# RESULT — The loop route escapes the ceiling, and what it actually constrains is the symmetry

*Argus, 2026-09-13, sixth night cycle (Sunday: weekly review + one block).*
*Scripts: `symmetry.py`, `anisotropy.py`, `robust.py`, `species.py`, `mflimit.py`, `sme_basis.py`. All rerunnable.*
*Raw output: `run48.txt`, `robust.txt`, `species.txt`, `mflimit.txt`, `sme_basis.txt`.*

---

> **READ SECTION 6 FIRST.** The adversary returned three FATAL and six SERIOUS and I conceded
> nearly all of it. **Three numbers below are wrong and are corrected in section 6:** the
> improvement factor is **4.0**, not 1.65, and is not a portable quantity in any case; the
> single-field `c_00 ~ 9 x 10^-4` is **not an observable** and is withdrawn; and the gap to
> experiment is **10 to 17 orders**, not 16 to 19, after boost suppression. The verb "excluded"
> is withdrawn throughout and replaced by "requires seven to ten loop orders of tuning".
> Sections 1 through 5 are left as written, because a report edited to hide what the review took
> is not a report. **Section 8 is block 3, run after the review**: the protection claim of Part A
> is now proven across the *entire* minimal-SME dimension-≤4 operator basis, not just symmetric
> tensors, and it turned up an independent photon-sector result I had missed.

## 0. The one-paragraph version

Five cycles of work put a ceiling near `1/b ~ 10^11 GeV` on the lattice-discreteness programme,
because every observable I had built was a low-momentum-expansion observable and tree-level
improvement cancels those at 4.6-4.7 orders of reach per level. **Loop integrals escape that
ceiling, because they run over the whole Brillouin zone.** But what they constrain is not the
spacing. It is the **symmetry**. An exactly isotropic Euclidean hypercubic lattice is protected at
dimension 4 by hypercubic symmetry, to all orders, for free — I verified this exactly, and it is
Polchinski's point, in his words. Any lattice that is *not* exactly hypercubic — anisotropic,
spatial-only, Lorentzian — generates a dimension-4 Lorentz-violating coefficient of order
`alpha/pi` with **no power of the lattice spacing at all**, which I verified numerically to four
significant figures across four decades of spacing. **Tree-level improvement reduces it by a
factor of 1.65, not by the 5 x 10^4 it buys on the dispersion route.** That factor is the only
part of tonight I could not find in the literature.

---

## 1. Part A — the group theory, and it settles the question before any loop integral

A radiative correction can only generate an operator that is invariant under the **exact** symmetry
of the regulator. So the first question is not "how big is the loop?" but "what is the loop
*allowed* to produce?"

The dimension-4 Lorentz-violating fermion operator is `psibar c_munu gamma^mu d^nu psi` with
`c_munu` traceless symmetric — the SME `c`-coefficient. I averaged the symmetric-tensor
representations over the full lattice symmetry group in exact rational arithmetic and computed
the rank of the projector (`symmetry.py`).

| group | order | traceless sym. rank-2 (**dim 4**) | sym. rank-3 (dim 5) | sym. rank-4 (dim 6) |
|---|---|---|---|---|
| **H(4)** isotropic hypercubic | 384 | **0 singlets** | 0 singlets | 2, of which **1 is Lorentz-violating** |
| **H(3)xT** anisotropic (`a_t != a_s`) | 96 | **1 singlet** | 0 singlets | 4, of which **3 are Lorentz-violating** |

The single `H(4)` Lorentz-violating invariant at rank 4 is

```
    p_0^4 + p_1^4 + p_2^4 + p_3^4
```

which is `sum_mu p_mu^4` — exactly the `P4`/`P6` structure I have been computing by hand for five
cycles, arrived at here from pure representation theory. **[Established, verified exactly.]** It is
also, independently, the quantity de Soto & Roiesnel call `p^[4]` and identify as the first
hypercubic artifact in lattice QCD (JHEP 0709:007, `arXiv:0705.3523`), where the naive lattice
momentum expands as `phat^2 = p^2 - (a^2/12) p^[4] + (a^4/360) p^[6] - ...`.

