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The Single Monad Model of the Cosmos Inner Time and Outer Time
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SMONAD · Gallery

A visual path through the research

Photographs, manuscripts, book artwork and connected research diagrams. Explore the origins of the Single Monad Model and follow its questions into the mathematics of time, space and observation.

Browse the collection Take a visual tour
39
archive entries
78
research figures
4
guided tours
The repeating green forms of a Romanesco inflorescence.Patterns in nature Red Single Monad spiral symbol.A symbol and its historyA group gathered at a teacher training workshop in Ajman.Teaching and exchange

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Begin with a visual tour

Follow a small sequence of images, then open the arguments behind them.

01

A first visual route through SMM and DTT

Follow source histories, six nested inner levels, outer completion, spatial readout and complex phase through five connected diagrams.

5 stops 02

From relations to geometry and physical clocks

Read six diagrams connecting local moves, causal cones, calibration, proper time and retained memory, with links to the underlying calculations.

6 stops 03

From structural counts to physical tests

Follow receiver modes, the conditional count 137, response normalization and hydrogen benchmarks toward a reproducible comparison with observations.

6 stops 04

Books, symbols and intellectual origins

Connect the restored gallery’s symbols, manuscript photographs and book artwork with the current reading routes through the SMM and DTT programme.

5 stops

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Search titles, descriptions and topics. Every result links to more context.

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117 of 117 entries

From the original gallery

39 entries

Research figures

78 entries
Four successive stages require definitions, assumptions, calibration, and a discriminating comparison. A feedback arrow sends failed comparisons back to revise the model.View full size

Methods & tests

Turning an analogy into a testable model

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

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A horizontal history track traverses four intermediate positions. At the completed position, one branch stores a memory result and another turns on a completion flag.View full size

Quantum theory & measurement

A record and its completion flag

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

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Four book themes are arranged as a two by two map: Time and Creation, Duality of Time, Ultimate Symmetry, and Time Chest. Annotations distinguish foundations, formal proposal, organization, and accessible synthesis.View full size

Philosophy & interpretation

A reading map through the book series

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

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A curve in a two-parameter plane represents models sharing the same calibrated observable. A second contour crosses it at a point and removes that illustrative ambiguity.View full size

Methods & tests

Calibration can leave hidden parameter directions

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

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Two example potentials have two wells. In one, the left well is lower; in the other, the right well is lower. A local minimum persists even when it is not the global minimum.View full size

Gauge theory

Local stability and global selection need separate tests

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

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A commuting square shows that a branch operation before or after an admitted clock test gives the same statistic. Resolution evolution changes the clock.View full size

Time & clocks

A protected clock ignores branch operations

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.

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Two paths split and rejoin on the left. On the right a sinusoidal normalized intensity runs from maximum at phase zero to zero at pi and back to maximum at two pi.View full size

Quantum theory & measurement

Interference depends on retained phase

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.

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Two cones have the same opening angle but different metric length standards. A calibrated ruler and a volume cell illustrate the extra scale information.View full size

Relativity & causal order

Causal order fixes a cone; volume fixes scale

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

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Four columns represent hydrogen, deuterium, a Rydberg probe, and antihydrogen. Each is linked to one fixed baseline and then to a residual comparison.View full size

Matter & atomic spectra

Freeze the baseline, then compare residuals

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

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The left panel shows a cubic mode-counting curve. The right panel has two nodes with no metric connection and two nodes joined by a weighted edge, emphasizing independent questions.View full size

Space & geometry

Counting modes does not determine distance

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

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Three base scales, action, length and duration, connect to example velocity, energy and mass conversions. A separate box notes dimensionless couplings do not require a unit conversion.View full size

Constants & response

A shared dimensional dictionary

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.

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A chain runs from a finite regulated model through uniform bounds and reconstructed physical theory to a full mass-gap statement. Separate arrows show where removal of regulators and zero modes must be controlled.View full size

Gauge theory

A finite gap is one step in a continuum proof

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

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A positive cone is shown as a triangle in the time-space section. An interior point is positive definite; boundary points have zero determinant.View full size

Relativity & causal order

Positive Hermitian records form a Lorentz cone

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

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An observed variable and a relaxing hidden variable exchange influence. Eliminating the hidden variable leaves an exponential memory kernel plotted against delay.View full size

Time & clocks

Eliminating a hidden variable produces memory

Eliminating a specified relaxing variable yields a history-dependent equation with an exponential memory kernel. The resulting reduced memory follows from the supplied coupling and initial data, even when the full system is local in time.

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A sphere carries a selected point. Above the point a small circle represents the family of spinors differing only by common phase, with a projection arrow from that circle to the point.View full size

Phase & complex structure

A phase circle over each Bloch-sphere point

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.

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A positive inverse-square kinetic curve, a negative inverse-distance attraction, and their total energy are plotted against size. The total has a negative minimum.View full size

Matter & atomic spectra

A hydrogen-like state balances two scales

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.

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Horizontal bound-state energy levels lie at minus one, minus one-quarter, minus one-ninth and subsequent inverse squares, approaching the zero-energy continuum threshold.View full size

Matter & atomic spectra

The Coulomb spectrum is conditional on its Hamiltonian

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.

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A large sample space contains scattered points and a small highlighted subregion with a square lattice. The captioned rate represents the conditional asymptotic bound, not measured probabilities.View full size

Space & geometry

Local structure can be rare in a relational ensemble

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

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Three parallel bands distinguish generative order, physical clock duration, and experienced or theological time. Each band lists its own kind of evidence.View full size

Philosophy & interpretation

Separate the meanings of time

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

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Six nested rounded enclosures represent successive inner functional roles. Their ranks are 1,1,2,2,3,3. An arrow leaves the outermost inner enclosure toward a published record.View full size

Source, generation & records

Six nested inner roles; one outer completion

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

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Five history nodes map to three classes. Two pairs merge under the admitted tests, while one history remains distinct.View full size

Source, generation & records

Different histories can yield one observation

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.

