A minimum action, and a mass gap
A research repository on two questions that turn out to share a shape. Does anything in physics force a positive unit of action, and does SU(3) Yang–Mills in four dimensions have a mass gap? Every note below states its result first, keeps its constants explicit, and says which premises it uses.
The primary sources are collected and annotated alongside the mathematics, from Newton's Opticks and Berkeley's Analyst to the Vaiśeṣikasūtra, the Abhidharmakośabhāṣya, the Tattvārthasūtra, the Mohist Canons and the Laozi. Comments and corrections go to the repository issues, which is the right place to dispute a dating, a reading or a proof.
Where to start
A recorded trajectory has a floor of order ℏ
Every protocol of marks whose resolution and recoil obey δΔ ≥ κ needs τΔE ≥ 9z²κ to tell free motion from forced. A momentum-transfer mark has κ ≥ ℏ/2 exactly, because its error operator and the impulse it delivers are canonically conjugate.
SU(3) is gapped at strong coupling, with an explicit threshold
For the Wilson transfer matrix the gap is explicit for g² ≥ 176, and the Kogut–Susskind Hamiltonian is gapped for g² ≥ 388 uniformly in the volume.
The conjecture is six named statements
T1 finite-lattice gap, proved; T2 strong-coupling gap uniform in volume; T2′ no Coulomb phase; T3 the scaling limit; T4 existence with the axioms; S the small-volume corner. The conjecture is T2′ with T3, given T4.
A universal action floor needs a fixed action unit
Two elementary criteria decide the recorded countertests before any calculation, and they identify ke/c = αℏ as the only mass-independent action unit classical electrodynamics admits.
Classical mechanics permits action arbitrarily close to zero
Across the tested classes, positive bounds appear only when the class supplies an excitation floor, a fluctuating reference or restricted measurement information. None of them is a universal quantum phase parameter.
Why the ancients argued about the arrow and not the sling
Newton defines centripetal force with a stone whirled in a sling, and the ancient debate is almost entirely rectilinear. The two sit on different rungs of one ladder, and the second rung cannot be stated until straight-line motion is held to need no account. Part of a collection of 50 primary sources, Greek, Chinese, Sanskrit and Latin, each with a companion recording its dating and what is contested.
The relativistic Kepler problem has an excluded action infimum
Regular bound orbits exist exactly for |L| > k/c, a mass-independent threshold equal to αℏ for two elementary charges, and the Sommerfeld–Dirac collapse condition is the same inequality.
The Planck gap
Newton reads a force off a trajectory by letting the sagitta and the enclosed area go to zero. Once the comparison must be recorded, it has a floor of order ℏ. These notes carry the theorems, the Newton-age premises and the ancient dispute about the cut.
- PDFThe cost of a mark: Newton's vanishing sagitta and a floor of order ℏ
- PDFThe mark cost is exactly ℏ/2, and the worst-case theorem survives as a statistical one
- PDFThe Planck gap as a theorem: every marking protocol needs F²τ³ > 9mκ
- PDFThe probabilistic Planck gap: τΔE ≥ (1-2ε)²ℏ²/(4A), and why it is resource-relative
- PDFThe cost of a mark: a Newton-age floor for the Galileo comparison
- PDFGalileo's falling parabola, Newton's refinement and the Planck-scale question
- Newton's vanishing areas and the proposed action scale
- The surviving area is the matched-endpoint arc–chord defect
- Newton on discovery, demonstration and limiting ratios
- The cut paradox in two signatures: sections of a solid and instants of a motion
- Before the arrow: can the parts make a whole?
- Ancient cuts and modern readjustment: provenance
- From cone sections to time refinement
- I003: finite propagation, internal state and resolution
- PDFWhy the ancients argued about the arrow and not the sling
The Yang–Mills mass gap for SU(3)
The conjecture turned into a finite list of named theorems, with explicit dependence on the box size, the lattice spacing and the coupling. The strong side is proved; the weak side and the region between are where the work stands.
