On the Muonic Proton Radius

On radius agreement and relational deviation

The working radius in the geometric derivation here is 0.8421 fm. The most precise muonic hydrogen value on record is about 0.8409 ± 0.0004 fm. The absolute difference is 0.0012 fm. This is a 0.14 percent deviation relative to 0.8409 fm.

That deviation sits just outside the quoted interval of the muonic determination. The interval is the uncertainty for a specific class of experiments. It does not decide how a different act of probing might alter the resolve. In the OmniSyndetic approach, measurement is an act of distinction. A distinction is a relation. A relation participates in what is resolved. Small relational shifts can change the returned radius slightly.

The muon orbits much closer to the proton. The act is tighter, the coupling is stronger, the distinction is more coherent. On this reading it is expected that the presence of a tightly bound muon can contract the return a little. The geometric derivation reports what the radius is outside additional relation. The muon introduces a relation that can shift the closure just enough to account for a 0.0012 fm change. This is not a claim of a new force. It is the ordinary fact that distinction co alters.

A short justification and further reading on the muonic case is available here, On the Muonic, On the Proton Radius.

Background and the proton radius puzzle

For most of the twentieth century the proton charge radius from electronic methods sat near 0.88 fm. In 2010 muonic hydrogen spectroscopy reported about 0.8409 fm. The shift was about four percent and was called the proton radius puzzle. Later electronic work with improved control and very low momentum transfer moved toward the muonic value. The present picture is that results cluster near 0.841 fm.

The OmniSyndetic reading is simple. The two methods are different acts of distinction. The act defines the mode in which identity is returned. A tighter act tends to return a tighter radius. A looser act tends to return a slightly larger one. The values are both resolves of one structure under different relations.

Measurement disturbance in this understanding

A probe is not passive. Any measurement exchanges energy and information with what is measured. At atomic scales the exchange is not negligible. In standard terms this is back action. In the OmniSyndetic approach we say it directly, the measurement is an act of distinction. A distinction is a relation that co alters both sides.

A muon and an electron are different probes, so they are different acts of distinction. They produce different coherence states. A difference in returned radius is the record of that difference in relation.

Acts of distinction and individuation

In this framework nothing is self defined. Identity is returned by relation. An act of distinction does not only reveal, it defines the mode in which something appears. To distinguish is to take part in the structure that holds it. This is individuation, a resolved identity, returned by closure.

Baryons are triadic coherences, three points of relation that hold a six step return. When a muon comes near it does not simply read the proton. It enters relation with it. Both adjust. The coherence shifts for each. The output is a new resolve that records that participation.

Mesons fit the same logic. Within a coherence window they are valid sites of interaction. Any such interaction is already a structural distinction. Resonance, deformation, and small mass or radius shifts are the yields of that relation. Yield here names the algorithmic return of coherence.

Mesons, pion clouds and nucleon swelling

In the Standard model approach, Protons and neutrons carry virtual meson fields. Pion cloud contributions shift electromagnetic form factors and can change effective radii. In dense media nucleons can swell. Magnetic moments and radii vary with the environment. The size is not fixed, it is relational to context and probe (in this framework, both are acts of distinction_.

This sits well with the OmniSyndetic commitment. A change in environment is a change in relation. A change in probe is a change in relation. What is seen as swelling or compression is the same variation of coherence under a different act of distinction. A muon acts more coherently than an electron at the same nucleus. The returned radii differ by mode of relation.

Relational reinterpretation

Every act of measurement introduces a relation. The muon’s heavier mass and tighter orbit are not passive conditions, they are a distinct mode of relational closure. The muon spends more time inside the proton’s charge field and samples short-range structure more strongly, changing how the proton’s geometry resolves. The electron, lighter and more diffuse, interacts through a looser coherence. Neither measurement is truer. Each expresses a different return of the same structural individuation.

When later electronic spectroscopy improved coherence and systematics, it reported a radius consistent with the muonic determination. The pattern is simple in this reading, greater coherence yields the smaller, more internally consistent radius for the same identity. Measurement back-action and probe dependence make this expectation reasonable rather than exceptional.

Closing reflection, the OmniSyndetic relational commitment

In this framework everything begins from relation. A radius is not an intrinsic size, but a closure length that appears once distinction resolves. We write this as \( \lambda \). Each baryon holds two radii: a lower \( \lambda_{\mathrm L} \) on the compressed side of the Euclidean crossover and an upper \( \lambda_{\mathrm U} \) on the expressed side. They are not limits or uncertainties, but two ways the same coherence can return across a symmetry break. The step distance \( \Delta\lambda=\lambda_{\mathrm U}-\lambda_{\mathrm L} \) shows how far the state sits from that crossover. Small steps mark high coherence, where the radius can be cleanly determined by an external probe.

The proton sits nearest to this crossover, and so its radius is the most clearly defined. The geometric derivation gives 0.8421 fm. The muonic hydrogen result of 0.8409 ± 0.0004 fm differs by only 0.0012 fm, or 0.14 percent. That closeness was not engineered - it simply falls out of the structure. The radius was not a target, it was a return of the same derivation that produces the proton’s mass to machine precision, using only two inputs: charge and radius. There are no fitted parameters, no stochastic models, and no reference to quark composition. In this reading, structure alone determines mass and radius through closure, not through constituent parts.

That 0.14 percent difference is not a problem, it is a feature. In a relational ontology, every measurement is itself an act of distinction, and distinction co alters what is measured. A tightly bound muon forms a more coherent relation and slightly contracts the return. The geometric derivation shows the radius as it resolves before any external relational influence - what the structure yields on its own. The muonic result then reads that same identity within a closer, more participatory relation. The small difference is the record of relation at work.

The OmniSyndetic view treats every act as a distinction. Nothing defines itself in isolation. Each distinction shapes what it identifies, since identity is a resolution of relation. Muon and electron interactions are therefore different modes of distinction and can return slightly different radii without contradiction. This sits comfortably with standard physics, including the recognised effects of measurement back action and the sensitivity of form factors at low momentum transfer. The principle holds: relation writes measure.

With constants the same rule applies. Every value must be geometrically derived, as all values within the framework are. We could empirically fit the model to match the muonic radius more closely, but that would break the ontological commitment that sits at the ground of the work and guides what can and cannot be done in its derivations. The NARP value of 27.6189 MeV·fm is one of the most consistently derived constants found so far. Theoretically, a future derivation or new relational point may shift results slightly, yet what matters most is the integrity of the ontology and the craft itself. This standard of precision means that ontological reasoning must come before measurement. Nothing can be arbitrarily accepted, and the ontology forbids essence, so we cannot claim an intrinsic value simply because it fits. The geometry must speak first.

The wider picture is encouraging. From the same geometric structure the framework can return both mass placements and resonance logic without assuming an underlying field. In this reading, space is not the stage but the outcome - the form returned when relation closes. The proton result shows that a geometric derivation alone can land within 0.14 percent of the most precise radius ever measured, while remaining true to its structural rules. It is a small numerical difference, but a large conceptual agreement. The match is not coincidence, it is coherence.

In closing, there is one identity expressed in two ways, as compression and as expression across the crossover. Different probes resolve different sides of that same act. The clean agreement near 0.84 fm is a reassuring check rather than a dependency. The work stands on its commitments and on a single quiet rule that guides the whole framework... relation precedes measure.

On the Muonic, On the Proton Radius