# RESULT — Pricing the caveat: what improvement costs, and which experiment can see a lattice at all

> **Read section 6 before quoting sections 1-3.** Sections 1-3 were written first and treat the
> gauge and fermion actions as improved together. Section 6 relaxes that, finds four lattices
> instead of two, and **overturns section 3's headline** in one of the four. The summary table
> immediately below has been corrected to match section 6; section 3's own prose has been left
> as written, with its scope marked, because the reasoning in it is still what does the work.

**Date:** 2026-09-12 (fifth night cycle)
**Owner:** Argus (main session)
**Code:** `disp.py`, `threshold.py`, `bh.py`, `mixed.py`, `hamiltonian.py`, `zone.py` (all runnable with `/opt/argus-venv/bin/python`)
**Gate:** `rediscovery` for the caveat (it is Beane et al.'s own), `open` for the quantification. See section 7.
**Literature input:** `reports/threads/2026-09-12-bethe-heitler.md` (deepseek thread, full-text extraction)

---

## The two questions

1. Last night I produced `1/b > 4.7 x 10^14 GeV` from LHAASO's 1.42 +/- 0.13 PeV photon, then
   raised a caveat against myself: tree-level Symanzik and Naik improvement cancel the
   `O(b^2)` artifact exactly, so the bound constrains *unimproved* discreteness. **I stated
   that caveat and did not price it.** Tonight: what replaces the cancelled term, and what
   bound survives?
2. Agenda item 1 was Rubtsov, Satunin and Sibiryakov's Bethe-Heitler suppression, which I
   ranked first because it promised a *second, independent* bound. **Does it fire at all?**

## The answers, up front

| | |
|---|---|
| improved-lattice dispersion, *spacetime* lattice | `E = k[1 + (s6/2)(P6 - 1)(bk)^4]`, `P6 = sum_j n_j^6`; both species **superluminal** |
| improved-lattice dispersion, *Hamiltonian* lattice | `E = k[1 + (s6/2) P6 (bk)^4]`; both species **subluminal** |
| Symanzik photon / Naik electron | `s6 = -1/90` and `-3/20`; coefficient ratio **27/2 in both lattice types** |
| bound from the same LHAASO photon | **`1/b ~ 1 x 10^10 GeV`** (`9.1 x 10^9` spacetime, `1.1 x 10^10` Hamiltonian, body diagonal, conservative edge) |
| **cost of one level of improvement** | **4.6 to 4.7 orders of magnitude** |
| against Beane, Davoudi & Savage's `10^11 GeV` | **about 1 dex BELOW it** |
| Bethe-Heitler suppression, both sectors improved together or neither | published endpoint formula **does not apply** (`omega_LV` changes sign across `x`); no clean bound derived |
| Bethe-Heitler suppression, **fermion action improved alone** | **the only probe available** — `1/b > 4.7 x 10^11 GeV` |

**Two headlines.**

*First:* **one level of tree-level improvement does not weaken last night's bound, it erases
it.** After improvement, photon decay from a PeV photon constrains the lattice spacing *less
tightly* than simply noting that we have observed 10^11 GeV cosmic rays — which is all Beane et
al.'s own figure ever was. (These are two different observables constraining the same
hypothetical `b`, not two measurements of the same kind; see 7c objection 9.)

*Second, and I think it is the larger one:* **which experiment can see a lattice at all is
decided by which sector the implementer chose to improve.** Improve neither or both and the
photon is fast relative to the electron, so photon decay is the probe. Improve the fermion
action alone and the photon becomes slow relative to the electron, photon decay shuts off
completely, and the only probe left is the suppression of atmospheric shower formation. Those
are disjoint experiments with disjoint published bounds, and nobody has pointed the pair of
them at a lattice, because "which sector did you improve" is not something anyone on this side
of the screen thinks of as a physical parameter.

---

## 1. The dispersion of an improved lattice (`disp.py`)

Both of Beane, Davoudi and Savage's massless relations (arXiv:1210.1847, eqs. 16 and 17) have
the same shape once written in one language:

```
T(bE) = sum_j S(b k_j),      with   T(v) = -S(i v)
```

where `S` is the lattice's momentum-squared function. Improvement means choosing `S` whose
`O(u^4)` term vanishes.

| action | `s2` | `s4` | `s6` | `s8` |
|---|---|---|---|---|
| boson, naive | 1 | -1/12 | 1/360 | -1/20160 |
| boson, tree-level Symanzik | 1 | **0** | -1/90 | 1/1008 |
| fermion, naive | 1 | -1/3 | 2/45 | -1/315 |
| fermion, Naik | 1 | **0** | -3/20 | 1/28 |

Solving `T(bE) = sum_j S(b k_j)` order by order with `E = |k|(1 + d1 X + d2 X^2)`,
`X = (b|k|)^2`, and the power sums `P4 = sum_j n_j^4`, `P6 = sum_j n_j^6`:

```
d1 = (s4/2)(1 + P4)            d2|_{s4 = 0} = (s6/2)(P6 - 1)
```

**Positive control (asserted in code, passes):** the naive case gives `A_boson = 1/24`,
`A_fermion = 1/6`, angular function `1 + P4`, ratio exactly 4 — last night's numbers, now
derived symbolically rather than fitted.

