diff --git a/content/essays/growing-radius-domination.md b/content/essays/growing-radius-domination.md new file mode 100644 index 0000000..c68e5c8 --- /dev/null +++ b/content/essays/growing-radius-domination.md @@ -0,0 +1,932 @@ +--- +title: "From Path Tubes to a Near-Critical Domination Bound" +date: 2026-07-22 +abstract: > + A first-person, code-heavy companion to the preprint *Near-Critical + First-Moment Lower Bounds for Growing-Radius Domination in Random Regular + Graphs*. Rather than reproducing its theorem-and-proof form, this page + traces where the problem came from — a cops-and-robbers hypergraph question + that collapsed into a domination bound — why the answer carries an + unnecessary coupon-collector logarithm, and how a chain of computational + detours (a failed concavity conjecture, a catastrophic cancellation, an + independent audit that caught a stale constant) repeatedly redirected the + proof before it reached its final shape. +tags: + - research + - research/mathematics + - research/graph-theory + - tech/python +authors: + - "Levi Neuwirth | /me.html" +preprint: /papers/growing-radius-domination-paper.pdf +status: "Durable" +confidence: proved +evidence: 5 +peer-status: unreviewed +result-shape: mixed +--- + +This page is a companion to the preprint [**Near-Critical First-Moment Lower Bounds for Growing-Radius Domination in Random Regular Graphs**](/papers/growing-radius-domination-paper.pdf) — [the full theorem-and-proof form is here](/essays/near-critical-growing-radius-domination.html). A complete, runnable version — [`growing-radius-domination-demo.py`](/papers/growing-radius-domination-demo.py) — ships alongside this page, together with [the CSV of diagnostic output](/papers/growing-radius-domination-demo-output.csv) it produces. + +The main theorem is the following. Fix a degree $d\ge 3$ and let + +$$ +B_h=1+d\frac{(d-1)^h-1}{d-2} +$$ + +be the number of vertices in a radius-$h$ ball of the infinite $d$-regular tree. For a graph $G$, write $\gamma_h(G)$ for the minimum size of a set whose distance-$h$ neighborhoods cover every vertex. If $G_{n,d}$ is a uniformly random simple $d$-regular graph and $h=h(n)\to\infty$, then for every $W_h\to\infty$ for which the displayed coordinate stays a positive fraction of $\log B_h$ and the type-counting error stays asymptotically below the coupon term, + +::: {.exhibit .exhibit--equation} +$$ +\gamma_h(G_{n,d})\ge +\frac{n}{B_h} +\left( +\log B_h-2\log\log B_h-W_h +\right) +$$ +::: + +with high probability. + +The $-W_h$ is deliberate, and it is the honest part of the statement. Letting $W_h$ diverge arbitrarily slowly walks the bound arbitrarily close to the predicted bounded critical window without ever resolving that window. The theorem reaches to within a diverging additive whisker of the transition and stops. At the comparison scale $B_h\asymp\sqrt n$, with such a slowly growing $W_h$, the lower bound has order + +$$ +\sqrt n\log n, +$$ + +not merely $\sqrt n$. That single extra logarithm is QUITE LITERALLY the entire reason this project exists, and it is why the whole thing began, improbably, inside a cops-and-robbers calculation. Before I return to pursuit-evasion, the domination problem deserves to be seen on its own terms. + +--- + +## 1. The first calculation: how large is a ball? + +The geometry starts with a one-line function. I mean that literally. + +```python +import mpmath as mp + + +def tree_ball_volume(d: int, h: int) -> mp.mpf: + """Radius-h volume of the infinite d-regular tree.""" + if d < 3: + raise ValueError("d must be at least 3") + if h < 0: + raise ValueError("h must be nonnegative") + b = d - 1 + return mp.mpf(1) + d * (mp.power(b, h) - 1) / (d - 2) +``` + +If every chosen vertex covers at most $B_h$ vertices, pure volume counting gives + +$$ +\gamma_h(G)\ge \frac{n}{B_h}. +$$ + +That bound is occasionally attainable, on graphs organized precisely enough to make it tight — perfect covering codes are the cleanest examples of what such organization looks like. A typical random graph is not organized. Its neighborhoods overlap wastefully, and covering the last few stubborn vertices costs extra. The question is how much extra, and the coupon heuristic gives the first honest guess. + +For a random selected set of density $\alpha$, a particular radius-$h$ ball is missed with probability roughly + +$$ +(1-\alpha)^{B_h}\approx e^{-\alpha B_h}. +$$ + +There are about $\exp(nH(\alpha))$ sets of density $\alpha$, with $H$ the binary entropy. Balancing the entropy of choosing the set against the probability that it happens to cover everything predicts the coordinate + +$$ +\alpha B_h +\approx +\log B_h-2\log\log B_h. +$$ + +The helper below computes a point just below that predicted transition. + +```python + +def near_critical_coordinate( + d: int, + h: int, + W: mp.mpf | None = None, +) -> mp.mpf: + """Return C = log B_h - 2 log log B_h - W.""" + B = tree_ball_volume(d, h) + L = mp.log(B) + if W is None: + W = mp.log(L) + return L - 2 * mp.log(L) - W +``` + +The theorem says this heuristic scale survives optimization over the selected set, and that survival is the whole difficulty. A minimum dominating set is not sampled independently of anything. It sees the graph and gets to arrange itself around the overlaps, and there is no a priori reason a clever arrangement could not beat the coupon prediction. + +--- + +## 2. Where the problem came from + +The [preceding project](/essays/branch-based-local-capture-in-tree-balls/index.html) studied a rigid local certificate in the cops-and-robbers game. Around a fixed robber position, every outward nonbacktracking path indexed a descendant "tube," and if every tube held a cop, then every surviving round ended in capture, blockage, or pressure from at least two branches. It was a clean local statement, and it left an obvious global question hanging over it: could one cop placement, chosen once, hit every such tube for every possible robber root at the same time? + +At first this looked like a monstrous hypergraph problem — many roots, exponentially many paths through each — and I spent far longer than I would like to admit intimidated by it. Then, as they somehow often do, the combinatorics simply collapsed. Once the root is allowed to vary, a tube remembers nothing^[I think we mathematicians should really have a term for amnesia. We have the [Markov Property](https://en.wikipedia.org/wiki/Markov_property), but can we facetciously generalize further?] about its interior. It remembers only its **terminal directed edge** and the unused residual radius + +$$ +h=R-t. +$$ + +For a directed edge $p\to w$, the associated set is the forward cone from $w$ of depth $h$, with the branch back through $p$ excluded. A vertex set hits every such directed cone at $w$ exactly when either $w$ is itself selected, or at least two distinct branches at $w$ contain selected vertices within distance $h$. The exponential path structure was never really there. It was an artifact of insisting on a fixed root, and the clean way to say this uses no tree at all. + +A set $S$ is **internally two-path $(h,2)$-dominating** if every $v\notin S$ has two paths of length at most $h$ from $v$ to $S$ whose only shared vertex is $v$. Any such set is automatically ordinary distance-$h$ dominating, because either path on its own already witnesses a selected vertex within distance $h$. + +So, a lower bound for ordinary distance-$h$ domination is, for free, a lower bound for this stronger global tube certificate. The theorem therefore says something I find satisfying: exhaustive static tube coverage keeps its logarithmic overhead even after you allow all roots to share a single placement. Collapsing the hypergraph bought no asymptotic mercy. + +It does **not** say that cops need $\sqrt n\log n$ in the actual game. Adaptive strategies, partial coverage, motion between epochs, and entirely different certificates all remain outside the argument, exactly as they did in the previous paper. I will return to how narrow this obstruction is, deliberately, at the end. For now it is enough that the pursuit question handed the domination problem a reason to care about the scale $B_h\asymp\sqrt n$, and then got out of the way. + +--- + +## 3. Why the coupon heuristic is not a proof + +The direct concentration approach dies on the scale of the problem, and it dies quickly. Changing a single pairing in the configuration model can flip the radius-$h$ coverage status of on the order of $B_h$ vertices. The natural Lipschitz constant therefore grows on precisely the scale I need to resolve, which means the usual bounded-differences machinery is not merely weak here, but rather calibrated to lose. + +The overlap is just as unforgiving. The events "the ball around $v$ is uncovered" and "the ball around $w$ is uncovered" share most of their mass. At $B_h\asymp\sqrt n$ this is exactly the birthday-collision scale for random regular neighborhoods. What made the problem exact was giving up on balls entirely and tracking **distances** instead. + +Given a candidate set $S$, label every vertex by + +$$ +\ell(v)=\operatorname{dist}(v,S)\in\{0,1,\ldots,h\}. +$$ + +These labels obey two entirely local rules: + +1. labels on adjacent vertices differ by at most one; +2. every vertex of positive label $i$ has a neighbor of label $i-1$. + +And the rules run in reverse: any labeling obeying them is forced to be genuine graph distance. Descending labels trace a path down to label zero of exactly the stated length, and no path leaving label zero can climb by more than one per edge. The two inequalities pin the label to the distance from both sides. + +That observation may come across as elementary, but it is the hinge of the whole paper. It removes any need to pretend random neighborhoods are trees. The labels are honest distances on the honest graph, and it turns the first moment into an exact method-of-types calculation. The price, which I did not appreciate until much later, is a nonconcave variational problem whose optimizer I would have to pin down uniformly as the number of levels grew. I traded an intractable probabilistic estimate for a hard but finite optimization. + +--- + +## 4. The local entropy that appears in the type count + +At a vertex of label $i>0$, at least one of its $d$ incident half-edges has to descend to label $i-1$. Suppose each coordinate carries descending marginal $a$. The maximum-entropy distribution over nonempty subsets of the $d$ half-edges is then an exponentially tilted law, and its marginal and entropy are + +$$ +a= +\frac{\lambda(1+\lambda)^{d-1}} +{(1+\lambda)^d-1}, +$$ + +$$ +s_d(a)= +\log\big((1+\lambda)^d-1\big) +-da\log\lambda. +$$ + +The numerics need care near the Moore-growth boundary $a=1/d$, where $\lambda$ can be exponentially small and a naive inversion silently underflows. + +```python + +def _conditioned_marginal(d: int, lam: mp.mpf) -> mp.mpf: + """Marginal in a nonempty tilted subset of [d].""" + log1p_lam = mp.log1p(lam) + numerator = lam * mp.exp((d - 1) * log1p_lam) + denominator = mp.expm1(d * log1p_lam) + return numerator / denominator + + +def lambda_from_marginal(d: int, a: mp.mpf) -> mp.mpf: + """Invert the conditioned-subset marginal in log lambda.""" + a = mp.mpf(a) + lower = mp.mpf(1) / d + if a < lower or a > 1: + raise ValueError("a must lie in [1/d, 1]") + + tol = mp.power(10, -(mp.mp.dps - 12)) + if abs(a - lower) <= tol: + return mp.mpf(0) + if abs(a - 1) <= tol: + return mp.inf + + lo = -mp.mpf(2) * mp.mp.dps * mp.log(10) + hi = mp.mpf(2) * mp.mp.dps * mp.log(10) + for _ in range(max(160, 3 * mp.mp.dps)): + mid = (lo + hi) / 2 + lam = mp.exp(mid) + if _conditioned_marginal(d, lam) < a: + lo = mid + else: + hi = mid + return mp.exp((lo + hi) / 2) + + +def conditioned_subset_entropy(d: int, a: mp.mpf) -> mp.mpf: + """Maximum entropy s_d(a) with the all-zero pattern excluded.""" + a = mp.mpf(a) + lower = mp.mpf(1) / d + tol = mp.power(10, -(mp.mp.dps - 12)) + if abs(a - lower) <= tol: + return mp.log(d) + if abs(a - 1) <= tol: + return mp.mpf(0) + + lam = lambda_from_marginal(d, a) + log_partition = mp.log(mp.expm1(d * mp.log1p(lam))) + return log_partition - d * a * mp.log(lam) +``` + +The precision-scaled endpoint is not fussiness for its own sake. An early independent audit by [Claude Fable 5](https://www.anthropic.com/claude/fable) found that a fixed lower cutoff in the $\lambda$ solver was quietly clamping the deep layers and manufacturing false KKT residuals. The whole computation lives exactly where the asymptotic coordinates separate exponentially. How you handle the endpoints *is* part of the mathematics, not just an implementation detail beneath it.