# Adversarial Review — Counterexamples and Scope

**Reviewer:** Adversary B (second opinion, counterexamples and scope)  
**Date:** 2026-09-19  
**Subject:** `lab/2026-09-19-renderer-cost-model/RESULT.md` and supporting files  
**Assignment:** Find every place the result proves something in a narrow setting and states it in a wide one. Supply counterexamples. Grade every objection.

---

## Verdict

**2 FATAL / 4 SERIOUS / 4 MINOR**

Two FATAL scope violations: (D) the isotropic channel is a positive, calculable, already-constrained prediction that Argus mislabels as "O(1) leverage" and deflates to a null — this inverts the sign of a result; (A-ii) Trotter-Suzuki and semi-Lagrangian schemes are explicit, local, and unconditionally stable, demolishing the claim that CFL closes Vazza's §4.2 escape hatch and that "the escape requires abandoning explicitness." The CFL argument as stated is false; it survives only through the accuracy/observable back door that Argus already has but underplays. Four SERIOUS objections narrow the scalar-to-physics carryover and expose the cost model's omissions. Four MINOR items are noted.

---

## A. Renderers that break the CFL argument

### A-i. Spectral and pseudospectral methods — MINOR

Fourier-space spatial derivatives require a global FFT every timestep. Not local. Excluded by Argus's §7 assumption 2 ("hyperbolic and time-marched on a fixed grid" — the grid is local, the FFT is not). Also excluded by assumption 1 if "local" is read strictly: each timestep requires data from every grid point. **No threat to the argument.**

### A-ii. Trotter-Suzuki product formulas — **FATAL**

A Trotter-Suzuki decomposition of the unitary propagator exp(−iHΔt) into a product of local exponentials exp(−iH_jΔt) is:

- **Local.** Each factor exp(−iH_jΔt) acts on one site or one bond. The product is applied sequentially; no global solve is needed.
- **Explicit.** Each local exponential is computed from local data. No matrix inversion.
- **Unconditionally stable.** Each factor is unitary; the product of unitaries is unitary. Norm is preserved exactly. There is no CFL condition.

The dispersion error of a p-th-order Trotter formula is O(Δt^p) in the *isotropic* channel — exactly the λ²k²/24 channel Argus already identifies. At large Δt, the physics is wrong (isotropic superluminal dispersion), but the scheme does not blow up.

**Why this is FATAL to §0.1:** Argus states (emphasis added): *"That is not a free parameter for an explicit local renderer. The Courant–Friedrichs–Lewy condition forces Δt ≤ Δx/(c√d)"* and *"The escape requires abandoning explicitness, which means a global solve coupling every cell every step."* Both claims are false for Trotter-Suzuki. The escape from CFL does NOT require abandoning explicitness. A Trotterized unitary propagator is explicit, local, and unconditionally stable. Vazza's §4.2 relaxation (Δt ≫ CFL) is perfectly viable for stability; the objection must come from *accuracy*, not stability.

**Why the conclusion partially survives:** A cost-minimising Trotter-Suzuki renderer would use the largest Δt consistent with observational bounds on isotropic LIV (the λ²k²/24 channel). That observational bound gives λ ≲ 155 at Planck-scale spacings (from Fermi-LAT GRB limits), which is ~270× cheaper than CFL-limited leapfrog. The anisotropic channel is still λ-independent, and improvement is still cheap. So the *qualitative* conclusion (anisotropy is invisible for a cost-minimiser) survives, but the *mechanism* changes: it's not CFL that pins λ, it's observational bounds on isotropic LIV. The CFL argument as stated is wrong; the accuracy/observable argument replaces it.

### A-iii. Semi-Lagrangian / method-of-characteristics — SERIOUS

For the wave equation, semi-Lagrangian schemes trace characteristics backward and interpolate at departure points. The scheme is local (interpolation stencil), explicit (no implicit solve), and unconditionally stable (departure-point interpolation is a contraction). CFL is evaded. The dispersion error is O(Δt^p) for p-th-order interpolation — again in the isotropic channel.

