Assumptions, results and research status
A guide to the conditional gauge research programme.
What is the research question of this section?
The section asks which relational information a completed record must retain and how specified retention rules can support gauge transport, symmetry groups and explicit interaction models. Its distinctive organizing question concerns the passage from generative histories to observable records. Standard gauge geometry supplies the comparison language; the source-level work is to construct the carriers, references, dynamics and physical matching rather than assume the observed theory. Start with completed records and the status ledger.
Is a gauge transformation a new physical event?
A local change of frame relabels the same underlying states and comparisons when every object transforms consistently. Changing only one matrix while leaving the others fixed can instead change the physical configuration. The transport explorer separates these operations: the two link angles define the loop, while the local-frame parameters change its description. The transformed loop must be conjugate to the original at the same base point.
Why are endpoint descriptions insufficient?
Two paths can share the same endpoint carriers while carrying different transports. If a future coherent comparison distinguishes the paths, a record retaining only the endpoints loses predictive information. This is a sufficiency statement relative to the declared tests. It does not assert that every possible apparatus can detect every phase. The predictive-sufficiency guide states the test-family dependence and the role of a shared reference.
Does the sevenfold structure mean seven independent time axes?
No. The stated SMM/DTT interpretation uses six nested inner generative levels leading to seventh, outer completion. The mathematical carrier construction adds explicit source, recurrence, cut and scalar-extension rules. Numerical controls such as cycle-site count or projection-step count belong to separate examples and can vary. A seven-sample projection loop illustrates one discretization; it does not prove or simulate the full temporal hierarchy. See sevenfold recurrence.
Has the complete Standard Model been derived?
The later supplied manuscripts present conditional reconstructions and exact model results, with explicit remaining tasks. The rank-three/rank-two stabilizer follows once the carriers and parallel product-volume tensor are supplied. A complete physical derivation must also construct the source update, spacetime setting, chiral matter, statistics, dynamics, scales and observed parameters. The group guide and matter guide identify which inputs each result uses.
Why use S(U(3) × U(2)) rather than only a product Lie algebra?
The special product group retains the condition that the product of the two determinants is one. It has Lie algebra su(3) ⊕ su(2) ⊕ u(1), but its global quotient controls which representations are allowed. In the convention q = 6Y, triality t₃ and weak parity t₂ must satisfy q + 2t₃ + 3t₂ ≡ 0 mod 6. The symmetry explorer tests that rule without confusing the group with only its infinitesimal algebra.
Does zero trace curvature fix the determinant reference?
Zero trace curvature makes the determinant connection locally flat under the relevant conventions. It does not necessarily remove nontrivial global flat holonomy. The stabilizer argument uses an actual global parallel product-volume tensor, not only a curvature condition. The source also treats determinant orientation as a reference resource: a covariant process cannot create a noninvariant reference from invariant inputs without supplying the required resource. See group reconstruction.
Does averaging over frames force a singlet?
Coherent averaging of representation amplitudes gives the invariant-vector projector P = ∫R(g)dg. Conjugation averaging of a density matrix gives a different map, ∫R(g)ρR(g)†dg. The second map can leave a mixed state supported on a nontrivial irreducible representation, so it does not generally select the singlet-vector subspace. The singlet-loop article derives both operations and explains which appears in the model.
What is exact about the singlet explorer?
For the declared finite four-position Hamiltonian and the selected holonomy family, the secular equation determines the ground energy and the excitation gap. The tool solves the physical root in a rigorous analytic bracket, up to floating-point numerical precision. The leading weak-hopping curve is displayed separately as an approximation. The exactness refers to the specified model, not to a source-level derivation of its Hamiltonian or to the real-world gauge vacuum. See the full derivation.
Are the response sliders measured physical couplings?
They are declared model inputs or calculated finite-model responses in stated units. To identify them with a physical gauge coupling, the action, generator traces, matter convention, electric terms, matching scale and quantum corrections must be fixed consistently. A relation among block responses can be a restrictive result within a model class without fixing a measured coupling. The coupling article explains the normalization and matching steps.
If P(dP)P = 0, how can a projected connection have curvature?
The connection acts on a moving section s = Ps by ∇s = P ds. The derivative includes how the section changes with the varying subspace. A local orthonormal frame u gives A = u†du, while two derivatives involve excursions through the complementary subspace. The curvature is P(dP ∧ dP)P, which can be nonzero although each sandwiched single derivative vanishes. See spectral bundles and geometric phases.
Why can the phase be undefined in the projection tool?
A phase requires a nonzero complex amplitude. At the selected equatorial two-step example, consecutive states are orthogonal and their overlap vanishes, so the postselected sequence has zero survival probability. Assigning a numeric phase would give meaning to the argument of zero. The projection explorer therefore displays “undefined” while retaining the zero probability and the continuous-loop comparison as separate quantities.
Does anomaly cancellation derive the particle spectrum?
