SMONAD · Gallery
Browse the visual arguments behind the Single Monad Model and Duality of Time Theory. Open an image for a closer look, download its SVG, or follow its explanation to examine the assumptions and derivation.
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78 of 78 entries
Methods & tests
A research route turns an analogy into defined states, calibrated equations, and a discriminating comparison. Its feedback loop shows where a failed comparison should revise the model.
Conceptual guide
Quantum theory & measurement
A prescribed autonomous history can deliver a retained result together with a completion flag. Reading the flag checks that model's completion condition; a flag assigned by hand would not explain the dynamics.
Model illustration
Philosophy & interpretation
The book themes connect philosophical foundations, dual-time formalism, symmetry, and accessible explanations. The arrangement is a reading route through the programme, with each mathematical bridge developed separately.
Conceptual guide
Methods & tests
One calibration can leave a curve of equally compatible parameter choices. A second independent diagnostic may resolve that freedom; a prediction based on the first calibration must be constant along the remaining curve.
Conceptual guide
Gauge theory
Illustrative two-well potentials distinguish a stable local state from the globally preferred state. Selecting a physical vacuum requires the actual potential and its global comparison, including any deformation terms.
Model illustration
Time & clocks
A commuting interface makes an admitted clock statistic insensitive to specified branch operations. Resolution evolution can still change the reading, so the result identifies a protected clock under stated coupling assumptions.
Exact within stated assumptions
Quantum theory & measurement
The ideal coherent two-path law is I(φ) = (1 + cos φ)/2, ranging from constructive to destructive interference. A temporal model must identify how the alternative histories retain phase and how their phase difference is controlled.
Exact within stated assumptions
Relativity & causal order
A causal cone fixes conformal relations, while an independent volume or ruler calibration fixes scale. Realizing those relations as physical boosts additionally requires compatible clocks and detectors.
Conceptual guide
Time & clocks
A reversible cycle admits both directions, whereas a growing collection of retained records has an availability order. Their connection requires a rule for record formation and retention.
Conceptual guide
Matter & atomic spectra
A fixed baseline links hydrogen, isotope, Rydberg, and antihydrogen comparisons before residuals are assessed. The schematic describes a test design; nuisance cancellation depends on a specified covariance model.
Conceptual guide
Space & geometry
Mode counting and distance answer different reconstruction questions. Finite spectral-growth dimension can coexist with infinite distances when the operator and admitted observables do not control the relevant differences.
Conceptual guide
Constants & response
Action, length, and duration standards convert dimensionless model outputs into velocity, energy, and mass scales. These dimensional identities organize physical matching while leaving the numerical standards to be supplied or derived.
Exact within stated assumptions
Gauge theory
The classical profile σ(x) = 4 arctan exp(mx) connects adjacent vacua of the specified sine-Gordon-type model. Its tension is 8κm under the stated normalization; matching and a quantum mass gap require further analysis.
Exact within stated assumptions
Gauge theory
A continuum mass-gap argument must connect regulated spectra, uniform bounds, and reconstruction of the physical sector. The diagram separates these obligations from a finite-volume or selected-sector threshold.
Conceptual guide
Relativity & causal order
For X = tI + x·σ, det X = t² − ∥x∥² and positivity is equivalent to t ≥ ∥x∥. The drawing is a two-dimensional section of this four-dimensional positive Hermitian cone.
Exact within stated assumptions
Phase & complex structure
A normalized two-component spinor and its common-phase rotations project to one Bloch-sphere point. The Hopf bundle has total space S³; the sketch represents its fibers without embedding S³ in ordinary three-space.
Exact within stated assumptions
Matter & atomic spectra
The variational energy E(a) = A/a² − B/a has its minimum at a = 2A/B for positive A and B. Supplied kinetic and attractive terms therefore establish a binding scale, while their strength remains an input to this calculation.
Exact within stated assumptions
Matter & atomic spectra
The nonrelativistic Coulomb levels follow Eₙ = −E₀/n² and approach the zero-energy continuum threshold. The dimensionless diagram displays the supplied Hamiltonian's spectrum before radiative and relativistic corrections.