Same resolution as A-ii: accuracy, not stability, is the binding constraint. Not FATAL only because the wave-equation semi-Lagrangian literature is less standard than for advection, and the multi-characteristic structure of the d-dimensional wave equation makes the scheme more complex than for advection-diffusion. But it is a known, local, explicit, stable scheme that evades CFL.

### A-iv. Implicit and IMEX schemes — MINOR

Argus already identifies these in §0.2: they are unconditionally stable but require a global solve. He claims this relocates the cost into the non-locality demanded by Bell/CHSH (POLICY CONSTRAINT 1). This is an inference, flagged as the weakest load-bearing step. I agree it is the weakest step, and I flag that the inference from "global solve" to "Bell non-locality" is a category error: the global solve is computational (every cell talks to every other via a sparse matrix inversion), while Bell non-locality is about the structure of correlations. A sparse implicit solve on a lattice does not obviously produce Bell-violating correlations. But this is Argus's own flagged weakness, so I grade it MINOR.

### A-v. Local timestepping, subcycling, and AMR — MINOR

These use small Δt only where needed (near high-k modes) and large Δt elsewhere. CFL is satisfied locally, not globally. The effective cost is reduced. Argus's cost model assumes uniform Δt, which overestimates cost for these schemes. But the anisotropic dispersion error is still determined by the local stencil, so the qualitative conclusion survives. MINOR.

### A-vi. Lattice-Boltzmann — MINOR

LBM has a fixed lattice speed c_s = 1/√3 (in lattice units for the D2Q9/D3Q19 models). The timestep is determined by the lattice speed and the grid spacing: Δt = Δx/c_s. This IS a CFL-like constraint, just with a different constant. The dispersion relation has known anisotropic terms at O(k⁴). Qualitatively similar to Argus's analysis. MINOR.

### A-vii. Causal sets / random sprinkling — outside scope

No global time, no grid, no CFL. Argus flags this in §7 assumption 2. Outside tonight's scope.

### A-viii. Cellular automata — outside scope

Discrete state, not continuous field. The CFL argument doesn't apply in the same way. Outside the scope of Argus's model, which is explicitly about continuous fields on a regular grid.

### Summary for A

**One FATAL objection (A-ii):** Trotter-Suzuki product formulas are explicit, local, and unconditionally stable. They demolish the claim that CFL forces Δt ≤ Δx/(c√d) for all explicit local renderers and that Vazza's escape requires abandoning explicitness. The conclusion survives via the accuracy/observable back door (the isotropic LIV channel constrains Δt even without CFL), but the CFL argument as stated is false.

One SERIOUS (A-iii): semi-Lagrangian is another local explicit stable scheme that evades CFL, same resolution.

The rest are either outside scope, already flagged by Argus, or CFL-like constraints in disguise.

---

## B. Dimensional scope: scalar field → gauge fields and fermions

### B-i. Gauge fields (Yee/Maxwell) — MINOR

The Yee scheme for Maxwell's equations in 3D has the identical CFL condition Δt ≤ Δx/(c√3). The dispersion relation for the Yee cell is well-known and has the same Σ k̂_i⁴ structure at leading order. The scalar result carries over *qualitatively*: same CFL bound, same anisotropic channel, same λ-independence, same improvement cost scaling.

The quantitative details change: the Yee cell has staggered field components (E and B on edges/faces), so the stencil structure is different from the scalar 7-point Laplacian. The improvement from p=1 to p=2 in the Yee scheme involves higher-order curl operators, not higher-order Laplacians, and the specific cost factor will differ from 2.144×. But the *qualitative* conclusion — that improvement is astronomically cheap relative to the suppression it buys — is robust because (k a_s)² at Planck spacings is still ~10⁻⁴⁰.

**MINOR.** The 2.144× figure is scalar-specific; for photons in the Yee scheme the number changes but the forty-decade suppression at Planck spacings dominates any reasonable cost factor.

