The Strong Interaction and Hadron Structure in One Medium

Abstract

A companion paper placed a gap of the medium at the pion mass using onlyelectron-anchored quantities, and stopped there. This paper says what sits atthat gap, and builds hadrons out of it. WHAT THE MEDIUM PROVIDES The framework treats a single continuous medium described by one complex field,written as an amplitude times a phase. The field has two branches and theybehave differently. The amplitude branch is gapped, short-ranged and alwaysattractive; we propose it as the carrier of the strong interaction, and withinthis framework it is the only reason several defects can hold together at all.The phase branch is gapless and long-ranged, and carries charge and light. Bothfollow from the same functional; neither is added by hand. THE PION The excitation sitting at the gap is the pion: a defect of the same medium andthe same functional as the electron, in a different class of configuration. Itis not a different kind of object, and not a component of something else in asecondary sense. It is the basic object of this layer, and everything below isbuilt from it. Charge is the winding number of the phase, so a charged pion mustcarry winding +1 or -1, the sign distinguishing the two charge states, while theneutral pion carries winding 0. The neutral pion has no winding to protect itscore, which is why its lifetime is shorter by many orders of magnitude. THE PROTON Five such defects — four carrying winding +1 and one carrying -1 — lock into aconfiguration that cannot be undone. That configuration is the proton. Its massof 1836 electron masses is the sum of five component energies, 5 x 273 electronmasses, plus one locking term, and that term is POSITIVE: the proton is a lockedstate held together by topology against its own repulsion, not a minimum ofenergy. Since the components are fixed by the companion paper, the locking term is theonly quantity left to compute. With no free parameters it gives a proton mass of1771 electron masses against the measured 1836, a deviation of 3.5 percent. Nothing in that chain was adjusted after the target was known. The medium'sconstant is fixed by its own internal relation, the component energy by theclosure count and the frequency sieve, the geometry by the single length theconfiguration possesses, the domain by the configuration span, and the radialprofile by the field equation. The magnetic moment, the charge radius and thepion mass were not used as constraints. CONFINEMENT Confinement dissolves rather than being explained, and the claim is narrowerthan it may sound. We do not account for confinement phenomenology — thespectrum, the string tension, the lattice results. What we claim is that thisframework does not require a mechanism whose purpose is to prohibit fractionalasymptotic particles, because it has no fractional object to prohibit: thecomponents carry integer winding. Pulling on a vertex costs a pair created outof the medium rather than a missing constituent, and what emerges is a pion —which is what the proton is made of. THE NEUTRON The neutron is the complete proton knot with one electron-level -1 passengerlocked into it. This returns the sign of the measured charge radius squared andidentifies the lock energy with the beta-decay Q value. No calculation of that Qvalue is offered here. QUARKS Reading the four vertices two at a time, with the centre on its own, returns theuud charge table term by term with nothing fitted. This is a correspondence andnot an independent result: we show that the composite's own bookkeepingreproduces the table, not that the quark assignment must arise this way. WHAT IS NOT SETTLED One coupled field problem in the overlap region remains unsolved — five bodies,six for the neutron, in three dimensions, under topological constraints. Fourquantities wait behind it: the proton's deformation term, the neutron's magneticmoment, the neutron's charge radius, and the exact parton shares. Its totalcontribution is bounded at order tens of electron masses. They will returntogether, which makes them mutual insurance: four numbers arriving at onceagainst four targets is a stronger test than any of them alone. The framework is a classical field theory throughout. At the classical level thelocked configuration is a true minimum at fixed winding and the second variationis positive under the same inequality that gives existence. Stability againstquantum fluctuations is a different question and is not addressed. The precision, as distinct from the mechanism, is limited by tooling rather thanby the framework: the integration itself is stable to better than 0.05 percent,checked against two independent quadrature schemes, but the profiles entering itare single-defect profiles and the components cannot relax against each otheruntil the coupled problem is solved. A percent-level figure is what this levelof tooling supports, and no claim is made beyond it. KEYWORDS Pion structure; topological defect; proton structure; confinement; neutron;emergent hadron mass; classical field theory. DECLARATION Derivations and numerical work were carried out with substantial assistance froma large language model (Anthropic Claude), under the author's direction andreview. Numerical results reported here are reproducible from scripts archivedwith the research record; those for the proton mass were rebuilt in August 2026,the originals having been lost, and the rebuilt scripts together with the recordof what could and could not be reproduced are archived alongside the rest.