An Angle-Free Lorentz Embedding

Reading Position

Read this after circular phase geometry and before the hopfion-like topology, propagation, and beat-phenomena notes. It is not a physical law and does not derive spacetime. Its role is technical: it shows how angular variables, rapidity parameters, and torus angles can be treated as optional coordinates once the primitive data are vectors, constraints, updates, and inner products.

Core Claim

The paper's discipline is simple: do not make an angle primitive when the relation can be expressed by a vector equation. Direction is carried by a unit vector; alignment is carried by a dot product; Lorentzian separation is carried by an inner product. A coordinate chart may still be useful, but the chart is not the object.

primitive data:
embedded vector X
unit direction u
Lorentzian inner product eta

optional charts:
angle theta
rapidity
torus parameters

This makes the note a guardrail for the rest of the site. It supports angle-free recurrence without pretending that familiar coordinate functions have disappeared; they have simply been demoted from primitives to readouts.

No-Angle Replacement

The starting replacement is simple. Instead of representing relative direction by an angle difference, assign each direction a unit vector and use the dot product between the unit vectors.

instead of:
cos(theta_2 - theta_1)

use:
u_1 dot u_2

with:
u_i dot u_i = 1

This is not a weaker approximation to the angular formula. It is the same directional information expressed algebraically, without committing the model to an angular coordinate chart.

Lorentzian Invariant

Directional separation is then placed inside a Lorentzian embedding. Write an embedded state as a scalar part plus a radial spatial part multiplied by the unit direction.

X_i = (C_i, S_i u_i)

C_i^2 - S_i^2 = 1

_eta = 1

The pairwise comparison is carried by the Lorentzian inner product:

chi_12 = C_1 C_2 - S_1 S_2 (u_1 dot u_2)

If a standard unit-hyperboloid distance convention is later introduced, the same invariant may be read as cosh(d_12) = chi_12. If inverse functions are being avoided, chi_12 can be used directly as the distance-ordering invariant.

Two-Dimensional Direction Space

The note then rewrites the construction with notation that avoids collision with survival weights, counts, or other RSG variables. Let u_i be a unit direction in R^2, and let alpha_i and rho_i be scalar embedding coordinates.

u_i = (u_i1, u_i2)
u_i dot u_i = u_i1^2 + u_i2^2 = 1

alpha_i^2 - rho_i^2 = 1

X_i = (alpha_i, rho_i u_i1, rho_i u_i2)

With eta = diag(1, -1, -1), each embedded vector lies on the unit Lorentzian surface.

_eta
  = alpha_i^2 - rho_i^2 (u_i dot u_i)
  = 1

For two states, relative direction enters only through u_1 dot u_2:

chi_12 = _eta
       = alpha_1 alpha_2 - rho_1 rho_2 (u_1 dot u_2)

Higher-Dimensional Version

The construction immediately generalises to n spatial embedding dimensions. This is one reason it is useful: no special trigonometric chart is needed for each dimension.

u_i = (u_i1, ..., u_in) in R^n
sum_a u_ia^2 = 1

X_i = (alpha_i, rho_i u_i1, ..., rho_i u_in) in R^{1,n}

eta = diag(1, -1, ..., -1)

_eta
  = alpha_i^2 - rho_i^2 sum_a u_ia^2
  = 1

chi_12 = alpha_1 alpha_2 - rho_1 rho_2 sum_a u_1a u_2a

Coordinate-Only Form

The note also gives the same result without separating radial and direction data. This makes the invariant especially compact.

X_i = (X_i0, X_i1, ..., X_in)

X_i0^2 - sum_a X_ia^2 = 1
X_i0 > 0

chi_12 = X_10 X_20 - sum_a X_1a X_2a

Unit directions can be recovered from the spatial components when the spatial magnitude is nonzero, but the invariant itself does not require that recovery.

Coordinate Time

The new version adds an important clarification about time. In the dimensionless Lorentz embedding, coordinate time is not an angle. It is the zeroth coordinate of the Lorentz vector after a frame and scale have been chosen.

X_i = (T_i, Y_i)
T_i^2 - Y_i dot Y_i = 1

Z_i = c tau_0 X_i = (c t_i, x_i1, ..., x_in)
t_i = tau_0 T_i

If the spatial part is written Y = R u, with u dot u = 1, then the future-sheet condition gives T = sqrt(1 + R^2). Direction and spatial size are separate: u gives orientation, R gives spatial embedding magnitude, and T gives the coordinate-time projection. No rapidity angle has to be primitive.

