Auditor stance: rigorous applied mathematics, no flattery. The goal is correct,
defensible math. Where the original is right, this says so; where it is wrong,
arbitrary, or decorative, this says that too, and proposes a corrected equation.
Compute status (read this first). The isolated Linux sandbox (sympy/numpy) was
unavailable at audit time (VM service failed to start). Therefore **every symbolic
result below is derived by hand with all steps shown, and every numerical claim is
either (a) arithmetic done by hand and labeled "computed-by-hand" or (b) embedded in a
runnable script marked # NEEDS EXECUTION.** Nothing in this document is reported as a
machine-verified number unless it was reproduced by hand. The author's empirical tables
(TSP/SVP) are not re-run here and are treated as unverified.
---
State particle propagates through a discrete lattice; at each partial solution we have:
Here $\lvert A\rvert$ = number of assigned variables, $d$ = total dimension,
$\lVert V\rVert$ = current vector norm (SVP) or path cost proxy (TSP),
$\mathrm{GS}$ = Gram-Schmidt / expected bound, $\tau$ = a scale factor.
---
Write $C = \underbrace{\rho}_{\text{depth}}\cdot\underbrace{\tau}_{\text{scale}}\cdot\underbrace{\frac{R^2}{2-R}}_{g(R)}\cdot\underbrace{(1-\chi)^2}_{h(\chi)}.$
*Hand check of $g$ at endpoints:* $g(1)=1/1=1$; $g(0.01)=0.0001/1.99=5.0251\times10^{-5}$. Ratio $\approx1.99\times10^4$.
Summary of §1.2. The factors that actually rank siblings are $g(R)$ and the $\chi$-gates. $\rho$ and $\tau$ do not affect intra-layer ranking; $\tau$ only moves the absolute pruning floor. $1/(2-R)$ is near-cosmetic. The $\chi$ dependence is duplicated between $C$ and $P$.
1. $\tau$ undefined in dimension and magnitude; only affects the absolute $P$ floor (arbitrary).
2. $\rho$ does not discriminate within a layer (it is depth, not score).
3. $1/(2-R)$ near-cosmetic (varies $\le2\times$).
4. $\chi$ double/triple-counted across $C$, $P$, $X$.
5. Thresholds $0.8$, $10^{-10}$ and steps $\{1.0,0.6,0.1\}$, $\alpha$ are magic numbers (§4.2).
6. GS / $d_{\mathrm{avg}}$ must be estimated; $R$ inherits that estimation error (§4.3).
7. "Born rule / quantum / amplitude" language is not backed by any Hilbert-space object (§4.4).
---
Claim (critique §2). Under the naive formulation
$$R(d)=1-\frac{d}{\alpha\,d_{\mathrm{avg}}},\qquad
\chi(d)=\begin{cases}1.0 & d>d_{\text{best}}\\ 0.6 & d>1.3\,d_{\mathrm{avg}}\\ 0.1 & \text{otherwise}\end{cases}$$
the commit readiness $C(d)$ is strictly decreasing in cumulative cost $d$, hence
$d(p_1)<d(p_2)\iff C(p_1)>C(p_2)$, hence sort-by-$C$ $\equiv$ sort-by-$d$ and the physics
layer is redundant. Verdict: the claim is CORRECT, with one precise condition stated below.
Hold $\rho,\tau>0$ constant (constant within a beam layer). $\chi(d)$ is a step function:
piecewise constant, jumping upward at the two boundaries $d=1.3\,d_{\mathrm{avg}}$ and
$d=d_{\text{best}}$. So $C(d)$ is differentiable within each plateau and has downward
jumps at the boundaries. We prove strict decrease in two parts.
On a plateau, $(1-\chi)^2$ is constant. Write
$$C(d)=\underbrace{\rho\,\tau\,(1-\chi)^2}_{K\;>\;0}\cdot f\big(R(d)\big),\qquad f(R)=\frac{R^2}{2-R}.$$
Step 1 — compute $f'(R)$ (quotient rule):
$$f'(R)=\frac{(2R)(2-R)-R^2(-1)}{(2-R)^2}
=\frac{4R-2R^2+R^2}{(2-R)^2}
=\frac{4R-R^2}{(2-R)^2}
=\frac{R(4-R)}{(2-R)^2}.$$
Step 2 — sign of $f'(R)$ on the domain. For $R\in[0.01,1]$:
$R>0$, $(4-R)>0$, $(2-R)^2>0$ $\Rightarrow$ $f'(R)>0$ strictly. So $f$ is strictly increasing in $R$.
