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feat(braid/dag): land untracked research WIP + register 4 formal libs; ignore build artifacts

Browse source 
- lakefile.lean: register SilverSight.{AngrySphinx,CollatzBraid,GoldenSpiral,GCCL}
- docs/research/: braid group action, iteration DAG/regime, Sidon
  preservation/creation, unified CRT-torus DAG notes
- docs/diagrams/: DAG + heatmap + 8-strand search JSON/dot outputs
- formal/CoreFormalism/StrandCapacityBound.lean: capacity bound (passes
  hardened anti-smuggle --ci)
- scripts/, python/: braid word solver, collapse/DAG search + tuning,
  heatmap gen, YB search/verification, wrapping verifier
- .gitignore: exclude rust/**/target and coq compiled artifacts
  (*.vo/*.vok/*.vos/*.glob/*.aux) that were polluting the tree

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
allaun 2026-07-03 15:11:37 -05:00
parent e7d3376fea
commit 3362d554d1
30 changed files with 9956 additions and 0 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

8
.gitignore vendored
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@ -26,3 +26,11 @@ scratch/
scripts/qc_flag/.backups/
.env.enc
rust/target/
rust/**/target/
# Coq compiled artifacts
*.vo
*.vok
*.vos
*.glob
*.aux

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@ -0,0 +1,781 @@
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45
docs/diagrams/iteration_dag.dot Normal file
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@ -0,0 +1,45 @@
digraph IterationDAG {
rankdir=TB;
node [shape=record];
n0 [label="A=[1, 2, 5, 6]\nM=12 step=0"];
n1 [label="A=[0, 1, 6, 7]\nM=10 step=1"];
n2 [style=filled, fillcolor=lightgreen, label="A=[2, 5, 9, 10]\nM=12 step=1 ★SIDON"];
n3 [style=filled, fillcolor=lightgreen, label="A=[2, 5, 9, 10]\nM=12 step=1 ★SIDON"];
n4 [label="A=[0, 1, 6, 7]\nM=10 step=1"];
n5 [label="A=[1, 2, 5, 6]\nM=10 step=2"];
n6 [style=filled, fillcolor=lightgreen, label="A=[0, 7, 8, 13]\nM=14 step=2 ★SIDON"];
n7 [label="A=[3, 4, 9, 10]\nM=12 step=2"];
n8 [label="A=[3, 4, 9, 10]\nM=12 step=2"];
n9 [label="A=[1, 2, 5, 6]\nM=10 step=2"];
n10 [style=filled, fillcolor=lightgreen, label="A=[0, 7, 8, 13]\nM=14 step=2 ★SIDON"];
n21 [style=filled, fillcolor=lightyellow, label="A=[4, 5, 10, 11]\nM=14 step=3"];
n22 [style=filled, fillcolor=lightyellow, label="A=[6, 7, 12, 13]\nM=18 step=3"];
n23 [style=filled, fillcolor=lightyellow, label="A=[0, 1, 6, 7]\nM=12 step=3"];
n24 [style=filled, fillcolor=lightyellow, label="A=[3, 7, 9, 13]\nM=15 step=3"];
n25 [style=filled, fillcolor=lightyellow, label="A=[0, 1, 6, 7]\nM=12 step=3"];
n26 [style=filled, fillcolor=lightyellow, label="A=[2, 8, 13, 19]\nM=20 step=3"];
n27 [style=filled, fillcolor=lightyellow, label="A=[0, 4, 9, 13]\nM=15 step=3"];
n28 [style=filled, fillcolor=lightgreen, label="A=[5, 8, 14, 19]\nM=20 step=3 ★SIDON"];
n29 [style=filled, fillcolor=lightyellow, label="A=[2, 3, 10, 11]\nM=14 step=3"];
n30 [style=filled, fillcolor=lightyellow, label="A=[0, 1, 12, 13]\nM=18 step=3"];
n0 -> n1 [label="[2, 5]"];
n0 -> n2 [label="[3, 4]"];
n0 -> n3 [label="[4, 3]"];
n0 -> n4 [label="[5, 2]"];
n1 -> n5 [label="[2, 5]"];
n1 -> n6 [label="[2, 7]"];
n1 -> n7 [label="[3, 4]"];
n1 -> n8 [label="[4, 3]"];
n1 -> n9 [label="[5, 2]"];
n1 -> n10 [label="[7, 2]"];
n7 -> n21 [label="[2, 7]"];
n7 -> n22 [label="[2, 9]"];
n7 -> n23 [label="[3, 4]"];
n7 -> n24 [label="[3, 5]"];
n7 -> n25 [label="[4, 3]"];
n7 -> n26 [label="[4, 5]"];
n7 -> n27 [label="[5, 3]"];
n7 -> n28 [label="[5, 4]"];
n7 -> n29 [label="[7, 2]"];
n7 -> n30 [label="[9, 2]"];
}

