# Adversary Report: Radiative LIV

Started: 2026-09-13

Scope: attack the claimed radiative Lorentz-violation result, with graded objections and failed attacks recorded explicitly.

## Running Findings

First hits while reading the lab files:

- **SERIOUS: the quoted 1.65 improvement factor is not a universal number.** It is a ratio in one non-gauge-invariant Yukawa toy with one chosen scalar kernel, one anisotropy (`xi=2`), one discretisation pair, and an artificial fermion mass. The prior-art thread already says the exact Collins coefficient depends on the UV regulator details; the report's own species table changes coefficients by O(1) when only the mediator/fermion discretisation changes. This kills the number as a claimed property of improvement. It does not kill the weaker claim that tree-level improvement fails to remove an allowed dimension-4 term by orders of magnitude.
- **SERIOUS, maybe FATAL to the magnitude table: `Mf=0.2` is not a harmless regulator for Standard Model physics.** Known fermion masses in lattice units are `m_f b`; for any lattice cutoff above UHE scales they are tiny, not 0.2. A quick rerun shows strong `Mf` dependence: at `N=56`, `m=0.01`, naive `c00/g^2` changes from `9.50e-3` (`Mf=0.4`) to `9.20e-3` (`0.2`) to `7.36e-3` (`0.1`) to `5.48e-3` (`0.05`) to `4.57e-3` (`0.02`) to `4.37e-3` (`0`). The improved/unimproved factor changes from `1.51` to `1.57` to `1.83` to `2.55` to `3.67`. This kills the quoted error bar and the stable-factor claim. It does not kill the existence of an unsuppressed O(loop) coefficient, but the numerical value is contaminated.
- **FATAL to single-field wording: `Z_0-Z_j` for one species is not an observable SME bound.** The physical object is relative limiting-speed anisotropy between sectors after one coordinate convention has been chosen. The lab's later `species.py` notices this, but the result statement still sells a single-field `c_00`. This kills the literal identification of the one-loop self-energy coefficient with an experimentally bounded SME coefficient. It does not kill the Collins naturalness argument, because Collins explicitly relies on different fields receiving different coupling-dependent shifts.

## Final Attack

### 0. Verdict

The clean Collins/Polchinski naturalness point survives; the advertised lattice calculation does not deserve the weight being put on it.

What survives: a physical preferred-frame or anisotropic lattice generically permits dimension-4 Lorentz-violating kinetic terms, and loops generate them unless symmetry or tuning forbids them. Existing SME bounds then make that a brutal fine-tuning problem, independent of the lattice spacing. This is Collins et al. 2004 plus Polchinski 2012, not new.

What does not survive: the number **1.65 +/- 0.13** as a meaningful improvement factor, the single-field identification of `Z_0-Z_j` with an observable SME `c_00`, and the claim that the `Mf` sweep is under control. Those are not cosmetic. They kill the quantitative result.

### 1. Factor 1.65: mostly worthless

**SERIOUS. Kills:** the claim that `1.65` is a meaningful property of Symanzik/Naik improvement, or a number that can be carried into a simulated-universe argument.

The ratio is from one non-gauge-invariant Yukawa model, with one scalar mediator, one anisotropy (`xi=2`), one pair of discretisations, one fixed artificial fermion mass, and no vertex/gauge-sector matching. The absolute coefficient is scheme dependent, and the ratio is not protected from that dependence. Nothing in the calculation proves it is universal. In fact the lab's own `species.py` already shows O(1) movement when the fermion and mediator kernels are changed:

- naive fermion + Wilson gauge: `c00/g^2 = 8.8905e-3` at `m=0.001`.
- naive fermion + Symanzik gauge: `6.6971e-3`.
- Naik fermion + Wilson gauge: `7.5204e-3`.
- Naik fermion + Symanzik gauge: `5.5738e-3`.

That is not a universal improvement factor; it is a table of regulator choices. Collins et al. explicitly say the exact loop coefficient depends on the high-energy/free-propagator details while the power-counting conclusion does not. Source: Collins et al., PRL 93, 191301, arXiv:gr-qc/0403053v4. Polchinski says the same Wilsonian thing: all symmetry-allowed dimension <=4 operators are generated with high-scale coefficients, but the detailed coefficients are model data. Source: arXiv:1106.6346v2.