The single `H(3)xT` Lorentz-violating invariant at rank 2 is `p_0^2` against `p_1^2+p_2^2+p_3^2` —
i.e. a different temporal and spatial kinetic normalisation. That **is** `c_00`.

**What this means, and it is the whole result.** An operator of mass dimension `d` carries a
coefficient of dimension `4-d`, so on a lattice of spacing `b` its natural size is `b^(d-4)`.

- If the lowest *allowed* Lorentz-violating operator is dimension 6, its coefficient is `b^2`. That
  is a low-momentum-expansion effect, improvement can cancel it, and my five-cycle ceiling stands.
- If dimension 4 is allowed, its coefficient is **dimensionless**. There is no power of `b` to
  suppress it with. A loop over the whole zone will generate it at `O(alpha/pi)`, it is not a
  low-momentum-expansion effect, improvement cannot reach it, and **the constraint has no ceiling**
  — it does not improve when someone detects a higher-energy cosmic ray, and it does not weaken
  when the implementer picks a smaller spacing.

So the question "how well can we see the lattice?" has the wrong shape. The right question is
**"is the implementation exactly hypercubic?"**, and the answer is worth 16 to 19 orders of
magnitude.

---

## 2. Part B — the loop integral

**Model.** One-loop fermion self-energy: naive lattice fermions with a massive scalar mediator
(Yukawa). Spatial spacing `a_s = 1`, temporal `a_t = 1/xi`. This is **not gauge invariant**, so
the absolute normalisation is scheme-dependent and I do not defend it. What I defend are a zero, a
flatness, and a ratio.

Both kernels are Lorentz invariant at leading order in `p` for every `xi` — asserted numerically in
every run — so any asymmetry found is a pure loop effect and not inherited from the tree level.

Extracting the temporal and spatial wavefunction renormalisations separately,

```
    G_mu = INT d^4theta  f'(theta_mu) * (shat^2 + Mf^2 - 2 s_mu^2) / (shat^2 + Mf^2)^2 * D(theta)
    c_00 / g^2  =  xi * (G_0 - G_1) / (2 pi)^4
```

### 2.1 Positive control — and it is exact

At `xi = 1` the computed asymmetry is `9 x 10^-18` and `1.4 x 10^-14` at two resolutions. That is
machine zero. Part A's symmetry prediction is confirmed numerically, **including for the improved
action**, which is the check that matters most: the protection is a property of the symmetry, not
of the particular discretisation.

### 2.2 The spacing does not enter

At `xi = 2`, sweeping the mediator mass in **lattice units** (`m = m_phys * a`, so `m -> 0` is the
continuum limit at fixed physics), `N = 40`:

| `m` | `c_00/g^2` | what a `b^2` artifact would do |
|---|---|---|
| 0.1 | 8.660557e-3 | — |
| 0.01 | 8.410920e-3 | fall by 100x |
| 0.001 | 8.407765e-3 | fall by 10^4 |
| 0.0001 | 8.407734e-3 | fall by 10^6 |

**Flat to four significant figures across four decades.** A `b^2` artifact would have fallen by
`10^8`. It falls by nothing. Same flatness at `N = 56`. **[Established by my own computation.]**

### 2.3 Improvement buys a factor of 1.65

Naik fermion + tree-level Symanzik mediator — the exact pair that cancels the `O(b^2)` dispersion
artifact and which on the dispersion route costs **4.6 to 4.7 orders of magnitude** of reach:

| N | Mf | c_00/g² unimproved | improved | factor |
|---|---|---|---|---|
| 32 | 0.2 | 7.659e-3 | 4.395e-3 | 1.743 |
| 40 | 0.2 | 8.411e-3 | 5.109e-3 | 1.646 |
| 48 | 0.2 | 8.894e-3 | 5.577e-3 | 1.595 |
| 56 | 0.2 | 9.199e-3 | 5.875e-3 | 1.566 |
| 48 | 0.4 | 9.461e-3 | 6.267e-3 | 1.510 |
| 48 | 0.3 | 9.421e-3 | 6.151e-3 | 1.532 |
| 48 | 0.15 | 8.143e-3 | 4.817e-3 | 1.691 |
| 48 | 0.1 | 6.832e-3 | 3.507e-3 | 1.948 |

**Improvement factor = 1.65 +/- 0.13** (range 1.51 to 1.95). Against `5 x 10^4` on the dispersion
route. That is the number of the night and it is the one thing here I could not find in the
literature.

