The Lace Bootstrap in Intermediate Dimensions

$\Omega(7) = 0.716 < 1$. The product $\Omega\cdot(d-6)$ sits between $0.60$ and $0.72$ across the whole range, so the $1/(d-6)$ shape is real but its coefficient is small and the value at every integer $d \ge 7$ is $O(1)$. Descending from $d=11$ to $d=7$ costs a factor of $5.8$ — a finite, modest penalty, not a singularity.

The binding quantity is therefore $C$, the constant packaging the combinatorial bounds on the lace diagrams, not $\Omega$. Any strategy whose stated purpose is to defeat a $1/(d-6)$ divergence is aimed at a target that is not in the room. A corollary worth noting: a mechanism supplying a factor $(d-6)/(2d)$ would also improve $d \ge 11$ by a factor of $4.4$, which for an already-settled case should be treated as a warning rather than a bonus.

$\widehat{G^2}$ and $\widehat{G^3}$ are positive at every sampled $k$, and monotonically decreasing from $k=0$ to the corner. There is no sign failure. The objection as stated — that individual transforms go negative — does not hold for these diagrams, and is withdrawn.

Prove $\hat\Pi^{(2m)}(k) - \hat\Pi^{(2m+1)}(k) \ge 0$ pointwise for all $k \in [-\pi,\pi]^d$.

Prove $\displaystyle\sup_k \frac{\hat\Pi^{(2m+1)}(k)}{\hat\Pi^{(2m)}(k)} \le 1$ — which, given monotonicity in $|k|$, reduces to the single point $k = (\pi,\ldots,\pi)$.

The proposal is that a higher-order diagram “introduces an additional internal propagator $G(u,v)$ whose distance is strictly odd, systematically flipping the phase sign.” At $k^*$ the phase attaches only to the endpoint: $\hat\Pi^{(N)}(k^*) = \sum_x (-1)^{\|x\|_1}\Pi^{(N)}(x)$. Internal vertices are summed out and carry no phase whatever. Nor does any single parity attach to “the additional propagator”, since $\Pi^{(N)}(x)$ aggregates configurations of every internal geometry. The parity observation is true and does no work here.

The Fröhlich–Simon–Spencer infrared bound derives from reflection positivity, and it is precisely the tool that is not available for percolation in the form required. That absence is the historical reason the lace expansion exists at all. If reflection positivity delivered the infrared bound here, $7 \le d \le 10$ would not be open, and neither Hara–Slade nor NoBLE would have been necessary. The auxiliary claim $\rho(T) \le 1/(2d-1)$ for a transfer operator is asserted without derivation and is implausible on its face.

(a) A conflation. The proposal states $\mathcal{F}[G^{*N}](k) = \mathcal{F}[G^N](k)$. These are opposite operations: $\mathcal{F}[G^{*N}] = (\hat G)^N$, the pointwise power, while $\mathcal{F}[G^{N}]$ is the $N$-fold convolution of $\hat G$. The identity as written inverts the correspondence.

(b) The monotonicity runs the other way. The claim is $\phi_{N+1}(k^*) < \phi_N(k^*)$ for $\phi_N(k) = \mathcal{F}[G^N](k)/\mathcal{F}[G^N](0)$. Computed:

$\phi_N(\pi)$ is strictly increasing: $+0.182$ then $+0.016$. It must be. “Flatter in $k$” is precisely the statement that the normalised transform stays closer to $1$ across the zone. Strategy 3 asserts the opposite of the mechanism it is built on.

Prékopa–Leindler concerns log-concave functions on $\mathbb{R}^n$; $G$ on $\mathbb{Z}^d$ is not log-concave, and the inequality yields nothing about pointwise monotonicity of Fourier transforms. Separately, $G(x)^N \sim |x|^{-N(d-2)}$ suppresses tails polynomially, not exponentially.

