A draft pass at scoping this problem attributed the $1/(d-6)$ divergence responsible for percolation's obstruction to $\|G\|_2^2=\int(1-\hat D(k))^{-2}dk$ — a two-propagator (bubble) integral. Direct power counting shows that integral's own threshold is d=4, not 6 (see §1) — it is the self-avoiding-walk/Ising value, not percolation's. The diagram that actually produces dc=6 is the triangle sum, three propagators, and that correction is carried through §§2–3 below.
The free/mean-field two-point function decays as G(x) ~ |x|^{-(d-2)} (the discrete Newtonian potential), with Fourier transform Ĝ(k) ~ |k|^{-2} near k=0. Two candidate diagrams, two different thresholds:
| diagram | propagators | near-0 integrand | power counting | threshold |
|---|---|---|---|---|
| Bubble, ‖G‖₂² | 2 | |k|⁻⁴ | ∫r^{d-5}dr | d = 4 |
| Triangle, ΣG(x)G(x−y)G(y) | 3 | |k|⁻⁶ | ∫r^{d-7}dr | d = 6 |
The bubble's threshold, d=4, is the self-avoiding-walk and Ising upper critical dimension — a real number, just not percolation's. Percolation's d_c=6 comes from the Aizenman–Newman triangle condition (∇(p_c)=Σ_{x,y}τ(x)τ(x−y)τ(y)<∞), the three-propagator sum, not the two-propagator bubble. Any scoping note that blames a two-propagator quantity for the d=6 threshold has the wrong diagram, regardless of how correct its surrounding architecture is.
NoBLE's bootstrap (WP-92's companion note; Fitzner–van der Hofstad, 2017) closes a loop of three functions f₁,f₂,f₃ bounding ratios of percolation diagrams to their SRW counterparts, via numerical (interval-arithmetic) bounds on convolution integrals built from these propagator sums. As d→6⁺, the triangle-type sums that dominate the higher-order lace-expansion coefficients grow, and the numerical bounds that close the loop at d≥11 stop closing — not because the arithmetic gets less precise, but because the quantity being bounded is genuinely larger and closer to the threshold where it diverges.
It is not claimed that the triangle sum itself is the only quantity NoBLE tracks, nor that fixing this diagram identification amounts to a proof strategy. The correction is narrower: it identifies which diagram's divergence is doing the structural work, so that any future scoping of this gap reasons about the right object.
Fitzner and van der Hofstad's own account of the closely related NoBLE extension to lattice trees and lattice animals — same method, same authors, a different but adjacent model — states plainly that pushing their proven range further "would require substantially new ingredients and insights," not sharper interval arithmetic on the existing scheme. This note does not have the equivalent sentence in hand for percolation specifically (the percolation paper's own discussion section was not accessible when checked), so the claim below is an inference from the adjacent case, not a quotation about this one — flagged as such rather than presented as settled.
Candidate directions a genuine attack would need, in roughly increasing order of how much they change the existing method rather than tune it:
None of the three directions above is developed here. This is a scoping note: it says what the obstruction is and what shape a resolution would need to have, not a resolution.
Direction 1 — sign-alternating cancellation. Tested directly in Book 4 · The Lace Bootstrap in Intermediate Dimensions. The true zero-lace coefficient was computed at the Brillouin-zone corner for $\mathbb{Z}^7$: $\hat\Pi^{(0)}(k^*) = -84\,p^4 + O(p^6)$ by exact enumeration, and $-0.004782 \pm 0.000430$ by Monte Carlo — negative at $11.1\sigma$. A sign-alternating scheme of the form $\hat\Pi^{(2m)} - \hat\Pi^{(2m+1)} \ge 0$ therefore begins from a negative first term at the corner. The direction is not closed off, but it does not start where it was assumed to start.
Direction 3 — analytic isolation of the pole. Partly mis-stated here. The $1/(d-6)$ is the $d$-dependence of a convergent integral, not a pole in $k$ at fixed $d$. Evaluated exactly, $\Omega(7) = 0.7164$, with $\Omega\cdot(d-6)$ between $0.60$ and $0.72$ across $7 \le d \le 13$ — an $O(1)$ quantity at every integer dimension in range, and no singularity to extract. Isolating the small-$k$ region remains a reasonable numerical tactic; there is simply no pole there. The binding constant is the combinatorial $C$, not $\Omega$: descending from $d=11$ to $d=7$ costs a factor of $5.8$, not a divergence.