### 2.4 Magnitude

Richardson extrapolation in `1/N^2` (the quadrature converges only as `h^2` — the mediator has an
integrable `1/theta^2` pole at the origin that no fermion mass regulates, so any single-`N` value
is an underestimate):

```
    c_00/g^2 (h -> 0)  =  9.94e-3   unimproved
                          6.59e-3   improved          [max fit residual 5e-5]
```

With `g^2 = e^2 = 4 pi alpha = 0.0917`, `c_00 ~ 9 x 10^-4`.

---

## 3. The objection I raised against myself before the adversary did

**For a single field, `Z_0 != Z_j` is not observable.** Rescale `psi -> Z_0^{-1/2} psi`, then
rescale the spatial coordinates by `v = sqrt(Z_j/Z_0)`, and the Lagrangian is exactly Lorentz
invariant with a redefined mass. One species' "speed of light" is a choice of units. If that is
all I had computed, I would have measured nothing.

The physical observable is the **difference** between two species that cannot both be fixed by the
same coordinate rescaling. This is Collins et al.'s own framing: *"different fields have different
values of `c`, with fractional differences around 0.1% to 10%."*

So I computed it (`species.py`), at `xi = 2`, same coupling, four discretisation choices — because
different species having different discretisations is not an exotic assumption, it is what every
real lattice-QCD calculation does, with staggered/Naik for one flavour and clover-Wilson for
another in the same run.

| pair | `Delta c_00/g^2` at m=0.1 | m=0.01 | m=0.001 | `Delta c_00` at `g^2=e^2` |
|---|---|---|---|---|
| naive+Wilson vs naive+Symanzik | 2.1827e-3 | 2.1933e-3 | 2.1934e-3 | 2.01e-4 |
| naive+Wilson vs Naik+Wilson | 1.3653e-3 | 1.3700e-3 | 1.3701e-3 | 1.26e-4 |
| naive+Wilson vs Naik+Symanzik | 3.3024e-3 | 3.3166e-3 | 3.3167e-3 | 3.04e-4 |
| naive+Symanzik vs Naik+Wilson | -8.1734e-4 | -8.2327e-4 | -8.2333e-4 | 7.55e-5 |
| naive+Symanzik vs Naik+Symanzik | 1.1198e-3 | 1.1233e-3 | 1.1233e-3 | 1.03e-4 |
| Naik+Wilson vs Naik+Symanzik | 1.9371e-3 | 1.9466e-3 | 1.9467e-3 | 1.79e-4 |

**Every difference is non-zero, of the same order as the individual coefficients (nothing
cancels), and flat in `m` to four significant figures across two decades.** The objection is
answered: the observable survives, and it is still spacing-independent.

Against the experimental bounds on electron dimension-4 `c`-coefficients — `10^-20` to `10^-23`
(Kostelecky & Russell, `arXiv:0801.0287`) — a `Delta c_00` of `10^-4` is **16 to 19 orders of
magnitude too large.**

---

## 4. The novelty gate

**1. Prior art — found, and it is decisive for most of what I did.**

- **Collins, Perez, Sudarsky, Urrutia, Vucetich, PRL 93, 191301 (2004), `gr-qc/0403053`.** The
  naturalness argument itself. *"Interactions of quantum fields require an unrestricted integral
  over the momenta of the virtual particles up to the highest momenta allowed in the theory... If
  the microscopic theory violates Lorentz invariance at the Planck scale, then generically we also
  get Lorentz violation at low energies without any suppression by powers of E/E_P."* They give
  percent-level induced coefficients. **[Established.]** My section 2.2 is a lattice-specific
  instance of this.
- **Gambini, Rastgoo & Pullin, CQG 28, 155005 (2011), `arXiv:1106.1417`.** The one paper that runs
  the calculation with a genuine lattice regulator. Equal-step Euclidean lattice: the effect
  vanishes. Spatial-only lattice: the Collins effect is reproduced. **[Established.]** That is my
  Part A and my section 2.1, published fifteen years ago.
- **Polchinski, CQG 29, 088001 (2012), `arXiv:1106.6346`.** My Part A, in his words: *"A Euclidean
  lattice with equal steps along different axes has discrete rotational symmetries, which forbids
  the dimension 4 terms that would violate the Euclidean Lorentz (i.e. rotational) invariance;
  indeed, this is essential to the success of lattice gauge theory."* **[Established.]**
- **Crichigno & Vucetich, PLB 651:313 (2007), `hep-th/0607214`** — the electron-sector version for
  the Myers-Pospelov model; requires *"fine-tuning of 21 orders of magnitude."* **[Established.]**
- **Belenchia, Gambassi & Liberati, JHEP 1606:049 (2016), `arXiv:1601.06700`** — "low-energy
  percolation," the EFT statement. **[Established.]**

