PUH Theorem 378 — A Static Core Stays at Rest: Massive Rest Is Fixed by the Cartan Involution, the Wall Bends Against Every Descent, and T342's Gravity Result Covers Every Static Core at Rest

Photonic Universe Hypothesis (PUH) — Theorem paper. THE QUESTION. T377 found that the massive states at rest meet T175's Casimir wall on a closed shell, and that T342's Theorem 342.1 — the bound that excludes the multiplier lambda as the gravity — holds for states at rest but is unproven for moving ones. Its OPEN (1) asked whether a static core's field stays at rest. T326 and T327 give the static core: the field's Laplacian equals minus lambda times the constraint gradient, the constraint holds at every point, and in spherical symmetry |grad phi|^2 = C/r^4. Every number in T378 is computed on the E8(−24) build of T373, with T377's sign, T238's couplings and T376's positive metric K_theta. RESULT 378.1 — MASSIVE REST IS THE FIXED SET OF A SYMMETRY. The Cartan involution theta keeps the eight invariants and T376's energy unchanged (computed exactly), so it is a symmetry of the wall. The states it fixes are those of E7 × SU(2): elliptic, with every invariant a sum of even powers of real numbers, so each meets the wall exactly once — the wall there is T377's closed massive shell. No state of the (56,2) meets the wall (60 of 60 checked). The fixed set of a symmetry is totally geodesic (Kobayashi). RESULT 378.2 — A STATIC CORE IS A GEODESIC OF THE WALL. The part of T326's radial equation along the wall says the curve's acceleration within the wall points along its velocity: a spherical static core is a geodesic of the wall, run at speed √C/r² — T340's geodesic reading, with the wall in place of T340's sphere. T327's first integral is that speed, the core outside its edge r0 is an arc of length L = √C/r0, and so C = (r0 L)². Checked from r = 1 to 12: C held to 5.2 × 10⁻¹¹, the constraint to 2.2 × 10⁻¹¹. RESULT 378.3 — A CORE THAT STARTS AT REST NEVER LEAVES. If the core's edge state and direction are massive and at rest, theta maps the solution to itself, so the core stays in E7 × SU(2). Computed in all 248 directions over an arc of length 2: the descent part of the field never exceeds 2.4 × 10⁻¹⁸. By Palais's principle of symmetric criticality, any configuration built only of massive rest states that is stationary among them is a genuine static solution. T329's integrations, on the compact Cartan, are exact rest cores of the full equations. RESULT 378.4 — THE WALL BENDS AGAINST EVERY DESCENT. Proved: at every massive state at rest each of the eight invariants curves up along every rest direction (E7 × SU(2)) and down along every descent direction (the (56,2)), with no mixing — the second change is −d Σ Y_ab Y_ba h_{d−2}(μ_a, μ_b), with Y skew for rest and symmetric for descent. Checked degree by degree at 10 states (200 random directions of each kind per degree, finite differences to 2.9 × 10⁻⁸). The wall's curvature runs +0.0112 to +0.2343 along rest and −0.2250 to −0.0112 along descent at 40 states; every rest–descent curvature is negative (Gauss), so a core at rest has less energy than any nearby core that leans into descent, while a descent tilt of 10⁻³ grows to 2.0 × 10⁻³ by arc length 2. The constraint's Hessian is never singular there (signature 136, 112): T331's exclusion of a rank drop holds in all 248 directions, its stated reason (positivity) along rest directions only. RESULT 378.5 — T342'S GRAVITY RESULT COVERS EVERY STATIC CORE AT REST. Along a spherical static core at rest, lambda = (C/r⁴) × H(t,t)/|G|², a positive factor bounded by T342's own compactness argument, which holds on the massive rest set; over the states computed it lies between 0.0019 and 0.0571. lambda falls exactly as 1/r⁴, so T342's conclusion — "1/r⁴ is not gravity whatever multiplies it" — covers every static core at rest. THE CONDITION. The core's edge must be massive and at rest. A uniform static vacuum must be at rest (T376/T377: only non-precessing states are static), but whether it is massive or one of T377's mixtures at rest is not settled. WHAT CHANGES: T377 OPEN (1) ANSWERED FOR MASSIVE CORES, its T331 check COMPLETED; T342 CONCLUSION COVERS EVERY STATIC CORE AT REST (compactness proof valid on the massive rest set); T331 COMPLETED; T327 EXTENDED (C = (r0 L)²); T329 CONSISTENT; T326 EXTENDED; T340 GEODESIC READING KEPT, TARGET CORRECTED (its 'the constraint pins the field's magnitude to a constant' fails even at rest); T376 EXTENDED. KILL-CONDITIONS: if T377's sign were reversed (no massive state would meet the wall); if the field metric is not K_theta (T376's coset alternative); if static cores are not spherical (Results 378.2 and 378.5 must be redone; 378.3 still holds); if any of T238's couplings were negative. NOT CLAIMED: that a core's edge must be massive and at rest; that the vacuum is uniform or massive; anything about cores between mixtures at rest; that moving cores obey T342's bound; that the rest core is the global energy minimum, or the only core joining its ends; what fixes the end states. OPEN: whether core formation leaves the edge massive and at rest, and whether the vacuum is massive; whether T342's ratio is bounded over moving states; what fixes the end states, and so C; whether a static core between mixtures at rest stays at rest.