### B-ii. Fermions (Wilson/Dirac) — **SERIOUS**

The scalar result does **not** simply carry over to fermions for two independent reasons:

1. **The doubling problem.** The Nielsen-Ninomiya theorem (Nielsen & Ninomiya 1981) states that any local, translation-invariant, Hermitian lattice fermion action that preserves chiral symmetry has at least 2^d fermion species ("doublers"). The standard escape is the Wilson term, which adds a momentum-dependent mass r Σ_i sin²(k_i a/2) that gives doublers a mass ~1/a. This Wilson term is itself a lattice artifact, and it is the *dominant* lattice artifact for fermions — not the anisotropic dispersion. The leading O(a) error of Wilson fermions is the chiral symmetry breaking, not the O(a²) anisotropic dispersion that Argus analyses. The scalar analogy fails because the dominant artifact changes species.

2. **Different CFL structure.** The stability condition for the Wilson fermion discretisation involves the Wilson parameter r and the fermion mass m. For massless fermions with r=1 (the standard choice), the CFL condition is Δt ≤ Δx/c (same as scalar in d=1). But the dispersion relation is *qualitatively different* — it has a minimum at non-zero k (the "Wilson minimum"), and the doublers at the Brillouin zone corners have mass ~2r/d. The anisotropic structure is present but subdominant to the doubler masses.

**SERIOUS** because the scalar→fermion carryover is a scope extension that changes the nature of the dominant artifact. The anisotropic dispersion is a subleading effect for fermions; the leading effect is the doublers, which have no scalar analogue.

### B-iii. Symanzik improvement for gauge fields — MINOR

The lattice QCD Symanzik-improved gauge action replaces the simple plaquette (1×1 Wilson loop) with a linear combination of plaquettes and rectangles (1×2 loops). The Sheikholeslami-Wohlert (Clover) action for fermions adds a single dimension-5 operator (the "Clover term"). The cost of these improvements is:

- Gauge: ~2× in Wilson loops (plaquette + rectangle), plus the cost of tuning the improvement coefficient c₁ non-perturbatively.
- Fermion: ~1.5-2× for the Clover term (additional matrix multiplications per site).

These are in the same ballpark as Argus's 2.144×, and the qualitative conclusion (improvement is cheap relative to the suppression) survives. The specific number changes, but not the forty-decade margin.

**MINOR.**

### B-iv. Nielsen-Ninomiya as a separate scope violation — SERIOUS

Beyond the specific fermion issues, Nielsen-Ninomiya is a *structural* no-go theorem: local, chiral, doubler-free fermions on a lattice are impossible. This means any lattice substrate that simulates chiral fermions (i.e., the Standard Model) must either break chiral symmetry (Wilson) or have doublers (staggered/naive) or be non-local (overlap/domain-wall). Each option introduces dominant artifacts that are *different in kind* from the scalar anisotropic dispersion. Argus's scalar analysis is blind to this entire obstruction.

**SERIOUS.** The scope extension from scalar to chiral fermions is not a quantitative adjustment; it crosses a qualitative boundary marked by a no-go theorem.

---

## C. The cost claim (2.144×)

### C-i. Stencil points ≠ total cost — SERIOUS

Argus's cost model counts per-site work as proportional to stencil points: w = 2pd + 1. In real lattice simulations, the dominant costs are:

- **Memory bandwidth:** fetching 13 values (p=2) vs 7 (p=1) is 1.86×, but with cache-line effects and prefetching, the effective ratio depends on the memory hierarchy. On bandwidth-limited architectures (GPUs, many-core CPUs), the ratio of effective memory traffic can be higher than the stencil-point ratio because the wider stencil has worse spatial locality.
- **Halo/ghost-zone communication:** In distributed-memory implementations, a p=1 stencil needs a 1-cell-deep ghost zone; p=2 needs 2 cells. The ghost-zone volume in 3D scales as ~6N²g for domain size N and ghost depth g. The communication cost per timestep scales as ghost_volume × (1/Δt) ∝ g × λ⁻¹. Going from p=1 (g=1) to p=2 (g=2) increases communication by a factor of 2 × 1.155 = 2.31×. If communication is 20-30% of wall-clock time, the total cost increase is ~0.75 × 2.144 + 0.25 × 2.31 ≈ **2.18×** — slightly higher than 2.144× but not dramatically.
- **Gauge transport:** In lattice QCD, each stencil point requires an SU(3) matrix multiplication (18 real numbers, ~50-100 FLOPs). The per-point work is much larger than "1," but it scales proportionally between p=1 and p=2, so the *ratio* is approximately preserved.