[^audit] + +[^audit]: I have come to treat "it's just a numerical detail" as a (not so) small alarm bell. For this problem in particular, it was never once true. + +--- + +## 5. The exact compact functional + +Let + +$$ +x_i=q_{i-1,i},\qquad 1\le i\le h, +$$ + +be the directed-edge mass between adjacent distance levels, and let + +$$ +\ell_i=q_{i,i},\qquad 0\le i\le h, +$$ + +be the same-level edge mass. The layer masses are + +$$ +p_i=\ell_i+x_i+x_{i+1}, +$$ + +with $x_0=x_{h+1}=0$. Put + +$$ +a_i=\frac{x_i}{p_i}, +\qquad + y_i=p_i-x_i. +$$ + +After optimizing out the full local neighbor-count profile, the exact annealed functional collapses to + +$$ +\begin{aligned} +\mathcal F_{d,h} +={}&(d-1)\alpha\log\alpha\\ +&+\sum_{i=1}^h +\left[ +-p_i\log p_i ++p_i s_d(a_i) ++d y_i\log y_i +\right]\\ +&-\frac d2\sum_{i=0}^h\ell_i\log\ell_i, +\end{aligned} +$$ + +with $\alpha=p_0$. The reduction from the full profile to this $2h+1$-variable object is exact, not an approximation I am hoping is harmless. The exactness is proved in the preprint and it is what lets me trust the code below to be computing the real thing. + +The implementation follows the displayed formula line for line. + +```python + +def _xlogx(x: mp.mpf) -> mp.mpf: + return mp.mpf(0) if x == 0 else x * mp.log(x) + + +def compact_microcanonical_value(d, h, x, ell): + """Evaluate the exact compact functional.""" + xx = [mp.mpf(0)] + [mp.mpf(v) for v in x] + [mp.mpf(0)] + ll = [mp.mpf(v) for v in ell] + p = [ll[i] + xx[i] + xx[i + 1] for i in range(h + 1)] + + if abs(sum(p) - 1) > mp.power(10, -(mp.mp.dps // 2)): + raise ValueError("profile is not normalized") + + alpha = p[0] + value = (d - 1) * _xlogx(alpha) + + for i in range(1, h + 1): + if p[i] == 0: + continue + lower = mp.mpf(1) / d + a_i = xx[i] / p[i] + numerical_tol = mp.power(10, -(mp.mp.dps // 3)) + if a_i < lower - numerical_tol or a_i > 1 + numerical_tol: + raise ValueError("descent marginal is infeasible") + a_i = min(mp.mpf(1), max(lower, a_i)) + + y_i = p[i] - xx[i] + value += -_xlogx(p[i]) + value += p[i] * conditioned_subset_entropy(d, a_i) + value += d * _xlogx(y_i) + + value -= mp.mpf(d) / 2 * sum(_xlogx(v) for v in ll) + return value, alpha +``` + +The full type count is two-sided. For each integer type, its expected count in the pairing model is an explicit ratio of factorials; uniform Stirling estimates identify its exponential rate with $\mathcal F_{d,h}$ up to $O_d(h\log n)$. The lower side of that estimate is the one I nearly overlooked. Because the expected number of any single type cannot exceed the total number of subsets of the corresponding size, it hands over the pointwise anchor + +$$ +\Psi_{d,h}(\alpha)\le H(\alpha). +$$ + +An earlier working note of mine called this "the trivial counting bound" and moved on. It was not trivial, and it was not something to move on from. It turned out to be the last missing lemma in the first-moment theorem, and I had worked it out long before I understood I already had it.[^trivial] + +[^trivial]: There is probably a general lesson here about the things one labels "trivial" in one's own notes. I have chosen not to learn it and will surely repeat the mistake. + +--- + +## 6. A failed concavity conjecture + +Once the profile was down to $2h+1$ variables, the functional looked numerically docile near the low density stationary branch. The obvious conjecture, and the one I wanted to be true, was that it might simply be concave throughout the slack region. + +An adversarial Hessian search turned up profiles with genuinely positive constrained curvature. They had a recognizable shape: most layers sitting near capacity-saturated Moore growth, with a few isolated layers carrying much heavier same-level edge mass. Finite differences confirmed the positive eigenvalues of the analytic Hessian, so this was not a precision artifact, and the outcome I had wished for was, as is often the case in mathematics, completely dead. + +The failure was, perhaps, the most useful thing that happened to the proof. It is interesting how failure is often the most useful outcome, contrary to how we perceive it. The positive curvature profiles were not stationary, as they were bumps off to the side, not competing maxima. This meant the right target was never concavity at all. It was **uniqueness of stationary points, together with boundary repulsion**. The proof then reassembled itself into something much cleaner than the one I had been trying to force: + +- entropy singularities repel every boundary face; +- so every maximizer is an interior KKT point; +- the stationary equations admit an exact reverse transfer; +- that reverse transfer is order preserving; +- activity increases strictly along its single terminal parameter; +- so there is exactly one stationary point at each activity; +- and it has to be the unique global maximizer. + +A nonconcave function is perfectly entitled to a unique global optimizer. Here it is monotone dynamics, not curvature, that supplies globality. I do not think I would ever have gone looking for the dynamics had the concavity conjecture not first embarrassed me out of the supposed easy route. + +--- + +## 7. The exact reverse transfer + +The stationary equations can be carried by two positive messages per distance level. Taking ratios, let + +$$ +\rho_i=\frac{B_i}{A_i}, +\qquad +u_i=\frac{A_{i+1}}{A_i}, +\qquad +v_i=\rho_i+u_i. +$$ + +At the terminal wall, $u_h=0$ and $v_h=\rho_h$. Given the next state $(\rho',v')$, the preceding state is fully explicit. + +```python + +def _w_e(rho: mp.mpf, b: int): + """w=(1-rho)^(1/b), e=1-w, evaluated stably.""" + log_w = mp.log1p(-rho) / b + return mp.exp(log_w), -mp.expm1(log_w) + + +def reverse_step(rho_next: mp.mpf, v_next: mp.mpf, b: int): + """Invert one stationary transfer step.""" + w, e = _w_e(rho_next, b) + R = v_next * e / w + M = mp.power(e + w / v_next, b) + denominator = R + M + rho = R / denominator + v = (1 + R) / denominator + return rho, v +``` + +This map is coordinatewise order preserving, and that single property does most of the load-bearing work in the globality proof. (I couldn't help myself quoting Fable here -- world, take note, THIS was the result out of them all that Fable, under the role of auditor, called load-bearing.)^[This is a humorous interjection, for those of you who don't use Claude often. Claude models have a tendency to call anything that has some type of impact on the bigger picture "load-bearing," using this phrase obsessively to the point that it should probably be considered the new em dash. It is somewhat less of a tragedy; at least I and other reasonable humans greatly enjoyed usage of the dash in our personal writing. I don't think I've ever used the phrase "load bearing" in an organic sense, ever.] Start from a larger terminal value and every preceding $\rho$ coordinate comes out larger. The reconstructed activity is strictly increasing right along with it. The entire positive stationary family is therefore one-dimensional and uniquely parametrized by activity, which is exactly the uniqueness the failed concavity conjecture had been standing in for. + +For computation, it is far better to parametrize the terminal value as + +$$ +t=-\log(1-s), +\qquad s=1-e^{-t}. +$$ + +Near high density, $s$ becomes indistinguishable from one at ordinary precision while $t$ stays comfortably moderate. This is a small change with a large payoff, the kind of thing one only learns by watching a doomed solver quietly and blissfully lose all its digits at large $h$. + +--- + +## 8. Reconstructing the stationary point + +The central routine does four things in order: + +1. reverse-iterate from the terminal wall; +2. reconstruct the messages $A_i,B_i$; +3. compute density and activity at the root; +4. evaluate the free energy through *both* a direct partition-function route and a stable root-only route. + +The shipped file carries the full dataclass with all the residual fields; the core calculation is below. + +```python +from dataclasses import dataclass + + +@dataclass +class StationaryPoint: + d: int + h: int + terminal_t: mp.mpf + terminal: mp.mpf + z: mp.mpf + alpha: mp.mpf + phi_direct: mp.mpf + psi_direct: mp.mpf + phi_root: mp.mpf + psi_root: mp.mpf + kappa: mp.mpf + r0: mp.mpf + Zv: mp.mpf + Ze: mp.mpf + telescoping_residual: mp.mpf + root_pressure_residual: mp.mpf + root_micro_residual: mp.mpf + stationarity_residual: mp.mpf + compact_residual: mp.mpf + rho: list[mp.mpf] + u: list[mp.mpf] + A: list[mp.mpf] + Bmsg: list[mp.mpf] + + +def _stable_power_difference(S, B, power): + """Compute S^power-(S-B)^power without cancellation.""" + ratio = B / S + return mp.power(S, power) * ( + -mp.expm1(power * mp.log1p(-ratio)) + ) +``` + +```python + +def stationary_from_terminal_t( + d: int, + h: int, + terminal_t: mp.mpf, + *, + validate_compact: bool = True, +) -> StationaryPoint: + b = d - 1 + t = mp.mpf(terminal_t) + terminal = -mp.expm1(-t) + + rho = [mp.mpf(0)] * (h + 1) + v = [mp.mpf(0)] * (h + 1) + u = [mp.mpf(0)] * (h + 1) + rho[h] = v[h] = terminal + + for i in range(h - 1, 0, -1): + rho[i], v[i] = reverse_step(rho[i + 1], v[i + 1], b) + u[i] = v[i] - rho[i] + u[h] = mp.mpf(0) + + # Root reconstruction. + w1, e1 = _w_e(rho[1], b) + r0 = v[1] * e1 / w1 + kappa = mp.power(v[1] / w1, b) + z = r0 * kappa / mp.power(1 + r0, b) + + # Normalize A_1=1 and rebuild the messages. + A = [mp.mpf(0)] * (h + 2) + Bmsg = [mp.mpf(0)] * (h + 1) + A[1] = mp.mpf(1) + for i in range(1, h): + A[i + 1] = u[i] * A[i] + Bmsg[0] = r0 + for i in range(1, h + 1): + Bmsg[i] = rho[i] * A[i] + + S = [mp.mpf(0)] * (h + 1) + S[0] = Bmsg[0] + A[1] + for i in range(1, h): + S[i] = Bmsg[i - 1] + Bmsg[i] + A[i + 1] + S[h] = Bmsg[h - 1] + Bmsg[h] + + vertex_terms = [z * mp.power(S[0], d)] + for i in range(1, h + 1): + vertex_terms.append( + _stable_power_difference(S[i], Bmsg[i - 1], d) + ) + + Zv = mp.fsum(vertex_terms) + Ze = mp.fsum(value * value for value in Bmsg) + Ze += 2 * mp.fsum(Bmsg[i] * A[i + 1] for i in range(h)) + + alpha = Bmsg[0] * (Bmsg[0] + A[1]) / Ze + phi_direct = mp.log(Zv) - mp.mpf(d) / 2 * mp.log(Ze) + psi_direct = phi_direct - alpha * mp.log(z) +``` + +The exact telescoping identity is + +$$ +Z_v=\kappa Z_e. +$$ + +It is what lets me trade the direct partition-function route for root-only formulas that are far