One executable framework. Every input disclosed.
Gravity, electromagnetism and nuclear interaction — one pressure mechanism read through different geometries in a discrete spatial medium. The benchmark ledger separates derived, computed, observed, calibrated, identity and pending results.
Space. Matter. Movement. The Ever-Present Now. Everything below is built from these four — nothing else is assumed.
You are being pressed — right now, from every direction at once.
At every point a convergence is arriving: an inward pressure from all around you and from every depth beyond, the same from one side as from the other. Perfectly answered on every side, it moves you nowhere — you feel nothing, and call the space around you empty. It is not empty. It is balanced. Now let a body settle nearby. It stands in the path of the convergence coming from its direction and casts a shadow — a cone where the inward push falls short. The push from the opposite side, with nothing left to answer it, has its way, and you drift toward the body.
You are not attracted. Nothing pulls you. You are pushed — by the pressure the other body fails to block.
Spatial Displacement Theory says it plainly: all of space is a discrete contact-relay medium — the spation lattice — built at Planck length. Movement passes neighbour to neighbour at exactly c, and there is no other way to travel. Light, gravity, electric fields — different gaits of the one medium. Every force is a pressure difference. All of them.
SDT is built from exactly four irreducible primitives. Everything else — mass, charge, time, temperature, entropy — is derived. The theory makes no claim about what existed before the Clearing, or whether the lattice is finite.
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Every prediction flows from these six principles. No additional postulates.
Every point in space receives convergent energy from N ≈ 5.89 × 10⁶¹ nested Planck-thick shells stretching to the Clearing. Each shell contributes exactly ε regardless of distance — the Shell Cancellation Identity. The total convergence burden:
Convergence pressure: Pconv ≈ 2.46 × 10⁴⁸ Pa — some thirty-seven orders of magnitude beyond the strongest pressure engineering has ever produced. Perfectly balanced at every point, so you feel nothing.
The Clearing — the epoch when every spation released its deformation content simultaneously — created the cosmic microwave background as a baseline operating pressure. The CMB at 2.7255 K is not a distant relic cooling from a primordial fireball. It is the current operating pressure of the universe. Stars are convergence processors — pressure-processing nodes in the lattice, not isolated furnaces.
Matter stands in the flow and casts a shadow. The unshadowed side pushes harder — that is the force, all of it. One formula carries Coulomb, gravity, and the nuclear force; the only thing that changes between them is the geometry of the shadow:
Mass is not a substance a particle carries. It is the price the medium charges to reorganise around a moving displacement — push the knot, and the lattice must restructure, and it resists in proportion to the volume displaced. Inertia and weight read the same geometry twice, so their equivalence costs nothing and explains itself. A century of principle, retired by bookkeeping.
Every particle runs its whole budget at c. Standing still, all of it goes to circulation; move, and the travel is paid out of the spin. That one constraint carries all of special relativity — dilation, contraction, E = mc², the speed limit, the energy-momentum relation. Eight theorems from three axioms, and nothing dilates but the bookkeeping.
The closure z · k² = 1 (z = v²/c², k = c/v) is an identity of the budget — true by construction, and labelled IDENTITY in the benchmarks. The content is the budget itself: eight theorems of special relativity recovered from three axioms, and the k-ladder it hangs on hydrogen (k = 137) through the Sun (k = 686).
Stable particles are persistent torus-knot windings of the lattice. The (1,1) unknot is the electron; the (2,3) trefoil is the proton — topologically protected, which is why the proton does not decay. The W+1 radius rule puts the proton boundary at:
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PEDAGOGICAL DISCRETE RAY-LATTICE — not GOM42-certified. The validated koppa trajectory solver is in the Atomicus 3D walkthrough. Its optional null markers are inertial-frame zeros of the displayed asin proxy, not co-rotating Lagrange points.
Stable particles are persistent vortex structures in the spation lattice — topological defects classified by winding number.
The topology of the lattice permits only certain self-sustaining vortex configurations, and the classification is knot theory, not a parameter choice. A (p,q) mode is a true knot only if gcd(p,q)=1 and min(p,q)≥2; its Alexander polynomial then protects it against any continuous deformation. The (1,1) electron is an unknot, confined by its own displacement. W=2 is an unknot too — merely metastable (≈0.1 GeV barrier, lifetime ~10⁻²¹ s), which is why it is never observed. The (2,3) trefoil is the first protected knot: the proton, which cannot decay. Higher odd knots (W = 5, 7, …) are protected as well and await discovery.
Open winding — no closed circuit
Translates at approximately c. No persistent loop, hence no electromagnetic wake. SDT predicts its magnetic moment is exactly zero — the Standard Model predicts a non-zero value. A direct test.
Simple torus — the (1,1) unknot
The simplest persistent loop. Its circulation generates the electromagnetic wake field. Antimatter (the positron) is the same torus circulating in the opposite sense. Annihilation is unwinding.