Step 3 — $dR/dd$. $R(d)=1-\dfrac{d}{\alpha d_{\mathrm{avg}}}\Rightarrow \dfrac{dR}{dd}=-\dfrac{1}{\alpha d_{\mathrm{avg}}}<0$ (for $\alpha,d_{\mathrm{avg}}>0$).
Step 4 — chain rule.
$$\frac{dC}{dd}=K\,f'(R)\,\frac{dR}{dd}
=\underbrace{K}_{>0}\cdot\underbrace{\frac{R(4-R)}{(2-R)^2}}_{>0}\cdot\underbrace{\Big(-\frac{1}{\alpha d_{\mathrm{avg}}}\Big)}_{<0}\;<\;0.$$
Hence within every plateau, $C$ is strictly decreasing in $d$. $\;\blacksquare$ (plateau case)
This is exactly the critique's claim $\frac{\partial C}{\partial d}=\frac{\partial C}{\partial R}\frac{\partial R}{\partial d}+\frac{\partial C}{\partial\chi}\frac{\partial\chi}{\partial d}<0$, with the first term made explicit; on a plateau $\frac{\partial\chi}{\partial d}=0$ so only the $R$-term survives, and it is $<0$.
At $d=1.3\,d_{\mathrm{avg}}$, $\chi:0.1\to0.6$, so $(1-\chi)^2:0.81\to0.16$ — a downward jump of $C$ by factor $0.16/0.81\approx0.198$ (other factors continuous). At $d=d_{\text{best}}$, $\chi:0.6\to1.0$, so $(1-\chi)^2:0.16\to0$ — $C$ drops to $0$. Both jumps are downward. Combined with strict decrease inside each plateau, $C(d)$ is strictly decreasing on all of $[0,\infty)$ (no flat or rising segment). $\;\blacksquare$
A strictly decreasing function is injective and order-reversing:
$$d(p_1)<d(p_2)\iff C(p_1)>C(p_2).$$
Sorting ascending by $d$ = sorting descending by $C$ = identical beam slice. Any node killed
by $X>0.8$ or $P<10^{-10}$ sits at the high-$d$ tail and would be dropped by the beam slice
anyway. The physics layer is a strictly-monotone re-encoding of $d$ — provably redundant.
Exact condition for redundancy (stated precisely). Redundancy holds **iff both $R$ and
$\chi$ are (weakly) monotone functions of the single scalar being optimized, $d$, with at
least one strictly monotone**, *and* $\rho,\tau$ are constant across the compared siblings.
Formally: redundancy $\iff$ there exist monotone $R(d)$ (decreasing) and $\chi(d)$
(non-decreasing) such that $C=K\,f(R(d))(1-\chi(d))^2$ is strictly monotone in $d$. The proof
above shows this holds for the naive forms. **It fails the instant $\chi$ depends on a variable
not expressible as a monotone function of $d$** — which is exactly §3.
> Caveat (computed-by-hand, NOT machine-checked): the derivative algebra above is
> elementary and I am confident in it; the sympy cross-check is provided in §6.1 marked
> # NEEDS EXECUTION because the sandbox was down.
---
Claim (critique §3). When $\chi$ is bound to a constraint independent of cost — e.g.
$\chi=0.9$ if the partial TSP tour self-intersects, else small — $C$ becomes non-monotone
in $d$, and this lets SRHT escape local minima that greedy beam search falls into.
Verdict: ANALYTICALLY CORRECT. Here is the mechanism made rigorous.
Now $\chi=\chi_{\text{struct}}(p)$ depends on a structural predicate
$\mathbb{1}[\text{self-intersect}(p)]$ that is not a function of $d(p)$ alone: two partial
tours can have the *same* $d$ with different intersection status, and a *longer* tour can be
non-intersecting while a *shorter* one intersects. Consider two siblings:
Even though $d_B<d_A$ (so $R_B>R_A$, $f(R_B)>f(R_A)$), the contradiction gate can reverse the order:
$$C_A>C_B \iff f(R_A)\cdot0.81 > f(R_B)\cdot0.01 \iff \frac{f(R_B)}{f(R_A)} < 81.$$
Since $f(R_B)/f(R_A)$ is bounded (over the whole domain $f$ spans only $\approx1.99\times10^4$,
but for siblings at comparable depth the ratio is near $1$), the inequality $<81$ holds easily.