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A large state-space rectangle contains a reachable region and an unobservable region. Their overlap is removed when forming the prepared minimal quotient.View full size

Source, generation & records

The prepared observable part is the minimal state

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.

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A quaternion a plus bi plus cj plus dk is decomposed into a central real component and three imaginary components. Three axes represent the resulting imaginary readout.View full size

Space & geometry

The center-insensitive quaternionic readout

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

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A staircase-like clock maps outer time to operational time. Three diffusion profiles on the right broaden as operational time increases.View full size

Time & clocks

A random operational clock changes observed diffusion

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

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Three receiver steps multiply the previous count by ten and add branch counts one, three, and seven, giving one, thirteen, and one hundred thirty-seven. The count expands as one hundred plus thirty plus seven.View full size

Constants & response

A conditional receiver count: 1 → 13 → 137

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.

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A central source connects to six relational nodes on a circle. A separate bracket marks the observer-facing descriptions of the network.View full size

Philosophy & interpretation

From a cosmological vocabulary to relations

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

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Bars show a geometric distribution with success probability one-quarter. The probability decreases with attempt number, while the cumulative success law approaches one.View full size

Quantum theory & measurement

Repeated fresh attempts: a geometric success law

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.

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Two equally sized two-mode systems have different stiffnesses. For J equal to one,one, stiffness matrix identity gives capacity two while diagonal one,four gives capacity one and one-quarter.View full size

Constants & response

Static response depends on more than rank

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.

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A square loop has four mobile positions with gauge link labels. The root is coupled to a stationary dual outside the loop; a loop arrow denotes holonomy.View full size

Gauge theory

The four-position singlet-bound loop

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

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A log return-probability plot against log diffusion scale contains two negative-slope straight lines. Slopes minus one and minus two correspond to spectral dimensions two and four.View full size

Space & geometry

Spectral dimension is a return-probability slope

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.

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A single nonrelativistic n equals two level branches into fine-structure levels. A further radiative step splits the lower pair into 2S one-half and 2P one-half levels.View full size

Matter & atomic spectra

Fine structure and Lamb shift: distinct benchmarks

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

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An upper-triangular four by four grid shows ten independent symmetric pairs. Four diagonal cells and six off-diagonal cells are highlighted.View full size

Constants & response

Why a free symmetric quadratic receiver has ten modes

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.

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A hyperbola in the left plane represents split-complex structure and a circle in the right plane represents ordinary complex phase. The two algebraic units have opposite squared signs.View full size

Phase & complex structure

Hyperbolic temporal pairing and complex quantum phase

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.

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A triangle has edge weights two, three, and seven. The highlighted shortest route joins a to c through b and has total length five.View full size

Space & geometry

A normalized graph block recovers path distance

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.

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The retained states zero-plus and zero-minus both display zero. A later conditional bit flip distinguishes them, so present display equality does not guarantee future-test equivalence.View full size

Source, generation & records

Same display; different future behaviour

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.

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A quantitatively plotted normalized radial probability four x squared times exp minus two x starts at zero, peaks at x equal to one, and decays. The variable x is radius divided by the reduced-mass Coulomb length. Its integral from zero to infinity equals one.View full size

Matter & atomic spectra

The Coulomb ground-state radial profile

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.

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A normalized two-component input is measured by projectors onto zero and one. The two recorded outcomes have probabilities p and one minus p, and their Kraus effects sum to the identity. Selecting outcome zero succeeds with probability p and produces state zero when p is positive.View full size

Quantum theory & measurement

Completion instruments and records

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.

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The upper graph is a five-node chain from A to E, whose graph distance is four. The lower graph has the same chain plus an edge joining A directly to E, reducing that graph distance to one.View full size

Space & geometry

Local moves determine graph distance

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.

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Six ordered inner roles supply signed channels plus e1, minus e1, plus e2, minus e2, plus e3 and minus e3. Their successive span ranks are one, one, two, two, three and three. A paired difference defines a three-component record.View full size

Space & geometry

Six roles and a paired readout

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.

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For a planar skew generator A equal to omega times J-zero, positive omega produces counterclockwise phase and negative omega clockwise phase. Polar normalization removes the magnitude of omega but retains its sign, and the resulting J squares to minus the identity.View full size

Phase & complex structure

Polar phase retains orientation

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.

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Four vertical stages show zero becoming one under multiplication by ten and addition of one; then thirteen under multiplication by ten and addition of three; then one hundred thirty-seven under multiplication by ten and addition of seven.View full size

Constants & response

The structural sequence 0 → 1 → 13 → 137

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.

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A Minkowski diagram compares a vertical worldline from (0,0) to (0,5) with a two-leg worldline through (1.5,2.5). The straight clock records five units of proper time; the traveling clock records four.View full size

Relativity & causal order

Elapsed time depends on the path

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.

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Three stages contain new labelled instances of A, B and C while the chain relation between them remains the same. Dashed arrows identify correspondence between successive instances.View full size

Philosophy & interpretation

Renewed instances, persistent relations

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

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Joint source dynamics branches to locality and physical clocks, which feed matched propagation and composition. Those structures feed observable predictions with error bounds and independent comparison.View full size

Methods & tests

A connected programme of open problems

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

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Five panels compare temporal order, measured duration, lived or remembered time, geometric time and generative time, each with its own guiding question.View full size

Philosophy & interpretation

Five questions about time

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

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