Trefoil knot — (2,3) torus knot
The simplest irreducible knot. The W+1 radius conjecture, R = (W+1)ℏ/(mc), predicts a boundary radius of 0.84124 fm — matching the muonic hydrogen measurement to 0.02% with no adjustable parameter.
The pressure disturbance around each vortex sorts itself by how many lobes it carries. The historically "separate" forces are faces of the same wake field — and the trefoil's own three-fold face carries the native selection rule m₃ = 3k, with no imported quantum numbers.
| face | Physical role | Potential falloff |
|---|---|---|
| monopole | Occlusion — the Coulomb-like push (and gravity, at body scale) | r⁻¹ (force 1/r²) |
| ℓ = 2 | Rotational wake — the magnetic face | r⁻³ |
| ℓ = 3 | Three-fold trefoil harmonic — the nuclear reach; the m = 3 fingerprint measured in the proton's own wake | r⁻⁴ |
The 10³⁶ ratio between electromagnetic and gravitational strength — the hierarchy problem — dissolves into a ratio of geometric cross-sections. A proton screens πRp² ≈ 10⁻³⁰ m²; a star screens πR²☉ ≈ 10¹⁸ m². Same pressure, different geometry.
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Every stable nucleus with Z ≥ 2 decomposes into exactly one alpha core plus some number of deuterons and tritons. And the grammar admits two distinct lineages depending on whether the core is complete.
The standard grammar builds on a complete He-4 alpha — a closed tetrahedral core. Electron capture isotopes use a He-3 core instead: an incomplete tetrahedron missing one neutron. This structural deficit is the geometric reason EC occurs — the nucleus seeks completion by capturing an orbital electron to convert a proton into a neutron, closing the tetrahedron.
The triton is the grammar's unbalanced cluster — the deuteron's proton–neutron pair sits symmetric, the triton carries one neutron more. One triton enters per excess neutron (nt = N − Z), and each triton holds two neutrons — the full ledger reads N = 2 + nd + 2nt. The discipline cuts both ways: because the count itself is bookkeeping of (Z, N), any magnetism rule written in triton counts merely restates the neutron ledger, so SDT claims it as nothing more. What the grammar buys is structure — the construction map below, its lineages, and the two-grammar split.
When a nucleus is built on a He-3 core — an incomplete tetrahedron — it is structurally predisposed to capture an inner-shell electron. The capture converts p → n, completing the tetrahedral alpha. The EC isotope transitions from the He-3 lineage to the α-core lineage. This is not random decay — it is geometric self-repair.
High-energy collisions can break structural bonds within the nucleus, knocking a deuteron or triton loose from the scaffold. If the damage exposes or creates an incomplete core, the resulting fragment becomes eligible for electron capture — spallation opens a recombination pathway between construction lineages that thermal processes alone would not access.
The map below plots each element by its nuclear grammar — deuteron count horizontally, triton count vertically. Elements fall into natural construction lineages visible as horizontal families (shared triton count) and vertical families (shared deuteron count). Toggle the overlay to see magnetic character — a view of the same counts, not an independent law.
Gold is the first element where the triton count (39) exceeds the deuteron count (38). Beyond this threshold, the nuclear construction becomes triton-dominated — magnetically saturated and topologically unstable. The grammar predicts the onset of radioactivity from construction geometry alone.
A candidate account priced nuclear binding by the overlap of projected proton boundaries: each isolated boundary contributes πRp², while a packed union counts shared projected regions once. NSEQ04 tested that rule on the active dual-tetra packing. It correlates with total binding energy, but it does not reproduce binding energy per nucleon and is not promoted as the binding mechanism.
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The grammar map below plots every known isotope in construction-grammar space. Each dot is one isotope: its horizontal position is the deuteron excess (nd − nt), its vertical position is the total subunit count (nd + nt). Stable isotopes glow; unstable positions fade by half-life tier. Toggle overlays to reveal electron shells, magnetic character, decay vectors, and the golden nd = nt boundary. Pan with middle-drag · zoom with scroll · hover any dot for detail.
A C++20 header-only library encoding the complete theoretical framework. Zero external dependencies. Scales sit as they are: ℓP the dimensional seed, unit bridges, one mass/action seat, hydrogen’s koppa rung, a FIRAS clock. sdt::laws::measured is the engine’s CODATA/FIRAS lookup.
The single source of truth is laws.hpp — a constexpr-heavy header containing all six laws, the bridge equations, atomic and nuclear domain functions, and the koppa derivation. The engine computes orbital mechanics without G, nuclear structure without quarks, and cosmological observables without dark components.
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SDT tests a non-expanding pressure-and-depth account against the same observations, with its failed routes retained in the record.
In SDT the "tension" is a conflation: one number, H₀, is being asked to carry two different mechanisms — the emission-depth stamp and the propagation squeeze. Probes weight the two differently, so they land in a spread rather than on a point:
Status: hypothesis. The old isotropic CR09 route was falsified (0/122 transition depths). The two-component ordering remains a live test; supernova Hubble-diagram fits currently favour ΛCDM by Δχ² ≈ 90–226.