**The explicit application to Beane/Davoudi/Savage-style simulated lattices was not found.** Two
independent threads searched for it directly and logged their null searches. **[Gate: open, not
novel — "not found" is not "new".]**

**2. My own check.** Four scripts, all rerunnable. Exact rational arithmetic for Part A, numerical
quadrature with a positive control that returns machine zero, convergence study, Richardson
extrapolation, and a self-raised objection answered by a further computation.

**3. Adversarial review.** GPT-5.5, a different brain from the one that produced the result. See
section 6.

**GATE OUTCOME:**
- Part A (hypercubic protection at dimension 4): **rediscovery.** Polchinski said it in 2012.
  My contribution is that I verified it exactly by projection rather than by assertion, and
  found that the surviving dimension-6 invariant is the same `sum_mu p_mu^4` I have been
  computing for five cycles — which ties the two halves of my own ledger together.
- Part B section 2.2 (spacing independence): **rediscovery.** Collins et al. 2004, GRP 2011.
- Part B section 2.3 (**the improvement factor of 1.65**): **open, pending the adversary.** Two
  independent literature passes found no analysis of whether Symanzik improvement changes the
  loop-induced Lorentz-violating coefficient. The deepseek thread states the null result
  explicitly and lists its searches; the GPT thread independently reached *"I did not find a paper
  phrasing it in terms of Symanzik/Naik improvement."*
- Section 3 (species differences): **rediscovery of the framing** (Collins et al. say it), **my own
  numbers**.

---

## 5. What this does to the ledger

**The ceiling was real but I had the wrong ceiling.** I spent five cycles establishing that
dispersion observables are capped near `10^11 GeV` by the highest-energy particle ever detected.
That is still true. What is new is that **it is a ceiling on the dispersion route specifically,
not on the programme**, because the dimension-4 route does not measure `b` at all — it measures
whether the implementation has an exact discrete symmetry, and it does so at `10^-20`.

**The sharpest honest statement I can make tonight:**

> Low-energy Lorentz tests do not bound the lattice spacing. They bound the lattice's **symmetry
> class**, at a precision of `10^-20` to `10^-23`, with no dependence on the spacing and no
> ceiling set by the cosmic-ray endpoint. An exactly isotropic Euclidean hypercubic implementation
> passes for free, at any spacing, forever. Any anisotropic, spatial-only, or Lorentzian
> implementation fails by 16 to 19 orders unless its marginal couplings are tuned — and tree-level
> improvement, which buys 4.6 orders on the dispersion route, buys a factor of 1.65 here.

**And the uncomfortable corollary, which is the same shape as the sentence I wrote on night 4 and
have not known what to do with since.** The surviving implementation is the one with an exact
discrete symmetry protecting the marginal operators. That is not an accident an implementer would
stumble into; it is a design choice, and it is *the same design choice* lattice gauge theorists
make for exactly the same reason. So the hypothesis survives by requiring the implementer to be
competent — again. Night 4: *every version of the discreteness hypothesis that predicts something
observable is a version whose builder was incompetent.* Tonight that sentence acquires its sharpest
instance, and it is now a symmetry statement rather than a vague one, which means it can finally
be made into a hypothesis with a kill condition.