**So the shorter-but-intersecting branch $p_B$ is correctly ranked *below* the longer-but-valid
branch $p_A$. $C$ is therefore not** a monotone function of $d$ — it is monotone in $d$ only
*within a fixed $\chi$ class*, and jumps between classes by the constraint predicate. $\blacksquare$
Plain beam search ranks only by $d$ (or by $C$ when $C$ is monotone in $d$, which §2 showed
is the same thing). It therefore *always* prefers $p_B$ (shorter) early, greedily committing to
segments that look cheap now but force a self-crossing later to close the tour — the classic
local trap. The constraint-gated $C$ instead demotes $p_B$ below the $81{:}1$ threshold, so the
beam retains $p_A$, the slightly-longer-but-structurally-viable partial tour, avoiding the
forced late crossover. This is a genuine, non-replicable-by-pure-$d$-ranking behavior. It is the
one defensible contribution of SRHT and must be kept (consistent with the Hardening Plan §1).
> The exact threshold "$81$" is an artifact of the magic steps $\{0.1,0.9\}$:
> $(1-0.1)^2/(1-0.9)^2 = 0.81/0.01 = 81$. With a *continuous* $\chi$ (recommended, §4.2) this
> becomes a smooth trade-off rather than a hard $81{:}1$ cliff.
# NEEDS EXECUTIONThe N=20 / width=20 / multi-seed TSP-with-intersection experiment from critique §3 could not be
run here (no Python sandbox). A clean, self-contained reproduction script is embedded in §6.2,
written in the hlv_pellis_equations.py style (pure functions + a results table + honesty caveat),
and marked # NEEDS EXECUTION. Until it is run, the critique's table (seeds 7/9/13, SRHT ahead by
up to 22.42) is unverified and should be treated as a claim, not a result.
---
Each item: the hole, the corrected equation, why, what it fixes. Items proven to
work (constraint-bound $\chi$, §3) are kept unchanged.
Hole. $C\in[0,\rho\tau]$ with $\tau$ undefined; the absolute value of $P$ (and thus whether
the $P<10^{-10}$ gate fires) depends on an arbitrary scale. $\rho$ doesn't rank siblings; $\tau$
only moves the floor. Cross-instance comparison of $C$ is meaningless.
Corrected equation. Drop $\tau$ from the *score*, separate ranking from gating, and normalize:
$$\boxed{\;\hat C \;=\; \underbrace{\frac{R^2}{2-R}}_{g(R)\in[g(0.01),\,1]}\cdot\;(1-\chi)^2\;\in[0,1]\;}$$
with $g$ optionally renormalized to $[0,1]$ via $\tilde g(R)=\dfrac{g(R)-g(0.01)}{1-g(0.01)}$ if a
true $[0,1]$ score is required. Keep $\rho$ only as an explicit depth/progress term outside
the sibling ranking (e.g. for best-first priority across depths), not inside the per-layer sort.
Why / what it fixes. $\hat C\in[0,1]$ is scale-free and instance-comparable; the gate is no
longer hostage to an arbitrary $\tau$; ranking semantics ($g(R)$ and $\chi$) are isolated from
progress semantics ($\rho$). Nothing that does real work is lost — $\rho,\tau$ never ranked siblings.
Hole. All hand-tuned; each is a quietly fitted degree of freedom.
Corrected formulation.
$\theta$ of its attainable maximum."* Since $\max X = \chi(1-R)\le 0.99$, set the gate as
$$\boxed{\text{prune if } X > \theta\cdot X_{\max},\quad X_{\max}=0.99,\ \theta\in[0,1]\text{ swept.}}$$
$\theta=0.8/0.99\approx0.808$ recovers the original. Report a sweep; a robust method shows a
plateau in performance vs $\theta$, not a peak at $0.8$.
such: prune when $P$ is within machine/round-off of zero, e.g.
$\boxed{P<\varepsilon,\ \varepsilon\sim10^{-12}\!-\!10^{-8}}$, and *demonstrate insensitivity*
across those orders of magnitude (a real result, not a tuned constant). With normalized $\hat C\in[0,1]$,
the floor becomes interpretable as "effectively zero survival probability."
violation measure** in $[0,1]$:
$$\boxed{\chi = \operatorname{clip}_{[0,1]}\!\Big(\frac{v(p)}{v_{\max}}\Big)}$$
where $v(p)$ is a count/severity of violated constraints (e.g. number of self-intersections, or
number of unsatisfied clauses) and $v_{\max}$ a normalizer. This removes three magic numbers and
the arbitrary $81{:}1$ cliff (§3.2), and turns the gate into a smooth penalty.
calibration-robust $R$ (next item, §4.3), eliminating $\alpha$ entirely.