- PDFThe mass gap: current position of this programme
- PDFThe mass gap as a finite list of theorems: the Hamiltonian lattice route
- PDFThe map as one conditional theorem: two hypotheses, both finite in kind, and an explicit lower bound m ≥ ℏ c γ'/a*
- PDFSU(3) in four dimensions: every constant of this programme, evaluated
- PDFThe strong-coupling gap of the Kogut–Susskind Hamiltonian is uniform in the volume
- PDFAn explicit strong-coupling gap for the Wilson transfer matrix: SU(3) is gapped for g² ≥ 176, with Δ ≥ (ℏ c/a) 4log(g²/176)
- PDFContinuous-time expansion for Kogut–Susskind: SU(3) gapped for g² ≥ 388, gap approaching (8/3)g² ℏ c/a
- PDFThe strong-coupling threshold made explicit: Yarotsky's proof gives g₀²~10¹⁰⁰, and only a direct expansion can give g₀²~10²
- PDFThe target box: the strong-coupling gap survives local perturbations, so the renormalization group has an explicit finish line
- PDFUpper bounds on the lattice gap: the Feynman–Bijl inequality, the abelian structure factor and the non-abelian dressing obstruction
- PDFThe gap is bounded by the variance of the averaged spatial Polyakov loop, with the small-volume scaling built in
- PDFThe weak-coupling gap in a small box reduces to three inequalities through the Schur complement
- PDFThe one-loop potential along the abelian valley of the torus is periodic, reproduces C133's linear term at the origin, and saturates at order 1/L
- PDFThe Schur error of the small-volume reduction is an ultraviolet problem: a fixed-lattice theorem is available, the renormalized regime is not
- PDFThe finiteness half of the mass gap is the nonvanishing of one smeared susceptibility
- PDFThe flowed bound in free field theory: the coupling cancels, and the bound is informative only at the confinement scale
- PDFIn three dimensions the mass gap is the positivity of one function at infinity
- PDFT2' is the absence of a zero-temperature phase transition, and the missing input is closedness of the gapped set
- PDFWhy the abelian theory has no gap, in the language of the target box: its coupling does not run
- PDFDobrushin's uniqueness condition for the Wilson action: a two-line strong-coupling gap at g² > 444, and the block criterion that is the finite verification
- PDFA Lieb–Robinson bound for the Kogut–Susskind Hamiltonian, with velocity ∝ Nc/g²
- PDFThe intermediate region as a finite verification: complete analyticity, the transfer matrix, and the two numbers whose meeting closes the proof
- PDFThe confinement scale in three bands, with published numbers, and the verification restated gauge-invariantly
- PDFWhat would unblock the confinement-scale region, and what was checked and fails
- PDFThe two reasons to stop, as research problems: what a correlation inequality would buy, what a certified verification cannot, and where the research is
- Navier–Stokes and Yang–Mills: comparison and bridges
- The two Millennium problems
Blocking, flow and the large-field obstruction
What happens to a renormalization step on the lattice, why the obstruction is the large-field tail rather than the block size, and how much the weak side is short by.
- PDFThe blocking criterion is monotonically worse in the block size: blocking gains nothing without a change of coupling
- PDFOne blocking step: the obstruction is the variation, not the size, of the inter-block coupling
- PDFOne small-field blocking step for SU(3), part 1: the Gaussian fluctuation integral with block averaging, with explicit constants
- PDFOne small-field blocking step, part 1b: the decay rate and the shape of the threshold, and why the weak side's deficit is ten orders of magnitude
- PDFFlowing before decimating leaves every norm-based criterion unchanged: what is needed is a statement about the state
- PDFConjugation by the flow preserves the spectrum, so the renormalization step is a truncation
- PDFThe flow Jacobian's growth factor is sharp: it is the Nielsen–Olesen mode, so the large-field region cannot be flowed
- PDFThe truncation error of a flow step is Gaussian times e2t\|G\|∞, so the renormalization step is small exactly on small-field configurations
- PDFOn the lattice one flow step has a uniformly bounded truncation error: the obstruction is decimation
- PDFOn typical configurations the flow truncation is controlled at any coupling; the whole obstruction is the large-field tail
- PDFThe large-field action lower bound is immediate in the natural variable, and the constant field saturates it
- PDFThe large-field region costs more action than it has unstable directions, by the factor 2π²/g²
- PDFThe large-field region in the Hamiltonian route needs an operator inequality, and the measure statement is weaker than that
- PDFThe Agmon bound controls the global excess only: the local large-field estimate does not follow
- PDFAn Agmon bound for the Kogut–Susskind ground state: large fields are suppressed at rate 1/g² per plaquette
- PDFTwo exact identities for the magnetic energy, a ground-state sum rule, and why pointwise Gibbs domination fails as Agmon does
- PDFThe ground-state measure is the time-slice marginal of the Euclidean measure, and that imports the local large-field estimate
- PDFA monotone hierarchy of upper bounds from one flowed correlator, and why no bound of this kind can give m > 0
Gaps that are solved, and what a gap costs
Solved low-dimensional gaps, the spin-chain gaps, and the exact sense in which a positive action floor and a mass gap are the same kind of statement.
