Proton, electron, neutrality, and binding

Four requirements for a hydrogen construction

A hydrogen model must distinguish four tasks. First, identify the constituent sectors. Second, explain how they form a neutral composite. Third, specify the dynamics of their relative motion. Fourth, show that the combined system possesses a stable state below the appropriate dissociation threshold. Each task constrains the model, but none can simply stand in for the next.

Paper 11 organizes the early construction around proton and electron record complexes, compact phase information, compatible neutral records, and a regulated operator acting on these records. The desired physical resonance and its relation to the familiar hydrogen Hilbert space are stated as further targets. [Paper 11, §§5–9 and theorem ledger]

This separation is useful for readers and collaborators. A theorem about a neutral representation can be valid and valuable without already proving an atomic spectrum. Conversely, a successful Coulomb calculation can describe the effective atom while leaving its proposed temporal origin unresolved.

The question on this page is how to connect these ingredients without using a later hydrogen measurement to choose the earlier construction.

Charge cancellation is a representation statement

Consider the elementary compact phase group U(1). Its integer-labelled characters can be written χ_q(e^iθ) = e^iqθ. If a proton sector has q = +1 and an electron sector q = −1, their product transforms trivially under a common phase:

This is a precise expression of total charge neutrality for the supplied representation. It fixes a relative sign and an integer charge unit. It does not determine the conversion between that unit and a physical electromagnetic response, nor does it specify how an interaction depends on separation.

A neutral composite can still contain charged constituents and interact with applied fields. Total charge describes one global response; dipole moments, polarizability, and internal transitions require additional operators and state information. An atomic model must therefore preserve more than the single number zero.

Paper 21 explicitly places the boundary-response correspondence between the neutral charge-character construction and the Coulomb potential. That correspondence must establish the physical normalization of the interaction. [Paper 21, Appendix F.2]

What phase opposition can establish

Paper 11 introduces a pair-phase convention and identifies phase-opposition support by a condition of the form cos Φ_pe < 0. Strong locking restricts the pair phase to a small neighborhood of π, with the tolerance controlled under refinement. These definitions identify compatible portions of the proposed temporal record structure. [Paper 11, §6]

A phase relation becomes a dynamical statement only when a specified update preserves, attracts, or destabilizes it. If a coupling term assigns lower energy to an antipodal configuration, that term and its coefficient must be supplied or derived. Merely labeling antipodal phases as compatible does not establish an energy minimum.

The phase convention must also remain fixed when comparing different states. A phase selected after looking at the desired energy is an input to the calculation. A physical relative phase should be invariant under the allowed changes of local description, or else accompanied by a declared reference that makes it measurable.

Exact opposition also need not select an orientation sign: sin π is zero. An orientation observable needs its own definition and stability analysis rather than an implicit choice of which side of π to use.

Build the joint state space and its interaction

If the constituent spaces are H_p and H_e, the initial joint quantum space is their tensor product. A neutral-sector constraint can select an invariant subspace of this product. One must specify whether this selection is a representation restriction, an energetic penalty, or a measurement outcome; these choices have different dynamics.

The first two terms describe independent constituent evolution. The interaction V_pe is the new ingredient that can alter their relative spectrum and produce correlations associated with binding. Projecting onto a compatible sector may restrict states, but the restriction alone does not supply a physically normalized attractive potential.

A regulated record operator may have a finite set of isolated eigenvalues simply because its state space is finite. To interpret one as a physical atomic resonance requires a mapping to energy, a separated-particle comparison, and stable behavior under refinement. Paper 11 distinguishes its temporal spectral projectors from physical hydrogen energy projectors before the required correspondence is established. [Paper 11]

Separate centre-of-mass and relative motion

Now supply an ordinary nonrelativistic two-body model with masses m_e and m_p and an interaction depending only on r = |r_e − r_p|. Define total mass M, centre-of-mass position R, and relative position r:

The inverse relations are r_e = R + (m_p/M)r and r_p = R − (m_e/M)r. Substituting the corresponding velocities into the total kinetic energy cancels the mixed term. The result is:

The same separation holds for the corresponding quantum differential operators. Hydrogen’s internal nonrelativistic spectrum therefore uses μ, not an exactly immovable proton. This reduced mass is included in Paper 12’s effective Coulomb model. [Paper 12, §§2 and 4]

Define binding against a physical threshold

Set the potential to approach zero at infinite separation. After subtracting the constituent rest energies and separating the centre-of-mass motion, the free relative continuum then begins at zero. A negative, normalizable eigenstate of the internal Hamiltonian lies below this dissociation threshold and represents binding within the supplied model.

There are two different energy scales to keep track of. The ionization energy is the distance from a bound level to the separated-particle continuum. The excitation gap is the distance to another internal level. For the leading Coulomb model, if E₁ = −B, then E₂ = −B/4: the ionization energy is B and the first excitation gap is 3B/4.

A source-level spectral gap has to be connected to one of these physical quantities through the same clock and action normalization used elsewhere. A gap counted in update steps or record-operator units cannot be identified with electronvolts by choosing a scale separately for each atom.

Stability under refinement is equally important: an apparent bound mode should not drift into a regulator artifact as the retained description becomes finer.

Worked example: neutrality with and without attraction

Consider a normalized radial trial state ψ_b(r) = exp(−r/b)/√(πb³), where b is any positive length. In the free relative model, its mean kinetic energy is ħ²/(2μb²). This value stays positive and approaches zero as the packet spreads without limit. Adding opposite charge labels to the free state does not change that conclusion.

Now supply an attractive Coulomb potential −C/r with C > 0. The same trial state gives:

The attraction and gradient cost now select a finite scale. The ground-state proof establishes that this exponential attains the global energy minimum, not merely the minimum within the trial family.

The result also reveals a limit on the inference: every positive C yields this kind of binding. Neither neutrality nor the existence of an atom selects a particular value of C. Determining the physical coupling requires an independent response calculation, as emphasized in the newer hydrogen sequence. [Paper 12]

What the temporal construction must add

A successful SMM/DTT account should specify one relation between constituent records, neutral compatibility, and physical interaction. The same prescription should determine which degrees of freedom survive in the relative state space and how their kinetic and potential coefficients are normalized. The resulting atom could then be compared simultaneously through its energy, size, momentum variance, and transition intervals.

The proton mass cannot be silently replaced by infinity in one calculation and restored only where a comparison becomes sensitive. Likewise, using a neutral phase convention to set charge signs does not derive the magnitudes of the constituent masses. Paper 21 keeps these inputs separate so their dependence can be traced. [Paper 21]

Further extensions require their own state spaces and interactions: proton spin for hyperfine structure, spatial charge distribution for finite-size effects, and a radiation field for emission and absorption. The leading two-body problem remains valuable because it gives a common starting point from which these extensions can be assessed.

Continue to the Coulomb-emergence problem for a precise account of the operator and response theorem that would connect neutral temporal structure to this effective atom.

Continue reading

• Nested generation and an observable atom
• Deriving the effective interaction
• The exact Coulomb ground state
• The existing matter guide to hydrogen