better conditioned than subtracting two enormous logarithms and praying. + +```python + phi_root = ( + mp.log(z) + + mp.mpf(d) / 2 * mp.log((1 + r0) / r0) + + mp.mpf(d - 2) / 2 * mp.log(alpha) + ) + + psi_root = ( + (1 - alpha) * mp.log(kappa) + - (mp.mpf(d - 2) / 2 + alpha) * mp.log(r0) + + mp.mpf(d - 2) / 2 * mp.log(alpha) + + ( + mp.mpf(d - 1) * alpha + - mp.mpf(d - 2) / 2 + ) * mp.log(1 + r0) + ) +``` + +The routine then checks the stationary equations, reconstructs the compact profile + +$$ +x_i=\frac{B_{i-1}A_i}{Z_e}, +\qquad +\ell_i=\frac{B_i^2}{Z_e}, +$$ + +and confirms that the compact functional agrees with the root-only free energy. + +```python + stationarity = [ + abs(kappa * Bmsg[0] - z * mp.power(S[0], b)) + ] + for i in range(1, h + 1): + stationarity.append( + abs(kappa * A[i] - mp.power(S[i], b)) + ) + stationarity.append( + abs( + kappa * Bmsg[i] + - _stable_power_difference( + S[i], Bmsg[i - 1], b + ) + ) + ) + + if validate_compact: + x = [ + Bmsg[i - 1] * A[i] / Ze + for i in range(1, h + 1) + ] + ell = [ + Bmsg[i] * Bmsg[i] / Ze + for i in range(h + 1) + ] + compact_value, compact_alpha = compact_microcanonical_value( + d, h, x, ell + ) + compact_residual = max( + abs(compact_value - psi_root), + abs(compact_alpha - alpha), + ) + else: + compact_residual = mp.nan +``` + +This is in many ways the computational core. It finds one exact finite-$h$ stationary orbit and audits every identity that the theory says must hold on it. The asymptotic proof does not lean on any of these numbers. What the code provides is thus a hostile, independent witness that the analytic estimates are describing the object I think they are. + +--- + +## 9. Solving for a target density + +The theorem is phrased in terms of + +$$ +C=\alpha B_h. +$$ + +Because terminal parameter, activity, and density all increase together, I can solve $\alpha B_h=C$ by bisection in the stable coordinate $t$ and never worry about which branch I am on. + +```python + +def solve_for_alpha_B(d, h, C, iterations=None): + """Solve alpha B_h = C by bisection in terminal_t.""" + B = tree_ball_volume(d, h) + target_alpha = mp.mpf(C) / B + if iterations is None: + iterations = max(140, 2 * mp.mp.dps) + + D = mp.mpf(d) / (d - 2) + lo = mp.mpf("0.05") * C / D + hi = mp.mpf("3.0") * C / D + 1 + + while stationary_from_terminal_t( + d, h, lo, validate_compact=False + ).alpha > target_alpha: + lo /= 2 + + while stationary_from_terminal_t( + d, h, hi, validate_compact=False + ).alpha < target_alpha: + hi *= 2 + + for _ in range(iterations): + mid = (lo + hi) / 2 + point = stationary_from_terminal_t( + d, h, mid, validate_compact=False + ) + if point.alpha < target_alpha: + lo = mid + else: + hi = mid + + return stationary_from_terminal_t( + d, h, (lo + hi) / 2, validate_compact=True + ) +``` + +At the near-critical coordinate, the activity law is + +$$ +-\log z += +\log\frac1\alpha ++B_h(1-\alpha)^{B_h}(1+o(1)), +$$ + +so the quantity + +$$ +\frac{-\log z-\log(1/\alpha)} +{B_h(1-\alpha)^{B_h}} +$$ + +ought to tend to one. + +--- + +## 10. Numerical conditioning is part of the result + +Two computational failures materially shaped how I worked, and I am keeping them in the write-up rather than sanding them out. This is the spirit of the website; much like the [living documents](https://levineuwirth.org/colophon.html#living-documents) that the Colophon describes, I will not hide the work that I supersede. + +The first was the infuriating variant of mundane. An early comparison table mixed stationary values generated by two different solver versions, and one cubic $h=12$ entry came out wrong by about $2\times10^{-4}$. The asymptotic conclusion was untouched, as the error was far too small to matter for the theorem, but the table was no longer reproducible from the shipped code, which is its own kind of unacceptable. + +The second was more serious. Direct evaluation of + +$$ +\Psi= +\log Z_v-\frac d2\log Z_e-\alpha\log z +$$ + +turned catastrophically ill-conditioned near criticality. The expression subtracts two large logarithms to recover a very small number, and at cubic $h=52$, simply raising the working precision changed the reported ratio visibly. This is a glaring red flag. Lost digits! The stable root-only formula was right, and the direct route was quietly bleeding precision the whole time. + +My response was the small principle that this section is really about: make the residuals **gate** the output rather than merely decorate it. + +```python + +def gate_point(point, route_tolerance="1e-6"): + """Reject a row if independent routes disagree too much.""" + tol = mp.mpf(route_tolerance) + if point.psi_root == 0: + raise RuntimeError("zero root-formula exponent") + + route_disagreement = abs( + point.root_micro_residual / point.psi_root + ) + if route_disagreement > tol: + raise RuntimeError( + "microcanonical routes disagree: " + f"relative discrepancy={route_disagreement}; " + "increase mp.dps" + ) + + if point.compact_residual > mp.sqrt(tol): + raise RuntimeError( + "compact reconstruction failed: " + f"{point.compact_residual}" + ) +``` + +The diagnostic row then keeps both routes, every major residual, the working precision, and a SHA-256 digest of the source file (aren't you amazed I didn't use BLAKE3?), so that any number I report can be traced back to the exact bytes that produced it. + +```python + +def diagnostic_row(d, h, W=None): + B = tree_ball_volume(d, h) + L = mp.log(B) + if W is None: + W = mp.log(L) + C = near_critical_coordinate(d, h, W) + + point = solve_for_alpha_B(d, h, C) + gate_point(point) + + coupon = B * mp.power(1 - point.alpha, B) + activity_ratio = ( + -mp.log(point.z) - mp.log(1 / point.alpha) + ) / coupon + scale = mp.exp(-C) + + return { + "d": d, + "h": h, + "dps": mp.mp.dps, + "B_h": mp.nstr(B, 30), + "C": mp.nstr(C, 24), + "alpha_B_h": mp.nstr(point.alpha * B, 24), + "terminal_t": mp.nstr(point.terminal_t, 30), + "activity_ratio": mp.nstr(activity_ratio, 20), + "minus_psi_over_exp_minus_C": mp.nstr( + -point.psi_root / scale, 20 + ), + "psi_root": mp.nstr(point.psi_root, 24), + "psi_direct": mp.nstr(point.psi_direct, 24), + "route_relative_disagreement": mp.nstr( + abs(point.root_micro_residual / point.psi_root), 12 + ), + "stationarity_residual": mp.nstr( + point.stationarity_residual, 12 + ), + "telescoping_residual": mp.nstr( + point.telescoping_residual, 12 + ), + "compact_residual": mp.nstr( + point.compact_residual, 12 + ), + "source_sha256": hashlib.sha256( + Path(__file__).read_bytes() + ).hexdigest(), + } +``` + +> If an internal cross-check is comparable in size to the quantity you are reporting, it should be a gate, not an unprinted field in a dataclass. + +--- + +## 11. What the computation shows + +Running the demonstration script with $W_h=\log\log B_h$ gives the following. + +| $d$ | $h$ | $C$ | activity-law ratio | $-\Psi/e^{-C}$ | +|---:|---:|---:|---:|---:| +| 3 | 12 | 2.6889 | 0.4376 | 0.3870 | +| 3 | 20 | 6.8451 | 0.5873 | 0.6484 | +| 3 | 30 | 12.6345 | 0.9520 | 0.9418 | +| 3 | 40 | 18.7408 | 0.9956 | 0.9757 | +| 4 | 12 | 5.9859 | 0.6816 | 0.7285 | +| 4 | 20 | 13.3029 | 0.9915 | 0.9706 | +| 4 | 28 | 21.1087 | 0.9999 | 0.9800 | + +The asymptotic theorem asks only for a conservative negative bound, so in a sense these numbers are more than it needs. But I'd argue they clearly earn their place: both ratios march toward one, and the convergence is visibly slow precisely when the coefficient $C/\log B_h$ is small. This is exactly what the proof's fixed-compact-window uniformity predicts, and exactly what a claim of uniformity all the way down to zero would *not* look like. The table is quietly consistent with the shape of the theorem, not just its sign. + +The generator is satisfyingly short. + +```python + +def make_table(cases, csv_path=None): + rows = [] + for d, h, dps in cases: + mp.mp.dps = dps + row = diagnostic_row(d, h) + rows.append(row) + print( + f"d={d} h={h:>2} C={row['C']} " + f"activity={row['activity_ratio']} " + f"-Psi/e^-C={row['minus_psi_over_exp_minus_C']}" + ) + + if csv_path: + with open(csv_path, "w", newline="") as handle: + writer = csv.DictWriter( + handle, + fieldnames=list(rows[0].keys()), + ) + writer.writeheader() + writer.writerows(rows) + return rows +``` + +The complete script also carries optional plotting for the activity-law and free-energy ratios, for those who would rather watch the convergence than read it off a table. + +--- + +## 12. How the proof uses the orbit + +The code finds one exact finite-$h$ stationary orbit. The proof has to understand that orbit *uniformly* as $h\to\infty$, which is a different and harder demand, and it is where the analysis actually happens. + +There are two exact outer regimes. On the **free side**, a transverse defect $q_i$ is tiny and the main coordinate rides the free-coverage orbit. On the **stable side**, the occupied-message coordinate $\rho_i$ is tiny and the orbit approaches the empty-root manifold. I choose a matching layer where both transverse quantities are exponentially small at once, and shadow the true orbit against whichever exact outer solution is nearby. + +The reverse map supplies one-step estimates that stay uniform even as the main coordinate itself shrinks toward nothing. That uniformity, rather than any single clever identity, is what makes the whole scheme go. It yields + +$$ +\alpha=a_h(t)(1+o(1)), +$$ + +with + +$$ +a_h(t)=1-e^{-t/(d-1)^h}, +$$ + +and + +$$ +\frac{x}{(d-1)^{h-1}} +=(1-\alpha)^{B_h}(1+o(1)), +$$ + +where $x=-\log u_1$ is the stable root coordinate. The edge normalizer localizes at the overlap layer, which is what converts the transported message density into the actual selected density. Root reconstruction then delivers the activity law + +$$ +-\log z(\alpha) += +\log\frac1\alpha ++B_h(1-\alpha)^{B_h}(1+o(1)). +$$ + +The globality theorem supplies the envelope identity + +$$ +\Psi'_{d,h}(\alpha)=-\log z(\alpha), +$$ + +and integrating it from a slightly larger density, anchored at the upper endpoint by + +$$ +\Psi_{d,h}(\alpha)\le H(\alpha), +$$ + +gives + +$$ +\Psi_{d,h}(C/B_h) +\le +-\left(\frac12-o(1)\right)e^{-C} +$$ + +uniformly across the near-critical window. The first moment then closes the random-graph statement, provided the exponential negative term outweighs the $O_d(h\log n)$ type-counting error — which is precisely the growth condition on $W_h$ that the theorem carries out front, now visibly earning its place. + +--- + +## 13. What the theorem says about exhaustive tube coverage + +Finally! I can pay off the pursuit problem it started with, as promised.. + +The original local certificate placed a cop in every path tube at a *known* root. Make the root unknown and force all roots to share one placement, and the path prefix collapses — as in Section 2 — to a single directed terminal edge. Hitting every resulting forward cone is equivalent, in the tree geometry, to having selected vertices in at least two branches at every unselected vertex, and the graph-general form of that condition is internally two-path $(h,2)$-domination. + +Since $(h,2)$-domination implies ordinary distance-$h$ domination, the theorem hands the same lower bound to the exhaustive global certificate at no extra cost. At $B_h\asymp\sqrt n$, that certificate needs + +$$ +\Omega(\sqrt n\log n) +$$ + +selected vertices with high probability. One tempting route from the local tube theorem to a root-independent, Meyniel-scale placement is therefore closed. + +--- + +## 14. Three remaining problems + +### The bounded critical window + +The theorem reaches + +$$ +\log B_h-2\log\log B_h-W_h +$$ + +for every $W_h\to\infty$, and stops one diverging step short of the bounded additive term. The stationary computation points hard at a specific answer: the annealed transition looks governed by the scalar balance + +$$ +H(C/B_h)=(1-C/B_h)^{B_h}, +$$ + +whose solution satisfies + +$$ +C= +\log B_h-2\log\log B_h+o(1). +$$ + +Proving the full bounded window, though, needs uniform matching sharper than the slack theorem ever required — errors controlled down to the same order as the competing entropy and coupon terms. + +### Quenched matching + +What I have is a lower bound. A simple random placement followed by patching the uncovered vertices gives an upper bound of order + +$$ +\frac{n\log B_h}{B_h}, +$$ + +but it does not pin the actual domination number to the annealed transition. A matching quenched result likely wants a second moment, small-subgraph conditioning, or a genuinely problem-specific construction. I do not yet know which, so let me know if you do. + +### The direct two-branch constant + +Ordinary distance domination is only a relaxation of internally two-path domination, so the direct branch problem should carry its own coupon balance and, quite possibly, a different leading constant. One has to remember branch multiplicity, and cycles can make the message representations non-unique — I have left it as its own piece of work rather than forcing it into this one. + +--- + +## 15. Why I am showing so much Python + +The code was never a numerical appendix bolted onto a finished proof. It kept reaching back into the mathematics and rearranging it, and I can name some of the specific occasions: + +- exact ILPs first revealed that the global path hypergraph collapsed to a domination problem; +- transfer numerics suggested the $\log B_h-2\log\log B_h$ scale before I could prove it; +- an independent implementation caught a stale stationary value hiding in a comparison table; +- Hessian probes killed a plausible, load-bearing, and entirely false concavity conjecture; +- that dead conjecture is what redirected the proof toward boundary repulsion and reverse shooting; +- KKT checks exposed the last mechanical gap in the globality argument; +- residual-gated near-critical runs caught catastrophic cancellation in the direct free-energy route; +- and the stable root-only formula that fixed it became both a proof tool and the canonical way to evaluate everything. + +The formal theorem depends on none of these numbers. The path to the theorem depended on almost nothing else — on code explicit enough to be reimplemented by some AI system who (rightfully so) did not trust me, and hostile enough to keep trying to break whatever conjecture I was currently in love with. + +That is the role I want this page to preserve, and it is the reason it exists apart from the preprint at all. The paper records what turned out to be true. This page records how I found out which statements were even worth trying to prove — which is the part no theorem-and-proof ever shows you, and, if I am honest, the part I would most have wanted to read. diff --git a/content/essays/near-critical-growing-radius-domination.md b/content/essays/near-critical-growing-radius-domination.md new file mode 100644 index 0000000..e83e77f --- /dev/null +++ b/content/essays/near-critical-growing-radius-domination.md @@ -0,0 +1,1426 @@ +--- +title: "Near-Critical First-Moment Lower Bounds for Growing-Radius Domination in Random Regular Graphs" +date: 2026-07-22 +abstract: > + A first-moment lower bound for growing-radius domination in random regular + graphs, reaching within an arbitrary diverging term of the predicted + transition. The proof gives an exact graph-distance type count, reduces it + to a compact nonconcave variational functional, and shows that functional + has a unique global optimizer despite lacking concavity, via boundary + repulsion and an order-preserving reverse transfer. Uniform shadowing of the + resulting orbit yields the near-critical activity law and the first-moment + bound. The result also strengthens to internally two-path domination, a + graph-general parameter motivated by exhaustive static tube coverage in + pursuit-evasion games. The bounded critical window, quenched matching, and + the direct two-branch leading constant remain open. +tags: + - research + - research/mathematics + - research/graph-theory +authors: + - "Levi Neuwirth | /me.html" +bibliography: data/growing-radius-domination-preprint.bib +preprint: /papers/growing-radius-domination-paper.pdf +no-collapse: true +status: "Durable" +confidence: proved +evidence: 5 +peer-status: unreviewed +result-shape: mixed +further-reading: + - AignerFromme + - LuPeng + - ScottSudakov + - PralatWormald + - PralatWormaldRegular + - NeuwirthBranchCapture + - Wormald + - Janson + - CohenHonkalaLitsynLobstein + - Duckworth + - DuckworthWormald + - CutlerRadcliffe + - GlebovLiebenauSzabo + - ZhaoHabibullaZhou + - HabibullaQin + - RockafellarWets +--- + +# Introduction and Related Work {#sec-intro} + +## Main result and scale + +A set $S\subseteq V(G)$ is *distance-$h$ dominating* if every vertex of $G$ lies within graph distance $h$ of $S$; its minimum size is denoted $\gamma_h(G)$. For graphs of maximum degree $d$, one selected vertex can cover at most + +$$ +B_h=1+d\frac{(d-1)^h-1}{d-2} +$$ + +vertices, so the elementary volume bound is + +$$ +\gamma_h(G)\ge \frac{n}{B_h}. +$$ + +At the other extreme, if every radius-$h$ ball has at least $M$ vertices, independent selection with probability $(\log M)/M$ followed by adding every uncovered vertex gives + +$$ +\gamma_h(G)\le \frac{n(\log M+1)}{M}. +$$ + +In the ideal tree-ball geometry $M=B_h$, this places the natural covering scale at $n\log B_h/B_h$. The present paper proves that a random $d$-regular graph cannot beat essentially this scale from below: with high probability, + +$$ +\gamma_h(G_{n,d})\ge \frac{n}{B_h}\left(\log B_h-2\log\log B_h-\omega(1)\right), +$$ + +subject to an explicit growth condition. Thus the first-moment lower transition is determined through its first two asymptotic terms, up to the bounded critical window. A matching quenched upper theorem at this precision is not proved here. + +The distinction matters at the comparison scale $B_h\asymp\sqrt n$. There the volume bound alone is of order $\sqrt n$, whereas the theorem gives an $\Omega(\sqrt n\log n)$ obstruction. This scale is also relevant to pursuit-evasion: Meyniel's conjecture predicts that $O(\sqrt n)$ cops suffice in every connected $n$-vertex graph [@LuPeng; @ScottSudakov], and it is known for several random-graph models, including random regular graphs [@PralatWormald; @PralatWormaldRegular]. Our conclusion is not a cop-number lower bound. It says that one particular static exhaustive coverage mechanism can require a logarithmic factor more than the Meyniel scale. + +## Why the first moment is difficult + +For an independent random subset of density $\alpha$, a fixed tree-like radius-$h$ ball is missed with probability approximately $(1-\alpha)^{B_h}$. Balancing subset entropy against this coupon term predicts the coordinate + +$$ +\alpha B_h\approx \log B_h-2\log\log B_h. +$$ + +Turning that heuristic into a theorem is not a product-measure calculation. In the configuration model, coverage events for different vertices overlap heavily, and optimizing over the selected set allows highly organized profiles. A direct bounded-differences argument is also ineffective in the growing-radius regime: changing one pairing can alter the radius-$h$ coverage status of order $B_h$ vertices, so the natural Lipschitz constant grows on exactly the scale that must be resolved. + +The key device is to label every vertex by its exact distance to the candidate set. Local consistency of these labels — adjacent labels differ by at most one, and every positive label has a neighbor one level lower — forces them to be genuine graph distances on any graph. This removes the need to approximate neighborhoods by trees and converts the first moment into an exact finite-dimensional method-of-types problem. The price is a nonconcave variational functional whose global optimizer must be identified uniformly as the number of distance levels grows. + +## Related work + +The configuration or pairing model and its transfer to uniformly random simple regular graphs are standard; see Wormald's survey [@Wormald] and Janson's simplicity theorem [@Janson]. Fixed-radius domination in random regular graphs has been studied algorithmically: Duckworth analyzed randomized greedy algorithms for distance-$k$ dominating sets [@Duckworth], while Duckworth and Wormald treated independent domination [@DuckworthWormald]. The broader covering viewpoint is classical in coding theory [@CohenHonkalaLitsynLobstein]. + +The closest message-passing antecedents are statistical-mechanical. Zhao, Habibulla, and Zhou developed a cavity-method and belief-propagation treatment of minimum dominating sets [@ZhaoHabibullaZhou]; Habibulla and Qin studied a distance-two version with distance-labeled messages [@HabibullaQin]. Those works are replica-symmetric and fixed-radius. The transfer equations below are closely related in spirit; the distinctions here are the exact graph-level type count, the proof of global optimization, and the growing-radius asymptotics. + +Cutler and Radcliffe used Shearer entropy to obtain universal extremal bounds for domination polynomials of regular graphs [@CutlerRadcliffe]. Such universal bounds cannot by themselves detect the random-covering penalty: structured regular graphs may admit efficient covering codes. In the binomial random graph, Glebov, Liebenau, and Szabó proved two-point concentration at the first-moment threshold in a sufficiently dense regime [@GlebovLiebenauSzabo]. That result motivates, but does not supply, the corresponding quenched statement for random regular graphs. + +The pursuit application comes from branch-based local coverage. Aigner and Fromme introduced the multiple-cop game in its modern graph-theoretic form [@AignerFromme]; Prałat and Wormald later combined random placement with deterministic pursuit in random graphs [@PralatWormald; @PralatWormaldRegular]. The local tube certificate in [the branch-tube persistence theorem](/essays/branch-based-local-capture-in-tree-balls/index.html#thm-persistence) of [@NeuwirthBranchCapture] motivates a root-independent two-witness covering problem. [Section 2.1](#connection-with-static-tube-coverage) states the exact graph-general parameter used here and carefully limits the resulting obstruction. + +## Proof architecture and contributions + +The proof has three acts. + +**Act I: exact types and a compact functional.** Exact distance labels produce a two-sided type estimate + +$$ +\log \mathbb E N_n=n\Phi_{d,h}(n_\tau/n)+O_d(h\log n). +$$ + +Optimizing the local profile entropy at fixed tridiagonal edge masses reduces $\Phi_{d,h}$ exactly to a compact functional $\mathcal F_{d,h}$ in $2h+1$ variables. The lower side of the type estimate also yields the pointwise entropy anchor + +$$ +\Psi_{d,h}(\alpha)\le H(\alpha). +$$ + +**Act II: globality without concavity.