X_1 = (T_1, R_1 u_1)
X_2 = (T_2, R_2 u_2)

chi_12 = T_1 T_2 - R_1 R_2 (u_1 dot u_2)

Recursive Time Steps

The coordinate-time section becomes recursive by updating the spatial embedding part and then recovering the next future-sheet time coordinate algebraically. This fits the site's broader discrete-first habit: the step is primary, and the time reading is recovered from the constrained state.

Y_{k+1} = F_k(Y_k)

T_{k+1} = sqrt(1 + Y_{k+1} dot Y_{k+1})
t_{k+1} = tau_0 T_{k+1}

Delta t_k = tau_0 (T_{k+1} - T_k)

Adjacent states are still compared by the same Lorentz product:

chi_{k,k+1}
  = T_k T_{k+1} - Y_k dot Y_{k+1}

  = T_k T_{k+1}
    - R_k R_{k+1} (u_k dot u_{k+1})

Differential Time

For a smooth path X(lambda) = (T(lambda), Y(lambda)), the same constraint gives the differential clock relation. Differentiating T^2 - Y dot Y = 1 yields:

T T_dot - Y dot Y_dot = 0

T_dot = (Y dot Y_dot) / T
t_dot = tau_0 (Y dot Y_dot) / T

In separated form Y = R u, with u dot u = 1, the orthogonality relation u dot u_dot = 0 gives:

Y dot Y_dot = R R_dot

t_dot = tau_0 R R_dot / T

This is one of the clearest intuitions in the update: changing direction at fixed R changes spatial orientation and pairwise invariants, but it does not by itself change the coordinate-time projection. Radial change changes T, and therefore changes the coordinate-time reading.

No-Angle Torus

The paper then moves from one closed direction to two closed directions. A torus usually appears with two angle parameters, but the same construction can be written using two unit direction pairs.

U = (U^1, U^2),  U dot U = 1
V = (V^1, V^2),  V dot V = 1

R_0 > r_0 > 0

T(U,V)
  = ((R_0 + r_0 V^1) U^1,
     (R_0 + r_0 V^1) U^2,
      r_0 V^2)

Here U carries the main ring direction and V carries the tube direction. In plain terms, a circle is one closed update; a torus is two closed updates tied together. The visible support point is recovered algebraically from those two directions.

Rational Closure

The torus becomes a recursive object when both directions update step by step. If the updates preserve unit length, a closed history occurs when both direction states return to their initial values after some number of frames.

U_{k+1} = A_k U_k
V_{k+1} = B_k V_k

A_k^T A_k = I_2
B_k^T B_k = I_2

P_k = T(U_k, V_k)

U_N = U_0 and V_N = V_0

If the main cycle closes after p turns and the tube cycle closes after q turns, the completed history can be read as a (p,q)-type torus path. The paper is careful here: the claim is not that an observer sees a literal rotating line. It says that an update sequence may be perceived or measured as one stable support trace when the observer resolves fewer frames than the process generates.

Hopfion-Like Step

The final extension adds linked fibres. The minimal topological model is a map h : S^3 -> S^2, where preimages of points in S^2 are closed loops in S^3 and distinct fibres are linked.

(z_1, z_2) in C^2
|z_1|^2 + |z_2|^2 = 1

z_1 = a U
z_2 = b V
a^2 + b^2 = 1

h(z_1,z_2)
  = (2 Re(z_1 z_2),
     2 Im(z_1 z_2),
     |z_1|^2 - |z_2|^2)

A preserved hopfion-like sector can be recorded by a class invariant:

Q_H(K_{n+1}) = Q_H(K_n)

The phrase "hopfion-like" matters. The paper does not claim a full physical hopfion. A physical claim would need a field, energy functional, boundary conditions, and a topological charge in that field theory. This note supplies the angle-free geometry for describing a linked-fibre support candidate.

Fit With RSG

This paper supports the site's angle-free mathematical spine. It pairs naturally with circular phase geometry, hopfion-like topology, and the beat-phenomena notes: all show how familiar trigonometric expressions can be replaced by algebraic state variables, recurrence, dot products, constraints, or invariants once the geometry has already been specified.

It also complements the v1.4 null-transport material. RSG v1.4 still imports the Lorentzian interval as a bridge structure; this note gives a clean embedding language for direction, time projection, toroidal closure, and linked support without promoting angles or rapidities to primitive data.