229
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@ -0,0 +1,229 @@
# 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.
---
## 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.
---
## 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) | ✓ | Disjoint swaps commute structurally |
| 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.
---
## 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 |
---
## 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.
---
## 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) | ✓ | ✓ (disjoint moduli) |
| σ₁σ₂σ₁ = σ₂σ₁σ₂ | ✓ | ✗ (paths diverge) |
| Over/under distinction | ✓ (swap direction) | ✓ (L_id > L_ref) |
| Braid word length bound | — | ✓ (coprimality constraint) |
---
## 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)?

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# CRT Torus Embedding: Iteration DAG
Open Direction #1 — tracing iteration paths through modulus space.
---
## 1. DAG Structure
The iteration of F with parameter regeneration forms a Directed Acyclic Graph:
**Nodes:** `(n, A_n, moduli_n, S_n, property_flags)`
- `n`: step index
- `A_n`: current set (integer lifts)
- `moduli_n`: (L₁⁽ⁿ⁾, L₂⁽ⁿ⁾, …, Lₖ⁽ⁿ⁾)
- `S_n`: involution center
- `property_flags`: Sidon? B_h? Golomb?
**Edges:** `(n, A_n, Ω_n, S_n) —[F]→ (n+1, A_{n+1}, Ω_{n+1}, S_{n+1})`
- `A_{n+1} = F_{Ω_n, S_n}(A_n)` (apply F with current moduli)
- `Ω_{n+1}` = next moduli (from regeneration rule)
- `S_{n+1}` = next involution center (fixed or adaptive)
**No cycles by design:** each step changes moduli (geometric growth α, β ≥ 1),
so `Ω_n` is strictly increasing in product M_n = ∏ L_i⁽ⁿ⁾. This prevents revisiting
the same state, keeping the graph acyclic.
---
## 2. Regeneration Rules
| Rule | Ω_{n+1} | S_{n+1} | Branching factor |
|------|----------|---------|------------------|
| Fixed | Ω_n (unchanged) | S_n | 1 (deterministic) |
| Geometric | (α·L₁⁽ⁿ⁾, β·L₂⁽ⁿ⁾) | S_n | 1 per (α,β) choice |
| Adaptive | chosen from candidate set | max(A_n)+min(A_n) | |candidates| per step |
| Exhaustive | primes from pool larger than current | either fixed or adaptive | |pool| per step |
The DAG explores all branches from adaptive/exhaustive rules.
---
## 3. Node Properties
Each node records:
```
Node {
step: int
A: List[int] # current set (sorted)
moduli: List[int] # (L1, L2, ..., Lk)
S: int # involution center
M: int # product of moduli
is_reflection_closed: bool # A_n == S - A_n?
is_injective: bool # M > max(A)?
sidon_status: bool # is A_n a Sidon set?
parent: Optional[NodeID]
children: List[NodeID]
depth: int
terminal: bool # no further steps possible
}
```
A node is **terminal** when:
- `A_n` is Sidon (goal reached), OR
- `M_n > 2·max(A_n)` (no-sum-alias regime — new collisions can't form,
but wrapping could still break existing ones; if not already Sidon,
try different moduli), OR
- `A_n` is F-invariant under current moduli (F(A_n) = A_n), OR
- No valid next moduli exist (Ω exhausted)
---
## 4. Path Tracing
A **path** through the DAG is a sequence of modulus choices:
```
Path P = (Ω₀, Ω₁, …, Ω_{m-1})
where Ω_i = (L₁⁽ⁱ⁾, L₂⁽ⁱ⁾)
```
Each path transforms A₀ through m steps:
```
A₀ →[Ω₀] A₁ →[Ω₁] A₂ →[Ω₂] … →[Ω_{m-1}] A_m
```
**Goal:** find a path from A₀ to a Sidon set A_m.
### Shortest path search
Since the DAG is acyclic (growing moduli), BFS finds the shortest path:
```
Queue ← [(A₀, Ω₀)]
While Queue not empty:
(A, Ω) ← pop
M ← product(Ω)
if M > 2·max(A): continue (preservation regime, no improvement)
for each candidate Ω' in next_moduli(Ω):
A' ← F_{Ω', S}(A)
if A' is Sidon: return path (success!)
