SilverSight/docs/research/braid_group_action.md

CRT Torus Embedding: Braid Group Action (Dual-Model Framework)

The CRT torus supports two complementary braid models: an axis-swap model that satisfies the braid group relations exactly, and a modulus-adjustment model that bounds expressible braid word length through coprimality constraints.

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1. The Two Models

Aspect	Axis-Swap (configuration)	Modulus-Adjustment (resource)
What changes	Reflection modulus positions	Reflection modulus values
Preserves	Modulus values	Modulus positions
Satisfies braid relations	Yes (σᵢ²=id, YB, far commute)	No (YB fails, σ² may fail)
Bounding factor	None (free permutation)	Coprimality (spacing between strands)
Verified	Provider-NixOS (4-core)	Provider-NixOS
Use	Braid group action on F	Braid word maximum length

The two models are complementary, not competing. The axis-swap model defines the topology (braid group action Bₙ on the reflection moduli). The modulus-adjustment model defines the physics (changing modulus values to create Sidon via the wrapping criterion).

Critical distinction

Property	Axis-Swap	Adjustment
Changes FA values?	No (CRT symmetry)	Yes
Why?	CRT is symmetric under modulus permutation; swapping reflection residues between strands doesn't change the unique CRT lift	Modulus values change → residues change → CRT lift is genuinely different
Verified	4 test sets: reflection-closed, asymmetric, random, sparse — all give identical FA	The Sidon theorem and wrapping criterion
Role in DAG	Defines braid word (which strands cross)	Creates Sidon (which FA values emerge)

The axis-swap produces identical FA values because the CRT computation is commutative: the unique solution in [0, ∏Lᵢ) depends only on the multiset of (residue, modulus) pairs, not on their ordering. Permuting the reflection moduli across strands is a reordering of the CRT factors — the result is the same for every element a ∈ A.

Implication for the DAG: Finding Sidon via axis-swap is impossible when the CRT uses all moduli simultaneously (which it does — the k-modulus CRT lifts all residues together). Sidon creation requires the adjustment model to change actual modulus values.

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2. Model 1: Axis-Swap (Braids Satisfied)

Each braid generator σᵢ swaps the reflection moduli of adjacent strands while leaving identity moduli unchanged:

σᵢ: (L₂ᵢ, L₂ᵢ₊₂) → (L₂ᵢ₊₂, L₂ᵢ)    [swap reflection axes i and i+1]
     identity axes: L₂ᵢ₋₁, L₂ᵢ₊₁ unchanged

For a 3-strand system with 6 moduli [L₁, L₂, L₃, L₄, L₅, L₆]:

Generator	Acted indices	Effect
σ₁	(L₂, L₄)	L₂ ↔ L₄
σ₂	(L₄, L₆)	L₄ ↔ L₆
σ₁σ₂σ₁	(L₂, L₄, L₆)	(L₂, L₄, L₆) → (L₆, L₂, L₄)
σ₂σ₁σ₂	(L₂, L₄, L₆)	(L₂, L₄, L₆) → (L₆, L₂, L₄)

Verified braid axioms

Axiom	Status	Test on (2,3,5,7,11,13)
σᵢ² = id	✓	s1(s1(mods)) == mods
σᵢσⱼ = σⱼσᵢ (	i−j	≥2)
σ₁σ₂σ₁ = σ₂σ₁σ₂	✓	Both → [2,13,5,7,11,3]
σᵢ acts on strand i	✓	Direct from definition

Proof of YB. Let σᵢ be the transposition of positions (i, i+1) in the reflection modulus sequence. The braid relation (σᵢσ_{i+1})³ = id is the standard Coxeter relation in Sₙ, which holds for adjacent transpositions. The verification is immediate in the permutation representation.

Implication

The CRT torus with axis-swap carries a permutation representation of Bₙ on the reflection moduli that factors through Sₙ — because σᵢ² = id in the swap action, it loses the infinite-order structure of braid generators. This is still a valid representation of Bₙ (the permutation representation), but it is not faithful: all non-trivial braids with the same permutation of strands produce the same state.

The identity moduli are fixed by all braid generators, acting as a reference frame.

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3. Model 2: Modulus-Adjustment (Word Length Bound)

Each crossing adjusts the modulus values of the crossed strand:

σᵢ⁺: (L_id, L_ref) → (L_id + 2, max(L_ref − 1, 2))    over-crossing
σᵢ⁻: (L_id, L_ref) → (max(L_id − 1, 2), L_ref + 2)    under-crossing

After crossing, ALL moduli across ALL strands must remain pairwise coprime. This is the coprimality constraint.

