Abstract
We treat the Greek letter τ as the symbol (codename illumina) for a rendezvous framework: mathematical optimality + anthropic selection push independent lineages toward the same short prime headers, simple ECC, dimensionless anchors, and optional topological flags. Using only plain HTML entities, we specify a minimal standard and show how to test its convergent inevitability.
1. Convergent Rendezvous Principle (CRP)
CRP: Under fixed physical law, mathematical optimality & anthropic selection induce a flow on design space with a small attractor set.
Choosing from that set maximizes cross-civilization discoverability per recipient τ.
Attractors (math)
- Prime-length headers with low sidelobes (Barker-like, Legendre, m-sequences).
- Compact ECC (Reed–Solomon; extended Golay).
- Dimensionless anchors: ratios any physics can in principle obtain.
- Optional topology flags (matched-circles template IDs).
Filter (anthropic)
- Inefficient codes waste τ and self-select out.
- Survivors converge on the same small toolkit.
- Asynchronous rendezvous (shared decoder without prior contact).
2. τ-Flow & Rendezvous Efficiency
Let X be a point in design space. Utility per τ-budget:
U(X;\,τ) = useful output / τ.
Gradient-like dynamics with constraints:
˙X ≈ Πconstraints ∇XU + noise.
Define rendezvous efficiency:
ηR = Pr(correct decode) per recipient-τ → attractor choices maximize ηR.
3. Illumina-1 Spec
3.1 Handshake header
M = { prime length p, Barker-like autocorrelation, checksum }.
- Recommended p: 31 or 47.
- Checksum: 16-bit parity or Golay(24,12,8) syndrome.
3.2 ECC
- RS(31,19) over GF(32) (short payloads; widespread tooling).
- Extended Golay (24,12,8) for ultra-compact, high-robustness payloads.
3.3 Dimensionless anchor tuple
a = ( α, me/mp, Q, Gℏ/c3 ).
Prefer symbolic reference or sub-noise perturbations; stay below structure-formation bounds.
3.4 Optional topology echo
One matched-circles template bit; harmless if null.
4. Predictions & Tests
4.1 Convergence predictions
- Independent high-reliability systems cluster on a few prime-length headers/ECC families.
- Maximum-likelihood decoders for those families recur across domains.
- Lockstep correlations appear across sensing → coding → scheduling to minimize wasted τ.
4.2 DIY checks
- Header audit: collect real preambles; test for prime lengths & sidelobes.
- ηR bake-off: simulate recipient cost; pick p, ECC maximizing decode per τ.
- Carrier sanity: plant a 31/47 header at <5σ and measure fixed-statistic detectability.
5. Safety
Cosmic safety charter: pre-register analyses; prefer diagnostic encodings (non-perturbative);
publish code & null ensembles; use multi-party review; encode pro-social norms alongside the handshake.
Appendix — Encoders & Capacity sketch
| Carrier | Handle | Noise | Pros / Cons |
|---|---|---|---|
| Spectrum | Low-ℓ phases; aℓm | Gaussian field + systematics | Global visibility / multiple-testing penalties |
| Constants | α, me/mp, Q, Gℏ/c3 | Lab error; drift | Simple & universal / very low capacity |
| Topology | Matched circles; compact flags | Cosmic variance; masking | Dimensionless / hard inference |
C ≈ ∑i log2(1 + τenc,i2/σi2) (orthogonal carriers i).