It tests the consistency of a supplied chiral matter list. In the initial ledger the non-Abelian representations, multiplicities and all-left-handed convention are already specified. The charge sums then vanish and the stated quotient rule holds. The arithmetic does not select three generations, construct chiral spacetime fields, determine masses or prove uniqueness of hypercharge from no assumptions. The anomaly explorer exposes those inputs so readers can vary them honestly.
Why is an auxiliary spectral gap not a physical mass gap?
The two gaps belong to different operators. A finite auxiliary fiber Laplacian sees a chosen holonomy and regulator. The physical Hamiltonian includes the full configuration space, electric dynamics, vacuum and mixing. Charged fiber minima can soften as holonomy approaches identity, while neutral modes remain. A physical lower bound requires an argument on the actual vacuum-orthogonal states and uniform control in the relevant limits. See the cycle guide and the gap guide.
What does puncturing a low window achieve?
It removes the component of the true vacuum projected into that window, preventing a positive excitation estimate from being demanded on a vector with a completed zero-energy continuation. It does not automatically create a positive bound on the remaining vectors. The punctured space can even be zero-dimensional, as in the smallest explorer example. A useful nontrivial result still needs a form-domain argument, high-block control and a relative mixing estimate. See the obstruction.
Can a very large high block guarantee a low-energy gap?
Only if mixing is controlled relative to it. In a two-block matrix, eliminating the high block subtracts b²/(d − z) from the low block. If b grows like the square root of d, that subtraction can remain order one even as d grows. Positive diagonal blocks alone can coexist with a negative full eigenvalue. The block explorer calculates both the exact eigenvalues and the Schur complement, making the missing condition visible.
Does this section solve the Millennium mass-gap problem?
It presents exact finite results and a conditional constructive programme. The official four-dimensional Yang–Mills target additionally requires a nontrivial continuum quantum theory with the specified physical properties and a positive mass gap. The Clay Mathematics Institute still listed the problem as unsolved at the 18 October 2026 check. The mass-gap guide and continuum guide compare the programme with the full official statement.
Why include earlier drafts and a manuscript skeleton?
They document how the programme developed and make the source material available for inspection. Their version notes state what each can support. A skeleton containing a theorem placeholder is a programme map, not a proof; an early strong claim is read alongside the later explicit hypotheses and counterexamples. The reference library keeps the PDFs unchanged while documenting the formulas and assumptions used in the website explanations.
How can I reproduce the calculations?
Each explorer states its equations and units, provides a default numerical example and exports the current inputs with results as JSON or CSV. The page links to the source derivation. The upload package also includes a numerical verification script and a validation record. Use the worked exercises to check simple identities first, then vary one input and compare with an independent matrix, integral or asymptotic calculation.
How can I contribute or follow updates?
I welcome focused contributions from physics, mathematics and philosophy. Begin with a particular manuscript section or article, state the claim you want to check, and identify a concrete output such as a proof, counterexample or reproducible calculation. The collaboration page lists starting projects and links to current contact details. Follow gauge updates, the section RSS feed, the research blog or email updates.
Do unchanged marginals mean the reference is preserved?
No. The Bell-mixture example has fixed maximally mixed marginals while its joint correlations vary. A theorem that a channel fixes full local algebras quantifies over all required operators and inputs; it is stronger than equality of reduced states in one experiment.
Can the same energy spectrum produce different curvature?
Yes, for the supplied graph models. H=Δ(I−P) always has spectrum 0,0,Δ,Δ, while the projector embedding changes its projected curvature. The example demonstrates that an eigenvalue list does not by itself reconstruct a bundle connection.
Is Kato transport automatically a physical time evolution?
Kato transport is defined from a smooth projector family. Relating it to a dynamical adiabatic limit, repeated projection, or observed temporal process requires an explicit evolution law and error estimate. The mathematical transport alone does not make that identification.
Does the new vacuum puncture leave a space to test?
Yes. The three-state example starts with a rank-two low window. Removing its projected vacuum leaves one low direction, and the corrected Schur form has a positive coefficient there under the stated transfer inequalities. This extends the rank-zero diagnostic in the earlier two-state example.
Do the new explorers supply new measured physical predictions?
They supply reproducible calculations of explicitly declared finite models. A physical prediction also requires a source-to-observable map, calibrated units, appropriate dynamics and controlled errors. The project briefs identify concrete work that could establish such a bridge.
Can I contribute a narrow result or a counterexample?
Yes. An independent calculation, a precise counterexample, an improved norm bound, or a well-specified account of the future-test algebra can be useful. State the exact source version, assumptions, method and result in the research note template, then use the contact page.
Sources for this guide
- Completed records, v21
- Temporal completion and exact loops, v20
- Sevenfold reconstruction, v18
- Vacuum obstruction and form Feshbach, v2
- Clay Mathematics Institute: current problem status
Source manuscripts, version notes, and standard references are listed in the reference library.