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For 0<τ≤10<\tau\le1 the phase sound bound is no larger than at τ=1\tau=1. This changes the action. The τ=0\tau=0 functional below is a variational comparison, not a valid substitution into the eliminated-density PDE. Write

Jτ(r)=∫Hr dμ+C2y2+τ∫b0(rlog⁡r−r+1) dμ,y=∫tr dμ,∫r dμ=1. J_\tau(r)=\int Hr\,d\mu+\frac C2y^2+ \tau\int b_0(r\log r-r+1)\,d\mu,\qquad y=\int tr\,d\mu,\quad \int r\,d\mu=1.

Let hcoh_{\rm co} be the lower convex envelope of HH on [0,T][0,T]. Relaxing densities to unit positive feature measures gives

G0=min⁡0≤y≤T[hco(y)+C2y2]=max⁡λ[min⁡0≤t≤T(H(t)+λt)−λ22C]. G_0=\min_{0\le y\le T}\left[h_{\rm co}(y)+\frac C2y^2\right] =\max_\lambda\left[\min_{0\le t\le T}(H(t)+\lambda t) -\frac{\lambda^2}{2C}\right].

The mean y0y_0 is unique. At least one minimizing measure uses at most two feature values, since the lower boundary of the planar convex hull is a point or line segment. This is a classical charge measure, not a position probability postulate or two-particle construction. One-sided replacement variations give λ0=Cy0∈[0,CT]\lambda_0=Cy_0\in[0,CT] and V0=H+λ0t−m0≥0V_0=H+\lambda_0t-m_0\ge0, where m0=min⁡(H+λ0t)m_0=\min(H+\lambda_0t). Consequently, including endpoint and corner cases,

Jτ(r)−G0=∫V0r dμ+C2(y−y0)2+τ∫b0(rlog⁡r−r+1) dμ≥0. J_\tau(r)-G_0=\int V_0r\,d\mu+ \frac C2(y-y_0)^2+\tau\int b_0(r\log r-r+1)\,d\mu\ge0.

Smooth radial bumps of width hh approximate the at-most-two feature values, with zero-entropy error O(h)O(h) and entropy O(1+∣log⁡h∣)O(1+|\log h|). Mixing a fraction δ\delta of r=1r=1 ensures strict positivity and adds O(δ)O(\delta) energy. Taking h=δ=τh=\delta=\tau gives

0≤min⁡Jτ−G0≤Kτ(1+∣log⁡τ∣),∣yτ−y0∣≤2Kτ(1+∣log⁡τ∣)C, 0\le\min J_\tau-G_0\le K\tau(1+|\log\tau|),\qquad |y_\tau-y_0|\le\sqrt{\frac{2K\tau(1+|\log\tau|)}C}, ∫V0≥arτ dμ≤Kτ(1+∣log⁡τ∣)a(a>0). \int_{V_0\ge a}r_\tau\,d\mu\le \frac{K\tau(1+|\log\tau|)}a\quad(a>0).

Constants depend on this fixed action and preparation. Concentration can occur in the joining layers; no integer-prefix selection or unique shell allocation follows. At each fixed positive τ\tau, compact recovery gives the same infimum over all finite compact-relative supports, without a uniform cutoff or lifespan. Uniform relative density r=1r=1 is optimal exactly when H+CyˉtH+C\bar y t is constant almost everywhere, with yˉ=∫t dμ\bar y=\int t\,d\mu. It fails for the nontrivial controls here and is distinct from uniform participating energy in ordinary spatial volume.

The 72 sampled convex envelopes use three nested grids through 32,768 intervals. Their last refinement is at most 2.618×10−72.618\times10^{-7} per QQ; these are finite sampled minima, not continuum certificates. A separate rigorous calculation instead bounds explicit witnesses. With FB(z)=21z5−35z6+15z7F_B(z)=21z^5-35z^6+15z^7, radial mass between squared radii a,ba,b is FB(b/(1+b))−FB(a/(1+a))F_B(b/(1+b))-F_B(a/(1+a)). Monotonicity of χ\chi gives rigorous first-moment lower and second-moment upper sums, hence j2/j12≤13.59888983367551j_2/j_1^2\le13.59888983367551. The smooth-step bound S′≤2S'\le2 gives Ewall≤ϵπ2μwalls/8E_{\rm wall}\le\epsilon\pi^2\mu_{\rm walls}/8.

For the exact rational candidate feature t∗t_* and share ww from each grid control, 192-bit Arb bounds

E0−EHQ≤ϵπ2μwalls8+w[H(t∗)−D]+Cupper2(wt∗)2<0. \frac{E_0-E_H}{Q}\le \frac{\epsilon\pi^2\mu_{\rm walls}}8+ w[H(t_*)-D]+\frac{C_{\rm upper}}2(wt_*)^2<0.

All 72 specified witnesses are negative; the least negative upper bound is below −9.8861×10−6-9.8861\times10^{-6}. An independent 256-bit calculation

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