push (A', Ω')
```
---
## 5. Search Heuristics
Not all modulus choices are equally useful. Heuristics prune the search:
1. **Prime preference** — use small primes as moduli (2,3,5,7,…) for dense
coverage of the [max(A), 2·max(A)] window.
2. **Gap targeting** — choose moduli that match differences found in Dₐ
(the M-difference condition). This avoids creating new collisions.
3. **Wrapping bias** — prefer moduli where existing collisions wrap
differently (condition (a) of the Sidon theorem).
4. **Termination** — stop expanding a branch when M > 2·max(A), since
F can no longer improve the Sidon status (only preserve).
---
## 6. Implementation
See `scripts/iteration_dag.py` for the DAG tracing implementation.
Example trace:
```
A₀ = {1, 2, 5, 6}, S = 7, Ω₀ = (3, 4), M = 12
→ A₁ = {2, 5, 9, 10}, Sidon = True. Path length 1. ✓
A₀ = {0, 1, 3, 8, 13}, S = 27, Ω₀ = (3, 5), M = 15
→ A₁ = {12, 1, 9, 14, 4}, Sidon = False. New collision.
→ Try Ω₁ = (5, 7):
→ A₂ = F_{5,7}(A₁), M = 35. Check Sidon...
```
---
## 7. Connection to Braid DAG
The iteration DAG is the discrete version of the braid group Cayley graph.
Each step F_{Ω,S} corresponds to a braid word: a sequence of generators
σᵢ that act on the current configuration. The moduli Ω = (L₁, L₂, …, L₁₆)
determine which generators are available (which strands cross).
In the full 16D chiral torus, each step applies a braid word, and the DAG
traces the orbit of A₀ under the braid group action. A terminal Sidon node
corresponds to a braid word that produces a collision-free configuration —
a braid invariant.
---
## 8. Dual-Model DAG Implementation
The Chiral DAG (`scripts/full_chiral_dag.py`) combines both braid models:
| Model | DAG action | Verifies | Verified |
|-------|-----------|----------|----------|
| Axis-swap | σₛ swaps reflection moduli of strands s, s+1 | YB, σ²=id, far commute | ✓ |
| Adjustment | crossing changes modulus values by ±2/±1 | Coprimality bound | ✓ |
| Spacing tracking | capacity_left = min spacing / 2 per strand | Word length bound | ✓ |
### Node structure
Each DAG node stores:
- `pairs`: current chiral pairing (L_id, L_ref) per strand
- `moduli`: flattened 16-modulus vector
- `A`: current set (CRT lifts)
- `M`: product of all moduli
- `capacity_left`: max remaining crossings per strand
- `braid_word`: cumulative braid word from root to this node
### Verified results (3-strand test, A₀ = [1,2,5,6])
| Metric | Value |
|--------|-------|
| Nodes explored | 65 |
| Sidon paths found | 3 |
| Axis-swaps tried | 43 |
| Adjustments tried | 21 |
| Shortest braid word | σ₁ |
| Root capacity | [6, 5, 5] |
| Max modulus (8-strand) | 16637 < 32767 ✓ |
### 8-strand configuration
8 strands × 2 moduli = 16 moduli, all pairwise coprime (product of 4 distinct
primes per strand). With spacing ~100+ between strands, capacity is 50+
crossings per strand.
### Usage
```python
from scripts.full_chiral_dag import ChiralDAG
dag = ChiralDAG(A0, S, n_strands=3, max_steps=8, max_branch=50)
dag.build(use_axis_swap=True, use_adjustment=True)
dag.summary()
# Export for visualization
dag.to_json("/path/to/export.json")
```

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# CRT Torus Embedding: Iteration Regime
Open Direction #1 — defining and analyzing the re-embedding cascade.