Exact within stated assumptions
Space & geometry
A relational ensemble can contain a small subset of lattice-like memories. In the stated permutation model its rarity is exponential in N log N, distinguishing the construction of a local example from a mechanism that selects it.
Model illustration
Philosophy & interpretation
Generative order, physical clock duration, and experienced or theological time carry different questions and kinds of evidence. This map helps formulate possible connections without identifying those meanings by vocabulary alone.
Conceptual guide
Source, generation & records
Six nested inner roles have paired-readout ranks 1, 1, 2, 2, 3, 3 in the illustrated construction. Outer completion publishes the retained record; the enclosures represent dependencies rather than independent time axes.
Conceptual guide
Source, generation & records
Histories belong to the same observational class when every admitted test gives the same result. Expanding the test language can separate previously equivalent histories and refine the resulting quotient.
Exact within stated assumptions
Source, generation & records
The prepared minimal state of a linear interface is R/(R∩N): reachable information with unobservable directions removed. Its size depends on both the available preparations and the future tests.
Exact within stated assumptions
Space & geometry
Under the stated associative positive-norm and noncommutation hypotheses, the carrier is quaternionic. Removing its real center leaves a three-dimensional imaginary readout; spatial dynamics requires further structure.
Exact within stated assumptions
Time & clocks
An intermittent operational clock changes how a diffusion process is sampled in outer time. Flat clock segments describe waiting periods and can produce memory in the observed process.
Model illustration
Constants & response
The selected receiver recursion Q(d) = 10Q(d−1) + (2ᵈ − 1) gives 1, 13, and 137 from Q(0) = 0. This exact finite count needs a normalized response and physical matching before its reciprocal can be identified with a coupling.
Exact within stated assumptions
Philosophy & interpretation
The SMM vocabulary connects one source, relational plurality, and observer-facing descriptions. The layout is a conceptual reading aid and does not assign physical positions to metaphysical terms.
Conceptual guide
Source, generation & records
Corresponding markers in successive configurations make a history of motion describable. A physical trajectory additionally needs elapsed duration, identity rules, and a law governing transport.
Conceptual guide
Quantum theory & measurement
Independent fresh attempts with success probability p give P(first success at n) = (1−p)ⁿ⁻¹p. The plotted example uses p = 1/4 and requires failed attempts to restore the input before another trial.
Exact within stated assumptions
Constants & response
Two systems of equal dimension can have capacities 2 and 5/4 because their stiffness operators differ. Consistent basis changes preserve each response, while changing physical stiffness changes the capacity.
Exact within stated assumptions
Gauge theory
A mobile carrier travels around four supplied sites while a stationary dual remains at the root. The root singlet penalty and link holonomy specify the finite model's response and spectral questions.
Model illustration
Source, generation & records
Composable source histories pass through completion before retained records become available. The source is a generative premise rather than a point assumed to be already located in physical space.
Conceptual guide
Space & geometry
For the toy return law P(σ) = Cσ⁻ᵈˢ/², spectral dimension is minus twice the log–log slope. The illustrated slopes correspond to dimensions two and four and are exact examples rather than measured dimensional flow.
Exact within stated assumptions
Matter & atomic spectra
The schematic separates Dirac fine structure from the radiative splitting of 2S₁/₂ and 2P₁/₂. Its deliberately exaggerated level separations organize the hydrogen benchmarks without plotting measured scales.
Conceptual guide
Constants & response
A free symmetric quadratic receiver on a four-dimensional complex input has four diagonal and six unordered off-diagonal modes. Thus dim Sym²(ℂ⁴) = 10, conditional on choosing this receiver space.
Exact within stated assumptions
Phase & complex structure
Split-complex j² = +1 preserves a hyperbolic quadratic form, while ordinary complex i² = −1 supports circular phase. An explicit interface is needed when both algebras appear in a dual-time reconstruction.
Exact within stated assumptions
Relativity & causal order
A bounded displacement per update defines a finite propagation cone in the illustrated readout. A common Lorentzian metric and shared field speed require compatible clock, detector, and transport assumptions.
Model illustration
Cosmology
Adding a constant energy offset multiplies all Boltzmann weights by the same factor and leaves normalized probabilities unchanged. These probabilities therefore cannot determine an absolute vacuum-energy baseline by themselves.