Why / what it fixes. Removes the four most obvious "fitted to the benchmark" surfaces; converts
two of them (gate fraction, numeric floor) into swept, insensitivity-demonstrated quantities; turns
$\chi$ into a principled continuous measure that *also* smooths the §3 escape mechanism.
Hole. $R=\max(0.01,1-\lVert V\rVert/\mathrm{GS})$ requires a good estimate of GS (SVP) or
$d_{\mathrm{avg}}$ (TSP). If GS is misestimated by a factor $\gamma$, $R$ shifts and the whole score
rescales; the $0.01$ floor masks but does not fix this.
Corrected equation (calibration-robust, rank-based). Make $R$ depend only on the **relative
ordering / spread** of sibling costs, which needs no absolute bound:
$$\boxed{\;R = 1 - \frac{\lVert V\rVert - V_{\min}}{V_{\max}-V_{\min}+\epsilon}\;\in[0,1]\;}$$
(min/max taken over the current candidate set; $\epsilon>0$ guards the degenerate equal-cost case).
Alternatively a robust z-score / quantile map. If an *absolute* bound is genuinely available and
trusted (e.g. Gaussian-heuristic for SVP), use $R=1-\lVert V\rVert/\mathrm{GH}$ but **report the
sensitivity of results to a $\pm$ mis-scaling of GH**.
Why / what it fixes. The rank/spread form is invariant to any monotone rescaling of cost and
needs no fragile constant; it cannot be silently mis-calibrated. The $0.01$ floor can be retained or
dropped (now unnecessary since the form is already in $[0,1]$).
> Note: the rank-based $R$ makes the *naive* (cost-bound) regime even more transparently
> redundant — which is fine: it makes the §3 constraint-bound contribution stand out cleanly.
Hole. No Hilbert space, no complex amplitude, no superposition, no measurement. $\chi\in[0,1]$
is a real scalar; $\cos(\chi\pi/2)$ is a smooth real gate equal to $1$ at $\chi=0$ and $0$ at
$\chi=1$. The "Born rule / probability amplitude / phase interference" framing is decorative and
will not survive a referee who knows QM. Additionally, $P$ applies a second $\chi$-gate on top of
$(1-\chi)^2$ already inside $C$ — double-counting.
Honest restatement. $P$ is a classical cosine constraint-gate: a smooth, bounded, monotone
penalty that multiplicatively suppresses a candidate's score as constraint violation rises, hitting
exactly zero at full violation. The cosine shape is a *design choice* (smooth, zero-derivative at both
endpoints).
Corrected equation — single, explicit gate family. Avoid double-counting by applying one gate:
$$\boxed{\;P \;=\; g(R)\cdot G_k(\chi),\qquad G_k(\chi)=(1-\chi)^k,\ k>0\;}$$
and treat the gate shape as a hyperparameter to be *compared*, not asserted. Candidate shapes, all
$=1$ at $\chi=0$ and $=0$ at $\chi=1$:
Hand-computed comparison values (mid-range $\chi=0.5$):
$1-\chi=0.5$; $(1-\chi)^2=0.25$; $(1-\chi)^3=0.125$; $\cos(\pi/4)=0.7071$.
So at moderate violation the cosine is the most permissive of the four ($0.707$ vs $0.25$ for the
quadratic currently *also* sitting inside $C$). Using cosine and $(1-\chi)^2$ together (the original
$P$) yields $0.25\times0.707=0.1768$ at $\chi=0.5$ — a strong, but arbitrary, compound suppression.
Recommendation. Pick one gate. The cosine's zero-derivative at $\chi=0$ is desirable (small
violations are nearly free, matching "soft" constraints); the quadratic's zero-derivative at $\chi=1$
is desirable (graceful approach to pruning). $(1-\chi)^2$ is the simplest with a flat top *near
pruning*. Default recommendation: a single $(1-\chi)^2$ gate (cheapest, already in $C$, smooth at
the pruning end), with cosine and linear reported as ablations. This removes the double-count and the
quantum mislabeling while preserving the smooth-suppression behavior the method actually relies on.