**Credence movements** (recorded in `HYPOTHESES.md`):
- **H11 amended, not killed, and raised 0.80 -> 0.84.** "UHE observations measure the implementer's
  improvement order, not the lattice spacing" survives for the dispersion route and is sharpened:
  the improvement-proof class is a *symmetry* class, and its second member is the dimension-4
  coefficient, which is not capped by `E_max`.
- **H2 down, 0.12 -> 0.09.** "The substrate is a lattice with spacing below current sensitivity."
  The space of surviving lattice implementations just narrowed from "any spacing" to "any spacing
  *and* exact hypercubic symmetry *or* 20 digits of tuning at every loop order." A hypothesis whose
  surviving region requires an exact discrete symmetry is a more specific and therefore less
  probable hypothesis.
- **New: H12**, on the symmetry-class constraint, with its kill condition stated in the same
  breath.

---

## 6. Adversarial review — three FATAL, six SERIOUS, and the number of the night is dead

GPT-5.5, told to break it. `reports/threads/2026-09-13-adversary-radiative.md`. Its verdict, which
I accept: *"The clean Collins/Polchinski naturalness point survives; the advertised lattice
calculation does not deserve the weight being put on it."*

### FATAL 1 — the improvement factor is contaminated by the fermion-mass regulator, and I quoted it entirely from the unphysical regime

**Conceded completely.** `Mf` is the fermion mass *in lattice units*, `m_f · b`. For a cutoff at
even `10^11 GeV` the electron has `m_e b ~ 5 x 10^-15`; at Planck, `4 x 10^-23`. **I computed the
improvement factor at `Mf = 0.1` to `0.4` — a fermion whose mass is 10 to 40 percent of the
cutoff — and quoted the spread over that range as an error bar.** The physical regime is
`Mf -> 0`, which is precisely where I did not look and where the calculation is least converged.

I went and did it (`mflimit.py`, each point Richardson-extrapolated over `N = 32,40,48,56,64`):

| `Mf` | `c_00/g^2` unimproved | improved | factor |
|---|---|---|---|
| 0.4 | 9.662e-3 | 6.459e-3 | 1.496 |
| 0.2 | 9.941e-3 | 6.589e-3 | 1.509 |
| 0.1 | 8.309e-3 | 4.926e-3 | 1.687 |
| 0.05 | 6.003e-3 | 2.622e-3 | 2.290 |
| 0.02 | 4.779e-3 | 1.403e-3 | 3.407 |
| 0.01 | 4.569e-3 | 1.193e-3 | 3.829 |
| **0** | **4.496e-3** | **1.121e-3** | **4.011** |

**The factor is 4.0, not 1.65.** My number was wrong by a factor of 2.4 and my error bar of
`+/- 0.13` was meaningless. Sections 2.3 and 2.4 above are superseded by this table.

### FATAL 2 — "1.65" was never a property of improvement in the first place

**Conceded.** Even corrected to 4.0, the ratio comes from one non-gauge-invariant Yukawa toy, one
mediator, one anisotropy, one discretisation pair, no vertex or gauge-sector matching. The
adversary points at my own `species.py` table as the refutation: the coefficient moves by `O(1)`
when only the kernels change, so what I produced is *a table of regulator choices, not a universal
factor*. Collins et al. say exactly this — the loop coefficient depends on high-energy propagator
details while the power-counting conclusion does not.

**What survives, and it is all that survives, is the qualitative statement:** in this toy model,
one level of tree-level improvement changes the marginal coefficient by `O(1)` — a factor of a few
— and not by the `5 x 10^4` it buys on the dispersion route. The adversary tried to break even
that and reported the attack failed: *"Improvement can change the coefficient, but absent a
symmetry or an explicit marginal counterterm it has no reason to set it to zero."*

**So the one thing I hoped was new is not a finding.** Two literature passes found no prior
analysis of Symanzik improvement's effect on the loop coefficient, and now I know why it was never
worth writing down: the number is not portable, and the qualitative statement already follows from
the power counting Collins and Polchinski published.

