08-ilang-propagation / 08-tilt
08tiltverified
Determines whether a push on one cell orients rotation-symmetric exchange cells far away, how small deviations match the anchored oscillator field, and which sources have a long-range mutual cost.
# Review: 08-tilt v5
- **Subproject:** 08-ilang-propagation
- **Package:** 08-tilt
- **Reviewed version:** v5
- **Date:** 2026-10-26
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VERDICT: minor issues
## Issues
### I1. The far-place check does not test the collective term with the h_eff it states, and the fixed-L statement has no remainder
- **Location:** Step 4, the paragraph on the far-site check (`spinwave_tilt.py`, decomposition of $\Delta S-\Delta S_3$); Result (8.7), sentence "At fixed $\eta$ and $L$ the collective rotation adds $\frac12\sqrt{2zJS/(Nh_{\rm eff})}$ to $\Delta S$ at the far place"; the same clause in the twin entry of 8.7.
- **Severity:** minor
- **Problem:** Step 4 defines $h_{\rm eff}=(1/h_p+\hat g_L(0)/(JS))^{-1}$ at finite $L$. The code computes the relaxed term with the limit value $\gamma_3$ in place of $\hat g_L(0)$ (`rel = 0.5*sqrt(12(1+eta*gam3)/(L^3 eta))`; output header "eta_eff = (1/eta + gamma_3)^-1"), and the text presents the result as the term with $h_{\rm eff}$. On these tori $\hat g_L(0)=0.22461,0.23394,0.23863$ for $L=8,12,16$, against $\gamma_3=0.25273$. With the $h_{\rm eff}$ of (8.6) the rest becomes $+0.0004,+0.0011,+0.0019$ at $L=8$ for $\eta=0.1,1,10$, and $+0.0001$ to $+0.0007$ at $L=12$: it grows with $\eta$. So the printed rest ($\le1.1\times10^{-4}$) does not test the stated formula at finite $L$. Its smallness comes from a near cancellation between $\hat g_L(0)-\gamma_3$ and the non-uniform part of the soft mode at the far place (Step 4 gives that part a relative norm $O(\eta_{\rm eff}/L)$; at the antipode $\hat g_L(\text{far})=-0.0084,-0.0054,-0.0040$), and the text mentions neither correction. For the same reason the far-place addition stated in (8.7) at fixed $\eta$ and $L$ holds only up to a relative correction of order $\eta_{\rm eff}/L$ (about $5\%$ at $L=8$, $\eta=10$), which the statement does not carry.
- **Fix:** Report the check as the code computes it, or compute it with the $h_{\rm eff}$ of (8.6). State the order of the finite-$L$ correction of the collective term at the far place in Step 4, in (8.7) and in its twin entry, and claim from the nine runs no more than they test.
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