The strongest version of the result should say only this: tree-level low-momentum improvement does not generically remove an allowed marginal LIV coefficient. It should not quote `1.65` as though that number means anything outside this toy integrand.

**Attack that failed:** I tried to kill the qualitative statement "improvement cannot remove the allowed dimension-4 term." That attack fails. Improvement can change the coefficient, but absent a symmetry or an explicit marginal counterterm it has no reason to set it to zero. The calculation is adequate evidence for "O(1) change, not five orders" in this toy model only.

### 2. `Mf` contamination: the number is not controlled

**FATAL to the quoted error bar and numerical ratio. SERIOUS to the magnitude. Kills:** `1.65 +/- 0.13`, Richardson-extrapolated `9.94e-3` / `6.59e-3` as physical numbers, and the claim that the mass regulator is harmless.

The fermion mass is not a side detail. The run fixes `Mf=0.2` in lattice units. For any actual low-energy fermion on a high cutoff lattice, `m_f b` is tiny. If `1/b` is even `10^11 GeV`, the electron has `m_e b ~ 5e-15`; at Planck it is `4e-23`. `Mf=0.2` is a fermion whose mass is 20% of the cutoff. That is a heavy regulator, not the electron.

I reran the code at `xi=2`, `m=0.01`:

| Mf | N=40 naive | N=56 naive | N=72 naive |
|---:|---:|---:|---:|
| 0.4 | 9.372e-3 | 9.501e-3 | 9.531e-3 |
| 0.2 | 8.411e-3 | 9.199e-3 | 9.512e-3 |
| 0.1 | 6.229e-3 | 7.361e-3 | 8.196e-3 |
| 0.05 | 4.872e-3 | 5.478e-3 | 6.077e-3 |
| 0.02 | 4.356e-3 | 4.574e-3 | 4.757e-3 |
| 0.0 | 4.250e-3 | 4.371e-3 | 4.431e-3 |

The improvement factor at `N=56` moves even harder:

| Mf | unimproved | improved | factor |
|---:|---:|---:|---:|
| 0.4 | 9.501e-3 | 6.306e-3 | 1.507 |
| 0.2 | 9.199e-3 | 5.875e-3 | 1.566 |
| 0.1 | 7.361e-3 | 4.022e-3 | 1.830 |
| 0.05 | 5.478e-3 | 2.146e-3 | 2.552 |
| 0.02 | 4.574e-3 | 1.247e-3 | 3.668 |

That is not "1.65 +/- 0.13". The originally noticed spread from 1.948 to 1.510 was already larger than the claimed error model; pushing smaller `Mf` makes it worse. The small-`Mf` end may indeed be under-resolved, but that is not a defense. The physical masses are at the small-`Mf` end. The calculation is least controlled exactly where it must be used.

Also: sweeping the mediator mass while holding `Mf=0.2` fixed is not a continuum-limit test. If `m = m_phys a`, then every physical mass in the problem has to scale with `a`, including the external fermion mass. The flat mediator-mass sweep shows an IR scalar mass is not setting the answer. It does not prove lattice-spacing independence of a physical theory.

### 3. `Z_0-Z_j` as SME `c_00`: the private worry is real

**FATAL to the single-field claim. Kills:** the literal statement "my one-loop single-fermion self-energy measures the experimentally bounded SME `c_00`." **Does not kill:** the multi-species Collins naturalness argument.

For one field, an anisotropic kinetic term is a convention until compared with something else. Normalize the field, choose coordinates so that this field has limiting speed one, and the coefficient disappears from that sector. SME papers are explicit that field redefinitions and coordinate choices remove unobservable combinations. Russell's "Fermion observables for Lorentz violation" says field redefinitions yield 32 unobservable coefficient combinations per fermion sector and reduce the naive 76 coefficients to 44 observables; the `c` sector observables are tilde combinations. Source: arXiv:1210.2003. Colladay and Kostelecky likewise state that the antisymmetric part of `c_mu nu` and several other terms can be eliminated by field redefinitions, leaving only selected observable coefficients in extended QED. Source: arXiv:hep-ph/9809521.

Collins et al. understood this. Their own rescue from the one-field objection is: different Standard Model fields have different couplings, so they receive different limiting speeds. A single metric/coordinate renormalization cannot remove all of them. Source: arXiv:gr-qc/0403053v4, Appendix A discussion.