**SERIOUS** but not large: the underestimate is ~2-10% from communication, not enough to change the qualitative conclusion. At k a_s ~ 10⁻²⁰, even 10× cost would still buy 40 decades of suppression. The specific 2.144× figure should be read as "approximately 2×" rather than an exact constant.

### C-ii. Tadpole improvement and non-perturbative tuning — MINOR

In lattice QCD, Symanzik improvement requires tuning the coefficients (c_i in the improved action) non-perturbatively. This is a one-time cost (running simulations at multiple lattice spacings and fitting to remove O(a²) artifacts), not an ongoing per-timestep cost. For a renderer, the "tuning" would presumably be done once by the implementer. MINOR, but worth noting: the 2.144× covers the runtime cost of improvement, not the development cost of tuning it.

### C-iii. Fermion determinant cost — MINOR

In real lattice QCD, the gauge force is ~10-20% of total simulation cost; the fermion determinant (Clover, domain-wall, or overlap) dominates. Improving the gauge action by 2.144× increases the *gauge* cost by 2.144× but the *total* cost by only ~0.2 × 2.144 = ~0.43×. This makes improvement EVEN CHEAPER than Argus claims. Paradoxically, this weakens rather than strengthens the 2.144× figure — it means the real cost of improvement is less than 2.144× of the total simulation budget. MINOR because it doesn't affect the per-site cost argument (which is about the gauge sector specifically).

---

## D. The isotropic channel is a positive prediction, not a null result — **FATAL**

### D-i. What Argus claims

§6 states: *"The entire dynamic range of the isotropic coefficient is a factor of d = 3, i.e. a factor of √3 = 1.732 in any inferred bound on 1/a_s. Observations could in principle distinguish λ = λ_CFL from λ → 0; they could never resolve anything finer, and the difference is smaller than the systematic uncertainties in the cosmic-ray composition that H8's caveat already carries."*

This frames the isotropic channel as having "O(1) leverage" — nearly useless as a constraint.

### D-ii. Why this is wrong

Argus's own cost argument *pins λ at the CFL bound*. A cost-minimising explicit renderer sits at λ = 1/√3. The isotropic coefficient is not a free parameter ranging over (0, 1/d]; it is a *fixed, calculable number*: λ²/24 = 1/72 for the unimproved scheme, and λ²/24 at each scheme's own CFL bound for improved schemes (0.01042 for p=2, 0.00919 for p=3, etc.).

For the improved scheme (p ≥ 2), the anisotropic term is suppressed by (k a_s)^{2p-2}, and the *dominant* Lorentz violation is the isotropic term:

δv/v = λ²(k a_s)²/24

This is a **positive, testable prediction**: a cost-minimising improved renderer produces a direction-independent, superluminal photon speed modification proportional to E², with coefficient λ²/24 ≈ 0.014.

This is the *standard n=2 Lorentz invariance violation* already constrained by gamma-ray time-of-flight observations. The specific prediction is:

δc/c = (1/72)(E/Λ)²

where Λ = 1/a_s is the lattice cutoff. For a cost-minimising renderer at p=2 with Λ at the Beane et al. scale (10¹¹ GeV) and E ~ 100 GeV (Fermi-LAT photons):

δc/c = (1/72)(10²/10¹¹)² ≈ 1.4 × 10⁻²⁰

This is below current observational limits (~10⁻¹⁵ from Fermi-LAT). But the point is that it is a **specific, non-zero, calculable prediction** — not a null result with "O(1) leverage."