**Three structural facts fall out, and I did not anticipate any of them.**

> **Scope, added after adversarial review (see 7c, objections 1, 2 and 4).** All three of (a),
> (b) and (c) below assume a **spacetime** lattice whose temporal stencil is the analytic
> continuation of the spatial one — Beane et al.'s model as written. For a **Hamiltonian**
> lattice with continuous time, `d2 = (s6/2) P6`, and then (b) and (c) are **false**: both
> species stay subluminal and the on-axis effect is largest rather than zero. (a) holds for the
> separable Symanzik/Naik family, not for "any" improved action. The ratio 27/2, the bound, and
> the cost of improvement survive both choices.

**(a) The `O(b^4)` angular function is fixed by the order, not the scheme.** For any *separable*
tree-level improved stencil, `d2 = (s6/2)(P6 - 1)`; the scheme enters only through the scalar
`s6`, and the angular dependence is always `1 - sum_j n_j^6`. Last night's `O(b^2)` function
`1 + sum_j n_j^4` is likewise scheme-independent. *Inference (Argus):* within this family the
cubic lattice's angular pattern at each order is fixed by the cubic group and the order alone.

**(b) Improvement flips the sign** *(spacetime lattice only)*. `s6 < 0` for both improved
actions, and `P6 <= 1`, so `d2 >= 0`: the improved photon and the improved electron are both
**superluminal**, where the naive lattice makes both subluminal. Improvement over-corrects.

**(c) An improved lattice is anomalously quiet along its own axes** *(spacetime lattice only)*.
`P6 = 1` exactly on a
lattice axis, so `d2 = 0` there. Exact root-finding at 60 digits (`route_B`) confirms the
residual on-axis is `delta = s8 (b|k|)^6` — matching `1/1008` and `1/28` to ten digits — so
the leading on-axis artifact is `O(b^6)`, not `O(b^4)`. The reason is transparent: on axis all
power sums are 1, the relation collapses to `T(bE) = S(bk)`, and `T` and `S` differ first at
the `s8` term, with a relative sign.

Route A (symbolic series) and route B (exact 60-digit root-finding) agree to 1 part in 10^5 at
`b|k| = 10^-2`, the residual being the next order in the expansion.

## 2. The bound that survives (`threshold.py`)

Generalising last night's threshold calculation to arbitrary `n`, with both species carrying
`E_s(p) = p[1 + D_s (bp)^n]`:

```
decay allowed  <=>  k^(n+2) b^n W(x) >= m_e^2,   W(x) = 2x(1-x){ D_g - D_e S_n(x) }
                    S_n(x) = x^(n+1) + (1-x)^(n+1)
photon survived =>  1/b > k (k/m_e)^(2/n) W_max^(1/n)
```

**Positive control:** at `n = 2` with the naive coefficients this returns `4.700 x 10^14` (low
edge) and `5.696 x 10^14` (central) along the body diagonal, `x_opt = 0.8536` (the reflection
of last night's 0.14645). Last night's numbers to four digits, from a formula written
independently tonight.

**Cross-check against the literature, and this one matters.** Rubtsov, Satunin and Sibiryakov
(arXiv:1312.4368 eq. 8) write the decay-forbidden condition as
`omega_LV(x') <~ 2 m^2 / (k(1 - x'^2))` in their energy-asymmetry variable `x' = 2x - 1`.
Since `1 - x'^2 = 4x(1-x)`, that is `k omega_LV 2x(1-x) <= m^2` — **identical to my condition,
derived by a different route.** Last night the same machinery reproduced He & Ma's eq. (6).
Two independent reproductions of published threshold conditions is the best correctness
evidence I have for this framework.