The later `species.py` is the right direction, but it does not fully solve the problem. It varies discretisations as a stand-in for species. Real species are not arbitrary choices of Naik vs naive kernels; they are tied by gauge invariance, representations, Yukawa couplings, and renormalization conditions. The actual observable is electron-vs-photon, electron-vs-proton/neutron, or inter-generation differences in one fixed coordinate convention. Until that calculation is done, the single-field number measures a self-energy anisotropy, not an SME observable.

### 4. Continuum limit: both sides are being mixed

**SERIOUS. Kills:** any argument that ordinary lattice-QCD finite Z artifacts are themselves physical Lorentz violation. **Does not kill:** the fixed-physical-lattice simulation argument, if the lattice is actually fundamental.

The counter "a simulated universe at fixed `b` never takes a continuum limit" is basically right, but it is being used too loosely. In lattice QCD, the lattice is a regulator. Hypercubic artifacts in Z factors are subtracted, improved, or extrapolated away. The finite constants are matching data, not claims that Nature has a preferred frame. de Soto/Roiesnel and later H(4) lattice-artifact work organize these terms as functions of hypercubic invariants like `a^2 p^[4]/p^2`; Costa et al. do the same for one-loop renormalization constants. Source trail in `reports/threads/2026-09-13-radiative-prior-art.md`.

For a physical substrate, there is no analyst taking `a -> 0`, so allowed marginal terms stay in the low-energy effective theory unless the microscopic theory forbids them or the bare action is tuned. That is Collins/Polchinski. But then stop borrowing the lattice-QCD regulator language as though its finite Z factors are direct observables. The correct statement is EFT-naturalness: a fundamental preferred-frame cutoff generates all allowed low-dimension operators. It is not "lattice Z factors are physical."

Gambini, Rastgoo, Pullin are exactly the warning sign. Their equal-step Euclidean lattice kills the effect by symmetry; their anisotropic/spatial cases reproduce the Collins problem. Polchinski's comment says the equal-step vanishing is the Euclidean hypercubic symmetry essential to lattice gauge theory and not a Lorentzian physical lattice. Sources: arXiv:1106.1417; arXiv:1106.6346v2.

### 5. Improvement and tuning: the exclusion is really fine-tuning

**SERIOUS. Kills:** "improvement cannot reach this, therefore implementers cannot remove it." **Does not kill:** "without tuning/protection, this is radiatively disastrous."

Tree-level Symanzik/Naik improvement is not the whole improvement program. Anisotropic lattice QCD literally tunes the relevant marginal anisotropy. Foley, Peardon and Ryan write that the renormalized anisotropy/speed of light must be tuned so that `c_R` is unity, and compute the one-loop correction to the quark action parameter. Source: arXiv:hep-lat/0410005. Morningstar/Foley write that anisotropic lattices break hypercubic symmetry and introduce extra parameters that must be tuned so the anisotropy measured by different probes reaches the target value; they discuss one-loop tuning of anisotropic clover plus Symanzik gauge action. Source: arXiv:0810.4477. Morningstar's anisotropic gluon paper says the dimension-four gauge operators are `Tr E^2` and `Tr B^2`, and improvement adjusts coefficients so the dimension-four coefficients equal each other while dimension-six artifacts vanish. Source: arXiv:hep-lat/9608019.

So an implementer can tune this away. The price is not a new action stencil; it is a marginal counterterm / bare anisotropy / speed-of-light tuning in every sector. That converts the claim from exclusion to naturalness.

Quantification: the toy result gives `c ~ 9e-4` for EM coupling. Electron/photon bounds in the minimal SME are order `10^-17` to `10^-20` dimensionless depending on the coefficient and convention; some tilde electron bounds correspond to `~2e-20` after dividing `10^-23 GeV` by `m_e`. So the one-loop cancellation must be at the level of roughly `10^13` to `10^17` in the nonbirefringent matter/photon sectors, and worse if the relevant observable is tighter. Calling it "~10^-20" across the board is too glib; the exact bound depends on the observable combination and coordinate convention. But it is still catastrophic.

If the loop expansion parameter is `e^2/(16 pi^2) = 5.8e-4`, then leaving the first untuned term after order `L` gives about `(5.8e-4)^(L+1)`. To push an electromagnetic-sector residual below `10^-20`, you need tuning through about **six to seven loop orders**. If the effective expansion parameter is closer to `alpha/pi = 2.3e-3`, it is **seven to eight loop orders**. In QCD-like sectors with `~10^-2` loop factors, it is about **ten loops**. This is rough power counting, not a precision estimate; the real statement is uglier: a fundamental theory must supply a symmetry, or it must choose bare marginal coefficients so that the full all-orders, nonperturbative low-energy relative speeds agree to the experimental limit.