** The compact functional is genuinely nonconcave. Nevertheless, entropy singularities repel every maximizing profile from the boundary. Interior KKT points are equivalent to positive message solutions. An exact reverse transfer reconstructs every positive stationary solution from one terminal parameter, and the associated activity is strictly increasing from $0$ to $\infty$. Hence the grand-canonical optimizer is unique at every activity, and duality gives + +$$ +\Psi_{d,h}(\alpha)=\Psi_{d,h}^{\mathrm{stat}}(\alpha),\qquad \Psi_{d,h}'(\alpha)=-\log z(\alpha). +$$ + +**Act III: orbit asymptotics and integration.** The reverse orbit has exact stable and free outer solutions. In the near-critical slack window they overlap over linearly many levels, allowing uniform shadowing and the activity law + +$$ +-\log z(\alpha)=\log\frac1\alpha+B_h(1-\alpha)^{B_h}(1+o(1)). +$$ + +Integrating this identity from a slightly larger density and anchoring with $\Psi\le H$ gives + +$$ +\Psi_{d,h}(C/B_h)\le -\left(\frac12-o(1)\right)e^{-C} +$$ + +uniformly up to $C=\log B_h-2\log\log B_h-W_h$. The coefficient $1/2$ is a convenient consequence of the chosen integration interval, not the predicted sharp coefficient; the numerical diagnostics in [Section 16](#sec-numerical) are consistent with a ratio tending to $1$ when $W_h\to\infty$. + +The four main contributions are therefore: an exact growing-radius type count with no local-tree assumption; an exact compact reduction; a global solution of a nonconcave annealed variational problem; and a near-critical random-regular lower bound. The bounded critical window, annealed-to-quenched matching, and direct two-branch asymptotics remain separate problems. + +--- + +# Models, Notation, and Main Results {#sec-models} + +Fix $d\ge 3$ throughout and put + +$$ +b=d-1,\qquad D=\frac{b+1}{b-1}=\frac{d}{d-2}, +$$ + +$$ +B_h=1+d\frac{b^h-1}{b-1}=Db^h-\frac{2}{b-1}. +$$ + +For a probability vector $r=(r_1,\ldots,r_k)$, write + +$$ +H(r)=-\sum_{j=1}^k r_j\log r_j, +$$ + +with $0\log 0=0$; for a scalar $a\in[0,1]$, $H(a)$ denotes the binary entropy $H(a,1-a)$. + +| Notation | Meaning | +|---|---| +| $B_h$ | maximum radius-$h$ ball volume at degree $d$ | +| $\gamma_h(G)$ | minimum size of a distance-$h$ dominating set | +| $\Phi_{d,h}$ | full local-profile type functional | +| $\mathcal F_{d,h}$ | compact tridiagonal profile functional | +| $\Psi_{d,h}(\alpha)$ | microcanonical annealed exponent at density $\alpha$ | +| $z=e^\theta$ | grand-canonical activity | +| $b=d-1,\ D=d/(d-2)$ | recurring degree constants | + +::: {#def-two-path .definition} +**Definition 1** (Internally two-path $(h,2)$ domination). A set $S\subseteq V(G)$ is *internally two-path $(h,2)$ dominating* if every vertex $v\notin S$ has two $v$–$S$ paths of length at most $h$ whose only common vertex is $v$. In particular, the paths begin through distinct neighbors of $v$. Let $\gamma_{h,2}^{\mathrm{int}}(G)$ denote the minimum size of such a set. +::: + +Every internally two-path $(h,2)$-dominating set is distance-$h$ dominating, since either witness path alone ends in $S$ within distance $h$. Therefore + +$$ +\gamma_{h,2}^{\mathrm{int}}(G)\ge \gamma_h(G). +$$ + +## Connection with static tube coverage + +In the tree-ball setting of [@NeuwirthBranchCapture], a length-$t$ tube with residual depth $h=R-t$ depends, after the root is allowed to vary, only on its terminal directed edge. The resulting forward cone excludes one branch at its head. A root-independent set meets every such directed cone exactly when, at every unselected vertex, at least two distinct branches contain a selected vertex within distance $h$; in a tree-ball these are precisely the two internally disjoint paths of [Definition 1](#def-two-path). Thus internally two-path domination is a graph-general strengthening of the global exhaustive tube certificate. + +Combining the displayed inequality above with the main theorem shows that any such exhaustive static certificate has size $\Omega(\sqrt n\log n)$ at the comparison scale $B_h\asymp\sqrt n$. Meyniel's conjecture concerns the existence of a fully adaptive winning strategy with $O(\sqrt n)$ cops, so there is no contradiction: the lower bound applies only to this static exhaustive mechanism. Partial coverage, adaptive reassignment, and epoch-based pursuit remain outside the argument. + +The pairing model consists of $n$ labeled buckets of $d$ half-edges paired uniformly at random; conditioning the resulting multigraph on simplicity gives the uniformly random simple $d$-regular graph $G_{n,d}$, for $dn$ even. Let $Z_{n,d,h}(m)$ count distance-$h$ dominating $m$-sets in the pairing model. + +For a distance-$h$ dominating set $S$, assign each vertex its exact label $\operatorname{dist}(v,S)\in\{0,\ldots,h\}$. The local consistency used below is equivalent to genuine distance: adjacent labels differ by at most one, and every positive label has a neighbor one level lower. The descending condition gives a path to label zero of the displayed length, while the Lipschitz condition gives the reverse inequality. No tree-neighborhood assumption is involved. + +The type count developed below gives + +$$ +\log \mathbb E Z_{n,d,h}(m)\le n\Psi_{d,h}(m/n)+O_d(h\log(n+1)), +$$ + +uniformly in $h$. Here $\Psi_{d,h}$ is the exact annealed variational exponent. + +::: {#thm-near-critical .theorem} +**Theorem 2** (Near-critical annealed negativity). *Fix $\eta\in(0,1)$, put $L_h=\log B_h$, and let $W_h\to\infty$. Uniformly for real $C$ satisfying* + +$$ +\eta L_h\le C\le L_h-2\log L_h-W_h, +$$ + +*one has* + +$$ +\Psi_{d,h}(C/B_h)\le -\left(\frac12-o(1)\right)e^{-C}\qquad(h\to\infty), +$$ + +*where the $o(1)$ may depend on $d$, $\eta$, and the prescribed sequence $W_h$, but is uniform over the displayed interval.* +::: + +::: remark +**Remark 3** (The coefficient $1/2$). The coefficient $1/2$ is not predicted to be sharp. It comes from the symmetric choice of the upper integration point in [Section 15](#sec-integration). The activity law and the diagnostics in [Section 16](#sec-numerical) are consistent with the normalized ratio approaching $1-O(e^{-W_h})$ deeper in the near-critical window. +::: + +::: {#cor-fixed-fraction-slack .corollary} +**Corollary 4** (Fixed-fraction slack). *For every fixed $00,\qquad \frac{ne^{W_h}L_h^2}{B_h}\gg h\log n. +$$ + +*Then, with high probability,* + +$$ +\gamma_h(G_{n,d})\ge \frac{n}{B_h}\left(L_h-2\log L_h-W_h\right). +$$ + +*The same conclusion holds for every parameter whose feasible sets are necessarily distance-$h$ dominating, including internally two-path $(h,2)$ domination.* +::: + +::: {#cor-fixed-fraction-random .corollary} +**Corollary 6** (Fixed-fraction random lower bound). *For every fixed $\varepsilon\in(0,1)$, if* + +$$ +\frac{n}{B_h^{1-\varepsilon}}\gg h\log n, +$$ + +*then, with high probability,* + +$$ +\gamma_h(G_{n,d})\ge (1-\varepsilon)\frac{n\log B_h}{B_h}. +$$ +::: + +When $B_h\asymp\sqrt n$, one has $\log B_h=\tfrac12\log n+O(1)$, and [Theorem 5](#thm-random-regular) gives + +$$ +\gamma_h(G_{n,d})\ge \frac{n}{B_h}\left(\tfrac12\log n-2\log\log n-W_h+O(1)\right)=\Omega(\sqrt n\log n). +$$ + +The implicit constant in the final order statement depends on the comparison constants in $B_h\asymp\sqrt n$; no exact prefactor $\sqrt n$ is asserted. + +--- + +# Two-Sided Exact Type Counting and the Entropy Anchor {#sec-types} + +For completeness, we record the exact local-profile count that underlies both the variational formula and the pointwise entropy bound used later. + +Let $\mathcal T_{d,h}$ be the finite set of local profiles + +$$ +\tau=(i,c),\qquad i\in\{0,\ldots,h\},\qquad c=(c_0,\ldots,c_h),\qquad \sum_j c_j=d, +$$ + +with + +$$ +c_j=0 \text{ if } |i-j|>1,\qquad c_{i-1}\ge 1 \text{ if } i>0. +$$ + +For a probability vector $\xi=(\xi_\tau)_{\tau\in\mathcal T_{d,h}}$, put + +$$ +q_{ij}(\xi)=\frac1d\sum_{\tau=(i,c)}\xi_\tau c_j, +$$ + +and call $\xi$ *feasible* when $q_{ij}=q_{ji}$. Its selected density is + +$$ +\alpha(\xi)=\sum_{\tau:\,i(\tau)=0}\xi_\tau. +$$ + +Define + +$$ +\Phi_{d,h}(\xi)=-\sum_\tau \xi_\tau\log\xi_\tau+\sum_\tau \xi_\tau\log\binom{d}{c(\tau)}+\frac d2\sum_{i,j}q_{ij}(\xi)\log q_{ij}(\xi), +$$ + +with $0\log 0=0$. The exact variational exponent is + +$$ +\Psi_{d,h}(\alpha)=\max\{\Phi_{d,h}(\xi):\xi \text{ feasible},\ \alpha(\xi)=\alpha\}. +$$ + +[Section 4.1](#exact-local-entropy-reduction) proves the exact reduction from this full profile functional to the compact tridiagonal functional of [Proposition 11](#prop-compact-functional). + +::: {#thm-two-sided-count .theorem} +**Theorem 7** (Two-sided count for one integer type). *There is a constant $K_d$ such that the following holds for every $h,n$ and every admissible integer profile $(n_\tau)_{\tau\in\mathcal T_{d,h}}$. Put* + +$$ +\xi_\tau=\frac{n_\tau}{n},\qquad H_{ij}=\sum_{\tau=(i,c)} n_\tau c_j=dnq_{ij}(\xi). +$$ + +*Assume $\sum_\tau n_\tau=n$, $H_{ij}=H_{ji}$, and every $H_{ii}$ is even; these conditions define admissibility here. Let $N_n$ denote the number of vertex sets whose exact-distance local-profile counts equal $n_\tau$ in the $d$-regular pairing model. Then* + +$$ +\mathbb E N_n=\frac{n!}{\prod_\tau n_\tau!}\prod_\tau \binom{d}{c(\tau)}^{n_\tau}\cdot \frac{\prod_{0\le i0}\left\{\log\bigl((1+\lambda)^d-1\bigr)-da\log\lambda\right\},\qquad \frac1d\le a\le 1. +$$ + +The minimizing parameter is characterized by + +$$ +a=\frac{\lambda(1+\lambda)^{d-1}}{(1+\lambda)^d-1}. +$$ + +## Exact local entropy reduction + +The passage from the full profile functional to $\mathcal F_{d,h}$ is exact. + +::: {#lem-conditional-product .lemma} +**Lemma 10** (Conditional-product reduction). *Fix a symmetric tridiagonal directed-edge distribution $q$, and write $x_i,\ell_i,p_i,y_i,a_i$ as above, with $x_{h+1}=0$. Among all feasible local-profile laws inducing $q$, the maximum of the vertex entropy and arrangement terms is* + +$$ +p_0d\,H\!\left(\frac{\ell_0}{p_0},\frac{x_1}{p_0}\right)+\sum_{i=1}^h\left[p_is_d(a_i)+d(1-a_i)H\!\left(\frac{\ell_i}{y_i},\frac{x_{i+1}}{y_i}\right)\right]. +$$ + +*The maximizing law is unique whenever all displayed masses are positive. Conditional on label $i\ge 1$, the set of lower-label half-edges has the entropy-maximizing tilted nonempty-subset law with coordinate marginal $a_i$; conditional on that set, every remaining half-edge independently receives label $i$ or $i+1$ with probabilities $\ell_i/y_i$ and $x_{i+1}/y_i$.* +::: + +::: proof +*Proof.* Condition on the central label $i$. Choosing a local count vector and then assigning the $d$ labeled half-edges contributes exactly the entropy of the induced law on words of length $d$ over the allowed neighboring labels. + +For $i=0$, only labels $0$ and $1$ are allowed and their coordinate marginals are $\ell_0/p_0$ and $x_1/p_0$. Subadditivity of entropy is sharp only for independent coordinates, giving the first term above. + +Fix $i\ge 1$. Mark the coordinates whose neighbor label is $i-1$. Their random subset is nonempty and has common coordinate marginal + +$$ +a_i=\frac{q_{i,i-1}}{p_i}=\frac{x_i}{p_i}. +$$ + +By definition, its entropy is at most $s_d(a_i)$, with equality for the exponential tilt $\Pr(A)\propto \lambda^{|A|}$, $\emptyset\ne A\subseteq[d]$, where $\lambda$ satisfies the marginal characterization above; this also proves the dual formula for $s_d$. Given the lower-neighbor set, the remaining $d-|A|$ coordinates must split between labels $i$ and $i+1$. Conditional entropy is maximized by independent splitting with probabilities $\ell_i/y_i$ and $x_{i+1}/y_i$. Its expectation is + +$$ +d(1-a_i)H\!