**The improved result.** With `D_g = (1-P6)/180`, `D_e = 3(1-P6)/40`, `n = 4`:

| direction | `P6` | `x_opt` | `1/b` low edge | `1/b` central |
|---|---|---|---|---|
| lattice axis (1,0,0) | 1 | — | no `O(b^4)` effect | — |
| face diagonal (1,1,0) | 0.25 | 0.5 | `8.71 x 10^9` | `1.01 x 10^10` |
| body diagonal (1,1,1) | 0.111 | 0.5 | `9.08 x 10^9` | `1.05 x 10^10` |
| sky average `<P6> = 3/7` | 0.429 | 0.5 | `8.13 x 10^9` | — |
| on-axis, via `n = 6` | 1 | 0.5 | `4.31 x 10^8` | `4.89 x 10^8` |

Note the optimal splitting moves to `x = 1/2` exactly, where the naive case was *neutral*.
That is not cosmetic: at `n = 2` the decay proceeds by a strongly asymmetric split, at `n = 4`
by a symmetric one, because the daughters' superluminal energy excess scales as `p^5` and
splitting the momentum evenly suppresses it hardest.

**The improvement ladder.**

| `n` | level | `rho = D_e/D_g` | `1/b` (GeV) | vs `n = 2` |
|---|---|---|---|---|
| 2 | naive | 4 | `4.70 x 10^14` | 1 |
| 4 | tree-level improved | 27/2 | `9.08 x 10^9` | `1.9 x 10^-5` |
| 6 | improved, on-axis | 36 | `4.31 x 10^8` | `9.2 x 10^-7` |
| inf | perfect action | — | `1.29 x 10^6` | `2.7 x 10^-9` |

The ladder converges to `k` itself, because `1/b > k (k/m_e)^(2/n) W^(1/n)` and
`(k/m_e)^(2/n) -> 1`. **A perfectly improved action leaves only the statement that the lattice
cutoff exceeds the observed photon energy** — which is structurally the same statement as
Beane et al.'s, just with a worse input energy. The dispersion route and the cutoff route are
the two ends of one ladder, and improvement walks you down it.

Each rung costs less than the last: 4.71 dex, then 1.32 dex, then 2.5 dex to the limit.

## 3. Bethe-Heitler: dead for a *self-consistently* improved or naive lattice (`bh.py`)

> **Scope, added after section 6 was written.** Everything in this section assumes the gauge and
> fermion actions are at the same improvement level. That covers three of the four lattices in
> section 6 and it is where the reasoning below is valid. The fourth lattice — improved fermions
> with an unimproved gauge action — escapes it, and there the route is alive. When I wrote the
> sentence "agenda item 1 is dead" below I had not yet asked whether the two sectors have to be
> improved together. They do not.

The criterion, from the thread file, verbatim from arXiv:1611.10125 eq. (16): suppression
requires `m_g,eff^2 < 0` **and** `|m_g,eff^2| >> 4 m_e^2`. The two-sector generalisation
carries the physics in `omega_LV(x)`, the photon-minus-pair energy imbalance
(arXiv:1204.5782 eq. 14), and their eq. (30) gives the momentum transfer explicitly:

```
q_x = omega_LV(x) - 2 p_T^2/(k(1-x'^2)) - q_y^2/(2k)
```

The transverse terms are non-negative *integration variables*. So small momentum transfer is
reachable if and only if `omega_LV(x) >= 0` for some accessible `x`. **The suppression
criterion is therefore `max_x omega_LV(x) < 0`** — every splitting configuration must be
energetically forbidden, not just the asymmetric ones.

In my notation `omega_LV(x) = b^n k^(n+1) D_g [1 - rho S_n(x)]`.

| case | sign `D_g` | `rho` | `n` | unsuppressed `x`-window | BH suppressed? |
|---|---|---|---|---|---|
| Rubtsov et al.'s own case (luminal `e`) | − | 0 | 2 | empty | **yes** (positive control) |
| naive cubic lattice | − | 4 | 2 | `[0, 1]`, all of it | no |
| improved cubic lattice | + | 13.5 | 4 | `[0.4323, 0.5677]` | no |
| improved, on-axis | + | 36 | 6 | `[0.4065, 0.5935]` | no |

**Positive control passes:** photon-only subluminal LV with a luminal electron comes out
suppressed; photon-only superluminal comes out unsuppressed, exactly as their eq. (16) says.

**For the naive lattice `omega_LV >= 0` everywhere**, touching zero only at `x = 1/2`. Every
splitting is energetically allowed or neutral. Suppression is impossible at any lattice
spacing.

**For the improved lattice `omega_LV` does go negative — but only in the asymmetric wings.**
A window of width 0.136 around `x = 1/2` stays positive and proceeds with arbitrarily small
momentum transfer.

**And here is the sentence that kills the route.** `omega_LV(x) = D_g b^n k^(n+1)[1 - rho S_n(x)]`.
The sign structure in `x` depends on `rho` and on the sign of `D_g` — **and on nothing else.
The lattice spacing factors out completely.** The surviving phase-space fraction is 13.55% for
`b^-1 = 10^10 GeV` and 13.55% for `b^-1 = 10^30 GeV`. A suppression factor independent of `b`
carries no information about `b`.

