An open chain has a calculable spectrum, but its lengths, couplings and boundary conditions matter. A useful temporal model must derive the observed mass-squared hierarchy and mixing from one formation law, while retaining the coherence required for neutrino interference.
What the neutrino data require
Neutrinos are produced and detected through flavour-dependent weak interactions, while their propagation is described by modes with different masses. The Super-Kamiokande atmospheric-neutrino result was a decisive demonstration of oscillation behavior. In the standard three-flavour framework, an initially prepared flavour is a coherent superposition of mass eigenstates. Relative phases change during propagation, producing a different distribution of detected flavours.
Oscillation measurements determine mass-squared differences and mixing parameters, not a common additive mass-squared offset. Absolute-mass experiments provide a complementary test. The KATRIN collaboration’s 2025 analysis reported an effective electron-neutrino mass below 0.45 eV at 90% confidence, with mass expressed in the usual c = 1 convention. These are different observables, and a structural model should calculate their connection.
The temporal interpretation of an open chain
The short original article in the Chapter III problem survey of Duality of Time suggests that neutrinos have open one-dimensional structures of different lengths. It contrasts them with closed or higher-dimensional particle configurations whose stability imposes different restrictions. This is an appealing way to connect a small mass to a weakly constrained formation pattern.
The chain should be understood as a sequence of dependent source relations, not automatically as a tiny material string stretched along an otherwise unexplained space. Its endpoints, adjacency and length need operational definitions. In the six nested inner stages, such relations can be assembled and retained before the seventh, outward completion makes their response available to an observer.
For a particle observed in three-dimensional space, an internal one-dimensional carrier is compatible with many propagation directions. Internal dimensionality and the trajectory through laboratory space are different properties. Explaining that distinction is essential if the open-chain idea is to reproduce neutrinos arriving from arbitrary directions.
A solvable spectral example
Take an unweighted path of N nodes with free endpoints. Its graph Laplacian LN measures differences between neighboring amplitudes. Introduce an illustrative mass-squared operator
mk²c⁴ = m0²c⁴ + 4ε² sin²(πk / 2N), k = 0, …, N − 1.
Here ε is an energy scale and m0 a possible common mass contribution. The constant-amplitude mode has zero Laplacian eigenvalue. Longer chains allow lower nonzero collective frequencies, illustrating how extended internal coordination can create small splittings without assigning every mode an unrelated parameter.
The smallest three-node example is especially informative: its Laplacian eigenvalues are 0, 1 and 3. Under the stated mass-squared identification it predicts a ratio of smaller to larger splitting equal to 1/3. The measured solar-to-atmospheric splitting ratio is about 0.03 in the NuFit-6.0 analysis. Thus a uniform three-node chain supplies a clear calculation but not the observed hierarchy. Changing the overall scale or adding m0 cannot repair that ratio.
How the same modes produce flavour oscillations
The next ingredient is the interface between formation modes and weak-interaction flavour. Let Uαk be the amplitude with which flavour α couples to internal mass mode k. In vacuum, for a relativistic neutrino with energy E traveling distance L, the standard propagation form is
A DTT chain model should derive U from how completed weak-interaction records couple to the source modes. Choosing arbitrary eigenvalues and an unrelated mixing matrix would retain the two freedoms the proposed geometry was meant to explain. A boundary-coupling rule is one concrete candidate: different flavour interactions could sample different endpoint or collective amplitudes.
The ordinary complex phase in this equation has i² = −1. Its use describes quantum interference; it does not replace the distinct split-complex unit j² = +1 in a hyperbolic temporal geometry. A complete theory must provide the maps relating those two structures.
Which chain structures could work?
Several precise extensions can be investigated. Nonuniform link weights can separate the two oscillation scales. A protected zero or almost-zero mode can make one state especially light. A source-imposed endpoint condition can change the eigenvalue pattern. Each option becomes an explanation only when its weights or boundary rule follow from the common formation dynamics rather than from fitting the desired masses.
Longer chains generally introduce additional modes. The model must say why only three have the measured weak couplings, or predict the extra states and their mixing. Calling the unobserved modes inner does not determine their experimental accessibility. Their coupling matrix does.
Sequential re-creation must also preserve the relevant relative phases. If each completion irreversibly records which mass mode occurred, interference is suppressed. A coherent completion map can retain mode amplitudes while releasing a flavour-sensitive detection record only during interaction. This gives the six-stage construction a definite task: preserve the internal superposition during free propagation.
From a geometric suggestion to a prediction
A focused advance would specify one source-derived chain and calculate its mass-squared splittings, mixing matrix and detector couplings together. The calculation should then determine the effective beta-decay mass mβ² = Σk|Uek|²mk². Whether neutrinos are Dirac or Majorana particles is a separate structural choice involving lepton number; an open chain by itself does not settle it.
The resulting tests are specific: the ordering of masses, the lightest mass, departures from three-flavour unitarity, additional oscillation frequencies, and any calculated loss of coherence. A temporal model that fixes even one of these beyond its fitted inputs would give the original open-chain idea a distinct and useful physical consequence.
Books, papers and further reading
- Duality of Time — Chapter III, neutrino-mass proposal
Original interpretation of neutrinos as open formation chains; Chapter V supplies the underlying temporal postulate.
- Super-Kamiokande Collaboration, Evidence for Oscillation of Atmospheric Neutrinos (1998)
Published in Physical Review Letters 81, 1562–1567; experimental oscillation evidence.
- I. Esteban et al., NuFit-6.0: updated global analysis of three-flavor neutrino oscillations (2024)
Published in JHEP 12 (2024) 216; global mass-squared differences and mixing parameters.
- KATRIN Collaboration, Direct neutrino-mass measurement based on 259 days of KATRIN data (2025)
Published in Science 388, 180–185; a direct absolute-mass constraint distinct from oscillation splittings.