Primitive triplet
QLM treats {ℏ, ℓP, tP} as primitive and derives Planck-sector quantities algebraically rather than beginning from mixed square-root dimensional forms.
QUANTUM LATTICE MODEL
A reduced-action framework in which phase, proper time, lattice transport, gravitational routing, and quantum wave dynamics are organized from the primitive triplet {ℏ, ℓP, tP}.
E = ℏ dθ/dτ
c = ℓP / tP
EP = ℏ / tP
Core idea
Physics organized as phase-action transport.
The QLM begins from a narrow primitive claim: reduced action is carried in radians of phase, and energy is that action-throughput measured against proper time. The foundational relation E = ℏ dθ/dτ is not treated as a derived convenience. It is the starting rule from which the Planck tick, the transport speed c = ℓP/tP, and the reduced-action Planck sector are built.
Once the saturated one-tick update is fixed, the framework can describe several sectors with one consistent vocabulary: density saturation for collapse, invariant transport for Lorentz kinematics, routing availability for gravitational throttling, and coherent transport for quantum wave dynamics. The five canonical QLM papers step through that buildout in sequence rather than presenting isolated identities.
Saturated tick
One coherent tick advances one radian of phase in one Planck proper-time interval, fixing the local throughput scale that later becomes EP, uP, and the density-cap sector.
Continuum emergence
Lorentz structure, Schwarzschild throttling, and the familiar quantum wave equations appear as coordinate or continuum descriptions of the same underlying transport architecture.
Canonical papers
QLM I through V
QLM I
Reduced-Action Foundations
Establishes the primitive triplet {ℏ, ℓP, tP}, the covariant phase-flow law, the saturated Planck tick, and the reduced-action derivation of Planck-unit quantities.
Open archiveQLM II
Planck Energy Density Cap
Promotes the Planck energy-density scale to a local action-throughput bound and derives a minimal saturated-core completion for gravitational collapse.
Open archiveQLM III
Invariant Transport and Lorentz Kinematics
Reconstructs relativistic interval structure, synchronization, and the energy-momentum relation from invariant lattice transport and proper-tick accumulation.
Open archiveQLM IV
Geometric Routing and Gravitational Throttling
Develops the exterior routing sector in which Schwarzschild redshift appears as geometric suppression of outward phase-action transport.
Open archiveQLM V
Quantum Phase Transport and Wave Dynamics
Organizes Schrödinger, Klein-Gordon, and Dirac dynamics as continuum transport laws emerging from coherent phase-action flow on the lattice.
Open archiveReference
Framework reference
1. Foundations
QLM I defines the primitive per-radian layer, the phase-flow law, the saturated Planck tick, and the reduced-action Planck-unit reduction chain.
2. Density cap
QLM II promotes the one-tick throughput scale into a local density cap, giving a finite saturated-core completion instead of singular collapse.
3. Kinematics
QLM III shows how invariant transport and proper-tick accumulation reproduce Lorentz interval structure and the energy-momentum relation.
4. Routing
QLM IV interprets exterior gravitational behavior as routing suppression, with Schwarzschild redshift appearing as geometric throttling of outward phase-action transport.
5. Wave dynamics
QLM V treats Schrödinger, Klein-Gordon, and Dirac dynamics as continuum transport layers built from coherent phase-action updates on the lattice.
Compiled from the notation sections and symbol definitions across the current QLM papers and companion notes, with repeated symbols merged into a single framework glossary.
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.