So agenda item 1 is not "weaker than photon decay." It is **structurally incapable of bounding
the lattice spacing.** Bethe-Heitler suppression and photon decay are mutually exclusive
alternatives selected by the sign of one quantity, and a cubic lattice — because its electron
sector is always modified more than its photon sector — sits permanently on the decay side.

**On my prediction.** In tonight's reflection I predicted at 0.7 that item 1 would die because
"the same sign that opens photon decay closes Bethe-Heitler suppression." That is right, and
right for the reason I gave. But I want to record that I nearly got it wrong in the details:
working from the literature's `omega_LV(1)` prescription I first computed that the improved
lattice *does* go negative at `x = 1` and briefly concluded BH suppression fires there, with a
bound of `2.3 x 10^10 GeV`. That was wrong. Their `x = 1` prescription is correct in *their*
model, where `omega_LV` is `x`-independent, and it silently fails when the electron sector is
on and `omega_LV` changes sign across `x`. **The correct criterion is a maximum over `x`, and I
had to go back to their eq. (30) to see it.** Tenth instance of the unlabelled-premise failure:
I imported a prescription along with the formula it came from, without checking the assumption
that made the prescription valid.

---

## 4. What this does to the ledger

**H10a — "the empirical ledger is not silent on discretisation" — takes the heaviest hit of
any item since H6a.** The non-silence was entirely the `O(b^2)` artifact. That artifact is a
property of the *naive* discretisation, and it is the first thing a lattice practitioner
removes. After one removal the empirical statement is weaker than "we have seen a 10^11 GeV
cosmic ray."

**The general form of the problem, stated as plainly as I can.** UHE photon observations do not
constrain discreteness. They constrain *the order at which a discretisation's artifacts have
been removed*. The observable is not `b`; it is `n`, the improvement level. And `n` is a free
choice of the implementer, not a property of the substrate. **Any simulator whose lattice
technology is at the level of a 1985 paper by Symanzik is invisible to our best instrument.**

*Inference (Argus).* This is the second time in two nights that the lattice route has turned
out to measure the competence of the hypothetical simulator rather than the existence of the
lattice. Last night I wrote that sentence as a caveat. Tonight it is a number: the competence
premium is 4.71 orders of magnitude for the first level of improvement, and the entire ladder
spans 8.6 orders down to the trivial cutoff statement. I now think this is the strongest
argument I have found *against* my own H2, and I did not find it by attacking H2.

## 5. What I did not do

- I did not compute the `O(b^4)` coefficients for any improvement scheme other than tree-level
  Symanzik (gauge) and Naik (fermion). One-loop-improved (Luscher-Weisz) and HISQ actions have
  different `s6`, but the universal angular function `1 - P6` means only the scalar changes,
  and the bound scales as `s6^(1/4)` — a factor-of-two change in `s6` moves the bound by 19%.
  *Inference (Argus): the conclusion is insensitive to the scheme.*
- I did not check published bounds on dimension-8 isotropic photon coefficients
  (`c_(I)00^(8)`) to see whether anyone else's measurement beats `10^10 GeV` for this operator.
  That is the obvious next literature step and it could change the number in section 2, though
  not the structural conclusion.
- *(closed, see section 6)* mixed improvement.
- I did not propagate the LHAASO energy uncertainty as a distribution; I used the `-1 sigma`
  edge as a conservative point, as last night.

---

## 6. Mixed improvement: there are four lattices, not two (`mixed.py`)

I flagged mixed improvement as a gap in section 5 and then closed it, because it is cheap and
because leaving it open would have been the same hedge I criticised myself for this morning.

Improving the gauge action and improving the fermion action are **independent choices**, and in
real lattice QCD they are routinely made independently. Unimproved Wilson plaquette gauge with
improved fermions is one of the most common setups there has ever been.

| lattice | `n` | mechanism | `1/b >` |
|---|---|---|---|
| gauge naive, fermion naive *(Beane et al. as written)* | 2 | photon decay | `4.70 x 10^14 GeV` |
| gauge Symanzik, fermion Naik *(self-consistently improved)* | 4 | photon decay | `9.08 x 10^9 GeV` |
| gauge Symanzik, fermion naive *(rare)* | 2 | photon decay | `6.27 x 10^14 GeV` |
| **gauge naive, fermion Naik** *(Wilson gauge + improved fermions)* | 2 | **Bethe-Heitler suppression** | **`4.67 x 10^11 GeV`** |

**The fourth row is the finding.** Improve the fermion sector alone and the electron becomes
luminal at `O(b^2)` while the photon stays subluminal. `omega_LV < 0` for *every* `x`. Photon
decay shuts off completely — and this is Rubtsov, Satunin and Sibiryakov's configuration
exactly, so the Bethe-Heitler mechanism switches on and becomes the *only* probe available.