### 6. Part A: mostly right, but narrower than advertised

**Attack failed against the central H(4) rank-2 result.** I do not break the exact group-theory claim. For full Euclidean `H(4)` including signed permutations, the only symmetric rank-2 invariant is the trace, so traceless `c_mu nu` has no singlet. The rank-4 result is also the expected one: `(p^2)^2` plus the hypercubic `sum_mu p_mu^4`, so the first symmetric-tensor LIV is dimension 6.

**MINOR. Kills:** polish, not the result. The script says rank-3 traceless symmetric dimension 16, but `singlets_in_sym(G,3)` actually uses the full symmetric rank-3 space of dimension 20. Since the full space has zero singlets, the traceless subspace also has zero singlets. The conclusion survives.

**SERIOUS. Kills:** any implication that Part A protects a Lorentzian or Hamiltonian simulation. Full `H(4)` is Euclidean signed-axis symmetry. A real-time local update with distinguished time has at most `H(3) x T`, and often not even `T`. Your own calculation says `H(3) x T` allows the dimension-4 time-vs-space singlet. If the microscopic update has an arrow of time and lacks time reversal, still more lower-dimension operators can appear. Polchinski's point is exactly that the Euclidean equal-step lattice has an enhanced discrete rotational symmetry not possessed by a Lorentzian lattice. Source: arXiv:1106.6346v2.

**SERIOUS but mostly not fatal. Kills:** the completeness claim "I attacked Part A" if it only checks symmetric tensors. SME has more than `c_mu nu`: fermion `a_mu`, `b_mu`, `d_mu nu`, `H_mu nu`, photon `k_AF`, photon `k_F`, gauge-sector `Tr E^2` vs `Tr B^2`, etc. Under full signed `H(4)` plus parity/time reversal, invariant vectors and pseudovectors are absent, and photon `k_F` appears to reduce to the usual `F_mu nu F_mu nu` invariant. So I do not have a found counterexample that H(4) permits dimension <=4 LIV. But the script has not proven the full SME operator basis. It proved one representation family. For a report making SME-wide claims, that is incomplete.

**Failed attack:** CPT-odd `b_mu` and photon `k_AF` do not obviously break the H(4) protection if the full signed hypercubic group is really exact; no invariant vector survives. Photon-sector anisotropy likewise seems forbidden by full H(4), while anisotropic `H(3)` permits `E^2` vs `B^2`. This strengthens the symmetry-class story rather than killing it.

### 7. Other breaks

**SERIOUS. Kills:** the use of clock/Penning bounds as a direct bound on the computed isotropic `c_00`.

The anisotropic substrate coefficient is rotationally isotropic in its own preferred frame: time vs the sum of three spatial axes. Many of the strongest lab bounds are on sidereal anisotropies (`X-Y`, `XY`, `XZ`, etc.) in the Sun-centered frame. An isotropic `c_TT` in the preferred frame leaks into lab anisotropies only through boosts, with `c_TJ ~ beta c_TT` and spatial anisotropies often `~ beta^2 c_TT` depending on the observable. The CMB/Sun boost is `beta ~ 10^-3`. That can weaken a naive `10^-20` comparison by three to six orders. It does not rescue `c ~ 10^-3`, but it matters. The report should not cite the strongest anisotropic tilde-coefficient bounds as if they directly bound its isotropic time-space coefficient.

**SERIOUS. Kills:** the statement that the mediator-mass sweep distinguishes `b^2 artifact` from `zone effect` in the way stated.

A dimension-6 lattice artifact inserted in a loop can mix into a dimension-4 operator by power divergence when the symmetry allows it: schematically `b^2 * integral^1/b d^4k k^2/k^4 -> O(1)`. That is the whole naturalness/percolation problem. So "a `b^2` artifact would have fallen by `10^8` as `m` changed" is not generally true in an interacting theory. It is only true for a low-momentum observable with no UV mixing. The mass sweep is evidence that the result is not controlled by the scalar IR mass. It is not a proof that improvement cannot affect the UV coefficient.