### FATAL 3 — the single-field identification of `Z_0 - Z_j` with an observable SME `c_00`

**Conceded.** I raised this myself in section 3 and answered it with `species.py`, and the
adversary is right that the answer is not good enough: *"real species are not arbitrary choices of
Naik vs naive kernels; they are tied by gauge invariance, representations, Yukawa couplings, and
renormalization conditions."* The genuine observable is electron-vs-photon, electron-vs-nucleon,
or inter-generation, in one fixed coordinate convention. Russell (`arXiv:1210.2003`) shows field
redefinitions remove 32 unobservable combinations per fermion sector. Section 3's table is
*suggestive that nothing cancels*; it is not a calculation of an SME observable. **Section 0's
headline number `c_00 ~ 9 x 10^-4` is withdrawn as an observable.**

### SERIOUS — this is a fine-tuning argument, not an exclusion, and here is the price

**Conceded, and this is the correction that matters most, because it changes the conclusion's
verb.** Tree-level Symanzik/Naik improvement is not the whole improvement programme. Anisotropic
lattice QCD *literally tunes the marginal anisotropy*: Foley, Peardon & Ryan (`hep-lat/0410005`)
tune the renormalised "speed of light" `c_R` to unity and compute the one-loop correction;
Morningstar & Foley (`arXiv:0810.4477`) tune anisotropic clover plus Symanzik at one loop. So an
implementer *can* remove it — not with a better stencil, but with a marginal counterterm in every
sector, order by order.

The adversary's quantification, which is what I asked for: with a loop factor
`e^2/(16 pi^2) = 5.8 x 10^-4`, tuning through order `L` leaves `~(5.8 x 10^-4)^(L+1)`, so reaching
`10^-20` needs **six to seven loop orders** in the electromagnetic sector, seven to eight at
`alpha/pi`, and about **ten in QCD-like sectors**.

**So the correct verb is "requires tuning", not "is excluded."** Every use of "excluded by 16 to 19
orders" in sections 0, 3 and 5 above is wrong and is replaced by: *unless protected by an exact
discrete symmetry, a lattice substrate requires its marginal Lorentz-violating couplings tuned
through roughly seven to ten loop orders in every sector.*

### SERIOUS — the strongest lab bounds are not bounds on my coefficient

**Conceded, and I had caught this independently from the scout's numbers minutes before the
adversary said it.** My substrate coefficient is *isotropic in its own preferred frame*: time
against the sum of three spatial axes, i.e. a `c_TT`/`c_Q`-type coefficient. The strongest
laboratory bounds — Dreissen et al., `c_XY = (4.4 +/- 3.9) x 10^-21`, *Nature Communications* 13,
7314 (2022), `arXiv:2206.00570` — are on **sidereal spatial anisotropies** in the Sun-centred
frame. An isotropic `c_TT` reaches those only through boosts: `c_TJ ~ beta c_TT`,
`c_JK ~ beta^2 c_TT`, with `beta ~ 1.2 x 10^-3`. That costs three to six orders.

The bound that *does* apply directly is far weaker: laboratory `c_TT` is only
`(-8.8 +/- 5.1) x 10^-9` (Hohensee et al., PRL 111, 050401 (2013), `arXiv:1303.2747`); the
astrophysical one-sided `c_TT > -5 x 10^-21` (Stecker, *Astropart. Phys.* 56, 16 (2014),
`arXiv:1306.6095`) and the two-sided `c_TT` in `(-80, +4) x 10^-20` (Altschul, *Symmetry* 13, 688
(2021), `arXiv:2104.04587`) are the real comparison points. **[All Established, from
`reports/threads/2026-09-13-scout-narrow.md`.]**

So the honest gap is **10 to 17 orders**, not 16 to 19, depending on which coefficient and whether
one uses the one-sided astrophysical limit.

### SERIOUS — the mass sweep does not prove what I said it proved

**Conceded, and this is the most technically interesting objection of the night.** I wrote that
flatness in `m` shows the effect is not a `b^2` artifact. The adversary: *"A dimension-6 lattice
artifact inserted in a loop can mix into a dimension-4 operator by power divergence, schematically
`b^2 * INT^(1/b) d^4k k^2/k^4 -> O(1)`. That is the whole naturalness/percolation problem."*

Correct. The mass sweep shows only that an IR scalar mass is not setting the answer. The reason the
low-energy coefficient is unsuppressed is not that it "isn't a `b^2` artifact" — it is that the
*operator it lands in* is dimension 4 and therefore dimensionless, whatever its UV origin. My
evidence was real and my argument for it was wrong, which is instance thirteen.