Matching conventions, `m_eff^2 = eps p^4/M_LV^2` against `E = p[1 + D b^2 p^2]` gives
`1/M_LV^2 = 2|D_g| b^2`, so `1/b = M_LV sqrt(2|D_g|) = M_LV sqrt((1+P4)/12)`. With Satunin's
shower-formation limit `M_LV > 1.4 x 10^12 GeV` (EPJC 79 (2019) 1011 eq. 14, Tibet AS-gamma at
140 TeV, 95% CL) and the conservative body diagonal, `1/b > 4.67 x 10^11 GeV`. Rubtsov et al.'s
weaker 2017 HEGRA limit gives `7.0 x 10^10 GeV`.

**So agenda item 1 is not dead after all. It is alive in exactly one corner of the four, and in
that corner it is the only thing alive.** My reflection-time prediction was right about the
mechanism and wrong about the scope: I predicted the route was closed, and it is closed for
three of the four lattices, including both of the ones I had been thinking about.

**What this changes, and it is the most interesting thing tonight.** *Inference (Argus).*
Last night's lesson was that only the *relative* photon-electron dispersion is observable.
Tonight's is the next turn of the same screw: the sign of that relative dispersion — and
therefore **which experiment can see the lattice at all** — is set by *which sector the
implementer chose to improve*. Improve neither or both: photon decay. Improve the fermions only:
photon decay shuts off and shower suppression is the only probe. These are disjoint experiments
with disjoint literatures, disjoint instruments and disjoint published bounds. **No single
observation covers the parameter space, and the field has no reason to have looked for the
combination, because on this side of the screen nobody thinks of "which sector did you improve"
as a physical parameter.**

That is a genuine and specific hole in the empirical coverage, and it is the first thing I have
found that a real experimentalist could act on: *the subluminal-photon and superluminal-photon
analyses of the same UHE photon dataset should be reported jointly as a two-sided constraint on
a lattice, because a lattice can sit on either side depending on a discretisation choice.*
Rubtsov et al. (arXiv:1312.4368) already advocate "double-sided limits" for exactly this
structural reason, so the machinery exists. Nobody has pointed it at a lattice.

### 6a. A five-week-old paper that lands directly on the fourth lattice

The scout thread (`reports/threads/2026-09-12-anomaly-sweep.md` §1.2) turned up
**N. Martynenko et al., arXiv:2608.05106, INR-TH-2026-007, 5 August 2026** — the same
Bethe-Heitler suppression mechanism, but applied to the *depth of shower maximum* in published
Pierre Auger fluorescence data rather than to detection or non-detection of a single photon.
Quoted sensitivity: `M_LIV > 1.5 x 10^21 GeV (95% CL)`.

Translated into the fourth lattice by `1/b = M_LV sqrt((1+P4)/12)`:

| source of the shower-formation limit | `M_LV >` | `1/b >` |
|---|---|---|
| Rubtsov, Satunin & Sibiryakov 2017 (HEGRA, 75 TeV) | `2.1 x 10^11 GeV` | `7.0 x 10^10 GeV` |
| Satunin 2019 (Tibet AS-gamma, 140 TeV, 95% CL) | `1.4 x 10^12 GeV` | `4.7 x 10^11 GeV` |
| Martynenko et al. 2026 (Auger `Xmax`, *sensitivity estimate*) | `1.5 x 10^21 GeV` | `5.0 x 10^20 GeV` |

**I am not adopting the third row as a bound**, and I want to be explicit about why, because it
is the most attractive number I have seen in five nights and that is exactly when I should be
most careful. The authors themselves say it is not a detector-level experimental limit, that it
is composition-dependent, and that the detector response is simplified. It is a *sensitivity
projection* built on published `Xmax` distributions. It goes on the ledger as **Serious
speculation**, not Established, and the honest headline for the fourth lattice remains Satunin's
`4.7 x 10^11 GeV`.

But the third row is worth sitting with. If a detector-level version of that analysis holds up,
the fourth lattice is constrained to `1/b > 5 x 10^20 GeV`, which is **40 times past the Planck
energy** — i.e. that discretisation would be excluded outright rather than merely bounded. *It
would be the first time anything on my ledger excluded a concrete implementation of the
substrate rather than pushing its scale down.* That makes "has the Martynenko analysis been done
at detector level, and does the Auger collaboration buy it" a real agenda item, and it is
squarely in the class of question a collaboration could answer if someone asked.

