Assumptions, results and research status
A guide to the mathematical research programme.
Four kinds of statement
A definition fixes a mathematical object. A theorem proves a consequence of specified hypotheses. A physical identification connects that object to an observation. An independent prediction uses an identified model whose adjustable inputs have been settled without the data being used to assess the prediction. These jobs can occur in the same paper, but they are not interchangeable.
For hydrogen, a normalized Coulomb eigenfunction is an exact result within a stated effective model. Its interpretation as an electron–proton state uses a physical dictionary. Showing that the same operator follows from a temporal source requires an additional construction and a correspondence theorem. Paper 12 explicitly locates its principal open obligation at that last step. [Hydrogen II, sections 2–4]
The summaries below describe the material used in this section. They are intended to help a reader identify a claim's scope and locate a productive next calculation. The source manuscripts remain available with their stated draft versions.
The current dependency map
| Step | Current content | Required advance |
|---|---|---|
| Nested generation | A proposed interpretation of inner generative roles and outer completed records. | A specified autonomous source update and operational readouts appropriate to the atom. |
| Neutral pair | Record variables, charge characters and candidate phase relations. | A physical neutral resonance and its stability under the same construction. |
| Effective state space | The benchmark uses square-integrable complex wavefunctions of relative position. | A justified map and limiting procedure from source records to that state space. |
| Coulomb interaction | An explicitly supplied attractive −αℏc/r potential. | A derivation of its shape and normalization from source response and matter coupling. |
| Ground state | Exact lower bound, normalized equality state, radial distribution and moments. | Explain physical preparation and the source correspondence, beyond stationary persistence. |
| Gross spectrum | The usual bound spectrum under the Coulomb and boundary-condition hypotheses. | Control the source-to-effective-model errors and determine inputs independently. |
| Fine structure | Standard relativistic and spin contributions at a declared order. | Derive the required matter and interaction operators from the temporal construction. |
| Radiative terms | A hierarchy of known contributions and proposed temporal-loop targets. | Calculate physical coefficients, finite terms and uncertainties from an explicit dynamics. |
| Distinctive corrections | Conditional level-shift examples and methods for comparing their shapes. | Obtain a correction operator, fix its parameters independently and establish distinguishability. |
Paper 11 supplies the construction programme, Paper 12 the conditional gross-spectrum chain, and Papers 13–14 the correction and comparison questions. Paper 21 assembles the dependency structure. [I] [II] [III] [Differential programme] [Synthesis]
A completed example inside a declared model
For H = −D∇² − A/r with D,A > 0, set a = 2D/A. Integration by parts gives the quadratic-form identity
The normalized exponential ψ = exp(−r/a)/√(πa³) makes the squared expression vanish. It therefore attains the global minimum E₁ = −A²/(4D). The proof establishes a ground state for every positive A and D. In particular, the existence of binding alone cannot select the physical value of the fine-structure constant.
This is the kind of result the site can present confidently and in detail: a complete statement, a proof that any reader can check and a clear explanation of its inputs. The ground-state article carries out the calculation and its probability consequences.
Keep the origin of constants explicit
An independently obtained dimensionless coupling, an SI unit convention and a measured constituent mass play different roles. The reproducibility record must say which role each input has. A derivation of a relation between observables can be valuable even when a numerical scale is externally calibrated, provided the calibration is stated.
The structural 137 construction supplies a counting or reconstruction result under its own hypotheses. Identifying it with part of an electromagnetic inverse coupling requires a physical matching calculation, including any correction. Choosing that correction to reproduce a hydrogen line would make the line a calibration datum. [Canonical complexification, sections 14–16]
A good source calculation fixes the relevant construction, normalization and parameters first. The subsequent atomic calculation uses that fixed input. This makes it possible to tell whether agreement is an independent consequence, a fitted description or a check of algebra and units. The reproducibility guide gives a practical record format.
The central next theorem
A substantive next result would specify a family of temporal source models, an atomic sector and maps into a physical state space, then establish convergence to a self-adjoint effective Hamiltonian. The same source should determine the clock calibration and interaction normalization used in that limit. Different arbitrary rules placed under one controller would leave their agreement unexplained.
One possible target is Heff = HC + δH, together with a quantitative error estimate for a specified range of states or times. The statement must say whether convergence concerns matrix elements, spectra, resolvents or real-time propagation. These are distinct claims. A finite example is useful evidence of a construction; it does not by itself establish a continuum limit.
A smaller result can still advance this target: prove a canonical neutral sector, obtain a controlled static response, establish a source-derived kinetic coefficient, or demonstrate why two physical preparations share a normalization. Each removes one independent freedom from the model. [Synthesis and theorem chain]
What a distinctive correction must contain
A proposed correction needs more than a small number with the right scale. It needs a specified mechanism, a map to level energies or a perturbation operator, a regime of validity and a comparison that can distinguish its effect from known contributions.
For a nondegenerate level, first-order perturbation theory gives δEa = ⟨a|δH|a⟩. In a degenerate subspace, the perturbation must first be diagonalized there. Every interval then uses the same δE values. An arbitrary independent correction for every observed line discards the shared-state constraints that make spectroscopy informative.
The synthetic residual explorer illustrates this discipline with an explicitly chosen contact-like level model. Its numbers are examples, not measured residuals or an established DTT prediction. The differential article explains how an otherwise correct correction shape can remain indistinguishable from a nuclear-size or other nuisance contribution.
Contribute at a definite point in the chain
Readers are invited to challenge a hypothesis, improve a proof, reproduce a calculation or develop a source mechanism. State the exact equation or section being addressed and identify the proposed output. A well-posed counterexample is useful because it tells the programme which condition is missing.
Visit collaboration opportunities for questions in physics, mathematics and philosophy. Revisions of the website are listed on section updates; source versions and editorial notes appear in the reference library.
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.