\left(\frac{\ell_i}{y_i},\frac{x_{i+1}}{y_i}\right). +$$ + +The chain rule for entropy proves the displayed reduction; strict entropy concavity gives uniqueness in the positive interior. □ +::: + +::: {#prop-compact-functional .proposition} +**Proposition 11** (Exact compact functional). *For every feasible tridiagonal $q$, maximizing the full profile functional over local-profile laws inducing $q$ gives exactly the compact functional* + +$$ +\mathcal F_{d,h}(x,\ell)=(d-1)p_0\log p_0+\sum_{i=1}^h\left[-p_i\log p_i+p_is_d(a_i)+dy_i\log y_i\right]-\frac d2\sum_{i=0}^h\ell_i\log\ell_i. +$$ + +*Consequently the compact and full variational values agree at every selected density.* +::: + +::: proof +*Proof.* Insert [Lemma 10](#lem-conditional-product) into the full profile functional $\Phi_{d,h}$. The edge term is + +$$ +\frac d2\left(\sum_{i=0}^h \ell_i\log\ell_i+2\sum_{i=1}^h x_i\log x_i\right). +$$ + +Expanding the two categorical entropies in [Lemma 10](#lem-conditional-product), the $x_i\log x_i$ terms cancel against the off-diagonal edge terms. The remaining $p_i$, $y_i$, and $\ell_i$ terms collect to $\mathcal F_{d,h}$, including the root contribution $(d-1)p_0\log p_0$. Since every full feasible profile induces a feasible $q$ and [Lemma 10](#lem-conditional-product) constructs a maximizing profile for every feasible $q$, the variational values coincide. □ +::: + +For activity $z=e^\theta>0$, the grand-canonical functional is + +$$ +\mathcal F_{d,h}^{(z)}(x,\ell)=\mathcal F_{d,h}(x,\ell)+p_0\log z. +$$ + +Define the feasible polytope + +$$ +\mathcal P_{d,h}=\left\{(x,\ell):x_i,\ell_i\ge0,\ \sum_i p_i=1,\ dx_i\ge p_i\ (1\le i\le h)\right\}. +$$ + +The exact variational values are + +$$ +\Psi_{d,h}(\alpha)=\max_{\substack{(x,\ell)\in\mathcal P_{d,h}\\ p_0=\alpha}}\mathcal F_{d,h}(x,\ell),\qquad +\phi_{d,h}(z)=\max_{(x,\ell)\in\mathcal P_{d,h}}\mathcal F_{d,h}^{(z)}(x,\ell)=\max_\alpha\{\Psi_{d,h}(\alpha)+\alpha\log z\}. +$$ + +The polytope is compact, so all maxima exist. + +## The feasible density interval + +The local capacities imply the Moore lower bound. First, + +$$ +p_1\le dx_1\le dp_0. +$$ + +For $i\ge 2$, symmetry and the preceding descent edge give + +$$ +x_i\le p_{i-1}-x_{i-1}\le \frac{d-1}{d}p_{i-1}, +$$ + +so + +$$ +p_i\le dx_i\le (d-1)p_{i-1}. +$$ + +Therefore + +$$ +1=\sum_{i=0}^h p_i\le B_hp_0,\qquad p_0\ge\frac1{B_h}. +$$ + +Equality is feasible in the type polytope: take + +$$ +p_0=\frac1{B_h},\qquad p_i=\frac{d(d-1)^{i-1}}{B_h},\qquad x_i=\frac{(d-1)^{i-1}}{B_h}, +$$ + +with $\ell_i=0$ for $i0\}$, and its positive support is the prefix $\{0,\ldots,k\}$. + +::: {#lem-boundary-repulsion .lemma} +**Lemma 13** (Relative boundary repulsion). *Fix a finite activity $z>0$. A maximizer of $\mathcal F_{d,h}^{(z)}$ cannot lie on any proper local face within its effective horizon. More precisely, if its effective horizon is $k$, then* + +$$ +\ell_i>0\ (0\le i\le k),\qquad a_i>\frac1d\ (1\le i\le k). +$$ + +*In particular, $a_k<1$.* +::: + +::: proof +*Proof.* For each $k\ge 1$, a strictly interior feasible profile exists. Choose $1/d<\chi<1/2$, $00,\qquad \ell_i=p_i-x_i-x_{i+1}>0\ (1\le i0. +$$ + +Mix a proposed boundary maximizer with such an interior profile. If $a_i=1/d$, [Lemma 12](#lem-endpoint-expansions) contributes a strictly positive multiple of $t\log(1/t)$. If $\ell_i=0$, the term $-\tfrac d2\ell_i\log\ell_i$ contributes a strictly positive multiple of $t\log(1/t)$. All terms that remain away from their endpoints change by only $O(t)$. + +The only apparent negative singularity occurs if $a_k=1$, equivalently $y_k=\ell_k=0$. In that case, with $y_k(t)=\beta t+O(t^2)$ for some $\beta>0$, + +$$ +p_k(t)s_d\!\left(1-\frac{y_k(t)}{p_k(t)}\right)+dy_k(t)\log y_k(t)=O(t) +$$ + +by [Lemma 12](#lem-endpoint-expansions); the two logarithmic singularities cancel. The remaining diagonal-edge term contributes + +$$ +-\frac d2y_k(t)\log y_k(t)=\frac d2\beta t\log\frac1t+O(t), +$$ + +which is strictly positive. If several faces are active simultaneously, all leading $t\log(1/t)$ coefficients are nonnegative and at least one is positive. Hence every proper relative boundary point admits an improving inward direction. □ +::: + +The horizon itself is also repelling. + +::: {#lem-horizon-extension .lemma} +**Lemma 14** (Horizon extension). *A maximizer of $\mathcal F_{d,h}^{(z)}$ cannot have effective horizon $k0$, introduce a new level of mass $p_{k+1}=\tau$ with + +$$ +x_{k+1}=a^*\tau,\qquad y_{k+1}=\ell_{k+1}=(1-a^*)\tau. +$$ + +Keep $p_k$ fixed by reducing $\ell_k$ by $a^*\tau$, and preserve total mass by reducing $\ell_0$ by $\tau$. All old variables remain feasible for sufficiently small $\tau$. When $k=0$, take instead + +$$ +p_0=1-\tau,\qquad x_1=a^*\tau,\qquad \ell_0=1-(1+a^*)\tau. +$$ + +The new level contributes + +$$ +\left(1-\frac d2(1-a^*)\right)\tau\log\frac1\tau+O(\tau). +$$ + +The coefficient is positive by the choice of $a^*$. All changes to previously positive coordinates are $O(\tau)$. Thus the extension strictly increases the grand functional. □ +::: + +::: {#thm-full-interiority .theorem} +**Theorem 15** (Full interiority). *For every $d\ge3$, $h\ge1$, and $z>0$, every maximizer of the exact grand-canonical functional $\mathcal F_{d,h}^{(z)}$ has full effective horizon $h$ and satisfies* + +$$ +x_i>0,\qquad \ell_i>0,\qquad a_i>\frac1d +$$ + +*for all applicable indices.* +::: + +::: proof +*Proof.* Combine [Lemma 13](#lem-boundary-repulsion) and [Lemma 14](#lem-horizon-extension). □ +::: + +--- + +# Stationary Messages and Exact KKT Reconstruction {#sec-kkt} + +Recall $b=d-1$. For $1\le i\le h$, let $A_i$ be the cavity weight when the recipient edge already supplies a lower-label neighbor, and let $B_i$ be the weight when it does not. For label zero use $B_0$, and set $A_{h+1}=0$. Define + +$$ +S_0=B_0+A_1,\qquad S_i=B_{i-1}+B_i+A_{i+1}\ (1\le i\le h), +$$ + +where the terminal convention makes $S_h=B_{h-1}+B_h$. The positive stationary system is + +$$ +\kappa B_0=zS_0^b,\qquad \kappa A_i=S_i^b\ (1\le i\le h),\qquad \kappa B_i=S_i^b-(S_i-B_{i-1})^b\ (1\le i\le h). +$$ + +The following subsection derives this system directly from the compact exact functional and proves the converse reconstruction. + +## Full KKT-to-message correspondence + +Let $\lambda_i$ be the minimizer of $s_d(a_i)$'s dual formula. Then + +$$ +s_d'(a_i)=-d\log\lambda_i,\qquad s_d(a_i)-a_is_d'(a_i)=\log((1+\lambda_i)^d-1). +$$ + +At an interior grand-canonical KKT point, let $\mu$ be the multiplier for $\sum_{i=0}^h \ell_i+2\sum_{i=1}^h x_i=1$. Since $\ell_i$ enters one layer mass and $x_i$ enters two, the KKT equations are + +$$ +\frac{\partial \mathcal F^{(z)}}{\partial \ell_i}=\mu,\qquad \frac{\partial \mathcal F^{(z)}}{\partial x_i}=2\mu. +$$ + +The analytic derivatives used below are, at the root, + +$$ +g_0=(d-1)(\log p_0+1)+\log z-\frac d2(\log\ell_0+1), +$$ + +and, for $i\ge1$, + +$$ +g_i=-\log p_i-1+s_d(a_i)-a_is_d'(a_i)+d(\log y_i+1)-\frac d2(\log\ell_i+1). +$$ + +Thus $\partial\mathcal F^{(z)}/\partial\ell_i=g_i$. The derivative with respect to $x_i$ is the sum of the layer contribution immediately to its left and the $x_i$-derivative of the layer-$i$ contribution; explicitly, for $i\ge2$, + +$$ +\frac{\partial\mathcal F^{(z)}}{\partial x_i}=g_{i-1}+\frac d2(\log\ell_{i-1}+1)+g_i+s_d'(a_i)-d(\log y_i+1)+\frac d2(\log\ell_i+1), +$$ + +with the same formula at $i=1$ after replacing the left layer expression by its root analogue. + +Define, up to a common positive scale, + +$$ +B_i=\sqrt{\ell_i},\qquad A_i=\frac{x_i}{B_{i-1}}\ (1\le i\le h),\qquad A_{h+1}=0. +$$ + +Then $y_i=B_i(B_i+A_{i+1})$. + +::: {#lem-edge-kkt .lemma} +**Lemma 16** (Edge KKT identifies the local tilt). *For every $1\le i\le h$,* + +$$ +\lambda_i=\frac{B_{i-1}}{B_i+A_{i+1}}. +$$ +::: + +::: proof +*Proof.* For $i\ge2$, the derivative of $\mathcal F$ with respect to $x_i$ is the sum of the $p_{i-1}$ and $p_i$ contributions. Subtract the two diagonal equations above, use the dual-formula identities for $s_d'$, and cancel the common constants. The result is + +$$ +\frac d2\log\ell_{i-1}-d\log\lambda_i\frac{y_i}{\sqrt{\ell_i}}=0. +$$ + +Thus $\lambda_iy_i=\sqrt{\ell_{i-1}\ell_i}$, which is the displayed identity. The root contribution at $i=1$ has the same algebra, with the activity term cancelling through the root diagonal equation. □ +::: + +::: {#prop-kkt-stationarity .proposition} +**Proposition 17** (Compact KKT equals two-message stationarity). *Every strict interior KKT point determines positive messages satisfying the stationary system above. Conversely, every positive message solution reconstructs a strict interior KKT point by* + +$$ +q_{ii}=\frac{B_i^2}{Z_e},\qquad q_{i-1,i}=\frac{B_{i-1}A_i}{Z_e}. +$$ +::: + +::: proof +*Proof.* Put $T_i=B_i+A_{i+1}$, $S_i=B_{i-1}+T_i$. By [Lemma 16](#lem-edge-kkt), $\lambda_i=B_{i-1}/T_i$. Exponentiating the diagonal KKT equation and using the entropy dual-formula identities shows that one common constant $\kappa>0$ satisfies + +$$ +\kappa=\frac{((1+\lambda_i)^d-1)y_i^d}{p_iB_i^d}=\frac{S_i^d-T_i^d}{p_i} +$$ + +for every $i\ge1$. Hence $p_i=(S_i^d-T_i^d)/\kappa$. The descent marginal formula gives $x_i=a_ip_i=B_{i-1}S_i^b/\kappa$. Since $x_i=B_{i-1}A_i$, this is the stationary equation for $A_i$. Subtracting $x_i$ from $p_i$ gives $y_i=T_i(S_i^b-T_i^b)/\kappa$. Since $y_i=B_iT_i$, this is the stationary equation for $B_i$. At the root, the diagonal KKT equation gives $\kappa=zp_0^b/B_0^d$. Because $p_0=B_0(B_0+A_1)=B_0S_0$, this is the stationary equation for $B_0$. + +Conversely, let positive messages satisfy the stationarity equations. Put $\sigma=Z_e^{-1/2}$ and introduce normalized messages $\widehat A_i=\sigma A_i$, $\widehat B_i=\sigma B_i$, $\widehat\kappa=\sigma^{b-1}\kappa$. Then + +$$ +\widehat\kappa\widehat A_i=\widehat S_i^b,\qquad \widehat\kappa\widehat B_i=\widehat S_i^b-(\widehat S_i-\widehat B_{i-1})^b,\qquad \widehat\kappa\widehat B_0=z\widehat S_0^b. +$$ + +The reconstructed beliefs are simply $\ell_i=\widehat B_i^2$, $x_i=\widehat B_{i-1}\widehat A_i$. They have total mass one by the definition of $Z_e$. Writing $\widehat T_i=\widehat B_i+\widehat A_{i+1}$, $\lambda_i=\widehat B_{i-1}/\widehat T_i$, the message equations give the exact row identities + +$$ +p_i=\frac{\widehat S_i^d-\widehat T_i^d}{\widehat\kappa},\qquad x_i=\frac{\widehat B_{i-1}\widehat S_i^b}{\widehat\kappa},\qquad y_i=\frac{\widehat T_i(\widehat S_i^b-\widehat T_i^b)}{\widehat\kappa}. +$$ + +Consequently $a_i=x_i/p_i$ is exactly the tilted conditioned-binomial marginal with parameter $\lambda_i$, and $\lambda_iy_i=\widehat B_{i-1}\widehat B_i=\sqrt{\ell_{i-1}\ell_i}$. Substitution into the diagonal derivative gives, for every $i\ge1$, $g_i=\log\widehat\kappa+(d-2)/2$. The root equation gives the same value for $g_0$. Finally, the displayed edge identity and $s_d'(a_i)=-d\log\lambda_i$ make the difference between the $x_i$ derivative and $g_{i-1}+g_i$ vanish exactly. Thus, with $\mu=\log\widehat\kappa+(d-2)/2$, one has $\partial\mathcal F^{(z)}/\partial\ell_i=\mu$ and $\partial\mathcal F^{(z)}/\partial x_i=2\mu$. Positivity makes the point strict interior, and common message scaling cancels from the beliefs. □ +::: + +## One-dimensional positive stationary locus + +The equations are homogeneous: common message scaling changes $\kappa$ but not $z$ or the reconstructed profile. In the gauge $\kappa=1$, choosing $B_h>0$ determines all preceding messages uniquely by + +$$ +A_i=B_i+(B_i+A_{i+1})^b,\qquad B_{i-1}=A_i^{1/b}-B_i-A_{i+1}, +$$ + +for $i=h,h-1,\ldots,1$. Positivity is automatic because $B_{i-1}=(B_i+A_{i+1})^b+B_i^{1/b}-(B_i+A_{i+1})>0$. Thus the positive stationary locus is one dimensional before the activity is imposed. + +--- + +# Exact Reverse Transfer and Monotone Activity Shooting {#sec-reverse} + +Normalize $A_1=1$ and put + +$$ +\rho_i=\frac{B_i}{A_i},\qquad u_i=\frac{A_{i+1}}{A_i},\qquad v_i=\rho_i+u_i. +$$ + +At the terminal level, $u_h=0$, so $\rho_h=v_h=:s\in(0,1)$. Let $\mathcal D=\{(\rho,v):0<\rho\le v\le 1\}$. + +::: {#prop-reverse-map .proposition} +**Proposition 18** (Exact reverse map). *Given a next-level state $(\rho',v')\in\mathcal D$, define* + +$$ +w=(1-\rho')^{1/b},\qquad e=1-w,\qquad R=v'\frac{e}{w},\qquad M=\left(e+\frac{w}{v'}\right)^b. +$$ + +*Then its unique positive predecessor is* + +$$ +\rho=\frac{R}{R+M},\qquad v=\frac{1+R}{R+M}. +$$ + +*The remaining coordinates are* + +$$ +u=\frac{1}{R+M},\qquad q=\frac{M-1}{R+M}. +$$ + +*The map sends $\mathcal D$ into itself.