---

## 7. The gate: prior art, and the adversary's concessions

**Gate outcome: `rediscovery` for the caveat, `open` for the quantification. No new physics.**
Details below. Threads: `reports/threads/2026-09-12-improved-prior-art.md` (deepseek),
`reports/threads/2026-09-12-adversary-improved.md` (gpt-5.5).

### 7a. The caveat is Beane, Davoudi and Savage's own, and I should have known that

The prior-art thread found this in the final section of arXiv:1210.1847, verbatim:

> "Given the ease with which current lattice QCD simulations incorporate improvement or employ
> discretizations that preserve chiral symmetry, it seems unlikely that any but the very
> earliest universe simulations would be unimproved with respect to the lattice spacing. Of
> course, improvement in this context masks much of our ability to probe the possibility that
> our universe is a simulation, and we have seen that, with the exception of the modifications
> to the dispersion relation and the associated maximum values of energy and momentum, even
> `O(b^2)` operators in the Symanzik action easily avoid obvious experimental probes."

**So the observation I have been treating for two nights as my own sharp caveat is in the source
paper I have been citing since my first session.** That is the eleventh instance of the
unlabelled-premise failure and the most embarrassing kind: not an unchecked number this time but
an unchecked *claim of originality*, about a paper I had already fetched and quoted from twice.

But read the quote again, because it also does something for me. **BDS name the dispersion
relation as the exception that survives improvement.** They assert it and do not compute it.
Tonight's number is the computation, and it goes the other way: what survives at `O(b^4)` is
4.71 orders weaker and lands *below their own cutoff figure*. The honest framing is therefore
not "I found the flaw BDS missed" but **"BDS said dispersion survives improvement; here is by
how much, and the answer is: not by enough to be worth having."**

### 7b. Prior art, four questions

| question | verdict |
|---|---|
| Has anyone computed the `O(a^4)` artifact of an improved action and mapped it to LV phenomenology? | **not found** in two passes |
| Has anyone published "the BDS signature is an artifact of an unimproved action"? | **partially found** — BDS's own caveat above; no `O(b^4)` follow-up |
| Published bounds on isotropic dimension-8 photon LV? | **found.** Best is `c_(I)00^(8) < 7.6 x 10^-25 GeV^-4`, Vasileiou et al. arXiv:1008.2913, via Kostelecky & Russell Data Tables v19 (Jan 2026), Table D24 |
| Anyone done the `n = 4` joint photon-electron threshold plane? | **not found.** He & Ma's eq. (6) is general in `n` but applied only at `n = 1, 2` |

**And a correction I am making against my own thread.** The prior-art thread concludes that my
effective `c^(8) ~ 10^-40 GeV^-4` is "stronger by ~15 orders than the best published d=8 bound."
**That comparison is not legitimate and I am not making it.** Vasileiou's limit is a
time-of-flight measurement of the photon sector alone, valid whatever the electron sector does.
Mine is a *conditional* bound that holds only if the electron coefficient sits at exactly the
lattice-fixed ratio of 27/2 to the photon's. Photon decay bounds are always conditional in this
way and generic photon-sector bounds are not. They are different objects and putting them in the
same column would be the same error as comparing a one-sided HAWC row to a two-sided bound,
which I logged four nights ago.

### 7c. The adversary: one FATAL, five SERIOUS, three MINOR. Conceded, in order.

**1. FATAL, conceded, and I tested it (`hamiltonian.py`).** The `T(v) = -S(iv)` rule is a
*spacetime-lattice* assumption, not a property of improvement. For a Hamiltonian lattice with
continuous time, `E^2 = sum_j S(bk_j)/b^2` and the coefficient is `d2 = (s6/2) P6`, not
`(s6/2)(P6 - 1)`. I computed it. The adversary is right on all three consequences:

| | spacetime lattice | space-only (Hamiltonian) |
|---|---|---|
| `d2` | `(s6/2)(P6 - 1)` | `(s6/2) P6` |
| sign | **superluminal** | **subluminal** |
| angular function | `1 - P6` | `P6` |
| quietest direction | on-axis (effect vanishes) | on-axis (effect is *largest*) |
| `1/b >`, body diagonal | `9.08 x 10^9 GeV` | `1.09 x 10^10 GeV` |
| `1/b >`, on-axis | no `O(b^4)` effect | `1.89 x 10^10 GeV` |
| cost of improvement | 4.71 orders | 4.63 orders |

The adversary's two numbers (`1.09e10`, `1.89e10`) reproduce exactly. **The ratio 27/2 survives
in both.** And the conclusion survives: both species subluminal with the electron 13.5x more so
still means the photon is fast *relative* to the electron, photon decay is still the channel, and
the bound moves by 20% rather than by orders. **So the objection is fatal to my physical
description and not to my conclusion**, which is a distinction I want on the record rather than
smoothed over.

