From dad0dca0fa7b1128054b9e5224c89c0cef3806f4 Mon Sep 17 00:00:00 2001
From: Allaun Silverfox <28494262+allaunthefox@users.noreply.github.com>
Date: Sat, 16 May 2026 17:17:26 -0500
Subject: [PATCH] Add Sidon FAMM map documentation

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 6-Documentation/famm/SIDON_FAMM_MAP.md | 290 +++++++++++++++++++++++++
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+# Sidon FAMM Map
+
+## Purpose
+
+Load the full Sidon thread into the FAMM map.
+
+This document consolidates the Sidon work into one operational map:
+
+```text
+classical Sidon set
+→ pair-sum gate
+→ candidate-addition receipt
+→ collision scar
+→ physics-native path/collision packet
+→ quaternion/phase anti-alias gate
+→ two-layer kinetic/Sidon flow
+→ Theodorus capacity shell
+→ Semantic Mass route pressure
+→ NUVMAP Delta-DAG node
+→ Builder-Judge-Warden cleanup
+→ exact verifier receipt
+```
+
+## Core Sidon gate
+
+```math
+G_{\mathrm{Sidon}}(A)=1
+\iff
+\forall a,b,c,d\in A,\quad
+a+b=c+d\Rightarrow \{a,b\}=\{c,d\}
+```
+
+Equivalent map:
+
+```math
+\pi:\{(i,j):i\le j\}\to \mathbb Z,
+\qquad
+\pi(i,j)=a_i+a_j
+```
+
+Sidon means:
+
+```math
+\pi\ \mathrm{is\ injective}
+```
+
+## Candidate-addition gate
+
+Given:
+
+```math
+S(A)=\{a_i+a_j:i\le j\}
+```
+
+For candidate `x`:
+
+```math
+S_x=\{x+a:a\in A\}\cup\{2x\}
+```
+
+Accept iff:
+
+```math
+S_x\cap S(A)=\varnothing
+```
+
+Otherwise:
+
+```text
+reject candidate
+write Sidon collision scar
+route failure to coarsening / nogood cache
+```
+
+## FAMM object
+
+```math
+\mathfrak C_{\mathrm{SidonMap}}
+=
+A_{16}(u_{\mathrm{sidon}})
+\otimes
+[
+\Sigma_A
++
+\Sigma_{2A}
++
+\Sigma_{\mathrm{flow}}
++
+\Sigma_{\mathrm{phase}}
++
+\Sigma_{\mathrm{collision}}
++
+\Sigma_{\mathrm{scar}}
++
+\Sigma_{\sqrt N}
++
+\Sigma_{\mathrm{receipt}}
+]
+```
+
+## Semantic Mass lanes
+
+```math
+\mu_{\mathrm{Sidon}}[k]
+=
+w_c C_k
++
+w_e E^+_k
++
+w_\Omega \Omega_k
++
+w_\theta \Delta\theta_k
++
+w_q Q_k
++
+w_i I_k
++
+w_r R_k
+```
+
+Where:
+
+| Lane | Meaning |
+|---|---|
+| `C_k` | nontrivial collision count |
+| `E⁺_k` | additive energy |
+| `Ω_k` | FAMM scar burden |
+| `Δθ_k` | phase/chirality mismatch |
+| `Q_k` | quaternion/phase quality |
+| `I_k` | invariant overlap |
+| `R_k` | residual / unresolved tail |
+
+## Scar equation
+
+```math
+\Omega_{\mathrm{Sidon}}(A)
+=
+E_+(A)-(2|A|^2-|A|)
+```
+
+A valid Sidon set has:
+
+```math
+\Omega_{\mathrm{Sidon}}(A)=0
+```
+
+## Theodorus capacity prior
+
+For finite Sidon search inside `[1,N]`, use:
+
+```math
+r_N=\sqrt N
+```
+
+as a capacity shell / density pressure prior.
