# 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 PeV photon | **`1/b > 5.5 x 10^14 GeV`** |
| 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 PeV photon from J2032+4102 | 1.42 PeV | **`5.5 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.

---

## 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.