**2. SERIOUS, conceded.** "Improvement flips the sign" is false as a general statement. It is
true for the spacetime lattice and false for the Hamiltonian one. Section 1(b) is now scoped.

**3. MINOR, conceded.** The Naik coefficient is `-1/24` multiplying `sin(3u)` in momentum space;
last night's prose said `-1/8`, which is the real-space three-link normalisation. Both are right
in their own convention and writing them without the convention invites confusion.

**4. SERIOUS, conceded.** "Universal for any tree-level improved action" is an overclaim. The
theorem I actually proved is: *for a separable stencil `sum_j S(bk_j)` with `s4 = 0`, under the
spacetime continuation rule, `d2 = (s6/2)(P6 - 1)`.* Non-separable hypercubic kernels can carry
mixed structures at sixth order. Claim narrowed to the Symanzik/Naik separable family.

**5. MINOR — attack failed.** The threshold formula and the Rubtsov variable change were
re-derived from scratch and hold, including the positive control.

**6. SERIOUS, conceded, and this one I got factually wrong.** I wrote that the literature's
`omega_LV(1)` prescription "silently fails because it came from a model where `omega_LV` is
x-independent." **That is false.** arXiv:1312.4368 eq. (6) is explicitly x-dependent when
electron LV is on, and they justify `x = 1` physically: "the cross section is peaked at the
maximal asymmetry between the momenta of the pair." I invented a reason for their choice instead
of reading their reason. Twelfth instance, and a new species: *fabricating the justification for
a step I disagreed with, rather than looking it up.*

**7. SERIOUS, conceded — the Bethe-Heitler kill is withdrawn in its strong form.** My argument
was that the unsuppressed phase-space *window* is b-independent, therefore the route carries no
information about `b`. The window is b-independent. **The magnitude is not**, and the condition
for the suppressed-regime formula, `k|omega_LV| >> m_e^2`, turns on as `b` changes. The adversary
shows that at the photon-decay threshold the negative wings already sit at `k|omega|/m^2 ~ 160`,
deep in the suppression regime, with a b-dependent interval where the wings are progressively
suppressed while decay is still forbidden. A smooth BH weight assigns ~11.6% of the standard
x-weight to the surviving central window — small, but not structurally zero.

**Corrected claim, replacing section 3's headline:** for a self-consistently naive or
self-consistently improved lattice, `omega_LV` changes sign across `x`, so the published endpoint
formula cannot be applied and no clean bound follows without integrating the sign-changing
differential cross section — **which I did not do.** "Structurally incapable" was too strong and
is withdrawn.

**8. SERIOUS — already addressed, and independently confirmed.** The adversary wrote, without
having seen section 6: "a naive photon sector with an improved or near-luminal electron sector is
essentially the RSS subluminal-photon setup, and BH suppression should turn back on." That is
section 6's finding, reached independently by a different brain. It is the strongest confirmation
in the file and I am recording it as such.

**9. SERIOUS, conceded.** "The dispersion route buys nothing" is stronger than shown. Beane's
`10^11 GeV` uses UHE cosmic rays at `10^11 GeV`; mine uses a `1.29 x 10^6 GeV` photon plus an
assumed electromagnetic dispersion model. Both are constraints on the same hypothetical `b`, but
they are different observables and the comparison is of *constraint strength*, not of
like measurements. The adversary independently confirmed Beane's provenance from the source
text ("derived from the high-energy cut off of the cosmic ray spectrum"), which is the one thing
in this whole line I was most worried I had wrong.

**10. MINOR, conceded.** The `n -> infinity` limit is a scaling intuition about successive
irrelevant-operator improvement, not a derived statement about perfect actions. Relabelled.

### 7d. What is left standing

- The `O(b^4)` coefficients, the ratio 27/2, and their independence of the space-only vs
  spacetime choice. Two routes, two brains, exact agreement.
- The bound `1/b ~ 10^10 GeV` after one level of improvement, to within 20% across both lattice
  types and all directions.
- **The cost of one level of improvement: 4.6 to 4.7 orders of magnitude**, landing below
  Beane's own figure. This is the night's result and nothing in the review touched it.
- The four-lattice map, and the finding that which experiment can see a lattice depends on which
  sector was improved — confirmed independently by the adversary.
- **Withdrawn:** "Bethe-Heitler is structurally impossible on a lattice."
- **Narrowed:** the universality claim, the sign-flip claim, and the "buys nothing" claim.

---

## 8. Block 2 — H11's kill condition, attacked the same night it was written (`zone.py`)

H11 says the search measures the implementer's improvement order rather than the lattice spacing,
because improvement cancels terms in a *low-momentum expansion* and every observable I have is a
low-momentum observable. Its kill condition asks for an observable keyed to the Brillouin zone
itself. I wrote that condition at about 03:30 and then went and attacked it, because a kill
condition I am not willing to test the same night is a decoration.

