Gauge research projects for collaboration

Start from an explicit question

I welcome collaboration with experts in physics, mathematics and philosophy. The most useful starting point is a specific question about the proposed gauge programme: a proof that closes a stated gap, a counterexample that sharpens an assumption, an independent numerical calculation, or a precise account of which completed records are sufficient for a declared class of observations.

The project briefs below are invitations to investigate; they do not describe results already achieved. Each separates the supplied source model from the additional structure that an investigator might propose. Read the linked guide and its source-version note before interpreting a numerical agreement as evidence for a broader physical claim. The research status ledger records the current scope of the finite calculations and the conditional reconstruction programme.

A first contribution can be small. Reproducing a matrix spectrum with a second method, finding the exact point at which a claimed implication fails, or specifying an operational protocol clearly enough to decide an ambiguity can materially improve the programme. Use the research note template to report the object, assumptions, calculation, result and limits together. A useful negative result deserves the same level of documentation as a positive one.

Project 1: specify and test completion of a shared reference

Question. Which algebra of future observations must a completion channel preserve, and what resources make those observations possible? The completed-record manuscript proves statements about unital completely positive maps that fix full carrier algebras. Preserving the marginals of one prepared state is a different, much weaker property. The channel derivation and Bell-reference explorer make the difference calculable. [Completed records, v21]

Starting calculation. Prepare a Bell pair and compare two declared operations: erase an independent classical label while leaving the pair alone, or depolarize one quantum member of the pair. Both reduced states remain maximally mixed in the second experiment, although the joint correlations change. Verify the full density matrix, the reduced states and at least one joint observable with an independent implementation.

Useful contribution. Specify a future-test algebra and a physical implementation of its reference. Then characterize which completion maps preserve that algebra, exactly or to a stated norm error. A theorem should quantify over the inputs it needs. A counterexample should identify a permitted input and test whose statistics contradict the proposed sufficiency. A conceptual contribution can clarify why a proposed observer is entitled to a shared reference or why that resource should be excluded.

Assessment. The result is ready for review when another reader can tell which subsystem is acted on, which preparation is assumed, whether the map is written in the state or observable picture, and which joint tests survive. Calling a process “erasure” without those specifications leaves the central question undecided.

Project 2: separate the information carried by loop readouts

Question. How much of a holonomy conjugacy class is recovered by a particular measurement protocol? A trace amplitude, its real part and its squared modulus generally retain different information. The loop tomography guide and two-angle SU(3) guide develop explicit examples beyond a single diagonal slice.

Starting calculation. Use a diagonal SU(3) matrix with eigenvalues exp(iα), exp(iβ), exp[−i(α+β)]. Compute its complex trace and the characteristic polynomial, then compare the proposed readouts on central elements and on conjugate phases. Check the corresponding singlet-loop energy with the secular equation rather than assuming the lowest-order trace term remains exact at larger hopping. [Temporal completion, v20]

Useful contribution. Design a finite set of permitted readouts that reconstructs the required invariant, or exhibit indistinguishable loops for a proposed incomplete set. Include the reference phase settings, normalization and repetitions needed by the protocol. For a numerical study, map the full two-angle family and report errors in both the weak-hopping approximation and the inverse reconstruction.

Assessment. A successful comparison identifies whether ambiguity comes from conjugacy, complex conjugation, an omitted phase reference, numerical resolution, or a genuine degeneracy of the chosen energy response. An exact finite result can stand on its own while the assignment of that readout to an experiment remains a separate task.

Project 3: reconstruct geometry beyond the spectrum

Question. Which source data select the embedding and transport of an isolated carrier subspace? Eigenvalues and a separating gap do not specify the way its eigenvectors vary. The rank-two graph example gives a fixed spectrum with tunable curvature, and the curvature explorer displays the projectors, Hamiltonian and loop holonomy. This develops the source programme's distinction between protection and geometric reconstruction. [Sevenfold reconstruction, v18]

Starting calculation. Hold the gap fixed while changing the graph embedding parameter. Verify P² = P and the spectrum of H = Δ(I − P). Compute the local curvature F=dA+A∧A from A=u†du, and the ambient component F_xy=P[∂_xP,∂_yP]P; compare the two descriptions through the frame u. Next vary Δ at a fixed embedding and check which geometric quantities remain unchanged. This pair of comparisons isolates the two types of data.