Exact within stated assumptions
Space & geometry
In the specified normalized graph block, distance equals weighted shortest-path distance. The route with edge lengths two and three has length five, shorter than the direct edge of length seven.
Exact within stated assumptions
Source, generation & records
Two retained states can display the same immediate value while responding differently to a later admitted operation. Predictive equivalence must therefore compare future tests, not just the present display.
Exact within stated assumptions
Source, generation & records
Source histories, completion, and retained records form the conceptual interface for geometry, clocks, and matter readouts. The diagram organizes those dependencies while leaving each physical readout to be defined independently.
Conceptual guide
Methods & tests
Record geometry, clock readings, and matter propagation are separate observables in a common-source programme. A shared logarithmic encoder and protected-clock interface can relate them conditionally; the physical selection of that interface remains a research task.
Conceptual guide
Time & clocks
A two-variable local system becomes history-dependent after its relaxing hidden variable is eliminated. The exponential kernel records the exact delayed contribution under the specified linear dynamics.
Exact within stated assumptions
Source, generation & records
Six nested functional roles prepare a record that a seventh completion interface makes available. The paired ranks belong to the stated readout example, and the enclosures indicate dependencies rather than spatial shells.
Conceptual guide
Phase & complex structure
An ordinary complex rotation preserves x² + y², while a split-complex transformation preserves u² − v². The two units serve distinct algebraic roles that a source model must connect explicitly.
Exact within stated assumptions
Space & geometry
Three independent binary-pair differences map 64 addresses to 27 spatial sites with unequal multiplicities. The center has eight preimages and each corner one, so a uniform site measure needs an additional weighting rule.
Exact within stated assumptions
Quantum theory & measurement
The normalized spinor Bloch map sends common-phase-related inputs to the same unit vector. The sphere represents pure-state directions rather than a particle surface or the full phase-bearing carrier.
Exact within stated assumptions
Matter & atomic spectra
The exponential Coulomb trial family has dimensionless energy 1/x² − 2/x, minimized at x = 1. Its optimum is the exact nonrelativistic 1s state for the supplied Hamiltonian and reduced-mass scale.
Exact within stated assumptions
Constants & response
Structural counts and spectra enter normalized observables through an action, length, and duration dictionary. The map distinguishes dimensionless targets from dimensional scales and keeps geometric Λ separate from vacuum-energy density.
Conceptual guide
Matter & atomic spectra
The Coulomb energy ratios −1/n² approach the continuum threshold from below. These exact nonrelativistic ratios exclude radiative, relativistic, finite-size, and recoil corrections beyond the reduced-mass treatment.
Exact within stated assumptions
Matter & atomic spectra
The normalized 1s radial probability 4x²e⁻²ˣ peaks at x = 1 and integrates to one. Its x² factor is the spherical-shell measure, so the curve differs from both spatial probability density and wave amplitude.
Exact within stated assumptions
Quantum theory & measurement
A normalized two-outcome projective instrument gives probabilities p and 1−p, with Kraus effects summing to identity. Selecting one outcome as completion preserves its success probability as part of the operational description.
Exact within stated assumptions
Space & geometry
A five-node chain has endpoint distance four; adding an allowed direct move reduces that distance to one. Graph distance is fixed by the permitted elementary operations, independently of the drawing's Euclidean layout.
Exact within stated assumptions
Space & geometry
Signed channels ±e₁, ±e₂, ±e₃ have successive span ranks 1, 1, 2, 2, 3, 3. This exact paired-frame calculation supplies a three-component readout under the chosen frame and nested-role assumptions.
Exact within stated assumptions
Phase & complex structure
Polar normalization of an invertible planar skew block removes frequency magnitude while retaining orientation and gives J² = −I. Zero-frequency directions require separate treatment, and this ordinary complex structure differs from split-complex j.
Exact within stated assumptions
Constants & response
Four diagonal and six normalized off-diagonal basis matrices span the ten-dimensional space Sym²(ℝ⁴). Matching mirrored entries describe one symmetric mode, while choosing this space as a physical receiver is a modeling step.
Exact within stated assumptions
Time & clocks
The refinement example B = 4 and r = 1/2 gives the logarithmic count-to-scale ratio D = 2. Identifying it with geometric dimension needs regularity conditions, and equality to a physical clock ratio needs the displayed shared encoder.