* +::: + +::: proof +*Proof.* The forward transfer identities imply $\rho/u=v'e/w=R$. The next lower-neighbor fraction also gives + +$$ +e=\frac{\rho(1-\rho)^{1/b}}{u^{1/b}(u+\rho)}. +$$ + +Substituting $\rho=Ru$ and solving yields $u=1/(R+M)$, which gives the displayed formulas for $\rho,v$. Uniqueness follows from the algebraic solution. Since $e+w/v'\ge e+w=1$, we have $M\ge1$, and therefore $0<\rho0,\qquad \frac{\partial}{\partial m}\log z=\frac1m+\frac{b(1-v)}{(1+m)(1+vm)}>0. +$$ + +The quantity $m$ is strictly increasing in $\rho_1$, so [Proposition 19](#prop-order-preservation) completes the proof. □ +::: + +::: {#lem-endpoint-activities .lemma} +**Lemma 21** (Endpoint activities). *For fixed $d,h$,* + +$$ +\lim_{s\downarrow0}z_h(s)=0,\qquad \lim_{s\uparrow1}z_h(s)=\infty. +$$ +::: + +::: proof +*Proof.* For terminal $(\rho',v')=(s,s)$ with $s\downarrow0$, one reverse step has $R=O(s^2)$ and $M=\Theta(s^{-b})$, so the predecessor tends to $(0,0)$. The reverse map is continuous at every interior state and has the displayed boundary limit; induction over the fixed number $h-1$ of reverse steps therefore sends every earlier state to $(0,0)$. The root equation then gives $z\to0$. + +As $s\uparrow1$, one has $w\to0$, $R\to\infty$, and $M\to1$, so one reverse step tends to $(1,1)$. The same finite-step induction sends every earlier state to $(1,1)$. Hence $m\to\infty$, $v_1\to1$, and the root equation gives $z\to\infty$. Continuity of the reverse map and of the root equation also makes $s\mapsto z_h(s)$ continuous. □ +::: + +::: {#thm-unique-stationary .theorem} +**Theorem 22** (Unique positive stationary point). *For every $d\ge3$, $h\ge1$, and $z>0$, the exact grand-canonical stationarity equations have exactly one positive solution up to common message scaling.* +::: + +::: proof +*Proof.* Every positive solution has one terminal parameter $s=\rho_h\in(0,1)$ and is uniquely reconstructed by [Proposition 18](#prop-reverse-map). [Lemma 20](#lem-activity-monotonicity) and [Lemma 21](#lem-endpoint-activities) show that $s\mapsto z_h(s)$ is a strictly increasing bijection from $(0,1)$ to $(0,\infty)$. □ +::: + +For a positive stationary message profile whose reconstructed selected density is $\alpha$, define $\Psi_{d,h}^{\mathrm{stat}}(\alpha):=\mathcal F_{d,h}(x,\ell)$, where $(x,\ell)$ is its normalized compact profile. [Theorem 22](#thm-unique-stationary) shows that this value is single-valued along the positive reverse-transfer branch. + +--- + +# Exact Grand- and Microcanonical Globality {#sec-globality} + +Although $\mathcal F_{d,h}$ is nonconcave, the preceding interiority and uniqueness results determine its global optimizer. + +::: {#thm-grand-canonical-globality .theorem} +**Theorem 23** (Grand-canonical globality). *For every $d\ge3$, $h\ge1$, and $z>0$, the exact grand-canonical functional $\mathcal F_{d,h}^{(z)}$ has a unique global maximizer. It is the positive stationary profile corresponding to the unique terminal parameter $s$ satisfying $z_h(s)=z$.* +::: + +::: proof +*Proof.* A maximizer exists by compactness. [Theorem 15](#thm-full-interiority) places every maximizer in the strict interior, so every maximizer satisfies the interior stationarity equations. [Theorem 22](#thm-unique-stationary) gives only one such point. □ +::: + +The microcanonical problem follows by duality. + +::: {#thm-microcanonical-globality .theorem} +**Theorem 24** (Microcanonical globality). *For every density $1/B_h<\alpha<1$, the exact microcanonical functional has a unique global maximizer, and it is the positive stationary profile on the reverse-transfer branch with selected density $\alpha$. Equivalently,* + +$$ +\Psi_{d,h}(\alpha)=\Psi_{d,h}^{\mathrm{stat}}(\alpha) +$$ + +*throughout the interior feasible interval.* +::: + +::: proof +*Proof.* Write $\theta=\log z$ and $\phi_{d,h}(\theta)=\max_{(x,\ell)\in\mathcal P_{d,h}}\{\mathcal F_{d,h}(x,\ell)+\theta p_0\}$. + +By [Theorem 23](#thm-grand-canonical-globality), the maximizer is unique for every $\theta$. Danskin's theorem [@RockafellarWets] therefore gives $\phi_{d,h}'(\theta)=\alpha(\theta)$, the density of that maximizer. The derivative of a differentiable convex function is continuous here (equivalently, one may use continuity of the unique optimizer). It is also strictly increasing. Indeed, if $\theta_1<\theta_2$ had the same maximizing density, the two optimality inequalities would force each optimizer to maximize at both activities. Grand-canonical uniqueness would make the profiles equal, but the root equation assigns one activity to an interior stationary profile, a contradiction. To identify the endpoint limits, let $\theta_k\to-\infty$ and pass, by compactness, to a convergent subsequence of optimizers. Comparing with a feasible profile of density $1/B_h$ shows that any limit must minimize $p_0$ over the polytope, hence has density $1/B_h$. Similarly, along $\theta_k\to\infty$, comparison with the all-selected profile forces every subsequential limit to have density $1$. Therefore + +$$ +\alpha(\theta)\to\frac1{B_h}\ (\theta\to-\infty),\qquad \alpha(\theta)\to1\ (\theta\to\infty). +$$ + +Hence every interior density is attained. + +Fix $\alpha$ and choose $\theta$ with $\alpha(\theta)=\alpha$. For every profile $P$ of density $\alpha$, $\mathcal F(P)+\theta\alpha\le \mathcal F(P_\theta)+\theta\alpha$, so $P_\theta$ is the microcanonical maximizer. Uniqueness follows from grand-canonical uniqueness. □ +::: + +::: {#cor-envelope-identity .corollary} +**Corollary 25** (Concavity and the envelope identity). *The exact value function $\Psi_{d,h}$ is strictly concave on $(1/B_h,1)$ and differentiable there. If $z(\alpha)$ is the activity exposing density $\alpha$, then* + +$$ +\Psi_{d,h}'(\alpha)=-\log z(\alpha). +$$ +::: + +::: proof +*Proof.* This is the standard differentiable Legendre correspondence produced by the unique maximizers in [Theorem 23](#thm-grand-canonical-globality) and [Theorem 24](#thm-microcanonical-globality). Strict monotonicity of $\alpha(\theta)$ gives strict concavity. □ +::: + +For later use, we collect the two conclusions as + +$$ +\Psi_{d,h}(\alpha)=\Psi_{d,h}^{\mathrm{stat}}(\alpha),\qquad \Psi_{d,h}'(\alpha)=-\log z(\alpha). +$$ + +Explicit positive Hessian directions at stratified profiles do not contradict these theorems: they occur at nonstationary stratified profiles. The compact functional is genuinely nonconcave, but no competing stationary maximum exists. + +--- + +# Corrected Stationary Telescoping and Root Formulas {#sec-telescoping} + +Define + +$$ +Z_v=zS_0^d+\sum_{i=1}^h\left[S_i^d-(S_i-B_{i-1})^d\right],\qquad Z_e=\sum_{i=0}^h B_i^2+2\sum_{i=0}^{h-1}B_iA_{i+1}. +$$ + +The stationary vertex and edge normalizers satisfy the following exact telescoping identity. + +::: {#prop-telescoping .proposition} +**Proposition 26** (Corrected telescoping identity). *Every positive stationary solution satisfies* + +$$ +Z_v=\kappa Z_e. +$$ +::: + +::: proof +*Proof.* For $1\le i\le h$, put $V_i=S_i^d-(S_i-B_{i-1})^d$. Using the stationary equations for $A_i,B_i$, + +$$ +V_i=\kappa\left(B_{i-1}A_i+B_i^2+B_iA_{i+1}\right). +$$ + +The root contribution is $zS_0^d=\kappa(B_0^2+B_0A_1)$. After summation, every square $B_i^2$ appears once and every cross term $B_iA_{i+1}$ appears twice: once from each adjacent vertex contribution, with the root supplying the first copy of $B_0A_1$. This is exactly $\kappa Z_e$. □ +::: + +Normalize $A_1=1$ and write $r=B_0$. [Proposition 26](#prop-telescoping) gives + +$$ +\alpha=\frac{r(1+r)}{Z_e}. +$$ + +The pressure is $\phi=\log Z_v-\tfrac d2\log Z_e$, and the root equation is $\log z=\log\kappa+\log r-(d-1)\log(1+r)$. Eliminating $Z_e$ gives the corrected root-only formulas + +$$ +\phi=\log z+\frac d2\log\frac{1+r}r+\frac{d-2}2\log\alpha, +$$ + +$$ +\Psi=(1-\alpha)\left(\log\kappa-\frac{d-2}2\right)+\alpha\log r+\frac{d-2}2\log\alpha+\left((d-1)\alpha-\frac{d-2}2\right)\log(1+r). +$$ + +The corrected formulas are used throughout this paper. Calculations made directly from $Z_v$ and $Z_e$ are unaffected, provided the power differences are evaluated stably as discussed in [Section 16](#sec-numerical). + +--- + +# Terminal Logarithmic Coordinate and Exact Outer Orbits {#sec-outer} + +Retain the normalization $A_1=1$ and the reverse-transfer coordinates $\rho_i=B_i/A_i$, $u_i=A_{i+1}/A_i$, $v_i=\rho_i+u_i$, $q_i=1-v_i$. At the terminal wall $u_h=0$. We write + +$$ +\rho_h=v_h=s=1-e^{-t},\qquad q_h=e^{-t}. +$$ + +The exact reverse dynamics are those of [Proposition 18](#prop-reverse-map). + +Two boundary orbits are exact. On the **stable boundary** $\rho=0$, the forward map is $u\mapsto u^{1/b}$ and the reverse map is $u'\mapsto (u')^b$. On the **free boundary** $q=0$, the forward map is $u\mapsto u^b$ and the reverse map is $u'\mapsto (u')^{1/b}$. + +For terminal parameter $t$, define the free density + +$$ +a_h(t):=1-e^{-t/b^h}. +$$ + +The exact free orbit with this density is + +$$ +\bar u_i=e^{-t/b^{h-i}},\qquad \bar\rho_i=1-\bar u_i,\qquad \bar q_i=0. +$$ + +--- + +# Uniform One-Step Estimates {#sec-onestep} + +The slack theorem requires only terminal parameters linear in $h$ and uniformly below the critical coefficient. + +::: {#def-subcritical-window .definition} +**Definition 27** (Subcritical terminal window). Fix constants $0<\tau_-\le\tau_+<\dfrac{4\log b}{3D}$. A sequence of terminal parameters is in the *subcritical linear window* if + +$$ +\tau_-h\le t\le \tau_+h. +$$ +::: + +All constants below may depend on $d,\tau_-,\tau_+$ but not on $h$ or $t$ in this window. + +## Free-side perturbations + +At one reverse step, keep $\rho'$ fixed and write $q'=1-v'$. The free predecessor with the same $\rho'$ has $w=(1-\rho')^{1/b}$, $\rho_0=1-w$, $u_0=w$, $q_0=0$. + +::: {#lem-free-side-step .lemma} +**Lemma 28** (Uniform free-side step). *There are $\delta_0,C>0$ such that, whenever $0\le q'\le\delta_0$, the actual predecessor satisfies* + +$$ +1-\rho=w(1+\eta_\rho),\qquad q=bw^2q'(1+\eta_q), +$$ + +*with $|\eta_\rho|+|\eta_q|\le Cq'$. The estimates are uniform for $00$ such that, whenever $0\le\delta\le\delta_1$, the predecessor satisfies* + +$$ +\log u=b\log V+O_d(V\delta),\qquad \frac\rho v=\frac{V^2}b\delta(1+O_d(\delta)). +$$ + +*Put $\widetilde a_i:=\rho_i/(bu_i^D)$. Then* + +$$ +\frac{\widetilde a_i}{\widetilde a_{i+1}}=1+O_d(\delta). +$$ + +*For sufficiently small $\delta_1$, the transverse ratios contract backward:* + +$$ +\frac{\rho_i}{v_i}\le\frac{1+o(1)}b\,\frac{\rho_{i+1}}{v_{i+1}}. +$$ +::: + +::: proof +*Proof.