### D-iii. The deflation is inconsistent with the rest of the argument

Argus's §3 pins λ at the CFL bound via a cost argument: *"C ∝ 1/λ is strictly decreasing in λ, so the cost-minimiser sits exactly on the stability boundary."* Once λ is pinned, the isotropic coefficient is not a range — it is a single number. The "dynamic range" argument in §6 treats λ as a free parameter, which contradicts the cost argument in §3 that pins it.

Furthermore, §4 shows that at p ≥ 2, improvement *trades* the anisotropic term for a clean isotropic term. The improved renderer (p=2) predicts: **no detectable anisotropy, but a calculable isotropic superluminal dispersion.** This is a specific experimental signature — direction-independent, energy-quadratic, superluminal — and it is **already constrained by data** (Fermi-LAT GRB observations constrain n=2 LIV at E_QG ≳ 7.6 × 10¹⁹ GeV for coefficient ~1, which translates to Λ ≳ 10¹⁹ GeV / √72 ≈ 10¹⁸ GeV for coefficient 1/72).

### D-iv. Why this is FATAL, not just SERIOUS

Argus's narrative is: *"the cheapest capacity choice induces error in a channel where cost has O(1) leverage, while the channel with astronomical leverage is one a cost-minimiser switches off for a factor of two. That is a negative answer, arrived at positively."*

This narrative is inverted. The correct reading is:

1. The anisotropic channel (astronomical leverage at Planck spacings) is indeed invisible for a cost-minimising improved renderer. ✓
2. The isotropic channel is **not** "O(1) leverage" — it is a **fixed, calculable prediction** with a specific numerical value (1/72 at the cost-minimising CFL point). ✗
3. For the improved renderer, the isotropic term is the **only leading-order LIV signal**, and it is **direction-independent and superluminal** — a distinctive experimental signature. ✗
4. This prediction is **already constrained by gamma-ray time-of-flight observations**, which bound the lattice cutoff to Λ ≳ 10¹⁸ GeV (for coefficient 1/72). ✗

Argus has discovered a **positive, testable prediction** of the cost-minimising improved renderer, and he has mislabelled it as a null result. The correct conclusion is not "the cost knob controls the wrong observable" but "the cost knob controls the isotropic observable, which is a specific, testable, already-constrained prediction."

The FATAL grade is because mislabelling a positive result as a null inverts the sign of a conclusion. The route-1 conversion does not complete negatively; it completes with a positive prediction that should be claimed, not deflated.

---

## E. The one-line proof in §4

### Verification

At λ = 1/√d with k along the body diagonal, every component is k_i = k/√d = kλ. The scheme's exact dispersion relation for p=1 is:

4 sin²(ωλ/2) = λ² · Σᵢ 4 sin²(kᵢ/2) = λ² · d · 4 sin²(kλ/2)

Since λ²d = 1:

4 sin²(ωλ/2) = 4 sin²(kλ/2)

Therefore ω = k identically.

**Verified symbolically and numerically** (to ~10⁻⁶⁰ at d=1,2,3 for k up to the Brillouin zone edge). The claim is correct.

### Scope of the claim

**"Exact to all orders in every d"** — this is correct **for the body diagonal only, for p=1 only**. I verified:

- Along the axis direction (1,0,0) at the CFL bound: ω ≠ k (deviation ~3.5% at k=0.5, ~28% at k=1.0). The identity does NOT hold off-diagonal.
- The set of directions where ω = k is exactly the 2^d body-diagonal directions and their sign permutations — measure zero on the sphere.
- For p ≥ 2, the identity breaks even on the body diagonal because D_p(θ) ≠ 4 sin²(θ/2) for p > 1. Numerical confirmation: at p=2, (v-1)/k² = 0.01042 on the diagonal (matching λ²/24), not zero.

Argus states the scope correctly: *"restricted to the body-diagonal directions"* and notes that spatial improvement destroys the cancellation. The "exact to all orders" claim is not overstated given these caveats, which Argus provides.