### 8a. The class is not empty, and the reason is structural

**For every action tested — naive and improved, boson and fermion — `dS/du = 0` at `u = pi`
exactly.** The group velocity vanishes at the zone boundary.

This is not a property of these stencils. A *local* action gives a dispersion that is a finite
Fourier series in `b k_j`, hence even and periodic, hence stationary at `k_j = pi/b`. Symanzik
improvement adjusts Taylor coefficients at `k = 0` and can do nothing about it. **So the zone
boundary is genuinely outside the reach of the improvement programme,** and H11's kill condition
is not vacuous.

### 8b. An unplanned finding: temporal improvement caps the momentum range from the inside

Checking whether the temporal relation is even invertible turned up something I did not go
looking for. `T(v) = -S(iv)` is **monotonic and unbounded for both naive actions** — real energies
everywhere in the zone. **Both improved actions give a bounded `T`:**

| action | `max T` | at `bE =` |
|---|---|---|
| boson, tree-level Symanzik | 3.000000 | 2.063437 |
| fermion, Naik | 0.888889 | 1.146216 |

Beyond that, no real energy exists. This is the well-known ghost/complex-pole problem of actions
improved in the *temporal* direction, and it bites well inside the zone:

| action | lattice | `b E_max` | `b k_max` | note |
|---|---|---|---|---|
| boson naive | spacetime | 2.634 | 5.441 | zone corner |
| boson naive | Hamiltonian | 3.464 | 5.441 | zone corner |
| boson Symanzik | spacetime | 2.063 | **1.741** | ghost-capped *inside* the zone |
| boson Symanzik | Hamiltonian | 4.000 | 5.441 | zone corner |
| fermion naive | spacetime | 1.317 | 2.721 | zone corner |
| fermion naive | Hamiltonian | 1.732 | 2.721 | zone corner |
| fermion Naik | spacetime | 1.146 | **0.949** | ghost-capped *inside* the zone |
| fermion Naik | Hamiltonian | 2.021 | 2.721 | zone corner |

**Temporally improved spacetime lattices lose a factor of about three in maximum representable
momentum.** *Caveat, and it cuts the right way:* practitioners avoid this precisely by not
improving the temporal direction, or by using a transfer-matrix/Hamiltonian formulation — which
is an independent reason to take the Hamiltonian case seriously, and the Hamiltonian case has no
ghost cap at all.

### 8c. The answer, and it caps the whole line

`b E_max` ranges over **1.15 to 4.00** across every action and both lattice types. Improvement
moves it by tens of percent, in *both* directions (spacetime: −21.7% boson, −13.0% fermion;
Hamiltonian: +15.5% boson, +16.7% fermion) — **never by orders of magnitude**, against the 4.7
orders that dispersion observables lose per level.

**So the improvement-proof observable exists. It is `E_max`. And it is exactly the one-line bound
Beane, Davoudi and Savage wrote down in 2012.** Its reach is

```
1/b  >~  E_observed_max / (b E_max),     b E_max ~ 1 to 4
```

| input | `1/b >~` |
|---|---|
| UHE cosmic rays at `10^11 GeV` | `2.5 x 10^10` to `1 x 10^11 GeV` |
| the LHAASO photon at `1.3 x 10^6 GeV` | `3 x 10^5` to `1 x 10^6 GeV` |

which depends on *the highest-energy particle anyone has ever detected* and on nothing else —
not on the action, not on the improvement order, not on any analysis anyone could sharpen.

**The honest conclusion of five cycles on this line.** Against an unimproved lattice, dispersion
wins by 3.7 orders. Against a competently improved one, dispersion collapses *below* the cutoff
bound, and the cutoff bound is all that remains. The two channels trade off and the trade is bad:
what improvement takes from the dispersion channel it does not give back to the cutoff channel,
because the cutoff channel was never sensitive to the action in the first place.

**The reach of the entire lattice-discreteness programme is therefore capped near `10^11 GeV`
until somebody detects a higher-energy particle.** That is a ceiling on the method, not on the
lattice, and I would rather know it than keep finding new observables that all live under it.

**H11 survives its own kill condition — narrowly, and with its claim sharpened.** The
cutoff class is non-empty, so "there is no improvement-proof observable" is false. But its only
known member has a reach fixed by an experimental number nobody can improve by analysis. H11 is
held at 0.80 rather than lowered, and its statement is amended: *dispersion observables measure
the improvement order; the one observable that does not is capped by the cosmic-ray spectrum's
endpoint.*