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VALIDATION-SCOPE REFRESH · 2026-08-19

The consolidated rerun in REVISED E47 MULTI-BRANCH FORMALISM reproduces $\beta_1=1$, $\Delta t_{\mathrm{safe}}=0.175$, $v^=0.07$, and a 9-dimensional admissible control subspace for the tested 8-agent geometry. In that geometry there are zero active close-pair collision rows, so $\|K_{\mathrm{safe}}u^\|=0$ is exact but vacuous with respect to an active collision barrier. Topological-control validation remains PASS; collision-avoidance enforcement remains unexercised in this configuration.

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Accuracy Update · Search-and-Rescue Swarm Visual

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Drive survey completed · 2026-07-25. The visual is retained as a mission-concept illustration, but its technical claims are now governed by the corrected Python formalism and evidence boundary below.

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Corrected canonical control identity

$E_{\mathrm{swarm}}(t)=\ker\!\begin{bmatrix}K_{\mathrm{safe}}(t)\\(I-P_{\mathrm{harm}}(t))W(t)\end{bmatrix}=\ker K_{\mathrm{safe}}(t)\cap W(t)^{-1}(\ker\Delta_1(\varepsilon^*(t))).$

Here, physical velocity commands live on vertices, topology lives on edge cochains, and the discrete de Rham midpoint map $W:C_0\otimes\mathbb{R}^d\to C_1$ supplies the type-correct bridge. The earlier product $P_{\mathrm{safe}}P_{\mathrm{harm}}$ is not the canonical projector because those operators act on different spaces.

What the image accurately represents

• Distributed local sensing, neighbor exchange, mission decomposition, cross-checking, and recombination.
• Persistence-selected topology, Hodge harmonic coordination, collision/admissibility constraints, execution gating, and certified fallback.
• Human mission supervision and per-epoch certificate logging.

Claims that remain illustrative rather than established

• Autonomous person detection, thermal/LiDAR fusion performance, all-weather capability, GPS accuracy of 1–3 m, energy-efficiency gains, scalability, redundancy under arbitrary drone loss, and faster rescue outcomes.
• Aerodynamic feasibility, radio robustness, hardware safety, regulatory compliance, and field superiority.

Operational gate

$G(t)=\mathbf{1}[\operatorname{pers}1\ge\tau_p,\ \lambda{\mathrm{gap}}(\Delta_1)\ge\tau_\lambda,\ d_{\min}\ge d_{\mathrm{safe}},\ \|K_{\mathrm{safe}}V\|\le\tau_K,\ \beta_1=\beta_1^{\mathrm{target}}].$

When $G=0$, execution returns to a certified fallback such as hover, controlled dispersion, safe landing, or a locally validated formation controller.

Certified timing relation

$\Delta t_{\mathrm{safe}}=\frac{\operatorname{pers}1-\tau_p}{4v{\max}},\qquad v_{\max}^{*}=\frac{\operatorname{pers}1-\tau_p}{4T{\mathrm{mix}}}.$

Evidence status: E0 formal mathematics + E1 Python implementation structure. Physical search-and-rescue claims remain E2+ pending simulation, HIL, and flight trials.

Source anchors: Google Drive documents Persistence-Gated Hodge Swarm Control — A First-Principles Proof from Murmuration Calculus and Murmuration Swarm Drone Python.