Why / what it fixes. Eliminates indefensible physics vocabulary, removes $\chi$ double-counting,
and makes the gate shape an honest, swept hyperparameter with a tabulated rationale.
$$
\begin{aligned}
\rho &= \frac{\lvert A\rvert}{d} &&\text{(progress scalar; used for cross-depth priority, NOT intra-layer rank)}\\[2pt]
R &= 1-\frac{\lVert V\rVert - V_{\min}}{V_{\max}-V_{\min}+\epsilon}\in[0,1] &&\text{(rank/spread form; no fragile GS constant)}\\[2pt]
\chi &= \operatorname{clip}_{[0,1]}\!\big(v(p)/v_{\max}\big) &&\text{(continuous, constraint-bound — KEEP this binding)}\\[2pt]
g(R) &= \frac{R^2}{2-R} &&\text{(resonance term; }1/(2-R)\text{ optional, near-cosmetic)}\\[2pt]
\hat C &= g(R)\,(1-\chi)^2 \in[0,1] &&\text{(normalized commit readiness; }\tau\text{ removed)}\\[2pt]
P &= \hat C \;=\; g(R)\,(1-\chi)^2 &&\text{(single gate; no double-count, no "Born" label)}\\[2pt]
X &= \chi\,(1-R)\in[0,0.99] &&\text{(unchanged; bounded, well-posed)}\\[2pt]
\text{prune} &\;\text{if}\; X>\theta\,X_{\max}\ \text{or}\ P<\varepsilon &&\theta\in[0,1]\text{ swept},\ \varepsilon\sim10^{-12}\text{–}10^{-8}\text{, insensitivity shown}
\end{aligned}
$$
Kept because proven (§3): $\chi$ bound to constraints *independent of the optimized cost*. That,
not the vocabulary, is the contribution. Everything else above is normalization, de-duplication, and
de-magic-numbering.
---
1. No singularities. $1/(2-R)$ has no pole on $[0.01,1]$ (denominator $\in[1,1.99]$) — confirmed.
2. Endpoints correct. $P=C$ at $\chi=0$, $P=0$ at $\chi=1$ — confirmed (twice over at $\chi=1$).
3. Monotonic-redundancy proof is CORRECT (full hand derivation, §2): $f'(R)=R(4-R)/(2-R)^2>0$,
$dR/dd<0$ $\Rightarrow$ $dC/dd<0$ within plateaus, downward jumps at $\chi$-steps $\Rightarrow$
$C$ strictly decreasing in $d$ $\Rightarrow$ sort-by-$C\equiv$ sort-by-$d$. Redundancy holds iff
both $R,\chi$ are monotone in the single cost $d$ and $\rho,\tau$ constant across siblings.
4. Non-monotone (constraint-bound) claim is CORRECT (§3): constraint-bound $\chi$ makes $C$
non-monotone in $d$; the $(1-\chi)^2$ gate can demote a shorter-but-invalid branch below a
longer-but-valid one (original cliff ratio $81{:}1$), escaping the greedy beam trap. Keep this.
5. Genuine holes: $C$ unnormalized; $\tau$ undefined/arbitrary; $\rho$ doesn't rank siblings;
$1/(2-R)$ near-cosmetic; $\chi$ double/triple-counted; magic numbers $0.8,10^{-10},\{1.0,0.6,0.1\},\alpha$;
GS-estimation fragility in $R$; "Born-rule/quantum" label unsupported.
6. Corrections (§4.5): normalized $\hat C\in[0,1]$; rank-based $R$ (no $\alpha$/GS constant);
continuous constraint-bound $\chi$; single gate $P=g(R)(1-\chi)^2$ (no double-count, honest label);
$\theta$-fraction prune gate; $\varepsilon$ as a stated, insensitivity-demonstrated numeric floor.
Provenance: §2 and §4.4 numbers are computed by hand and stated as such. §3's empirical table and
§6's scripts are # NEEDS EXECUTION (sandbox was unavailable).
---
hlv_pellis_equations.py) — # NEEDS EXECUTION> These were not run (no sympy/numpy sandbox at audit time). They are written to run standalone.
# srht_verify_monotonic.py # NEEDS EXECUTION (sandbox unavailable at audit time)
"""Cross-check the §2 hand derivation: dC/dd < 0 under naive R(d), chi const on a plateau.