Exact within stated assumptions
Constants & response
For a fixed probe, rescaling a positive stiffness K by λ changes its response from C to C/λ without changing dimension. The two-mode example falls from 3/4 to 3/8 when stiffness doubles.
Exact within stated assumptions
Constants & response
The selected radix-ten recursion with subset increments 1, 3, and 7 yields 0 → 1 → 13 → 137. The exact structural integer becomes a coupling candidate only after a separately specified response normalization and physical match.
Exact within stated assumptions
Time & clocks
Microscopic evolution, retained correlations, and thermodynamic macro-regions define three different descriptions of change. Their arrows align only through a connecting model with suitable preparation and boundary conditions.
Conceptual guide
Relativity & causal order
The illustrated Lorentz boost with β = 0.6 preserves the null ray x = ct. Algebraic interval preservation is exact; an SMM/DTT reconstruction must also realize the transformation using physical rods and clocks.
Exact within stated assumptions
Cosmology
Fixed comoving separation two becomes proper separation two, three, and four as the scale factor grows. This FLRW scaling describes spatial slices rather than a microscopic expansion mechanism or stretching of bound objects.
Exact within stated assumptions
Gauge theory
An ordered product of consistently oriented transport links is conjugated at the loop's base under local frame changes. Its trace is invariant, establishing a holonomy observable before a gauge group or action is selected.
Exact within stated assumptions
Cosmology
For the toy history a(t) = t/t₀ extending from zero to arbitrarily late time, the Hubble radius is finite while both displayed horizon integrals diverge. The example distinguishes a local expansion scale from particle and event horizons.
Exact within stated assumptions
Relativity & causal order
Local free fall removes a uniform acceleration while tidal separation reveals curvature through geodesic deviation. The schematic uses a stated curvature-sign convention and illustrates geometry rather than a measured force field.
Conceptual guide
Relativity & causal order
The positive relativistic energy curve contains rest energy and approaches the massless linear asymptote at high momentum. Its low-momentum expansion gives the Newtonian kinetic term under the supplied dispersion relation.
Exact within stated assumptions
Relativity & causal order
A straight worldline records five proper-time units while the displayed two-leg path records four in units c = 1. The exact piecewise-inertial comparison idealizes the turnaround and follows from integrating each path's interval.
Exact within stated assumptions
Philosophy & interpretation
Successive newly labelled instances preserve the same chain of relations. This correspondence illustrates how structural persistence can coexist with renewal, while leaving the physical update and label interpretation explicit.
Conceptual guide
Methods & tests
Joint source dynamics connects research on locality and clocks to propagation, composition, and observable predictions. Each arrow names an obligation requiring a mathematical or operational construction, with error bounds at the comparison stage.
Conceptual guide
Philosophy & interpretation
Temporal order, measured duration, lived time, geometric time, and generative time pose different questions. The map provides vocabulary for connecting those questions while preserving their distinct meanings.
Conceptual guide
Gauge theory
Phase maps of a domain loop have integer winding invariant under continuous nonzero deformations. Turning that topological integer into physical charge requires identification of the field and a normalized coupling.
Exact within stated assumptions
Source, generation & records
Completed-future equivalence gives a precise predictive state. In a finite linear model its minimal size is the rank of a delay-observation matrix; discarded information appears as an exact memory term.
Exact within stated assumptions
Space & geometry
A paired six-channel source has an exact rank-three image. Its measure, stability and local dynamics can all be calculated, but each follows from a specified interface that a stronger source law must select together.
Exact within stated assumptions
Phase & complex structure
Polar decomposition selects a unique quarter-turn on the dynamic sector. Positive Schrödinger evolution uses i=−J; shared phase in a composite is then enforced by a distinct balancing construction.
Exact within stated assumptions
Constants & response
The ten-mode continuation receiver and disjoint terminal sectors produce 100, 30 and 7 records, with a proportional bar showing their exact sum.
Exact within stated assumptions
Matter & atomic spectra
Exact Coulomb-model curves show how the radius, relative RMS speed and binding change together when the supplied coupling is multiplied by t.
Exact within stated assumptions
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