Spatial Displacement Theory · The Causal Chain · Technical Translation

The Dependency Spine in Formal Notation: Every Link, Its Inputs, Its Relation, Its Residual

J. C. Harvey

Melbourne, Australia · July 2026 · Domain-wide dependency spine

Notation. ϟ ≡ v²R/c² (koppa, a length) · z ≡ ϟ/r (convergence depth, dimensionless) · k ≡ c/v (gear) · W (winding number) · ℓ_P (spation, the grain of space) · Φ = Nε (Law I throughput). Certification labels follow the engine: DERIVED COMPUTED IDENTITY CALIBRATED OBSERVED OPEN.

0. Method: what a dependency spine asserts

A causal chain in this sense is not a narrative ordering but a partial order of derivational dependency. Link L_n depends on L_m when the relation at L_n consumes a quantity whose meaning is fixed at L_m. The discipline this imposes is severe and is the point: a result may not be cited at a link earlier than the link that defines its terms, and a circular route — one whose output is algebraically its own input — must be labelled IDENTITY and never DERIVED.

Three failure modes are policed explicitly throughout. Circularity: a relation returning its own input (e.g. e = √(αℏc/k_e), which is a restatement of α's definition) is labelled as such. Shared form: where an SDT relation is algebraically identical to a rival framework's at the tested order, the correspondence is disclosed and the relation is recorded as unable to discriminate. Count degeneracy: where a regularity is a pure function of quantities already fixed upstream, it is bookkeeping, not evidence (NP24-A).

The external input ledger

The stated whitelist has nine members. A single-pass re-derivation of each (derivelist/, 2026-07-30) with an independent audit reduces the set to five, chiefly by identifying which members were never physics but unit definitions:

The collapse is real but it relocates rather than removes: the mystery moves from a mass to a length. Closing a₀, or landing the 6π⁵ wake integral, is what would actually shorten this list.

L0 — AxiomThe seed: one grain of length

FLM06 attempted the derivation and returned a negative with a proof attached: dimensionless ratios of permitted cosmic quantities bottom out near 10³ where 10⁶¹ is required. Exactly one dimensional anchor is mathematically unavoidable. AXIOM

L1 — SubstrateThe substrate: a superfluid of grains

Space is a contact-relay medium of grains of one size, individually incompressible and collectively deformable. Icosahedral 12-around-1 coordination does not close; the residual packing fraction is a conserved slack.

The shaped voids are load-bearing downstream: they are the seats atoms keep electrons in (L14) and the reason the alpha core is a tetrahedron (L13). DERIVED (geometry) with the void-tier capacities owed to L13's completion law.

L2 — Derived frameThe tick: nothing but now

Time is not a primitive. It is the tally movement keeps; the present is the plane on which everything moves. The clock rate is a closure ratio, not a coordinate:

Repaired 2026-07-30 (GOM22). This relation previously read √(1−z), which returned half the measured solar gravitational redshift and contradicted the depth-differential relation at L17. See L17 for the measurement that settled it. DERIVED

L3 — Law IThe Clearing, and the push that never stopped

The convergence arrives from the Clearing boundary in every direction each tick. Perfect balance is why the medium reads as empty. Every force downstream is an imbalance in this one push. DERIVED from {ℓ_P, T_CMB, R_CMB}; R_CMB is an observed scale and is declared as conditioning the Law I chain.

L4 — Law IIIOcclusion: the only force

Two bodies each shadow part of the convergence arriving at the other; the unshadowed side wins and they are pressed together. Nothing pulls. The inverse square is not postulated — it is the solid angle a cross-section subtends. Gravity, the Coulomb force and the nuclear grip are this one relation read at three cross-sections; their famous hierarchy of strengths is a ratio of areas, not of kinds.

L5 — The one new ideaMatter: the knot that displaces

A particle is circulation wound on itself and locked by its own winding — a torus knot, not a cloud and not a point. It has a measurable boundary and a surface that runs faster than the relay. The single new premise of the framework is that matter displaces space; the displaced grains must go somewhere, and where they go is every force.

L6 — Law IVMass: the cost of moving a displacement

Mass is the throughput reorganisation cost of changing a displacement's vector — resistance of the field, not a substance carried by the body. FLM12 recovers Newton I and II, E = mc², and the equivalence of inertial and gravitational mass as a theorem rather than a coincidence.

L7 — Law VThe movement budget: one purse, two spends

Every particle spends its whole allowance at c at all times, divided between internal circulation and translation. Special relativity's kinematics follow as bookkeeping of that division: the slowed clock, the shortened rod and the unreachable ceiling are one budget being spent differently, with no dilation of anything but the tally.