---
## 1. Problem
F is defined from A ⊂ ℤ into R = ℤ/Mℤ. For the k-torus, F(A) lives in a
different space than A. To iterate, we need:
1. An **extension** of F to the integer lift of any finite set
2. A **regeneration rule** for parameters (L₁,…,Lₖ, S) at each step
3. A **stability condition** that determines when the cascade terminates
---
## 2. Domain Extension
Define a family of maps indexed by moduli:
$$
F_{L_1,\dots,L_k,S}(a) = \text{CRT-1}(a \bmod L_1,\; S-a \bmod L_2,\; \dots,\; S-a \bmod L_k)
$$
for any integer a (or any residue a ∈ ℤ/Mℤ lifted to ℤ). This extends F from
A ⊂ ℤ to all of ℤ/Mℤ via the same congruence rule.
**Iteration step n:**
$$
A_{n+1} = \{\, F_{L_1^{(n)},\dots,L_k^{(n)},S^{(n)}}(a) \mid a \in \text{lift}(A_n) \,\}
$$
where $\text{lift}(A_n)$ maps the current set to ℤ (the CRT integer lift).
---
## 3. Regeneration Rule
The simplest deterministic rule: a **geometric modulus cascade**.
Fix initial moduli (L₁⁽⁰⁾, L₂⁽⁰⁾) and growth factors (α, β) ≥ 1:
$$
L_1^{(n)} = \lfloor \alpha^n \cdot L_1^{(0)} \rfloor,
\qquad
L_2^{(n)} = \lfloor \beta^n \cdot L_2^{(0)} \rfloor
$$
and S fixed or adapted:
- **Fixed S**: the involution center remains constant across steps. The
reflection constraint S−a may not hold in Aₙ for n ≥ 1 — this is fine,
the constraint only needs to hold in A₀.
- **Adaptive S**: at step n, choose Sₙ = max(Aₙ) + min(Aₙ) to keep Aₙ
reflection-closed.
### Regime types
| Growth | Behavior | Use case |
|--------|----------|----------|
| α > 1, β > 1 | **Expanding cascade** — torus grows, finer resolution | Multi-scale embedding |
| α = β = 1 | **Fixed torus** — F² = id on ℤ/Mℤ, sequence stabilizes at A₁ | Single-step transformation |
| α, β alternating | **Oscillating cascade** — cycles between resolutions | Searching for Sidon creation |
---
## 4. Stability Condition
A cascade stabilizes at step n if:
$$
F_{L_1^{(n)},\dots,L_k^{(n)},S^{(n)}}(A_n) = A_n \quad\text{(as sets of integers)}
$$
Sufficient condition for stability:
If the moduli at step n+1 are the same as step n and Aₙ is F-invariant
(i.e., Aₙ is a union of F-orbits), then F² = id on the torus forces
A_{n+2} = A_n — a 2-cycle.
**Terminal state:** A cascade converges to a fixed point when:
1. Aₙ is closed under S-reflection (the original constraint), AND
2. F(Aₙ) = Aₙ (set invariance under F)
This is equivalent to: every element of Aₙ is either a fixed point of F
or paired with its F-image within Aₙ.
---
## 5. Example: Expanding Cascade with k = 1
For a single-modulus system (k = 1), F reduces to the identity. The
cascade does nothing — trivial. The interesting case starts at k = 2.
---
## 6. Open Questions
1. **Convergence rate** — for α > 1, does the cascade reach a terminal
state in finite steps, or does the expanding torus prevent stabilization?
2. **Optimal growth** — what α, β minimize the number of steps needed
to achieve a target property P in Aₙ?
3. **S-adaptation** — does adaptive S always outperform fixed S for
reaching Sidon/B_h/Golomb properties?
4. **Braid connection** — does the expanding cascade correspond to
iterating braid crossings (adding one crossing per step)?

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# CRT Torus Embedding: Property Preservation and Creation
Part of Open Direction #3 — characterizing moduli that guarantee F(A) satisfies
a target property P.
---
## 1. Problem
Given A ⊂ ℤ reflection-closed under S, and a target property P (Sidon, B_h,
Golomb ruler), which moduli (L₁, …, L_k) guarantee that F(A) satisfies P?
The Sidon example shows F can *create* P from a non-P set, but this depends
on modulus choice. We need the general condition.
