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

Sidon FAMM Map

Purpose

Load the full Sidon thread into the FAMM map.

This document consolidates the Sidon work into one operational map:

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

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:

\pi:\{(i,j):i\le j\}\to \mathbb Z,
\qquad
\pi(i,j)=a_i+a_j

Sidon means:

\pi\ \mathrm{is\ injective}

Candidate-addition gate

Given:

S(A)=\{a_i+a_j:i\le j\}

For candidate x:

S_x=\{x+a:a\in A\}\cup\{2x\}

Accept iff:

S_x\cap S(A)=\varnothing

Otherwise:

reject candidate
write Sidon collision scar
route failure to coarsening / nogood cache

FAMM object

\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

\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

\Omega_{\mathrm{Sidon}}(A)
=
E_+(A)-(2|A|^2-|A|)

A valid Sidon set has:

\Omega_{\mathrm{Sidon}}(A)=0

Theodorus capacity prior

For finite Sidon search inside [1,N], use:

r_N=\sqrt N

as a capacity shell / density pressure prior.

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

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:

kinetic state
→ pairwise kinetic relation
→ Sidon signature
→ anti-aliasing gate
→ feedback into kinetic update

Quaternion / phase Sidon mapping

Classical Sidon:

a + b = c + d ⇒ {a,b} = {c,d}

Quaternionic / phase-lattice generalization:

σ(i,j)=σ(k,l) ⇒ {i,j}={k,l}

where:

σ(i,j) = encoded pairwise relative phase / interaction signature

Physics-native Sidon packets

Pair sums become path packets:

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:

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.

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

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:

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.