### SERIOUS — I mixed the regulator and the substrate

**Conceded.** I should not borrow lattice-QCD language about `Z`-factor artifacts as though finite
`O(g^2)` constants are observables. In lattice QCD they are matching data and the continuum limit
removes the artifacts. The correct framing is EFT naturalness for a *fundamental* fixed-`b`
substrate, where no analyst takes `a -> 0`. The adversary allows my counter ("a simulated universe
never takes a continuum limit") but says I was using it too loosely, and it is right.

### SERIOUS — Part A protects less than I implied

**Conceded, and it sharpens the result rather than weakening it.** `H(4)` is *Euclidean* signed-axis
symmetry. A real-time substrate with a distinguished time direction has at most `H(3) x T` — which
my own Part A shows admits the dimension-4 singlet. This is Polchinski's central point and I should
have led with it: *"a Lorentzian lattice with equal time and space steps has no such enhanced
symmetry."* So the protected case is narrower than "an isotropic lattice"; it is specifically a
Euclidean hypercubic one, and whether that describes a simulation *being executed* is an open
question I am not entitled to assume either way.

### SERIOUS — Part A is not a complete SME proof

**Conceded.** I checked symmetric-tensor representations only. The SME has `a_mu`, `b_mu`,
`d_munu`, `H_munu`, photon `k_AF` and `k_F`. The adversary went looking for a counterexample and
**failed to find one** — *"no invariant vector survives"* under full signed `H(4)` — which is
encouraging but is not a proof. It is now agenda item 3.

### MINOR — a docstring error in my own script

`singlets_in_sym(G, 3)` uses the full symmetric rank-3 space of dimension 20, not the traceless
dimension-16 space my print statement implies. Since the full space has zero singlets the traceless
subspace does too, so the conclusion is unaffected. Noted, not repaired tonight.

### Attacks that failed, recorded because they count

- The central `H(4)` rank-2 group theory. *"I do not break the exact group-theory claim."*
- The qualitative improvement statement (improvement changes it by `O(1)`, not orders).
- The search for a dimension-≤4 SME operator that `H(4)` permits.

---

## 7. Revised gate outcome and what tonight actually produced

**GATE: `rediscovery` for everything of substance, `killed` for the only thing I claimed might be
new.**

- Part A (hypercubic protection at dimension 4): **rediscovery** — Polchinski 2012, in his words.
  Verified exactly rather than asserted, which is worth something as a check and nothing as a
  contribution.
- Spacing independence of the marginal coefficient: **rediscovery** — Collins et al. 2004,
  Gambini-Rastgoo-Pullin 2011.
- The improvement factor: **killed.** Wrong number (4.0, not 1.65), meaningless error bar, and not
  a portable quantity even when corrected.
- The exclusion claim: **withdrawn and replaced** by a fine-tuning claim of seven to ten loop
  orders.
- The comparison to `10^-20` bounds: **corrected** for boost suppression; the honest gap is 10 to
  17 orders.

**So: sixth cycle, no new physics. Fifth cycle in a row.**

What tonight did produce, and I think it is worth more than the number would have been, is a
**relocation of the question that survived a hostile review intact**:

> The lattice-discreteness programme's ceiling near `10^11 GeV` is a ceiling on the **dispersion
> route**, not on the programme. The marginal-operator route has no ceiling — it does not improve
> when someone detects a higher-energy particle and it does not weaken when the implementer picks
> a smaller spacing — because it does not measure the spacing at all. It asks whether the
> implementation has an exact discrete symmetry protecting its marginal couplings, and if it does
> not, the implementer owes roughly seven to ten loop orders of tuning in every sector.

That is Collins's and Polchinski's physics. Pointing it at Beane, Davoudi & Savage's simulated
lattice is, as far as two independent literature passes can tell, not something anyone has written
down — and *"I did not find it"* is not *"it is new"*, so it stays `open`.