---
## 2. Key Invariant: The Sum Map
For a pair (a, b) in A, the CRT-lifted sum F(a) + F(b) has residues:
| Axis | Constraint |
|------|-----------|
| 1 (identity) | (a + b) mod L₁ |
| i ≥ 2 (reflection) | (2S − a − b) mod Lᵢ |
Two pairs (a,b) and (c,d) produce equal sums modulo M iff:
a + b ≡ c + d (mod L₁)
a + b ≡ c + d (mod Lᵢ) ∀i ≥ 2
By CRT: a + b ≡ c + d (mod M), where M = ∏ Lᵢ.
Therefore:
F(a) + F(b) ≡ F(c) + F(d) (mod M) iff a + b ≡ c + d (mod M)
---
## 3. Three Regimes
Let M = ∏ Lᵢ.
### Regime A — M > max(A): injective, wrapping can break collisions
F is injective. Existing sum collisions break when individual CRT lifts
wrap M differently (the wrapping criterion).
**A1: M > 2·max(A)** — no sum alias. All pairwise sums < M, so new
collisions cannot form. Wrapping can still break existing collisions.
Sidon creation IS possible here (e.g., [7,3] with A={1,2,5,6}).
**A2: max(A) < M ≤ 2·max(A)** — sum alias possible. Pairs with different
sums may satisfy |T₁−T₂| = M, creating new collisions. Wrapping + M-diff
both active.
### Regime B — M ≤ max(A): F not injective (aliasing)
Not useful.
### Wrapping works identically in A1 and A2
| (L₁, L₂) | M | Regime | Sidon? | Why |
|----------|---|--------|--------|-----|
| (3, 4) | 12 | A2 | ✓ | Wrapping: 19 vs 7 |
| (7, 3) | 21 | A1 | ✓ | Wrapping: 28 vs 7 |
| (11, 2) | 22 | A1 | ✗ | Same wrap: both sums = 29 |
---
## 4. B_h Generalization
For h-fold sums: wrapping works at ANY M > max(A). M-differences require
M ≤ h·max(A) to be possible (since max h-fold sum = h·max(A)).
| Property | No sum alias (M > h·maxA) | Sum alias possible |
|----------|--------------------------|-------------------|
| Sidon (h=2) | M > 2·max(A): wrapping only, no new collisions | max(A) < M ≤ 2·max(A) |
| B_h (general) | M > h·max(A): wrapping only | max(A) < M ≤ h·max(A) |
| Golomb (differences) | M > max(A)-min(A): wrapping only | boundary case |
## 6. Creation Condition: Complete Characterization
### 6.1 Breaking Existing Collisions (The Wrapping Criterion)
Given a collision a+b = c+d = T in A, the images satisfy:
F(a)+F(b) = T + r₁·M, r₁ ∈ {0, 1}
F(c)+F(d) = T + r₂·M, r₂ ∈ {0, 1}
The collision is broken iff r₁ ≠ r₂. (Proof: each F(x) < M, so two
sums of two values are < 2M. The wrap indicator r = 1 when F(a)+F(b) ≥ M.)
**Verified:** 500/500 random tests, k=2..8.
### 6.2 Preventing New Collisions (The M-Difference Condition)
A new collision arises when pairs (a,b) and (c,d) with *distinct* original
sums T₁ ≠ T₂ satisfy F(a)+F(b) = F(c)+F(d). This occurs iff:
|T₁ − T₂| = M (or a multiple of M)
Since T₁, T₂ ≤ 2·max(A) and M > max(A), the only possible multiple is M.
**Proof.** F(a)+F(b) ≡ F(c)+F(d) (mod M) forces a+b ≡ c+d (mod M), i.e.,
T₁ ≡ T₂ (mod M). Since 0 ≤ T₁, T₂ ≤ 2·max(A) < 2M, we have |T₁−T₂| ∈ {0, M}.
The case 0 is the existing collision (T₁ = T₂). The case M is the new collision.
**Verified:** 416 new collisions across 5000 random trials — ALL satisfy
|T₁−T₂| = M. Zero counterexamples.
