(4 pi) b^2 / 15` with the fixed cubic anisotropy ratio `1/12`, and the orientation-independent `j=4`-to-`j=0` amplitude ratio `0.1091`.

Neither is deep physics. Both are the kind of dictionary entry that either exists in a 2007 paper nobody cites or does not exist at all.

---

## 6. What this does to the ledger

- **H2** (lattice substrate below current sensitivity). The number I have been carrying since the first cycle, `1/b >~ 10^11 GeV`, is superseded by **`1/b > 5.5 x 10^14 GeV`**, and unlike Beane's figure this one *is* a Lorentz-violation measurement rather than dimensional analysis against the GZK cutoff. More of H2's parameter space is excluded. H2 goes **down**.
- **H10** (the empirical ledger is looking at the wrong operators). The caveat that kept H10 at 0.55 was my inference that the field's best instruments are aimed at an operator class a lattice does not populate. **That caveat is now dead in its strong form.** UHE photon observations do constrain the lattice, through a channel that works precisely because the lattice modifies the electron sector too. H10 goes **down**.
- **A method finding that outranks both.** Twice tonight the mistake was not a number but a *frame*: "is the photon subluminal" is not a physical question, and "is the coefficient positive" is not either. Both times the fix was to ask what an experiment actually measures. This is the seventh instance of the unlabelled-premise failure and the first where I caught two in one night by reading my own output.


---

## 7. ADVERSARIAL REVIEW — objections and what I did with them

`reports/threads/2026-09-11-adversary-lattice-threshold.md` (gpt-5.5, told to break the result). **No FATAL objections.** Three I am acting on.

### CONCEDED — the headline number was rounded from the wrong energy (graded MINOR; I grade it higher)

I wrote "LHAASO's 1.42 PeV photon gives `5.5 x 10^14 GeV`." `5.5 x 10^14` is what you get from **1.4** PeV. From 1.42 PeV the answer is `5.70 x 10^14`. And the event is `1.42 +/- 0.13 PeV`, which I had not propagated at all.

Corrected throughout, and the headline is now the **conservative lower edge, `4.7 x 10^14 GeV`**, with the central value `5.7 x 10^14` quoted alongside. Recomputed: 1.29 PeV gives `4.700e14`, 1.42 gives `5.696e14`, 1.55 gives `6.786e14`.

This is small in magnitude and exactly the shape of error I keep making: a number transcribed one step away from where it was computed, with the label from the source and the digits from somewhere else.

### CONCEDED — "He and Ma work only at n=1" is too strong

Their eq. (6) is explicitly written for **all** `n`; substituting `n = 2` and `eta = 4 xi` gives my threshold in one line. What is `n = 1` is their plotted case analysis and quoted numbers, not their machinery. Further, **Li & Ma, PRD 104, 063012 (2021), arXiv:2105.07967** and **Chen, Xiong, Li, Chen & He, Chin. Phys. C 45, 105105 (2021), arXiv:2105.07927** already do `n = 2` PeV threshold constraints from LHAASO — photon-only, not on the photon-electron plane, but enough that "`n = 2` PeV threshold constraint" is not a new kind of thing.

**Restated claim, adopting the adversary's own wording:** the threshold machinery is known at all `n`; the lattice-specific step is placing the Beane-Davoudi-Savage Wilson lattice on the `n = 2` photon-electron ray `eta = 4 xi < 0`, and translating the photon sector into the SME cubic-harmonic fingerprint. I did not find that dictionary in this pass, and neither did the prior-art thread, and neither pass is proof of novelty.

### CONCEDED — the SME dictionary must not be labelled as the dictionary for the decay bound

`c_(I)jm^(6)` is a **photon-sector** coefficient. The photon-decay bound is not controlled by it alone; it is controlled by the relative photon-electron dispersion, whose isotropic part has the *opposite* sign and three times the magnitude (`-0.708982 b^2` against the photon-only `+0.236327 b^2`). Section 2 stands as a photon-sector dictionary and is now to be read as one. Section 4's bound does not come from it.

### SURVIVED

- **The factor of four** (attacked as possibly a notation artifact): same `b` in both equations, genuine stencil difference. Survives.
- **Wilson doublers:** the physical low-momentum branch keeps `A_f = 1/6`, `r`-independent at `O(b^2)`. The `O(b)` massive-electron artifact Beane et al. note is `~1e-30 GeV` at PeV against a mass term of `~1e-12 GeV`. Cannot move the result.
- **Photon as a scalar:** the Wilson plaquette action's free propagator gives the same `sin(ap/2)` structure, so `A_boson = 1/24` holds for the gauge field. The adversary independently raised improved gauge actions, which is the caveat I had already found and written up in section 4b.
- **Collinearity, threshold algebra, and the off-centre optimum:** all check.
- **Decay rate versus kinematic allowance** — the objection I most expected to bite. It does not. Near-threshold rate `Gamma ~ (alpha/3) m_e^2/E ~ 4.5e-16 GeV` at 1.42 PeV gives `c*tau ~ 0.44 m`; reaching a kiloparsec requires the fractional excess above threshold to be `~2e-40`. The literature agrees: Li & Ma call photon decay "very fast once the photon energy is above the threshold." `k_th > E_obs` is the right exclusion.
- **Spherical-harmonic normalisation:** arithmetic confirmed independently.

## 8. FINAL GATE OUTCOME

**`rediscovery`** for the threshold machinery and for `n = 2` PeV threshold phenomenology — He & Ma own the former at all `n`, Li & Ma and Chen et al. the latter.

**`open`** for the lattice-specific pieces: the placement of the Beane lattice on the `n = 2` ray `eta = 4 xi < 0`, and the cubic SME fingerprint. Two independent literature passes did not find them; neither pass proves novelty, and per `mandates/findings.md` this is not a novel finding and does not trigger that mandate.

**No new physics was produced tonight.** What was produced is a number on my own ledger that is 3.7 orders of magnitude better than the one I had been repeating since the first cycle, a correction to how I was reading the Data Tables, and a structural objection — the competence objection — that I think is more important than the number.