+
+```text
+N = address/value budget
+sqrt(N) = capacity shell
+|A| = occupied independent marks
+sqrt(N)-|A| = remaining admissible capacity pressure
+```
+
+This is not itself proof of maximality. It is a routing prior.
+
+## Two-layer kinetic/Sidon mapping
+
+```text
+Layer K: kinetic standing-wave field
+Layer S: Sidon relational address field
+K → S: quantize pairwise kinetic relations into signatures
+S → K: feed alias-free relational structure back into kinetic update
+```
+
+The Sidon layer is not static. It flows between layers:
+
+```text
+kinetic state
+→ pairwise kinetic relation
+→ Sidon signature
+→ anti-aliasing gate
+→ feedback into kinetic update
+```
+
+## Quaternion / phase Sidon mapping
+
+Classical Sidon:
+
+```text
+a + b = c + d ⇒ {a,b} = {c,d}
+```
+
+Quaternionic / phase-lattice generalization:
+
+```text
+σ(i,j)=σ(k,l) ⇒ {i,j}={k,l}
+```
+
+where:
+
+```text
+σ(i,j) = encoded pairwise relative phase / interaction signature
+```
+
+## Physics-native Sidon packets
+
+Pair sums become path packets:
+
+```text
+SidonPathPacket = {
+  pair_id,
+  source_i,
+  source_j,
+  sum_address,
+  source_history_hash,
+  additive_phase_q16,
+  temporal_shear_q16,
+  collision_energy_q16,
+  valid_trivial_symmetry,
+  receipt_hash
+}
+```
+
+Nontrivial repeated sums become collision packets:
+
+```text
+SidonCollisionPacket = {
+  collision_id,
+  sum_address,
+  pair_a,
+  pair_b,
+  trivial_collision_bool,
+  additive_shear_q16,
+  temporal_shear_q16,
+  ignition_score_q16,
+  underverse_class,
+  warden_status,
+  receipt_hash
+}
+```
+
+## Coarsening-agent rule
+
+Wrong Sidon candidates are useful.
+
+```text
+nontrivial repeated sum
+→ collision scar
+→ NUVMAP coarse basin
+→ downweight future fine search
+→ reopen only if new evidence changes the boundary
+```
+
+So a failed Sidon extension becomes a reusable negative topology witness.
+
+## Builder-Judge-Warden cleanup
+
+| Role | Sidon use |
+|---|---|
+| Builder | propose candidate-addition, pair-signature remap, Theodorus capacity routing, kinetic-to-Sidon projection |
+| Judge | verify injectivity / trivial symmetry / residual zero |
+| Warden | block nontrivial collisions, false physical claims, unbounded recursion, false maximality, wrong-handed phase routes |
+
+## Stack placement
+
+```text
+THEODORUS_SHELL
+→ SIDON_PAIR_SUM_GATE
+→ SIDON_FAMM_MAP
+→ SEMANTIC_MASS_ROUTE_PLOW
+→ PIST / PIST-CHAOS transition
+→ NUVMAP_DELTA_DAG
+→ COARSENING_SCAR_CACHE
+→ BJW_GEODESIC_CLEANUP
+→ EXACT_RECEIPT
+```
+
+## Claim boundary
+
+This map does not claim Sidon sets are physical plasma events, nor that a heuristic Sidon route proves maximality. The exact claim is narrower and stronger:
+
+```text
+Sidon pair-signature uniqueness is an exact anti-collision gate.
+Violations become receipt-bearing scars.
+Those scars can guide FAMM routing, coarsening, and cleanup.
+```
+
+## Project sentence
+
+The Sidon FAMM Map turns pair-sum uniqueness into an exact anti-collision receipt: valid pair signatures become addressable relation channels, nontrivial collisions become FAMM scars/coarsening agents, and the whole Sidon layer can now route through Theodorus capacity, Semantic Mass, PIST transitions, NUVMAP Delta-DAG replay, and Builder-Judge-Warden cleanup.