L8 — Law IIRelease: the cascade

Held throughput releases in a cascade through pressure domains; the blackbody suite follows from counted relay modes rather than from quantised oscillators.

L9 — The bridgeKoppa: the gear that is a length

Koppa is the framework's load-bearing bridge: a velocity-defined length that replaces the product GM without importing either G or M. One relation, v²r = c²ϟ, runs from the proton seat to Sgr A* — twenty orders of magnitude on one law. Disclosed: ϟ is numerically GM/c², so every first-order gravitational result below shares its form with the rival account and cannot discriminate on that basis; what SDT claims is the origin, not the number.

L10 — The hydrogen chainHydrogen: three lengths, one invariant

α is not a magic number in this account: it is the gear at which hydrogen's ground state sits, k = 137. The three lengths of hydrogen — a₀, r_e, ϟ_H — are one ladder read at three rungs.

L11 — EmissionEmission: a path times a gear

Seven independent emission floors return the same ratio to 1.0000 under this reading. Spectral structure is route geometry, not a probability amplitude.

L12 — Charge and EMCharge: a handedness, not a substance

There is no charge substance. What is called charge is a handed redirection of the convergence; attraction is a deficit (shadow) and repulsion a surplus (focus), which is why gravity has one sign and electromagnetism two. Relation (12.1) is printed as the tautology it is; the physical content sits in (12.2), which closes to 0.0431% against the value the force law requires.

L13 — The nuclear floorNuclei: castings from the stellar foundry

The closure schedule — the magic numbers derived

Four mesh facts force (13.4). A proton gears only with neutrons, so every tier is a bipartite double layer; the packing's two-dimensional order is triangular, so a layer holds a triangular number of rod seats — together, F(n), with F(0) = 2 being the alpha core under the same formula. Tritons are radial rods and the assembly must roll without jamming, so a rod runs slip-free only on the equator of the collective rotation — a one-dimensional antipodally-paired ring, B(n). No spin–orbit coupling is imported anywhere; the belt's location is geometric and carries no strength parameter. Forward prediction: closure(7) = 126 + R(6) + B(7) = N = 184.

L14 — SeatsAtoms and metals: exclusive seats

Electrons occupy the shaped voids of L1 — seats, not orbitals or probability clouds. Exclusivity is geometric occupancy rather than a symmetry postulate; the periodic trends are the seat ladder's own arithmetic, with core occlusion pulling the outer seat inward across a period and a new void tier opening at each shell.

L15 — FluidsFluids: one poloidal operation

All fluid movement is a poloidal operation upgraded to torus level, where it wraps. The intermittency correction to the Kolmogorov exponent comes from the packing gap of L1 with no new parameter: −1.6928 against a measured −1.70.

L16 — Heat and freedomFreedom: entropy with its costume off

Entropy is replaced by a freedom ledger — a count of available degrees of movement rather than a measure of disorder. Thermodynamics is inherited, not re-founded: across the domain no new primitive and no free parameter is introduced.

L17 — Gravity and orbitsGravity: depth, not warping

Gravity is a depth gradient in a medium, not curvature of a manifold. Orbits are closed least-resistance paths through that gradient: real motion, no pull. Nothing is attracted — the convergence of L3 presses everything together and matter takes the cheapest route.

The factor-2 repair (GOM22, 2026-07-30)

The Shapiro delay measures the relay-speed profile, because a propagation delay is an integral of the inverse local speed. Writing c_local = c(1 − K·z), the measured delay fixes K with no freedom:

The canon previously carried K = 1 in (2) and √(1−z) in (3). Both returned exactly half the measurement. The root was a mixed-frame slip: the derivation scaled the hop length by the local closure while leaving the hop duration at its far-frame value. Speed is length per time, so the closure enters twice — a native repair drawn from a relation already in canon, and the factor 2 in the delay is now derived rather than inserted.

L18 — Stars and galaxiesThe open frontier

This link fails and is printed as failing. The canonical three-dimensional eclipse rotation model returns RMS 66% against real SPARC data with a BTFR slope of 1.33; the earlier reported success was a circular mock and has been withdrawn. What stands: the derived floor (18.1), and an 8.8% cross-scale collapse re-earned on real rotation data against a control six times worse. The crossover's functional shape is still borrowed and is not claimed.

L19 — CosmologyThe cosmos: nothing recedes

Redshift is two native effects and no recession: a depth stamp at emission plus a propagation squeeze. Conflating them is what produces the Hubble tension. No expansion, no dark energy and no dark matter enters any relation on this page.

L20 — The ledgerWhat the framework owes

A dependency spine is only as good as its willingness to print its own gaps. The register, current at this build:

References