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SDT carries numbered experiment specifications (E13–E111) spanning nuclear physics, cosmology, orbital mechanics, and electromagnetism. The theory is designed to break.
The k-hierarchy returns 42.98 arcsec/century from ϟ = R/k², with G and M never written; the measured anomalous precession is 42.98. Convergent with GR — same number, half the furniture; the closure also recovers c itself to 0.0009%.
Light deflection at solar limb: 1.750 arcsec computed vs 1.751 measured. Derived from convergence gradient, not spacetime curvature.
Status: the eclipse rotation model FAILED real SPARC on direct rerun (GD05, RMS 66%). The derived floor a₀ = cH₀/2π and the 8.8% collapse stand; the rotation shape is OPEN.
SDT predicts mechanical signal propagation through vacuum between Casimir plates — sound through empty space via the lattice.
Mass-independent lateral drift exceeding 10 km — a direct consequence of pressure-gradient mechanics incompatible with general relativity.
The cosmological lithium problem resolved through α+d+t nuclear construction grammar constraints on light element formation.
Frequency-dependent speed of light for gamma rays — if the lattice is discrete at Planck scale, propagation velocity should vary with wavelength at extreme energies.
Displacement pressure predictions for seismic wave propagation through the inner core — testable against existing seismological data.
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Individual element of the spatial lattice. Individually incompressible, collectively deformable. Structured at Planck-length scale (1.616 × 10⁻³⁵ m).
ϟ is the ancient Greek k — same letter, same object. k = c/v is the gear; ϟ = R/k² = v²R/c² is that same gear read as a length on the boundary R. One number, in metres, carries a body's entire gravitational field — same formula for the proton and the Sun. Replaces GM/c² — no G, no kilograms, ever.
The epoch when every spation released its stored energy simultaneously. Replaces the Big Bang. SDT makes no claim about what existed before.
Matter blocking a fraction of convergent pressure from one direction, creating a net force toward the occluding body. The mechanism behind gravity, Coulomb force, and nuclear binding.
Inward-propagating pressure from all Planck shells to every point in space. Total burden: Φ = Nε ≈ 1.04 × 10⁻⁵⁶ J.
The volume of lattice a particle excludes. Determines both inertial mass (reorganisation cost) and gravitational mass (occlusion cross-section).
v²circ + v²trans = c². Every particle's internal circulation plus translational velocity sums to the speed of light squared.
A vortex's pressure disturbance sorted by angular order ℓ — the lobe count of the wake. ℓ=1 carries Coulomb, ℓ=2 carries the magnetic dipole, ℓ≥3 carries the nuclear reach.
Topological classification of vortex structures. W=0: neutrino (open winding). W=1: electron (unknot, confined). W=3: proton (trefoil, protected). W=2 is an unknot — metastable only (≈0.1 GeV barrier, ~10⁻²¹ s), never observed. Higher odd knots (W=5, 7, …) are protected and await discovery.
Historical model: a galactic disk's total occlusion saturating was proposed to flatten rotation curves. FAILED on direct rerun against real SPARC data (GD05: RMS 66%). Kept in the glossary because the theory keeps its assessments; galactic rotation is OPEN.
Z = 79 (Gold). The first element where triton count exceeds deuteron count (39t vs 38d). The onset of topological instability in nuclear construction grammar.
Ratio Peff / Pconv = 2.125 × 10⁻¹⁷. Maps convergence pressure to effective interaction pressure at atomic scale.
T = 3(W+1) = 12 for the proton: the mismatch between the angular rate the trefoil demands of a contact spation and the fastest the lattice can relay (ω_max = c/R). That slip is the mechanical origin of the wake's faces — Coulomb, magnetic, orbital entrainment. (Separately, the 1/3 in Law IV is the angular-averaging share: P_cf = P_conv/3.)
The decomposition of every nucleus Z ≥ 2 into 1 alpha core + nd deuterons + nt tritons, with nt = A−2Z and nd = 3Z−A−2. The counts are fixed by (Z, N) arithmetic — unique wherever both are non-negative — so the grammar's content is the structure built on it, not the counts themselves. ATOMICUS characterises 290 isotopes on this decomposition.
One proton + two neutrons — the grammar's unbalanced cluster. One triton per excess neutron (nt = N − Z), each holding two neutrons: N = 2 + nd + 2nt. The count is bookkeeping of (Z, N); magnetism rules written in it restate the ledger and are claimed as nothing more.
When a nucleus built on a He-3 core (incomplete tetrahedron) captures an inner-shell electron, converting p → n and completing the alpha tetrahedron. Not random decay — geometric self-repair. EC isotopes use the alternate grammar: 1 He-3 + nd + nt.
High-energy collision that breaks structural bonds within a nucleus, knocking deuterons or tritons loose from the scaffold. If spallation exposes an incomplete core, it opens an electron capture pathway — a recombination route between construction lineages.
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