Useful contribution. Derive a projector family from an explicit source operator instead of supplying it by hand. State regularity, rank stability and a quantitative gap over the intended parameter domain. Construct Kato transport, compare it with a stated dynamical or repeated-projection protocol, and bound the difference in the regime claimed.

Assessment. The result should identify the parameter space and explain what, if anything, maps it to observed spacetime. A connection on a laboratory control manifold is already a meaningful object; identifying it with a physical gauge field needs an additional observable and dynamical correspondence. The Kato transport guide gives a precise next calculation.

Project 4: audit global structure and matter assumptions

Question. Which globally defined tensors and representations survive completion? The stabilizer S(U(3) × U(2)) requires a retained parallel product-volume tensor in the stated reconstruction. A connection with zero trace curvature can still have nontrivial determinant holonomy around a noncontractible loop. Work through the determinant-line counterexample before treating a local curvature condition as a global reduction. [Completed records, v21, PDF pp. 12–15]

Starting calculation. Verify the kernel of the map (A,B,z) ↦ (z⁻²A,z³B), then test q + 2t₃ + 3t₂ modulo six on proposed representations. The representation examples and anomaly ledger expose the charge and chirality conventions used for these tests.

Useful contribution. Supply a global determinant reference from source dynamics, classify a precisely chosen family of allowed representations, or examine which additional hypotheses are needed for a uniqueness statement about the matter list. An anomaly calculation must specify its spacetime background assumptions as well as the representation content.

Assessment. Keep group reconstruction, representation descent, local anomaly cancellation and global anomaly questions as separately stated results. Their logical dependencies can then be inspected and improved without silently treating one calculation as a proof of all the others.

Project 5: obtain a useful bound after the vacuum puncture

Question. Can a chosen low window support a positive excitation estimate after its projected vacuum has been removed? The original two-state example exposes the obstruction but leaves a rank-zero punctured low space. The three-state derivation and punctured-window explorer retain one nonzero low direction, making the positive Schur coefficient inspectable. [Vacuum obstruction and form Feshbach reduction, v2]

Starting calculation. Compute every block of the vacuum-corrected threshold form before taking its Schur complement. Verify that the projected vacuum is annihilated, that the remaining coefficient is positive under the declared finite-model inequalities, and that direct diagonalization gives the expected two excited transfer eigenvalues.

Useful contribution. Extend this bookkeeping to a specified regulated Hamiltonian and low window. Give estimates for the high-block inverse and low/high mixing that remain controlled in a named parameter range. If a step fails, produce a counterexample with the same declared hypotheses, rather than changing the window or the operator without recording the change.

Assessment. Track dependence on volume, regulator and physical time step. A bound at one finite matrix size is an exact starting result; a field-theoretic gap requires the further uniformity and limiting construction described in the proof roadmap.

Project 6: make one limiting claim quantitative

Question. Which estimate survives when the cutoff, volume or time step changes? The spectral cutoff guide, character-sector estimates and resolvent guide provide distinct starting points. Choose one limiting operation and name the topology before combining it with another.

Useful contribution. Prove a uniform estimate, identify a necessary order of limits, or construct a family that separates pointwise convergence from the stronger claim under investigation. For a numerical contribution, include a convergence table over several resolutions and a reason that the measured quantity tests the stated operator claim. The source framework's continuum programme supplies the larger target, but each intermediate result must carry its own assumptions. [Auxiliary-time lattice framework, v4]

Assessment. Distinguish finite-dimensional evidence, a mathematical limit and identification with a continuum observable. Report both what the calculation establishes and the next assumption needed. Send a concise research note through the contact page, with the exact source version and any reproducible code or data attached or linked.

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