Foundations
QLM I defines the primitive per-radian layer, the phase-flow law, the saturated Planck tick, and the reduced-action Planck-unit derivation chain.
Quantum-Q · Independent theoretical research
Quantum
Lattice Model.
Exploring the foundations of physics through phase, action, and transport. A research framework built from three Planck-scale primitives.
THE PRIMITIVE SET
ℏ, ℓP, tP
PHASE-FLOW LAW
E = ℏ dθ/dτ
INVARIANT TRANSPORT
c = ℓP/tP
SATURATED THROUGHPUT
EP = ℏ/tP
What follows when reduced action, a spatial increment, and a proper-time tick are taken as the starting point?
The Quantum Lattice Model begins with the primitive set {ℏ, ℓP, tP}. It relates action to phase through dS = ℏ dθ, then defines energy as the rate of that action flow with respect to proper time.
From this starting point, the papers develop a connected account of Planck-scale quantities, density saturation, relativistic kinematics, gravitational routing, and quantum wave dynamics. Each sector builds on the same reduced-action vocabulary.
QLM is an independent theoretical research program. The papers document the framework’s assumptions, derivations, and extensions so readers can examine the work directly.
02 / The research series
One foundation.
Five connected papers.
Follow the development from the primitive layer to wave dynamics. Start with QLM I, then continue through the series in order.
QLM I QLM II QLM III QLM IV QLM V
Reduced-action foundations
The foundational QLM paper establishing the per-radian primitive triplet {ℏ, ℓP, tP}, the covariant phase-flow law, the saturated Planck tick, and the reduced-action derivation of Planck-unit quantities.
Density saturation & collapse
Develops the QLM local Planck energy-density cap and applies it to a minimal non-singular saturated-core completion for gravitational collapse.
Invariant transport & kinematics
Reconstructs Lorentz kinematics from invariant lattice transport, proper-tick accumulation, radar synchronization, and QLM phase-gradient relations, then connects invariant rest-phase energy to inertial response through the derived boost structure.
Geometric routing & gravitation
Develops the exterior routing-admittance sector of QLM, interpreting Schwarzschild redshift and gravitational throttling as geometric suppression of radial phase-action routing.
Phase transport & wave dynamics
Develops the quantum dynamical layer of QLM, organizing Schrodinger, Klein-Gordon, and Dirac equations as continuum descriptions of coherent phase-action transport.
Also in the archive: the Planck energy identity and temporal-spatial phase closure.
View all seven papersKeep the foundations in view
Framework reference.
Density cap
QLM II proposes a local energy-density cap based on the one-tick throughput scale and develops a finite saturated-core model for gravitational collapse.
Kinematics
QLM III develops Lorentz interval structure and the energy–momentum relation from invariant transport and proper-tick accumulation.
Gravitational routing
QLM IV interprets exterior gravitational behavior through routing availability, relating suppression of outward phase-action transport to Schwarzschild redshift.
Wave dynamics
QLM V develops Schrödinger, Klein–Gordon, and Dirac dynamics as continuum transport descriptions of coherent phase-action updates.
A shared glossary drawn from the QLM papers and companion notes. Symbols can have different meanings in different sectors; consult the relevant paper for its full definitions and assumptions.
Primitive Layer
- {ℏ, ℓP, tP}
- The primitive reduced-action, spatial, and temporal triplet used as the base layer of the framework.
- ℏ
- Reduced Planck constant; the primitive reduced-action quantum per radian.
- ℓP
- Planck length; the primitive lattice spatial increment.
- tP
- Planck time; the primitive lattice temporal increment or one lattice tick.
- c = ℓP/tP
- Invariant lattice transport speed and causal update bound.
- h = 2πℏ
- Full-cycle Planck constant, derived from the per-radian primitive.
Phase and Action
- θ
- Dimensionless physical phase measured in radians.
- τ
- Proper time along a worldline or local closure period, depending on context.
- S
- Action; at one saturated tick, ΔS = ℏ.
- dS = ℏ dθ
- Primitive phase-action identity.
- E = ℏ dθ/dτ
- Covariant phase-flow law defining energy as reduced-action throughput per unit proper time.
- ω, ν
- Angular and cycle frequencies, with ν = ω/(2π).
Planck and Density Sector
- ωP = 1/tP
- Planck angular frequency.
- EP = ℏ/tP
- Planck energy throughput for one saturated tick.
- pP = ℏ/ℓP
- Planck momentum scale.
- mP = ℏtP/ℓP²
- Planck mass in QLM reduced-action form.
- VP = ℓP³, V4 = ℓP³tP
- Planck spatial cell volume and Planck four-volume cell.
- uP = ℏ/(ℓP³tP)
- Planck energy-density scale used as the local throughput cap.
- ρmax = mP/ℓP³
- Maximum lattice mass density implied by the density cap.
- Rc(M)
- Saturated-core radius as a function of total mass in QLM II.
Relativity and Routing
- E, p
- Energy and momentum, with invariant transport giving E/p = c for null propagation.
- β, γ
- Usual relativistic speed ratio and Lorentz factor used in the kinematic sector.
- M, r
- Gravitating mass and radial coordinate in the exterior routing field.
- rs = 2(M/mP)ℓP
- Schwarzschild radius written in lattice variables.
- Y(r)
- Routing availability field, often written as 1 - rs/r in the exterior sector.
- Zg(r)
- Gravitational throttle or redshift factor associated with exterior routing suppression.
Quantum Transport
- χ(x, t)
- Generic lattice phase-transport amplitude used in discrete transport and Klein-Gordon development.
- ψ(x, t)
- Relativistic multi-component field used in the Dirac factorization.
- ϕ(x, t)
- Slow nonrelativistic envelope field after rest-carrier extraction.
- ρ = |ϕ|²
- Conserved envelope density associated with coherent phase transport.
- J = (ℏ/m)ρ∇θ
- Continuum transport current in the Schrödinger layer.
- Δd, ∇, ∇², ∂t, ∂i
- Discrete and continuum transport operators used to pass from the lattice update rule to wave dynamics.
- αi, β
- Dirac matrices used in the first-order relativistic factorization.
Electromagnetic and Calibration
- e, ϵ0, µ0, Z0
- Elementary charge, vacuum permittivity, vacuum permeability, and vacuum impedance.
- qP
- Planck charge, written in QLM as qP² = ℏ/ZP.
- ZP, YP
- Per-radian Planck impedance and admittance.
- α
- Fine-structure constant, used as the Coulomb-routing calibration factor.
- a0, vB
- Bohr radius and Bohr orbital speed used in atomic calibration identities.
- ω0 = mc²/ℏ
- Rest-phase angular frequency in the relativistic wave sector.
Open documentation
Begin with the derivation.
Read the foundational paper for the primitive definitions, assumptions, and algebra that the rest of the framework builds on.