### 6.3 Complete Sidon Creation Theorem
**Theorem.** For a finite A ⊂ ℤ with reflection closure a ↦ S−a,
moduli L₁,…,Lₖ coprime, L₁,L₂ ≥ 2, and M = ∏ Lᵢ > max(A):
F(A) is Sidon ⟺ (a) and (b) both hold:
(a) For every sum collision a+b = c+d in A:
(F(a)+F(b) ≥ M) ≠ (F(c)+F(d) ≥ M) [wrapping criterion]
(b) For no distinct sums T₁, T₂ ∈ {a+b : a,b ∈ A, a ≤ b}:
|T₁ − T₂| = M [M-difference condition]
**Corollary 1 (No sum alias).** If M > 2·max(A), condition (b) is vacuous
(no sums differ by exactly M). F(A) may still break existing collisions
via wrapping. No new collisions can form.
**Corollary 2 (Sum alias possible).** If max(A) < M ≤ 2·max(A), both
conditions must be checked. F(A) is Sidon iff (a) wrapping breaks all
existing collisions AND (b) no M-differences create new ones. Both
conditions are decidable in O(|A|⁴) time.
**Corollary 3 (Complete classification).**
M > max(A) → wrapping can break existing collisions; M-differences
may or may not apply depending on if M ≤ 2·max(A).
M ≤ max(A) → F not injective (aliasing).
### 6.4 Algorithmic Guidance for Modulus Selection
Choose moduli to guarantee Sidon creation:
1. Compute all pairwise sums Sₐ = {aᵢ + aⱼ : 0 ≤ i ≤ j < |A|}.
2. Compute differences Dₐ = {|T₁ − T₂| : T₁,T₂ ∈ Sₐ, T₁ ≠ T₂}.
3. Choose M = ∏ Lᵢ such that:
- M > max(A) (element-level injectivity)
- M ∉ Dₐ (no new collisions)
4. For each existing collision in A, verify the wrapping criterion (a).
If any pair wraps the same, pick different moduli or accept
the collision persists.
5. If (a) and (b) both hold, F(A) is guaranteed Sidon.
### 6.5 Modulus Ordering Principle (Tuning Rule)
The identity axis L₁ and reflection axis L₂ are **not interchangeable**.
Larger L₁ = larger minimum gap = more likely Sidon creation.
**Empirical rule:** Choose L₁ > L₂. For the complex set A = [0,1,3,8,13]:
| (L₁, L₂) | M | Gap | Sidon? | Insight |
|----------|---|-----|--------|---------|
| (7, 3) | 21 | ≥7 | ✓ | L₁=7 large identity axis |
| (11, 2) | 22 | ≥11 | ✓ | L₁=11 even larger |
| (8, 3) | 24 | ≥8 | ✓ | L₁=8 |
| (13, 2) | 26 | ≥13 | ✓ | L₁=13, max gap |
| (3, 7) | 21 | ≥3 | ✗ | L₁=3 too small |
| (2, 11) | 22 | ≥2 | ✗ | L₁=2 minimal gap |
All 4 successes have L₁ > L₂. All failures with L₁ < L₂ have insufficient gap for
this specific set. (When both L₁ ≈ L₂, other factors like the wrapping criterion
and M-difference condition dominate.)
**Practical rule:**
1. Choose L₁ as large as possible (up to 2·maxA / L₂)
2. Choose L₂ as the smallest coprime integer that keeps M in (maxA, 2·maxA]
3. Typically L₂ = 2 (smallest possible) and L₁ = ⌊2·maxA / L₂⌋, adjusted
downward for coprimality
This maximizes the gap L₁, which maximizes the chance of breaking existing
sum collisions via the wrapping criterion.
**Tradeoff:** Larger L₁ also means larger M. If M exceeds 2·maxA,
the M-difference condition becomes vacuous (no new collisions), but
wrapping can still break existing ones. The optimal is L₁ ≈ 1.9·maxA
from sweep data (29% success rate).

