PUH Theorem 340 (The Angular Eigenvalue Sources the Newtonian Tail) — The Constraint Makes the Field a Sigma Model, Which Rederives C/r⁴ Independently; and the Framework's Own Three-Fold A₂ Structure Forbids l < 3, Fixing the Tail Coefficient at 4

Photonic Universe Hypothesis (PUH) — Structural Resolution. THE CONFLICT. T299's metric identification requires the tension's departure from flatness to fall as 1/r, which reproducing Newtonian gravity demands. T329's integration of the lattice field equations gives r^(−3.17), reproduced across three independent starting configurations to within 0.036 and therefore structural. T335 located the source — T327's first integral |∇φ|² = C/r⁴ — showed it holds ONLY in spherical symmetry, and noted the framework's cores rotate, so the conflict sits in a configuration the framework does not admit. It left the rotating case unattempted, believing it blocked on T303's circulation sector being unparametrised. AN OBSTACLE THAT NEVER EXISTED. That belief was wrong. T303 states what the sector is: the order parameter has "modulus … the lattice pressure amplitude" and "argument … a U(1) phase," with cores as "topological point defects in that order parameter with integer winding number." A MODULUS, A PHASE AND AN INTEGER WINDING IS A COMPLEX FIELD WITH A VORTEX — Ψ(r,θ,φ) = A(r,θ)e^(inφ) — a fully specified object. What T303 records as unshown is quite different: that the sl(2,ℝ) located there "IS the Ehlers group … rather than merely occupying the correct slot," and "any derivation, as opposed to identification, of either." THE PARAMETRISATION WAS AVAILABLE THE WHOLE TIME. A WRONG ROUTE, RECORDED. The vortex ansatz appears to supply the missing source immediately: factoring the phase out of the Laplacian leaves a centrifugal term −n²A/(r²sin²θ), proportional to the squared winding number over the squared radius — exactly the r^(−2) power T335 identified as producing the required tail. THAT TERM CANCELS. The constraint Σζ_k I_k = 9E_P holds at every point and the Casimirs are homogeneous, so the constraint is one equation in one unknown for the magnitude, whose solution is a CONSTANT; hence ∂_r|φ|² = 0 and φ_r·φ = 0 identically. The centrifugal term multiplies φ rather than φ_θ, so it contracts against a vanishing quantity. THE WINDING NUMBER DOES NOT ENTER THE RADIAL EQUATION AT ALL. THEOREM 340.1. The same fact makes the problem tractable: |φ| pinned to a constant means the field takes values on a fixed-radius sphere in the 248-dimensional space — THE THEORY IS A SIGMA MODEL. Harmonic maps into fixed-radius targets have a standard spherically symmetric first integral: writing the field as a geodesic parametrised by arclength s(r) with |∇φ|² = (s′)², the harmonic-map equation reduces to (r²s′)′ = 0, hence s′ = C/r² and |∇φ|² = C²/r⁴ — WHICH IS T327'S FIRST INTEGRAL, REDERIVED INDEPENDENTLY. Two routes to the same law, and the second makes the conflict SHARPER: the C/r⁴ law is not a peculiarity of E8 but what every spherically symmetric harmonic map into a fixed-radius target does, so no adjustment of couplings, invariant basis or reference configuration can change the exponent — none of them enters the derivation. THEOREM 340.2. What survives the constraint is the SEPARATION CONSTANT. With φ(r,θ) = f(r)Y(θ) the radial equation becomes u′ + (4/r)u = l(l+1)/r² for u ≡ |φ_r|², whose general solution is u(r) = C₁/r⁴ + l(l+1)/(3r). THE PARTICULAR PART FALLS AS 1/r — the tail the identification requires and the spherical case cannot produce — with coefficient l(l+1)/3. This is NOT the term that cancelled: the winding number multiplies φ and dies against the constraint; the separation constant multiplies φ_θ, which the constraint does not annihilate — precisely the cross terms T332 identified. THEOREM 340.3. The archive states that the A₂ Weyl group — "the symmetry group of the equilateral triangle in the E8 root system" — "imposes a three-fold structure on the angular momentum distribution. The rebound produces three preferred directions, separated by 120°, corresponding to the three roots of A₂," and notes these are "the same Z₃ structure that produces the three lepton" generations. Three-fold azimuthal dependence means m = 3, which requires l ≥ 3, forbidding l = 0 (coefficient 0), l = 1 (0.667) and l = 2 (2). THE LOWEST ADMISSIBLE EIGENVALUE IS 12 AND THE TAIL COEFFICIENT IS EXACTLY 4 — not a free parameter but an integer fixed by the same symmetry that fixes the number of lepton generations. ONE CHECK THE STRUCTURE HAD TO PASS: azimuthal dependence is not axisymmetric, which might have invalidated the (r,θ) reduction. It does not — the linear azimuthal factor averages to zero over a full turn, but u = |φ_r|² is QUADRATIC and cos²(3φ) integrates to π rather than zero, so the structure survives with weight π instead of 2π; and m enters only through admissibility, since l(l+1) is independent of m. WHAT THIS SETTLES. THE POWER NOW MATCHES: T299 requires λ − 1 ~ 2GM/rc², and the lattice now yields a 1/r tail with coefficient 4 times the solution's overall normalisation. Before this note the falloff was r^(−3.17) conflicting with 1/r, the source unidentified, the coefficient a free function, and nothing fixed it; after, the 1/r tail is available, the source is the angular eigenvalue, the coefficient is the integer 4, and the A₂ three-fold structure fixes it. WHAT REMAINS IS THE NORMALISATION — whether the coefficient equals 2GM/c² without adjustment is A COMPUTATION, NOT A CHOICE, and is the named successor. Either outcome is useful: if it matches, the framework reproduces Newtonian gravity from its own field equation; if not, there is a definite mismatch with a definite number attached. KILL-CONDITIONS: (i) if the configuration carries several angular modes, the tail is a sum and the coefficient is not a single eigenvalue — not excluded, and the lowest mode dominating is an assumption; (ii) if the three-fold structure belongs to the rebound's ENERGY PARTITION rather than a core's angular dependence, m = 3 does not apply and l = 1 returns, giving 2/3 — the archive uses Z₃ in both senses and Section 5 relies on the angular one, quoted directly; (iii) if the field does not separate as f(r)Y(θ), no single eigenvalue appears; (iv) if the constraint does not pin |φ| to a constant, the sigma-model structure and the winding cancellation fail together. NOT CLAIMED: that the conflict is resolved, since only the exponent is settled; that the axisymmetric solution has been exhibited, only the structure of its radial equation; that l = 3 is the physical mode rather than the lowest admissible; that the sigma-model identification is derived from the constraint's exact form; or that T303's Ehlers-group question has been touched.