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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19656705 |
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| author | Manfredi, Frederick |
| author_facet | Manfredi, Frederick |
| contents | <div class="abstract-body"> <p>Paper I establishes numerically that the Euler–Lagrange minimum of the density-feedback energy E<sub>fb</sub> = K<sub>fb</sub> · J<sub>4</sub> satisfies K<sub>fb</sub>/J<sub>4</sub> = λ/2<sup>1/3</sup> to within O(h<sup>2</sup>) discretisation error (+0.020% on a 256×256 lattice), where λ = φ<sup>6</sup>. This paper addresses the statement analytically in three steps, introducing the generalised functional G[f; λ<sub>eff</sub>] = K<sub>fb</sub>[f] + λ<sub>eff</sub> J<sub>4</sub>[f] and the ratio map H(λ<sub>eff</sub>) := K<sub>fb</sub>[f<sub>λeff</sub>]/J<sub>4</sub>[f<sub>λeff</sub>].</p> <ul class="result-list"> <li><span class="label">Existence</span> For each λ<sub>eff</sub> > 0, G has a critical point f<sub>λeff</sub> in the Q = 2 topological sector, via a mountain-pass argument in the energy space X<sub>2</sub> (Theorem 2.1, subject to a concentration-compactness condition resolved in Paper III via Rellich–Kondrachov compactness).</li> <li><span class="label">Monotonicity and uniqueness</span> H is strictly decreasing with H(0<sup>+</sup>) > 0 and H(λ<sub>eff</sub>) − λ<sub>eff</sub> → −∞, so the fixed-point equation H(λ<sub>eff</sub>*) = λ<sub>eff</sub>* has exactly one solution (Theorem 3.1, subject to a non-degeneracy condition).</li> <li><span class="label">Fixed-point value</span> In the thin-torus approximation, restricting to the Battye–Sutcliffe ansatz f = 2 arctan(C/d), Beta-function integration gives I<sub>4</sub>/I<sub>2a</sub> = 3/(4C<sup>2</sup>) exactly, from which self-consistency gives I<sub>4</sub>/I<sub>2a</sub>|<sub>C=C*</sub> = 2<sup>4/3</sup>/φ<sup>5</sup> (Theorem 4.2, unconditional). For the full 2D EL equation the same fixed-point value is the Linking Scale Conjecture, confirmed numerically to 0.020% and proved completely in Paper III.</li> </ul> <p>The thin-torus proof of this paper is the result used by Paper I to promote its conditional predictions to unconditional status. The Linking Scale Conjecture is stated precisely and its algebraic context — including S-matrix self-duality S<sub>1/2,1/2</sub> = S<sub>0,0</sub>, twist pairing θ<sub>0</sub> θ<sub>1/2</sub> = 1, and fusion count N<sub>fus</sub> = 2 for all k ≥ 2 — is established as the foundation for the full 2D proof in Paper III.</p> </div> |
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| publishDate | 2026 |
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| spellingShingle | HopfionSeries Paper II - BPS Structure and Fixed-Point Theorem for the Density-Feedback Faddeev--Niemi Hopfion Manfredi, Frederick Hopfion Faddeev-Niemi BPS structure fixed-point theorem mountain-pass theorem WZW model fusion rules Linking Scale Conjecture calculus of variations topological soliton thin-torus approximation beta function integrals <div class="abstract-body"> <p>Paper I establishes numerically that the Euler–Lagrange minimum of the density-feedback energy E<sub>fb</sub> = K<sub>fb</sub> · J<sub>4</sub> satisfies K<sub>fb</sub>/J<sub>4</sub> = λ/2<sup>1/3</sup> to within O(h<sup>2</sup>) discretisation error (+0.020% on a 256×256 lattice), where λ = φ<sup>6</sup>. This paper addresses the statement analytically in three steps, introducing the generalised functional G[f; λ<sub>eff</sub>] = K<sub>fb</sub>[f] + λ<sub>eff</sub> J<sub>4</sub>[f] and the ratio map H(λ<sub>eff</sub>) := K<sub>fb</sub>[f<sub>λeff</sub>]/J<sub>4</sub>[f<sub>λeff</sub>].</p> <ul class="result-list"> <li><span class="label">Existence</span> For each λ<sub>eff</sub> > 0, G has a critical point f<sub>λeff</sub> in the Q = 2 topological sector, via a mountain-pass argument in the energy space X<sub>2</sub> (Theorem 2.1, subject to a concentration-compactness condition resolved in Paper III via Rellich–Kondrachov compactness).</li> <li><span class="label">Monotonicity and uniqueness</span> H is strictly decreasing with H(0<sup>+</sup>) > 0 and H(λ<sub>eff</sub>) − λ<sub>eff</sub> → −∞, so the fixed-point equation H(λ<sub>eff</sub>*) = λ<sub>eff</sub>* has exactly one solution (Theorem 3.1, subject to a non-degeneracy condition).</li> <li><span class="label">Fixed-point value</span> In the thin-torus approximation, restricting to the Battye–Sutcliffe ansatz f = 2 arctan(C/d), Beta-function integration gives I<sub>4</sub>/I<sub>2a</sub> = 3/(4C<sup>2</sup>) exactly, from which self-consistency gives I<sub>4</sub>/I<sub>2a</sub>|<sub>C=C*</sub> = 2<sup>4/3</sup>/φ<sup>5</sup> (Theorem 4.2, unconditional). For the full 2D EL equation the same fixed-point value is the Linking Scale Conjecture, confirmed numerically to 0.020% and proved completely in Paper III.</li> </ul> <p>The thin-torus proof of this paper is the result used by Paper I to promote its conditional predictions to unconditional status. The Linking Scale Conjecture is stated precisely and its algebraic context — including S-matrix self-duality S<sub>1/2,1/2</sub> = S<sub>0,0</sub>, twist pairing θ<sub>0</sub> θ<sub>1/2</sub> = 1, and fusion count N<sub>fus</sub> = 2 for all k ≥ 2 — is established as the foundation for the full 2D proof in Paper III.</p> </div> |
| title | HopfionSeries Paper II - BPS Structure and Fixed-Point Theorem for the Density-Feedback Faddeev--Niemi Hopfion |
| topic | Hopfion Faddeev-Niemi BPS structure fixed-point theorem mountain-pass theorem WZW model fusion rules Linking Scale Conjecture calculus of variations topological soliton thin-torus approximation beta function integrals |
| url | https://doi.org/10.5281/zenodo.19656705 |