Transporter Envelope v0.5 — Multi‑Substrate Arcs

Summary#

v0.5 extends the transporter envelope to support multi‑substrate paths:

$$ S_1 \to S_2 \to \dots \to S_n $$

while preserving identity and asymmetry along the entire chain.

1. Multi‑Substrate Arc#

Define:

$$ \gamma : [0,1] \to \mathcal{T} \times \mathcal{S} $$

with:

• $$\gamma(t) = (T(t), S(t))$$
• $$S(t)$$ piecewise constant, changing only at discrete waypoints

Waypoints:

$$ 0 = t_0 < t_1 < \dots < t_n = 1 $$

with:

$$ S(t) = S_i \quad \text{for } t \in [t_i, t_{i+1}) $$

2. Envelope Definition (v0.5)#

The Transporter Envelope is:

$$ E_T = { (T(t), S(t), A(T(t))) \mid t \in [0,1] } $$

Constraints:

• $$A(T(t)) > 0$$ for all $$t$$
• No branching, no duplication
• Substrate changes only at waypoints
• Each segment $$[t_i, t_{i+1}]$$ is continuity‑preserving

3. Reconstruction Windows per Segment#

Each segment may have its own window:

$$ W_i = [t_{i+1}-\delta_i, t_{i+1}] $$

Within $$W_i$$ :

• drift‑correction allowed
• local stabilization toward next substrate’s constraints

4. End‑to‑End Guarantee#

If all segments are legal:

• identity preserved from $$S_1$$ to $$S_n$$
• asymmetry preserved
• arrival substrate can be any $$S_k$$ where:
  • $$T(t_k) \approx T^*$$
  • $$A(T(t_k)) = 0.01$$

Claim#

Transporter Envelope v0.5 supports chained, multi‑substrate transport while maintaining a single, unbroken continuity path.