- PDFA gap is a scale the classical theory lacks: solved low-dimensional mass gaps and the positive-action question
- PDFAn action floor on transverse phase-space area produces the Yang–Mills quantum-mechanical gap, and a gap forces an action unit
- PDFHolography in the lowest dimensions: at d=0 it is the wave-function identity, at d=1 it is the Schwarzian
- PDFA uniform relaxation gap with nearest-neighbour interactions
- Foundational value of the established Ising gap
- PDFThe Ising gap gives a conditional local parent Hamiltonian
- PDFLocal entangling gates preserve an energy gap uniformly in chain length
- PDFWhen an action plateau controls a spectral gap
Does classical physics select an action scale?
A map of the premises a selection principle must add and the counterexamples it must exclude. The short answer is that the tested classical classes permit action-valued observables arbitrarily close to zero.
- PDFClassical action scales: obstructions, conditional bounds and quantum premises
- PDFA universal action floor needs a fixed action unit and no admitted similarity
- Dilation closure and the action-selection obstruction
- Relativistic Kepler: a singular-core angular-action threshold
- PDFA shared action coefficient from classical composition
- PDFOne action scale, two rules for composing paths
- PDFNonlinear feedback selects an action only while its energy source supplies power
- Radiation balance selects a spectral shape, but leaves its action amplitude free
- PDFA topological energy floor survives shrinking while the action cost vanishes
- PDFStabilizing a topological radius exposes an action coefficient
- PDFGlobal spin phases quantize a ratio, not an absolute action scale
- PDFThermal reliability selects a detector cost, not a universal action
- PDFA passive threshold creates events, but its scale belongs to the receiver
- PDFMechanical interference measures phase without fixing an action unit
- Closed trajectories: action from total turning and persistent excitation
- Fixed force ceiling: small bound circles and an excitation floor
- Bound motion: transport variance and a canonical action estimator
- PDFA sharp mechanical cost for a finite-duration reversal
- PDFReciprocal exchange fixes relative mechanical scales
Cuts, refinement and what survives insertion
Insert a cut into a motion and eliminate it again: which data survive, and what a classical cut has to retain.
- PDFA surviving action defect and consistency under inserting cuts
- What a classical cut must retain
- PDFA physical cut: elastic reversal, finite speed and memory
- PDFA finite-speed return bridge: cuts and midpoint crossover
- Midpoint crossover: consolidated manuscript
- Classical readout under cut refinement
- Reachable information across a classical cut
- Finite position precision and the reachable action area
- Tagged momentum and the memory of a third body
- Two regulators: what survives when the paths shrink?
- Composition, crossover and gap product: verified calculations for A08, A09 and G01
- Worst-case information scales under mechanical composition
- Variations, kernels and distinguishability
- A sharp information bound from indistinguishable motions
- New information behind a delayed record
- Exact finite-horizon phase recovery
Apparatus, records and reconstruction
Finite clocks, probes and pointers, with every preparation and record made explicit. The recurring outcome is that a canonical error product closes as the record precision improves, so no apparatus of this kind supplies a floor.
- PDFA conservative receiver and the origin of a correlation scale
- A finite mechanical clock and four persistent readout records
- Full pointer records recover receiver and unknown probe positions
- Complete final apparatus records replace the initial clock calibration
- Two position records recover the phase state
- Two calibrated probe positions give local recovery and two global branches
- Three calibrated positions give uniform global receiver recovery
- Fixed coupling permits precise calibrated readout
- Calibration tolerance sets the recovery crossover
- Calibrated two-energy records still hide both canonical coordinates
- One calibrated displacement still leaves full receiver ambiguity
- Correlated calibration errors suppress the canonical response
- One uncertain calibration leaves an exact receiver curve
- A known clock position permits stable local recovery
- Final clock momentum restores global receiver recovery
- Distinct clock speeds give identical full pointer records on the shell
- An unknown clock hides an exact fixed-energy receiver phase
- Exact apparatus energy leaves a receiver-sign ambiguity
- Full final apparatus records can hide the entire receiver energy shell
- Positive preparation width hides receiver states from exact records
- Unknown probe positions can hide the receiver after momentum calibration
- Independent apparatus blocks restore extensive composition
- A shared record budget changes the composition exponent
- A13: ordered-beam preparation
Excluding classical operational models
What it takes to rule out a classical account of a quantum experiment, and which premises do the work.
- Two ways to exclude a classical operational model
- Finite operational closure does not select quantum composition
- Product probabilities restrict reversible generators
- Reversible interaction excludes the minimal orientation composite
- The spin interaction admits no exact finite observable repair
- A reversible spin interaction separates identical operational states
- PDFIndependent monotone receivers cannot suppress coincidences
- PDFIndependent settings test shared readiness, while leaving action units free
- PDFA shared release produces exclusive fringe-weighted events