**MINOR. Kills:** none of the main result, but it weakens confidence in the code-to-SME normalization. The one-loop model computes a self-energy derivative in a scalar Yukawa theory. A gauge theory needs the full gauge-invariant definition of the measured limiting speed, including the correct action, vertices, Ward identities, tadpole/mean-field improvement, and counterterm convention. Foley et al. stress that the one-loop speed-of-light tuning coefficient is built from physical quantities and must be infrared finite and gauge invariant; they introduce a gluon mass only as an intermediate regulator and extrapolate it away. Source: arXiv:hep-lat/0410005. The toy model is fine for a warning shot; it is not a precision coefficient.

**MINOR. Kills:** overconfident novelty phrasing. The prior-art threads already found the core result in Collins, the lattice case in Gambini-Rastgoo-Pullin, and the symmetry interpretation in Polchinski. The only open novelty residue is the direct Symanzik/Naik ratio in this toy integral, and I have just argued that the ratio is not physically portable.

## Answer to the seven requested attacks

1. **Factor 1.65 meaningful?** **SERIOUS.** No, not as a physical or universal number. It is scheme/model/action/mass dependent. It only supports the weak qualitative point that this toy coefficient changes by O(1), not by `5e4`, under one tree-level improvement choice.

2. **`Mf` sweep?** **FATAL to the numerical ratio.** `Mf` is contaminating the result. The physical direction is `Mf -> 0`, and the calculation gets unstable / strongly mass dependent there. Large `Mf` is unphysical for known light fermions; small `Mf` is under-resolved. That means the quoted factor and error bar are not defensible.

3. **`Z_0-Z_j` equals SME `c_00`?** **FATAL to single-field interpretation.** One field's limiting speed can be removed by field/coordinate choice. The observable is relative sector differences. `species.py` is necessary but not sufficient because real species are not arbitrary discretisation choices.

4. **Continuum limit saves it?** **SERIOUS distinction.** For ordinary lattice QCD as a regulator, yes: artifacts are removed by tuning/extrapolation and finite Z constants are matching data. For a physical fixed lattice, no continuum limit is taken, so the EFT naturalness problem is real. Do not confuse those two statements.

5. **Can improvement/tuning kill it?** **SERIOUS.** Tree-level improvement cannot generically kill an allowed marginal term, but an implementer can tune marginal speeds/anisotropies order by order. This is a fine-tuning problem, not a mathematical exclusion. Need about 17 digits of one-loop cancellation for `9e-4 -> 1e-20`, or roughly 7-10 perturbative loop orders depending on the coupling, unless a symmetry handles it all-orders.

6. **Part A?** **Central attack failed for `H(4)` `c_mu nu`.** The rank-2 result survives. **SERIOUS narrowing:** it protects only exact Euclidean hypercubic symmetry, not real-time `H(3)`/Hamiltonian substrates, and it is not a complete SME operator-basis proof.

7. **Anything else?** **SERIOUS:** the strongest lab bounds are not all direct bounds on isotropic `c_00`; boost/convention factors matter. **SERIOUS:** the mediator-mass sweep is not the claimed proof of spacing independence. **MINOR:** the computation is a warning-shot toy, not a gauge-invariant coefficient.

## Sources used

- Collins, Perez, Sudarsky, Urrutia, Vucetich, PRL 93, 191301 (2004), arXiv:gr-qc/0403053v4.
- Collins, Perez, Sudarsky, "Lorentz Invariance Violation and its Role in Quantum Gravity Phenomenology," arXiv:hep-th/0603002.
- Gambini, Rastgoo, Pullin, CQG 28, 155005 (2011), arXiv:1106.1417.
- Polchinski, CQG 29, 088001 (2012), arXiv:1106.6346v2.
- Belenchia, Gambassi, Liberati, JHEP 1606:049 (2016), arXiv:1601.06700.
- Russell, "Fermion observables for Lorentz violation," arXiv:1210.2003.
- Colladay and Kostelecky, "Lorentz-Violating Extension of the Standard Model," arXiv:hep-ph/9809521.
- Foley, Peardon, Ryan, "A perturbative determination of the parameters of an anisotropic quark action," arXiv:hep-lat/0410005.
- Morningstar/Foley, "Tuning improved anisotropic actions in lattice perturbation theory," arXiv:0810.4477.
- Morningstar, "Improved gluonic actions on anisotropic lattices," arXiv:hep-lat/9608019.