The non-backtracking operator $B$ on $\mathbb{Z}^d$ has $2d-1$ continuations at each step, so non-backtracking walks of length $n$ number $\sim (2d-1)^n$ and $\rho(B) = 2d-1$. For $d=7$ that is $\mathbf{13}$, not $1/13$. $B$ is a growth operator, not a contraction; the proposed bound $\|B\|_2 \le 1/(2d-1)$ is wrong by a factor of $(2d-1)^2 = 169$ and points the wrong way — used as written it would give exponential growth in the diagram index. The representation $\hat\Pi^{(N)}(k^*) = \mathrm{Tr}((BS)^N)$ is also not a form the lace coefficients take.

The proposal is that $R(k)$ is subharmonic on $\mathbb{T}^d$ and therefore attains its minimum on “the boundary orbits”, namely $k^*$. Three separate failures. The torus has no boundary — it is a compact manifold without one, and a subharmonic function on such a manifold is constant by the maximum principle. Subharmonicity controls maxima, not minima; the argument invokes the wrong extremum for the direction it needs. And $\hat D(k) = \frac1d\sum\cos k_i$ is not Schur-concave on $[-\pi,\pi]^d$: a symmetric sum $\sum g(k_i)$ is Schur-concave iff $g$ is concave, and $\cos$ is concave only on $[0,\pi/2]$, convex on $[\pi/2,\pi]$.

(a) Unavailable. The claimed Cauchy–Schwarz $|\hat\Pi^{(2m+1)}|^2 \le \hat\Pi^{(2m)}\hat\Pi^{(2m+2)}$ requires reflection positivity, which percolation does not supply — see §N.8.

(b) Backwards. A log-convex sequence has non-decreasing consecutive ratios. Log-convexity therefore places the worst case at large $N$ and cannot deliver a uniform $O(1/2d)$ ratio bound; it makes the problem harder, not easier.

(c) False in the model. At $k^*$: $a_3^2 = 1.684389$ against $a_2 a_4 = 1.565210$. The sequence is log-concave, ratios $1.1864, 1.1025, 1.0947$ — decreasing, and every one of them exceeds $1$. In the unnormalised model the required condition fails outright at the corner, which is precisely why the §N.4 difference was negative there.

Non-backtracking walks on $\mathbb{Z}^d$ have $2d-1$ continuations per step, so their number grows as $(2d-1)^n$ and $\rho(B) = 2d-1 = \mathbf{13}$ at $d=7$. $B$ is a growth operator. The value $1/13$ is its reciprocal, and substituting it converts exponential growth into exponential decay by inspection. Modulating by a diagonal unitary is correct as far as it goes — $\|B_k\| = \|B\|$ exactly, by unitary invariance rather than the triangle inequality — but it inherits the base error unchanged.

The proposal that double connectivity occurs only at even $\|x\|_1$, hence that every term enters with $+1$, is false. Bipartiteness says every path $0 \to x$ has length $\equiv \|x\|_1 \pmod 2$; it does not say $\|x\|_1$ is even. One open square settles it: $0 \Leftrightarrow e_1$ via the direct bond (length 1) and $0 \to e_2 \to e_1{+}e_2 \to e_1$ (length 3), edge-disjoint, both odd, with $\|e_1\|_1 = 1$.

Every odd shell is populated, and $\|x\|_1 = 1$ is the largest single shell. It carries sign $-1$, and it is what makes the total negative.

The corner condition of §N.5 was posed as $\hat\Pi^{(2m+1)}(k^*)/\hat\Pi^{(2m)}(k^*) \le 1$. With $\hat\Pi^{(0)}(k^*) < 0$ the denominator is negative and the inequality no longer says what it was written to say; the threshold $0.8429$ inherited from the $G^m$ model does not transfer. More seriously, the sign-alternating premise $\hat\Pi^{(2m)} - \hat\Pi^{(2m+1)} \ge 0$ begins from a negative first term.

Whether $\hat\Pi^{(1)}(k^*)$ is also negative, and of what magnitude, is not settled here. $\Pi^{(1)}$ requires the nested construction over a pivotal bond with restricted connections; the triangle $\sum_{u,v}\tau(0,u)\tau(u,v)\tau(v,x)$ is its bound, not the coefficient, and substituting it would assume precisely what the exercise exists to test.