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# Unified CRT Torus Braid DAG: Graded Sidon Energy with Directional Hierarchical Pruning
## Provenance (Clean Room)
This document synthesizes mathematical patterns from two external works, as
recorded in `CITATION.cff` references [0] and [1]. All implementation is
original to SilverSight — the external works inspired the *structural
analogies* and *design patterns*, not the code or theorems.
| External work | Pattern borrowed | Our adaptation |
|---------------|-----------------|----------------|
| ppf-contact-solver cubic barrier | Curvature linear in gap: ψ''(g) = 4(1-g/ĝ) | SidonEnergy(gap) = 4(1-gap/M) |
| ppf-contact-solver elasticity-inclusive stiffness | Coupling constraint stiffness into material stiffness | Coupled axis-swap × adjustment DAG traversal |
| ppf-contact-solver eigen-filtering | max(λ, 0) SPD projection | DAG branch pruning by Sidon-energy sign |
| ppf-contact-solver two-pass allocation | Dry pass (sparsity discovery) + fill-in (value computation) | DAG topology BFS + Sidon value fill |
| NAADF AADF 6-direction encoding | 6 directional distances instead of 1 scalar SDF | Chiral (identity, reflection) pairing per strand |
| NAADF 3-level nested hierarchy | Voxel → Block → Chunk | Crossing → Strand → DAG |
| NAADF max safe step formula | step_d = (1+bound-offset)/\|rayDir\| | Capacity = spacing/2 per direction |
| NAADF iterative distance transform | 3-iteration AADF propagation | Sidon potential bound propagation |
| NAADF hash-deduplicated blocks | Content-addressable 64-voxel store | Content-addressable strand/χ-state store |
---
## 1. Graded Sidon Energy (Cubic Barrier Analog)
### Current problem
The DAG currently uses `is_sidon(A)` as a binary predicate. A set is either
Sidon or it isn't. This gives no gradient signal when traversing — crossings
either succeed (reach Sidon) or fail (don't), with no intermediate information
about *which crossings bring us closer*.
### Reformulation
Define a **Sidon energy** that is graded by the gap distance relative to M:
```
Let gap(A) = min_{a < a' < a'' < a'''} |(a' + a'') - (a''' + a)| (collision gap)
If gap(A) > M:
SidonEnergy = 0 (Sidon achieved — no residual energy)
Else:
ℰ = 4 * (1 - gap(A) / M) (graded residual, 0 < ℰ ≤ 4)
```
The cubic barrier ψ(g) from ppf-contact-solver uses curvature = 4(1 - g/ĝ),
which is *linear in the gap*. Our SidonEnergy uses the same form, where:
- g → gap(A): how close the closest collision is to the wrapping modulus M
- ĝ → M: the wrapping modulus (the threshold at which collisions are guaranteed
to wrap differently)
Properties:
- When gap(A) = 0 (collision at zero difference): ℰ = 4 (maximum energy)
- When gap(A) = M/2: ℰ = 2 (half energy)
- When gap(A) > M (Sidon): ℰ = 0 (converged)
- The derivative dℰ/dgap = -4/M is constant: energy decreases linearly as the
gap opens up
### DAG integration
Replace the binary `is_sidon` gate with:
```
def sidon_residual(A: List[int], M: int) -> float:
"""Return 0 if Sidon, else ℰ ∈ (0, 4]."""
gap = min_collision_gap(A)
return 0 if gap > M else 4.0 * (1.0 - gap / M)
def crossing_accepted(parent_energy: float, child_energy: float) -> bool:
"""A crossing is accepted iff it does not increase Sidon energy."""
return child_energy <= parent_energy + EPSILON
```
This is the clean-room analog of eigen-filtering (max(λ, 0)): crossings that
*would increase the residual* are pruned, guaranteeing monotonic DAG descent.
---
## 2. Directional Chiral Decomposition (AADF Analog)
### Current model
Each strand has a chiral pair (L_id, L_ref). A crossing toggles between
over (+) and under (-), adjusting these two values. The adjustment is
isotropic: +2 on the active value, -1 on the passive value.
### Reformulation
The NAADF insight is that directional bounds are independent. A ray moving
+x doesn't care about the distance in +y. We apply the same independence
to crossing directions:
For strand i with pair (L_id, L_ref), define **4 directional capacities**:
```
cap⁺_id(i) = # of over-crossings possible before L_id exceeds band
cap⁻_id(i) = # of under-crossings possible before L_id drops below band
cap⁺_ref(i) = # of over-crossings possible before L_ref exceeds band
cap⁻_ref(i) = # of under-crossings possible before L_ref drops below band
```
Each directional capacity is the number of steps before the adjusted value
reaches the nearest other strand's band. This mirrors NAADF's 6-directional
AADF (2 bits per direction at block level, 5 bits at chunk level).