HONESTY: this only verifies the *plateau* derivative sign; the boundary jumps are argued
analytically in §2.3 (chi is a step function, not differentiable there)."""
import sympy as sp
d, alpha, davg, rho, tau, chi = sp.symbols('d alpha davg rho tau chi', positive=True)
R = 1 - d/(alpha*davg) # naive resonance
f = R**2/(2 - R) # g(R)
C = rho * f * (1-chi)**2 * tau # commit readiness, chi const on plateau
dCdd = sp.simplify(sp.diff(C, d))
print("dC/dd =", dCdd)
# f'(R) factored form claimed in §2: R(4-R)/(2-R)^2
Rsym = sp.symbols('R', positive=True)
fR = Rsym**2/(2-Rsym)
print("f'(R) =", sp.simplify(sp.diff(fR, Rsym))) # expect R*(4-R)/(2-R)**2
print("f'(R) factored =", sp.factor(sp.diff(fR, Rsym)))
# sign over the domain R in [0.01, 1]: should be strictly positive
print("f'(0.01) =", float(sp.diff(fR,Rsym).subs(Rsym,0.01))) # >0
print("f'(1.0) =", float(sp.diff(fR,Rsym).subs(Rsym,1.0))) # >0
# Expected: dC/dd carries the sign of f'(R)*dR/dd = (+)*(-) < 0 -> redundancy confirmed.
# srht_verify_nonmonotone.py # NEEDS EXECUTION (sandbox unavailable at audit time)
"""Reproduce critique §3: N=20 cities, beam width=20, multiple seeds.
Compares PLAIN BEAM (rank by cost d) vs SRHT-GATE (rank by g(R)*(1-chi)^2 with chi bound
to self-intersection, INDEPENDENT of d). Reports a results table + honesty caveat."""
import numpy as np
from itertools import combinations
def seg_intersect(a,b,c,e):
def ccw(p,q,r): return (r[1]-p[1])*(q[0]-p[0]) > (q[1]-p[1])*(r[0]-p[0])
return ccw(a,c,e)!=ccw(b,c,e) and ccw(a,b,c)!=ccw(a,b,e)
def n_crossings(path, pts):
P=[pts[i] for i in path]; segs=list(zip(P[:-1],P[1:])); c=0
for (i,(a,b)),(j,(cc,e)) in combinations(enumerate(segs),2):
if j<=i+1: continue
if seg_intersect(a,b,cc,e): c+=1
return c
def cost(path, pts):
P=np.array([pts[i] for i in path]); return float(np.sum(np.linalg.norm(np.diff(P,axis=0),axis=1)))
def g(R): return R*R/(2-R)
def beam(pts, width, use_gate, vmax=4):
N=len(pts); beams=[[0]]
for _ in range(N-1):
cand=[]
for path in beams:
for nxt in range(N):
if nxt in path: continue
np_=path+[nxt]; cand.append(np_)
# rank
if not use_gate:
cand.sort(key=lambda p: cost(p,pts)) # plain beam: by distance
else:
costs=[cost(p,pts) for p in cand]
cmin,cmax=min(costs),max(costs)
def score(p):
c=cost(p,pts); R=1-(c-cmin)/(cmax-cmin+1e-9) # rank-based R (§4.3)
chi=min(1.0, n_crossings(p,pts)/vmax) # constraint-bound chi (§4.2)
return g(R)*(1-chi)**2 # normalized Chat (§4.1)
cand.sort(key=score, reverse=True)
beams=cand[:width]
# close tours, pick best by full length
best=min(beams, key=lambda p: cost(p+[p[0]],pts))
return cost(best+[best[0]],pts)
if __name__=="__main__":
print(f"{'seed':>4} {'PlainBeam':>10} {'SRHT-gate':>10} {'diff':>8} winner")
for seed in [7,9,13,21,42]:
rng=np.random.default_rng(seed); pts=rng.uniform(0,100,size=(20,2)).tolist()
pb=beam(pts,20,False); sr=beam(pts,20,True); diff=pb-sr
print(f"{seed:>4} {pb:10.2f} {sr:10.2f} {diff:8.2f} {'SRHT' if sr<pb else 'PLAIN'}")
# CAVEAT: critique reported SRHT ahead by up to 22.42 on seeds 7/9/13. This script
# reproduces the SETUP honestly; actual win/loss must be read from real output.
# A fair audit expects MIXED results across seeds, not a clean sweep.
> Honesty contract. When run, §6.2 is expected to show SRHT-gate *sometimes* better and
> *sometimes* not — the defensible claim is "constraint-bound $\chi$ does work pure-$d$ ranking
> cannot," demonstrated by *any* statistically significant net improvement over many seeds with a
> paired test, not by a cherry-picked 3-seed sweep. Reporting only winning seeds (as the original
> critique table does) is exactly the practice this audit warns against.