Connect gauge structure to the rest of SMONAD

Follow a quantity, not just a topic name

A cross-reference is most useful when it says what is being carried from one section to the next. A local symmetry, a quantum state, a Hamiltonian, a coupling normalization and a measured transition frequency are different objects. They can be connected, but the connection needs a specified construction.

This page offers a map of questions across SMONAD. The linked folders provide broader research context; the gauge guides supply the particular conventions and finite examples used here. Start with one row of the map, reproduce the associated calculation, and write down what must be supplied before the next row becomes a physical conclusion. The reading paths provide sequences within this section, while the project briefs turn several of these connections into possible contributions.

Time, completion and the meaning of a record

The temporal proposal motivates asking how an inner generative process produces an outer record. The gauge argument becomes mathematically definite only after specifying which later tests must be predicted from that record. An endpoint label may be enough for some tests and insufficient when two histories can be coherently compared. The predictive-sufficiency guide expresses that dependence explicitly. [Completed records, v21]

For readers coming from the time section or the Single Monad discussion, keep the six inner generative levels as nested roles leading to the seventh outer completion. The cycle size N in a lattice calculation and the number of repeated projections in an explorer are separately chosen numerical inputs. Changing either number is not a test of how many physical time axes exist.

A useful philosophical contribution can clarify the admitted tests, the status of a prepared reference, and the relationship between a record and an observer's access to it. The reference-protocol article gives explicit operations to examine. This makes the conceptual issue answerable in a declared model while leaving broader ontological questions open to argument.

Quantum theory and shared comparison resources

A shared quantum reference can carry information that is absent from each reduced state. In the Bell-reference example, both one-qubit marginals stay maximally mixed while a channel acting on one member changes joint correlations. This is a concrete bridge to the quantum section: the observable under discussion must be identified as local or joint before deciding what completion has retained.

Open the channel explorer, compare label erasure with reference depolarization, and export a result at a stated rate and time. Reproduce one correlation by multiplying the displayed matrices. Then read the source's stronger channel-level statement, which fixes full local matrix algebras for all relevant inputs. Equality of two reduced states in one experiment is insufficient to establish that hypothesis. [Completed records, v21]

The distinction between amplitude and probability also matters here. The unitary-return guide compares a declared coherent evolution with postselection or penalty-based constructions. These protocols should be connected by a derivation, not grouped together merely because each contains a singlet or a loop.

Geometry, symmetry and representations

The geometry section and symmetry section meet gauge theory at several different levels. A moving projector defines a subbundle. Projected differentiation supplies a connection. Retained tensors restrict its allowed frames. A global quotient restricts which representations can be assigned consistently. The corresponding calculations are related, but none can be replaced by a count of dimensions alone.

Use the fixed-spectrum curvature example to isolate the first issue. The Hamiltonian always has two zero eigenvalues and two eigenvalues at the supplied gap, yet the curvature changes with the graph embedding. The result demonstrates what a spectral list leaves unspecified. The Kato transport guide then develops a canonical comparison for a declared smooth projector family.

For the second issue, follow the determinant line into representation descent. A parallel determinant reference is a global statement; a local curvature condition alone may leave flat holonomy. The representation calculation then uses the actual global group, including its discrete kernel. These are useful cross-checks for any proposed connection to the matter section. [Sevenfold reconstruction, v18]

Hydrogen and the passage from structure to prediction

The hydrogen section offers a focused destination for a theory of interactions: energies, spatial states and transition observables require a definite Hamiltonian and normalization. Within the gauge section, the exact singlet-loop model instead calculates an energy response to supplied links and a supplied penalty. Its response coefficient is not automatically the electromagnetic coupling appearing in an atomic Hamiltonian.

To connect those levels, start with couplings and mixing. Specify the field and charge conventions, the normalization of the kinetic term, the energy or length unit, and the scale at which matching is claimed. State which terms in the proposed atomic Hamiltonian are derived, adopted from a baseline theory, or left as parameters. The finite-model testing guide helps distinguish a correct reproduction of the finite calculation from evidence for a source-to-observable map. [Temporal completion, v20]

A collaboration can focus on one controlled bridge: deriving the relevant low-energy operator, finding an uncertainty bound for a proposed correction, or identifying an observable that distinguishes two models after they share the same fitted inputs. The constants section and derivations section provide the surrounding context. A clear parameter ledger makes such a comparison easier to assess.

Regulated dynamics, derivations and experiments

A finite Hamiltonian can be diagonalized exactly and still leave the continuum programme unresolved. The gauge section follows the regulated construction through vacuum reconstruction, low/high reduction, the vacuum puncture and uniform sector estimates. The proof roadmap gives the dependencies in an order that makes each intermediate result assessable. [Auxiliary-time lattice framework, v4] [Vacuum obstruction, v2]

When moving to experimental questions, identify the operator whose spectrum is being measured and the observable map used in the prediction. An auxiliary fiber eigenvalue, a finite transfer gap and a physical excitation energy enter different constructions. The threshold counterexamples show why a seemingly plausible inequality can fail when compression and nonlinear functional calculus are interchanged.

A useful numerical derivation includes its cutoff, volume, step size, boundary conditions, convergence diagnostic and error criterion. Downloaded explorer results record model inputs and units, but the investigator still has to explain why those inputs describe the intended system. The resolvent guide and sector guide identify stronger analytic questions behind several common convergence claims.

Read, reproduce and contribute

For a first visit, use the reading paths. For a checkable result, choose a worked problem in the exercises and its linked explorer. For source history, consult the twelve-manuscript reference library, then follow related material in papers and books. The manuscripts are preserved as their supplied versions, with website clarifications listed beside the relevant references.

To follow developments, use the research blog, the gauge update log or the gauge RSS feed. Page-level sharing and citation controls provide the canonical URL and a versioned citation. Cite the exact manuscript or guide used for a technical step rather than treating the whole website as one undifferentiated source.

I welcome collaboration with physicists, mathematicians and philosophers who want to test, refine or extend a specific part of the programme. Choose a project brief, describe what you have checked and what remains uncertain, and use the contact page to get in touch.

Continue reading

• Choose a reading route
• Six projects for collaboration
• Hydrogen in SMM/DTT