From cc798b04382cdf9eec18c39245a4a0816cf03a1f Mon Sep 17 00:00:00 2001
From: Allaun Silverfox <28494262+allaunthefox@users.noreply.github.com>
Date: Sat, 16 May 2026 20:47:57 -0500
Subject: [PATCH] Add BraidStorm Sidon Crossing Anti-Alias Gate documentation

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+# 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:
+
+```text
+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:
+
+```math
+s_1,s_2,\dots,s_m
+```
+
+assign labels:
+
+```math
+a_i\in A\subset[1,N]
+```
+
+such that:
+
+```math
+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
+
+```math
+\operatorname{addr}(i,j)=a_i+a_j
+```
+
+Every crossing event becomes addressable:
+
+```math
+\beta_{ij}:(s_i,s_j)\to(s_i',s_j',r_{ij},\epsilon_{ij},\Omega_{ij})
+```
+
+with:
+
+```math
+r_{ij}=H(\operatorname{addr}(i,j),\Delta_{ij},\epsilon_{ij},\Omega_{ij})
+```
+
+## FAMM object
+
+```math
+\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
+
+```math
+R_{\mathrm{SidonCross}}
+=
+\left|\{(i,j,k,l):a_i+a_j=a_k+a_l,\;\{i,j\}\ne\{k,l\}\}\right|
+```
+
+Solved means:
+
+```math
+R_{\mathrm{SidonCross}}=0
+```
+
+## Theodorus/Sidon capacity prior
+
+For ordinary integer Sidon labels in `[1,N]`, use the rough routing prior:
+
+```math
+|A|\sim\sqrt N
+```
+
+So for `m` strands:
+
+```math
+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:
+
+```text
+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
+
+```text
+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:
+
+```text
+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.