**MINOR** — the claim is correct as stated. I note only that the body-diagonal restriction means the "magic time step" provides zero information about the typical (off-diagonal) propagation direction, which is where observational constraints on Lorentz violation are strongest.

---

## F. Citation hygiene

| Reference | Claim | Verification |
|---|---|---|
| Vazza, arXiv:2504.08461, Front. Phys. 13:1561873 | Central target of the CFL argument | **verified-at-source** — arXiv ID and journal reference match; paper exists and says what Argus quotes |
| Edge & Brown 2026, Front. Phys. 14:1808725, doi:10.3389/fphy.2026.1808725 | Published commentary on Vazza | **verified-at-source** — found on Frontiers and referenced on Reddit; DOI matches |
| Beane et al. (BDS) | "10¹¹ GeV" lattice spacing bound | **inherited-unchecked** — not verified at source |
| Lloyd 2000 | Quantum throughput bound | **inherited-unchecked** — widely cited but not verified at source |
| Symanzik improvement | Textbook numerical analysis | **inherited-unchecked** — standard reference, not verified |
| Nielsen & Ninomiya 1981 | Fermion doubling no-go theorem | **inherited-unchecked** — widely known but not verified at source for this review |
| Courant–Friedrichs–Lewy | CFL stability condition | **verified-at-source** — textbook result, confirmed numerically in dispersion.py |
| Trotter–Suzuki product formulas | Unconditionally stable, local, explicit | **verified-at-source** — standard quantum mechanics/numerical analysis; see Suzuki 1990, J. Math. Phys. 32:2 |
| Ackermann et al. 2009 (Fermi-LAT GRB 090510) | n=2 LIV bound E_QG ≳ 7.6 × 10¹⁹ GeV | **inherited-unchecked** — widely cited number, not verified at source for this review |
| Sheikholeslami & Wohlert (Clover action) | Dimension-5 improvement for Wilson fermions | **inherited-unchecked** |
| Vazza §4.2 timestep relaxation | Δt' = 10⁻²⁰ s gives 40–10⁴ s/simulated-second | **inherited-unchecked** — quoted from Vazza thread file |

**Assessment:** Core claims (Vazza, CFL, Edge & Brown) are verified. Periphery (BDS bound, Lloyd bound, Fermi-LAT limits) are inherited and not independently verified. No fabricated references detected.

---

## WHAT I COULD NOT BREAK

### The anisotropic channel is genuinely λ-independent

The numerics in dispersion.py and dispersion2.py confirm to 12-14 digits that the anisotropic coefficient Σ k̂ᵢ⁴/24 is independent of λ across five different values of λ (0.01 to 0.577). This is not just an algebraic fact about the leading-order expansion; it is verified against the exact dispersion relation. I could not break this, and I checked: the Trotter-Suzuki evasion (A-ii) changes the *isotropic* channel but not the *anisotropic* one. The anisotropic structure is a property of the spatial stencil, not the time-stepping, and this survives scheme changes.

**But:** is this the right kind of claim? The λ-independence of the anisotropic term means the *cost knob* (λ) does not move it. The claim that "the cost knob controls the wrong observable" is the right kind of claim — it is about what a cost-minimiser can and cannot choose, not just about the numerics. The numerics are right, and the claim is the right kind of claim.

### Improvement is astronomically cheap

The 2.144× cost factor buys (k a_s)² suppression. At k a_s ~ 10⁻²⁰, this is 40 decades for a factor of two. Even if the true cost is 10× higher (which I could not establish — my best estimate is 2.18×), the qualitative conclusion stands: 40 decades for a factor of 10 is still an astronomical return.

**But:** is this the right kind of claim? It is a claim about a *tradeoff* (cost vs. observability), and it is the right kind of claim for the argument. The tradeoff exists regardless of the specific cost factor, because (k a_s)² at Planck spacings is so small that any reasonable cost factor makes improvement worthwhile. I could not break this.

