Gauge research: assumptions, results and open problems

Read a result at the level where it is established

The section contains several kinds of statement. Some are standard identities of gauge geometry. Some are exact theorems of explicitly declared finite models. Others are conditional reconstructions whose input data must still be obtained from a source law. A constructive field-theory programme additionally needs uniform estimates and a controlled passage through limits.

These levels can support one another without becoming interchangeable. A correct link law does not choose a curvature. An exact finite ground-state energy does not determine a measured coupling. A protected auxiliary spectral band does not establish a physical mass gap. A common spectral-action language does not by itself derive the internal algebra or the chiral particle content. The later supplied manuscripts repeatedly make these distinctions explicit. [Exact loops, v20] [Conditional unification]

Throughout, “conditional” means that the conclusion follows once the named inputs and hypotheses are satisfied. The open problem is often to construct those inputs from a single completion law, or to prove that the necessary bounds persist in the required limit. A counterexample to an omitted condition can therefore be a positive contribution: it identifies the condition the next theorem must contain.

New worked examples and their exact scope

The associated advanced articles supply derivations, counterexamples and proof dependencies. Their exact finite statements do not change the status of the broader physical identification tasks. Proposed next contributions are listed in the research projects.

The central claim ledger

Read the detailed arguments in predictive sufficiency, group reconstruction, the exact singlet model, protected bands and continuum limits. [Completed records] [Vacuum obstruction]

A source-to-observable claim needs one connected construction

Suppose a proposed temporal update is intended to explain a particular gauge theory. The first obligation is to define the source state space and update, including what counts as a completed record. The next is to identify the observable carriers and prove that their comparison descends consistently to records. Carrier dimension, phase structure and any determinant reference must enter as derived objects or explicit assumptions.

The construction must then specify a dynamical operator. A desired loop energy can be engineered in more than one finite Hamiltonian. Showing that it follows from one particular Hamiltonian is valuable, but uniqueness or physical necessity requires an additional argument. Compression into a finite representation space must preserve the relevant links or include the boundary defects it creates.

Finally, define an observable and its unit convention. A spectral interval, coupling or correlation length must come from the same dynamics and normalization. Fitting a new coefficient separately to each requested output removes the predictive connection that a source-level construction is intended to establish. A convincing small result can instead derive one linked family of responses under fixed inputs and show where its corrections enter. [Conditional unification]

Keep each meaning of “gap” explicit

A spectral gap is a separation in a particular operator’s spectrum. In this section the word can refer to the penalty in a finite singlet model, an isolated inner spectral cluster, a charged auxiliary Laplacian, a physical transfer Hamiltonian at one regulator, or a continuum excitation threshold. Naming the operator is essential.

The auxiliary circle illustrates the issue sharply. At fixed twist a charged finite fiber may have a positive minimum. As the twist approaches identity that minimum tends to zero, while neutral centralizer modes can remain exactly zero. This defeats an unconditional floor inferred from compactness alone. The vacuum adds another obstruction: a window selected by an auxiliary multiplication operator need not preserve the physical vacuum complement.

The constructive programme therefore requires a correctly punctured low window, a form-domain argument, a high-block estimate, a relative mixing estimate and control of the regulator dependence. A neutral-sector Poincaré estimate gives a neutral-sector result unless a further estimate handles the full nonneutral complement. The mass-gap article develops these distinctions.

The Clay Mathematics Institute continued to list the four-dimensional Yang–Mills existence and mass-gap problem as unsolved when checked on 18 October 2026. The official statement is linked in the reference library so readers can assess the complete target against the results of a particular finite or regulated model. [Official status] [Problem statement]

What counts as progress

Progress can take several concrete forms. A proof can remove a hypothesis by deriving it. A counterexample can show that a proposed implication needs stronger conditions. A numerical construction can provide controlled evidence in a declared regulator regime, with convergence tests that address an actual limit. An exact solvable model can isolate a mechanism and expose which coefficients it fixes.

The seven explorers support the last of these roles. Their functions, inputs, equations, units and exports are inspectable. They calculate examples rather than simulate the complete SMM/DTT source. The source PDFs are included unchanged, and the reference notes identify formula clarifications or placeholders in earlier drafts. Website revision dates refer to these explanations; they do not confer a new publication status on a manuscript.

A useful research report should state the question, definitions, assumptions, precise conclusion, limiting regime and an independent check. For a negative result, give the smallest example that demonstrates the obstruction. For a numerical result, retain the code and raw parameters needed to repeat it. The collaboration page turns this approach into specific invitations for physicists, mathematicians and philosophers.