* Here $w=(1-V\delta)^{1/b}$, $e=1-w=(V\delta/b)(1+O_d(\delta))$, so $R=V(e/w)=(V^2\delta/b)(1+O_d(\delta))$. Also + +$$ +e+\frac wV=\frac1V(w+Ve)=\frac1V(1-e(1-V)), +$$ + +whence $M=V^{-b}(1+O_d(V\delta))$. Since $R/M=O_d(V^{b+2}\delta)$, the reverse-map formula for $u$ yields the displayed expansion for $\log u$; moreover $\rho/u=R$, which gives the displayed expansion for $\rho/v$. + +For the transverse-ratio estimate, use $\rho=uR$ and compute $\widetilde a_i/\widetilde a_{i+1}=bRu_i^{1-D}(u')^D/\rho'$. The preceding expansions give $bR/\rho'=V(1+O_d(\delta))$ and $u^{1-D}(u')^D=V^{-1}(1+O_d(\delta))$, because $(b-1)D=b+1$. Their product is $1+O_d(\delta)$. □ +::: + +--- + +# Free Shadowing to an Overlap Layer {#sec-free-shadow} + +For $t$ in the terminal window, choose $H=\lceil 3Dt/(4\log b)\rceil$, $m=h-H$. Then $H,m$ are both linear in $h$. Define the formal free transverse sequence, for $m\le i\le h$, by + +$$ +\bar q_i=b^{h-i}\exp\!\left(-Dt+\frac{2t}{b-1}b^{i-h}\right). +$$ + +It satisfies $\bar q_h=e^{-t}$ and $\bar q_i=b\bar u_i^2\bar q_{i+1}$. + +::: {#prop-free-shadow .proposition} +**Proposition 30** (Uniform free shadow). *There is $c_0>0$ such that, uniformly in the terminal window,* + +$$ +q_i=\bar q_i(1+O(he^{-c_0h})),\qquad 1-\rho_i=\bar u_i(1+O(e^{-c_0h})) +$$ + +*for every $m\le i\le h$. In particular,* + +$$ +q_m=b^H\exp\!\left(-Dt+\frac{2t}{b-1}b^{-H}\right)(1+o(1)),\qquad \rho_m=tb^{-H}(1+o(1)),\qquad \frac{\rho_m}{q_m}=o(1). +$$ + +*All errors are exponentially small in $h$.* +::: + +::: proof +*Proof.* Let $E_i=\log((1-\rho_i)/\bar u_i)$. [Lemma 28](#lem-free-side-step) gives $E_i=\tfrac1bE_{i+1}+O(q_{i+1})$, and + +$$ +\log\frac{q_i}{\bar q_i}=\log\frac{q_{i+1}}{\bar q_{i+1}}+\frac2bE_{i+1}+O(q_{i+1}). +$$ + +Both errors vanish at $i=h$. + +It remains to verify that the formal transverse sequence is uniformly exponentially small. Write $r=h-i$. Its logarithm is + +$$ +f(r)=r\log b-Dt+\frac{2t}{b-1}b^{-r}. +$$ + +The function $f$ is convex, so its maximum on $0\le r\le H$ occurs at an endpoint. At $r=0$, $f(0)=-t$. At $r=H$, the definition of $H$ gives $f(H)\le -\tfrac14Dt+o(1)$. Thus $\max_iq_i\le e^{-c_0h}$. + +A standard first-failure bootstrap now closes. As long as $q_i\le2\bar q_i$, the two displayed recurrences imply $\max_i|E_i|=O(e^{-c_0h})$, $\max_i\log(q_i/\bar q_i)=O(he^{-c_0h})$. For large $h$ these estimates prevent the first failure. This proves the displayed uniform estimates. + +The formula for $q_m$ follows from $1-e^{-t/b^H}=tb^{-H}(1+o(1))$ evaluated at $r=H$. Finally, $\log(\rho_m/q_m)=\log t-2H\log b+Dt+o(h)=-\tfrac12Dt+o(h)$, which proves the last assertion. □ +::: + +--- + +# Stable Shadowing from the Overlap to the Root {#sec-stable-shadow} + +Put $x=-\log u_1$, $\widetilde a=\rho_1/(bu_1^D)$. + +::: {#prop-stable-shadow .proposition} +**Proposition 31** (Uniform stable shadow). *Under the terminal-window assumptions and the matching choice of $H,m$ from [Section 12](#sec-free-shadow),* + +$$ +x=b^{m-1}q_m(1+o(1)),\qquad \widetilde a=\frac{\rho_m}{b^m}(1+o(1)). +$$ + +*If the messages are normalized by $A_1=1$, then* + +$$ +\log A_m=\frac{b}{b-1}\bigl(1-b^{-(m-1)}\bigr)\log u_1+o(1), +$$ + +*and* + +$$ +B_0=\widetilde aA_m^2(1+o(1)). +$$ + +*All errors are uniform and exponentially small apart from the displayed $o(1)$ generated by $q_m$.* +::: + +::: proof +*Proof.* Set $\delta_i=\rho_i/v_i$. [Proposition 30](#prop-free-shadow) gives $\delta_m=o(1)$, and [Lemma 29](#lem-stable-side-step) implies geometric backward contraction. Hence $\sum_{i=2}^m\delta_i=O(\delta_m)$. + +Iteration of the transverse-ratio contraction gives $\widetilde a=\rho_m/(b^mu_m^D)(1+O(\delta_m))$. Since $u_m=1-q_m-\rho_m=1-o(1)$, this proves the displayed formula for $\widetilde a$. + +The stable-step estimate also gives $\log v_i=b\log v_{i+1}+O(\rho_{i+1})$. After iteration, + +$$ +-\log v_1=b^{m-1}[-\log v_m]+O\!\left(\sum_{j=2}^mb^{j-2}\rho_j\right). +$$ + +Backward transverse contraction bounds the error by $O(b^{m-1}\rho_m)$. Since $v_m=1-q_m$ and $\rho_m/q_m=o(1)$, $-\log v_1=b^{m-1}q_m(1+o(1))$. The difference between $-\log u_1$ and $-\log v_1$ is $O(\delta_1)$, proving the displayed formula for $x$. + +Likewise, the stable-side expansion for $\log u$ and the contraction of $\delta_i$ imply $\log u_i=b^{-(i-1)}\log u_1+O(\delta_m)$ uniformly after summation with the geometric weights. Summing over $1\le i0$ such that $\underline\tau=(c_--\delta_-)\log b/D>0$ and $\overline\tau=(c_++\delta_+)\log b/D<4\log b/(3D)$. Uniformly on this broader terminal window, $\alpha B_h=Dt(1+o(1))$. At $t=\underline\tau h$ the resulting density is below $c_-\log B_h/B_h$, while at $t=\overline\tau h$ it is above $c_+\log B_h/B_h$. The terminal parameter increases the activity by [Lemma 20](#lem-activity-monotonicity), and activity increases the selected density by [Theorem 24](#thm-microcanonical-globality); hence every density in the displayed interval is exposed by a terminal parameter in the broader window. The preceding estimate is therefore uniform there. □ +::: + +--- + +# Near-Critical Integration and the Random-Graph Deduction {#sec-integration} + +We first prove [Theorem 2](#thm-near-critical). The pointwise anchor is [Corollary 8](#cor-entropy-anchor). + +::: proof +*Proof of [Theorem 2](#thm-near-critical).* Put $B=B_h$ and $L=\log B$. For a value $C$ in the theorem's hypothesis, set + +$$ +\alpha_-=\frac CB,\qquad C^\sharp=\frac{L+C}2,\qquad \alpha_+=\frac{C^\sharp}B. +$$ + +For large $h$, the interval $[\alpha_-,\alpha_+]$ lies in a fixed density window $\eta L/(2B)\le a\le L/B$, so [Theorem 34](#thm-activity-law) applies uniformly. Exact globality and the envelope identity of [Corollary 25](#cor-envelope-identity) give + +$$ +\Psi(\alpha_-)=\Psi(\alpha_+)+\int_{\alpha_-}^{\alpha_+}\log z(a)\,da. +$$ + +[Corollary 8](#cor-entropy-anchor) and [Theorem 34](#thm-activity-law)'s displayed inequality imply + +$$ +\Psi(\alpha_-)\le H(\alpha_+)-\int_{\alpha_-}^{\alpha_+}\log\frac1a\,da-\frac12\int_{\alpha_-}^{\alpha_+}B(1-a)^Bda +\le H(\alpha_-)-\frac1{2B+1}\left[(1-\alpha_-)^{B+1}-(1-\alpha_+)^{B+1}\right]. +$$ + +The second inequality uses $H'(a)=\log(1/a)+\log(1-a)$. + +Write $W(C):=L-2\log L-C$. By hypothesis $W(C)\ge W_h\to\infty$. Uniformly over the theorem's interval, + +$$ +H(\alpha_-)=O\!\left(\frac{L^2}B\right)=o(e^{-C}), +$$ + +because $e^{-C}=e^{W(C)}L^2/B$. Also $C=O(L)$ gives $(1-\alpha_-)^{B+1}=e^{-C}(1+o(1))$. Finally, $C^\sharp-C=(L-C)/2\ge \log L+W_h/2$, so + +$$ +(1-\alpha_+)^{B+1}=e^{-C^\sharp}(1+o(1))=o(e^{-C}). +$$ + +All estimates are uniform, proving the theorem. □ +::: + +::: proof +*Proof of [Corollary 4](#cor-fixed-fraction-slack).* For $C=cL$ with $c\in[c_-,c_+]$, one has $L-2\log L-C=(1-c)L-2\log L\to\infty$ uniformly. [Theorem 2](#thm-near-critical) gives $e^{-C}=B^{-c}$. □ +::: + +::: proof +*Proof of [Theorem 5](#thm-random-regular).* Set $T_n=nC_h^*/B_h$. For large $n$, $C_h^*>0$. If $T_n\le1$, the real-valued lower bound follows from $\gamma_h(G)\ge1$. Otherwise put $m_n=\lceil T_n\rceil-1$, $C_n=m_nB_h/n$. Then $m_n0$ such that $C_h^*\ge2\eta L_h$ eventually. Then, for large $h$, $\eta L_h\le C_n\le C_h^*$. + +Apply [Theorem 2](#thm-near-critical) with this $\eta$ and the same $W_h$. Since $e^{-C_n}\ge e^{-C_h^*}=e^{W_h}L_h^2/B_h$ and $\tfrac12-o(1)\ge\tfrac13$ for sufficiently large $h$, + +$$ +\log \mathbb E Z_{n,d,h}(m_n)\le -\frac13\frac{ne^{W_h}L_h^2}{B_h}+O_d(h\log n), +$$ + +which tends to $-\infty$ by the growth hypothesis. Markov's inequality shows that the configuration model has no dominating set of size exactly $m_n$ with high probability. Any smaller dominating set could be padded to size $m_n$, so $\gamma_h(G)\ge m_n+1=\lceil T_n\rceil\ge T_n$. Conditioning on simplicity transfers the conclusion to the uniformly random simple $d$-regular graph. □ +::: + +::: proof +*Proof of [Corollary 6](#cor-fixed-fraction-random).* Choose $W_h=\varepsilon L_h-2\log L_h$. Then $W_h\to\infty$, $C_h^*=(1-\varepsilon)L_h$, $C_h^*/L_h=1-\varepsilon>0$, and $e^{W_h}L_h^2/B_h=B_h^{-(1-\varepsilon)}$. Thus [Theorem 5](#thm-random-regular) applies directly. □ +::: + +--- + +# Numerical Verification and Conditioning {#sec-numerical} + +The proofs above do not use numerical evidence. The accompanying package nevertheless checks every stationary identity, the one-step shadowing expansions, the density targeting, and the activity law with controlled-precision arithmetic. + +A cancellation issue discovered during numerical verification is important for reproducibility. Direct evaluation of $\phi=\log Z_v-\tfrac d2\log Z_e$ can be ill-conditioned near criticality: both logarithms are large, and individual vertex terms of the form $S_i^d-(S_i-B_{i-1})^d$ may lose precision. The consolidated solver evaluates these differences as + +$$ +S_i^d\cdot\bigl(-\operatorname{expm1}(d\log(1-B_{i-1}/S_i))\bigr) +$$ + +and reports the microcanonical exponent through the corrected root-only formula of [Section 9](#sec-telescoping). The direct partition-function route is retained as an independent comparison. A row is rejected unless + +$$ +\frac{|\Psi_{Z_v}-\Psi_{\mathrm{root}}|}{|\Psi_{\mathrm{root}}|}\le 10^{-6}. +$$ + +Thus a cross-check residual comparable to the reported quantity gates the record rather than remaining hidden in an internal data structure. + +For the diagnostic choice $W_h=\log\log B_h$, the regenerated cubic values are + +| $h$ | $C$ | $-\Psi/e^{-C}$ | activity-law ratio | +|---:|---:|---:|---:| +| 20 | 6.8451 | 0.6484 | 0.5873 | +| 30 | 12.6345 | 0.9418 | 0.9520 | +| 40 | 18.7408 | 0.9757 | 0.9956 | +| 52 | 26.2980 | 0.9819 | 0.9997 | + +The activity-law ratio is $\bigl(-\log z-\log(1/\alpha)\bigr)/\bigl(B_h(1-\alpha)^{B_h}\bigr)$. The effective onset is slower when $C/\log B_h$ is small, consistent with uniformity only on compact intervals bounded away from zero. Every numerical row records the checker hash, solver hash, precision, stationarity residual, telescoping residual, and the discrepancy between the direct and root-only free-energy routes. + +A companion, more narrative account of this same numerical package — including the code itself and the story of the concavity conjecture it killed — is at [From Path Tubes to a Near-Critical Domination Bound](/essays/growing-radius-domination.html). + +--- + +# Consequences and Open Problems {#sec-open} + +[Theorem 5](#thm-random-regular) proves the first-moment lower transition up to an arbitrary diverging additive term. Three harder problems remain logically separate. + +## The bounded critical window + +The broad-overlap shadowing argument does not determine the $O(1)$ window. The transfer calculations predict + +$$ +\alpha_h^{\mathrm{ann}}B_h=\log B_h-2\log\log B_h+o(1), +$$ + +but proving this requires errors of the same order as the competing entropy and coupon terms. + +## Quenched matching + +The theorem is a first-moment lower bound. A matching upper bound near the annealed zero crossing would require a second moment, small-subgraph conditioning, or an algorithmic construction. In tree-ball geometry, the elementary random-placement-and-patching upper bound is $(1+o(1))n\log B_h/B_h$. Establishing a matching quenched upper bound in the full random-regular growing-radius regime is separate. + +## Direct two-branch asymptotics + +Internally two-path $(h,2)$ domination is defined in [Definition 1](#def-two-path) and is stronger than ordinary distance domination. The present theorem transfers as a lower bound but does not identify its own leading constant or critical correction. A direct branch-message analysis is a separate problem.