Research-Stack/6-Documentation/famm/BRAIDSTORM_SIDON_CROSSING_ANTIALIAS_GATE.md

BraidStorm Sidon Crossing Anti-Alias Gate

Purpose

Use Sidon pair-sum uniqueness to give every BraidStorm crossing a collision-free additive address.

This is the first concrete problem selected for the Sidon FAMM Map:

Given m BraidStorm strands,
assign each strand a Sidon label a_i,
so every unordered crossing {i,j} has a unique address a_i + a_j.

This solves the pair-address aliasing problem for BraidStorm receipts, scars, delta edges, coarsening agents, and NUVMAP handoffs.

Problem statement

Given strands:

s_1,s_2,\dots,s_m

assign labels:

a_i\in A\subset[1,N]

such that:

a_i+a_j=a_k+a_l
\Rightarrow
\{i,j\}=\{k,l\}

for unordered pairs, usually with i <= j if self-crossings are allowed or i < j if only distinct-strand crossings are allowed.

Crossing address

\operatorname{addr}(i,j)=a_i+a_j

Every crossing event becomes addressable:

\beta_{ij}:(s_i,s_j)\to(s_i',s_j',r_{ij},\epsilon_{ij},\Omega_{ij})

with:

r_{ij}=H(\operatorname{addr}(i,j),\Delta_{ij},\epsilon_{ij},\Omega_{ij})

FAMM object

\mathfrak C_{\mathrm{BraidSidon}}
=
A_{16}(u_{\mathrm{braid\_sidon}})
\otimes
[
\Sigma_{\mathrm{strand}}
+
\Sigma_A
+
\Sigma_{2A}
+
\Sigma_{\mathrm{crossing}}
+
\Sigma_{\mathrm{collision}}
+
\Sigma_{\mathrm{scar}}
+
\Sigma_{\mathrm{receipt}}
]

Residual

R_{\mathrm{SidonCross}}
=
\left|\{(i,j,k,l):a_i+a_j=a_k+a_l,\;\{i,j\}\ne\{k,l\}\}\right|

Solved means:

R_{\mathrm{SidonCross}}=0

Theodorus/Sidon capacity prior

For ordinary integer Sidon labels in [1,N], use the rough routing prior:

|A|\sim\sqrt N

So for m strands:

N\approx m^2

Examples:

Strands	Address budget prior
16	256
32	1024
64	4096
128	16384
256	65536

This is a capacity prior, not a constructive maximality proof.

Integer versus modular mode

Two operational modes are supported:

Mode	Meaning	Use
integer	pair sums are unique as integers	safest receipt; direct additive address
modular	pair sums are unique modulo N	useful for fixed-width address rings / hardware registers

The Warden must record which mode was used.

Coarsening behavior

If two distinct crossings alias:

crossing(i,j) and crossing(k,l) share address
→ collision scar
→ coarsen that address basin
→ downweight future fine routing through that label assignment
→ reopen only if labels or address mode change

BraidStorm placement

BraidStorm strand registry
→ Sidon label assignment
→ crossing address map
→ crossing receipt
→ FAMM scar ledger
→ NUVMAP Delta-DAG edge
→ coarsening cache if collision

Builder-Judge-Warden mapping

Role	Use
Builder	propose Sidon labels / address budget / modular or integer mode
Judge	verify pair-sum injectivity and emit zero-collision receipt
Warden	block aliasing, false maximality, wrong address mode, and unreceipted collisions

Claim boundary

This gate does not prove BraidStorm convergence. It solves a narrower infrastructure problem:

pairwise crossings get collision-free additive addresses.

Project sentence

The BraidStorm Sidon Crossing Anti-Alias Gate assigns every strand a Sidon label so each pairwise crossing has a unique additive address, letting FAMM receipts, NUVMAP Delta-DAG edges, scars, and coarsening agents attach to the correct interaction without pair-address collision.