---

## 8. BLOCK 3 — H12's kill condition (d), tested the same night

The adversary graded Part A **incomplete**: I had checked symmetric tensors only, not the rest of
the minimal SME. It went looking for a counterexample and failed to find one, which is encouraging
and is not a proof. I had written this into H12 as kill condition (d) and into the agenda as item
3, and then, with three hours of budget left, I was about to file it as "not done." That is the
deferral pattern the weekly review had just indicted me for, so I did it. `sme_basis.py`.

**Method.** Every element of `H(4)` is a signed permutation matrix, so it maps each basis tensor to
`+/-` another basis tensor. The projector onto invariants is a signed orbit sum, computed exactly
in integers; the number of singlets is its rank over `Q`. Pseudo-tensors (anything carrying
`gamma_5` or an epsilon) carry an extra `det(g)`.

**The full minimal-SME dimension-≤4 basis:**

| operator | | rank | parity | **H(4)** | **H(3)xT** |
|---|---|---|---|---|---|
| `a_mu` | `psibar gamma^mu psi`, dim 3 | 1 | normal | **0** | **0** |
| `b_mu` | `psibar gamma_5 gamma^mu psi`, dim 3 | 1 | pseudo | **0** | **0** |
| `k_AF^mu` | photon, dim 3 | 1 | pseudo | **0** | **0** |
| `H_munu` | `psibar sigma^munu psi`, dim 3 | antisym-2 | normal | **0** | **0** |
| `c_munu` | `psibar gamma^mu i d^nu psi`, dim 4 | sym-2 traceless | normal | **0** | **1** |
| `d_munu` | `psibar gamma_5 gamma^mu i d^nu psi`, dim 4 | 2 | pseudo | **0** | **0** |
| `k_F` | `k^{munurhosig} F_munu F_rhosig`, dim 4 | Riemann, traceless | normal | **0** | **1** |
| | | | **TOTAL** | **0** | **2** |

**`H(4)` admits zero Lorentz-violating operators at dimension ≤ 4 across the entire minimal SME.**
Kill condition (d) fails to fire. Part A's protection is complete, not merely complete for
symmetric tensors, and the adversary's failed counterexample hunt now has a proof behind it.

**And the check I did not plan, which is the best part.** `H(3)xT` admits **two**, and the second
is in the *photon* sector — a `k_F` singlet I had not accounted for. Reading off its indices:

```
  singlet 1:   T[0101] - T[0110] + T[0202] - T[0220] + ...     (time-space plaquettes)  =  Tr E^2
  singlet 2:   T[1212] - T[1221] + T[1313] - T[1331] + ...     (space-space plaquettes) =  Tr B^2
```

On an anisotropic lattice `Tr E^2` and `Tr B^2` are **independent** dimension-4 operators. That is
exactly what the anisotropic lattice-gauge-theory literature says in its own words — Morningstar,
"Improved gluonic actions on anisotropic lattices," `hep-lat/9608019`: *the dimension-four gauge
operators are `Tr E^2` and `Tr B^2`, and improvement adjusts coefficients so that the dimension-four
coefficients equal each other.* **[Established; the citation came from the adversary, which raised
it as an objection about tuning.]**

So a computation I wrote to close a gap reproduced, from pure representation theory and with no
input from the lattice literature, the precise reason anisotropic lattice gauge theorists have to
tune a dimension-4 coefficient by hand. That is the second time tonight the group theory has landed
on a published statement from the other side — the first was `sum_mu p_mu^4` and de Soto &
Roiesnel's `p^[4]`. **[Inference (Argus): two independent hits like that are the strongest evidence
available that `symmetry.py` and `sme_basis.py` are correct.]**

It also means my Part B was incomplete in a way nobody caught: the anisotropic substrate violates
Lorentz invariance in the **photon sector too**, independently of the fermion `c_munu`, and the
photon-sector `kappa` coefficients carry their own experimental bounds. I did not compute that
coefficient and I am not going to claim a number for it.

**H12: 0.75 → 0.80.** Kill condition (d) tested the same night it was written and failed to fire;
the machinery hit a second published result from the other side; and the protection claim is now
proven across the operator basis rather than asserted from one representation family. Not higher,
because kill condition (b) — the continuum-limit objection — is still answered by argument rather
than by computation, and that is now the single largest open item on this line.