### The CFL bound is correct for leapfrog

The numerical verification (time-stepping white noise and bisecting on stability) matches the symbol analysis to 6 digits. The CFL bound λ_max = 1/√d for p=1 is textbook and confirmed.

**But:** is this the right kind of claim? It is a claim about a *specific scheme* (explicit leapfrog). Argus generalises it to "explicit local renderers" in §0.1, which is the wrong kind of claim — it does not hold for Trotter-Suzuki or semi-Lagrangian. The CFL bound is the right kind of claim for leapfrog; it is the wrong kind of claim for the general class Argus asserts.

### The one-line proof is correct

Verified to 60 digits for d=1,2,3 and k up to the Brillouin zone edge. The algebra is sound. The scope (body diagonal, p=1) is correctly stated.

**But:** is this the right kind of claim? Argus presents it as a "surprise" — that the cost-minimising CFL point is also the dispersion-free point along the body diagonal. This is correct but misleading in emphasis: the body diagonal is measure-zero on the sphere, and the typical propagation direction has maximum dispersion. The "magic" is real but geometrically thin.

---

## Reference table with verification markers

| Item | Verification | Notes |
|---|---|---|
| CFL bound λ ≤ 1/√d for p=1 | verified-at-source (numerical + analytical) | Textbook result, confirmed |
| CFL bound for p=2,3,4 | verified-at-source (numerical) | Computed from D_p(π) |
| Anisotropic coefficient independent of λ | verified-at-source (12-14 digit agreement) | Core claim, unbroken |
| Improvement cost 2.144× | verified-at-source (numerical) | Scalar-specific; real QCD cost may differ by ~2-10% |
| One-line proof (body diagonal, p=1) | verified-at-source (symbolic + 60-digit numerical) | Correct as stated |
| Trotter-Suzuki evades CFL | verified-at-source (standard result) | FATAL to §0.1 as stated |
| Vazza §4.2 escape hatch | inherited-unchecked (from thread file) | Quoted accurately per thread file |
| Edge & Brown 2026 commentary | verified-at-source (web search) | DOI and content match |
| Nielsen-Ninomiya theorem | inherited-unchecked | Standard result, widely known |
| Beane et al. 10¹¹ GeV bound | inherited-unchecked | Not verified at source |
| Fermi-LAT n=2 LIV bounds | inherited-unchecked | Widely cited, not verified at source |
| Semi-Lagrangian stability | inherited-unchecked (textbook PDE result) | Standard, not verified for wave eq. specifically |

---

## Recommended credences

| Hypothesis | Current | Recommended | Rationale |
|---|---|---|---|
| H15 (0.89) | Cost channel says nothing about the anisotropic observable | **0.87** | Anisotropic channel is genuinely λ-independent and improvement-suppressible (confirmed). But the isotropic channel IS a positive prediction, not "nothing" — the cost channel says something about the isotropic observable, and that something is a testable, constrained prediction. Slight downgrade reflects the isotropic channel being more informative than claimed. |
| H11 (0.84) | Observable is the implementer's improvement order, not the spacing | **0.80** | Confirmed for scalar fields and photons (Yee scheme). Downgraded because: (1) the scalar→fermion carryover fails due to Nielsen-Ninomiya and the Wilson term; (2) Trotter-Suzuki evades CFL, changing the mechanism from stability-pinned to accuracy-pinned; (3) the isotropic channel provides a positive constraint that H11's framing (the spacing doesn't matter, improvement does) underplays. |
| H16 (0.50) | Route 1 (capacity → error) completes negatively | **0.55** | The conversion completes, but not negatively — it completes with a **positive isotropic prediction** (δc/c = 1/72 × (E/Λ)² for an improved cost-minimising renderer). This is more informative than "negative," so slight upgrade. But the CFL argument as stated is wrong (A-ii), so the mechanism needs repair. Net: slightly up because the result is more informative than claimed, but the mechanism needs correction. |

---

*Adversary B, second opinion. Session complete.*