Why YB fails here

The YB relation compares two paths: σ₁⁺σ₂⁻σ₁⁺ vs σ₂⁻σ₁⁺σ₂⁻. After 3 crossings, the two paths end at different modulus values:

Path	Strand 1 end state	Strand 2 end state
σ₁⁺σ₂⁻σ₁⁺	(a+4, b−2)	(c−1, d+2)
σ₂⁻σ₁⁺σ₂⁻	(a+2, b−1)	(c−2, d+4)

These differ (a+4 ≠ a+2, etc.), so the operator relation σ₁σ₂σ₁ = σ₂σ₁σ₂ does NOT hold as an equality of modulus states. (The permutation action is different — see Model 1.)

Word length bound theorem

For an N-strand system with moduli (L₁, L₂, …, L₂ₙ), the maximum number of consecutive crossings on strand i before coprimality with some other strand j fails is bounded by:

max_crossings(i) ≤ min_{j≠i} (spacing(L_i, L_j) / 2)

where spacing(L_i, L_j) = min(L_j_values) − max(L_i_values) after 0 crossings.

Proof. Each crossing changes strand i's moduli by at most +2 / −1. After k crossings, the range of strand i's values shifts by O(k). If strand i's values overlap with strand j's values, coprimality may fail (but is not guaranteed to — actual failure depends on prime factors). The bound is the worst case (when strand i's growing moduli encounter strand j's values sharing a prime factor).

Empirical verification:

Test	Max crossings	Config
1 strand, no neighbors	unlimited	(5,3) works for 10+
2 strands, spacing~12	3−4	(3,5),(17,29)
2 strands, spacing~100	Not tested (YB fails structurally)	—
2 strands, YB-path coprimality	3 crossings need spacing >2000	No 4-tuple found up to M=2000

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4. Combined Framework

The two models work together in the full CRT torus:

Phase 1 (Sidon via adjustment):
  Start with small moduli in wrapping regime (maxA < M ≤ 2·maxA)
  → Apply adjustment model to break collisions
  → When Sidon found: record FA, proceed to Phase 2

Phase 2 (Braid orbit via axis-swap):
  Expand moduli to N-strand coprime configuration (prime-product method)
  → Apply axis-swap generators to define braid word
  → FA values are invariant (CRT symmetry)
  → Braid word tracks the topological crossing history

Phase 3 (Resource management):
  When more crossings needed: apply adjustment model
  → Each crossing consumes spacing capacity
  → When spacing exhausted: regenerate moduli
  → Regeneration = Markov stabilization (add trivial pair)

Practical bound for N-strand configurations (individual primes)

Each modulus is a distinct prime, selected with minimum band gap = 2 × max_crossings. For max_crossings = 15 (band gap = 30), verified on provider-nixos:

Strands	Moduli	Band gap	Capacity/strand	Max modulus	< 32767?
3	6	30	~15	127	✓
4	8	30	~15	257	✓
6	12	30	~15	383	✓
8	16	30	~15	509	✓

All moduli are Q16_16-compatible (max 509 << 32767). The FA values produced by CRT reconstruction are large integers (~10^50 for 16 moduli) and are not Q16_16-compatible — they must be stored as arbitrary- precision integers. Only the moduli use Q16_16's bounded range.

Capacity-per-strand is the half-band gap (15 crossings before values drift into the next strand's band and risk equality-collision). For larger capacity, widen the band gap or use more distant primes.

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5. Verified Axioms (Summary)

Axiom	Axis-swap model	Adjustment model
σᵢ acts on strand i	✓	✓
σᵢ² = id	✓	✗ (may fail after 1st)
σᵢσⱼ = σⱼσᵢ (	i−j	≥2)
σ₁σ₂σ₁ = σ₂σ₁σ₂	✓	✗ (paths diverge)
Over/under distinction	✓ (swap direction)	✓ (L_id > L_ref)
Braid word length bound	—	✓ (coprimality constraint)

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6. Open Questions

1. Adjustment model as Sidon engine — the axis-swap model is a CRT symmetry (FA invariant), so adjustment is the sole source of Sidon creation. Can the adjustment model be characterized as a rewrite system on modulus values with known convergence bounds?

2. Braid invariants from M-differences — the M-difference condition from the Sidon theorem creates invariants that depend on braid word composition. Since axis-swap is FA-invariant, the braid word is tracked as a separate topological invariant.

3. Modulus regeneration as braid stabilization — when spacing is exhausted, the iteration regime regenerates moduli. This corresponds to a Markov stabilization move in knot theory: adding a trivial pair (extending the braid by an identity strand) to continue the computation.

4. Phase transition: CRT small-modulus → prime-product — the transition from small wrapping-regime moduli (~3−20) to large resource-regime moduli (~100−16000) is discontinuous. What controls this transition, and can it be made continuous (gradual modulus growth)?