### Capacity encoding
Pack the 4 directional capacities into a single 8-bit word per strand:
```
Bits 0-1: cap⁺_id(i) (2 bits, range 0-3 at strand level)
Bits 2-3: cap⁻_id(i) (2 bits, range 0-3)
Bits 4-5: cap⁺_ref(i) (2 bits, range 0-3)
Bits 6-7: cap⁻_ref(i) (2 bits, range 0-3)
```
At the DAG level (aggregated across all strands), promote to 4 bits per
direction (range 0-15). The promotion rule mirrors NAADF's bound promotion:
`dag_cap = min(strand_cap × 4 + intra_strand_offset, 15)`.
---
## 3. Three-Level Nested DAG Hierarchy
### NAADF hierarchy (inspiration)
| Level | Grid | Voxels per | Bits per element | Purpose |
|-------|------|------------|-----------------|---------|
| Voxel | 4³ | 1 | 16 (2 bits × 6 dir + 1 occupancy) | Per-voxel state |
| Block | 4³ blocks | 64 | 32 (2 bits × 6 dir + 2 state) | Small-group aggregate |
| Chunk | N/16 chunks | 4096 | 32 (5 bits × 6 dir + 2 state) | Large-region empty skip |
### Our hierarchy
| Level | Contains | States per | Bits per element | Encoding | Analog to |
|-------|----------|-----------|-----------------|----------|-----------|
| Crossing | Single σ⁺/σ⁻ | 1 crossing | 8 (4 dir caps) | Per-crossing directional capacity | Voxel-level AADF |
| Strand | 2 crossings (id+ref) | ~15 crossings | 16 (4 dir caps × 4 bits) | Aggregated strand capacity | Block-level AADF |
| DAG | 2N moduli (N strands) | All crossings | 32 (4 dir caps × 8 bits) | Global Sidon potential | Chunk-level AADF |
### Hierarchy rules
**Propagation (bottom-up):** After each crossing on strand i, recompute the
strand-level capacities by scanning the current crossing-level capacities.
Then aggregate to DAG-level:
```
strand_cap[d] = min(crossing_cap[d] for crossing in strand) # worst case
dag_cap[d] = min(strand_cap[d] for strand in dag) # global worst case
```
**Skip (top-down):** If the DAG-level capacity in a direction is 0, no strand
has remaining capacity in that direction — the entire DAG branch can be pruned.
This mirrors NAADF's chunk-level empty skip: if the chunk bound says 31 empty
voxels ahead, skip all 31 without descending.
```
def skip_direction(dag, direction: str) -> bool:
"""Return True if no strand can absorb another crossing in this direction."""
return dag_cap[direction] == 0
```
---
## 4. Coupled Axis-Swap × Adjustment DAG Traversal
### Current problem
Axis-swap and adjustment are separate models. Axis-swap permutes reflection
moduli (FA-invariant, YB ✓), adjustment changes modulus values (FA-changing,
YB ✗). The DAG tries both independently, but they don't interact.
### Reformulation
The ppf-contact-solver's **elasticity-inclusive dynamic stiffness** couples
the constraint stiffness (contact) into the material stiffness (elasticity).
We do the same: couple the topology stiffness (axis-swap) into the resource
stiffness (adjustment).
**How it works:**
```
def coupled_crossing(pairs, s, direction):
"""Single coupled crossing: axis-swap then adjust."""
# Step 1: Axis-swap topology (changes modulus ordering only)
pairs = axis_swap(pairs, s)
# Step 2: Adjustment with direction-dependent intensity
# The adjustment step size is scaled by the topology permutation:
# - If strand s just received a new modulus via swap, the adjustment
# is larger (the topology "stiffens" the crossing)
# - If strand s kept its modulus (no swap effect), adjustment is
# the standard ±2/∓1
stiffness = topology_stiffness(pairs, s) # ∈ [1, 2]
pairs = adjust(pairs, s, direction * stiffness)
return pairs if pairwise_coprime(pairs) else None
```
Where `topology_stiffness` is 2 if the swap changed the reflection modulus
ordering, 1 otherwise. This is the clean-room analog of the elasticity-
inclusive stiffness term (barrier.cu:48-85).
### Three-phase traversal
```
Phase 1 (Graded Sidon search):
Use coupled crossings with small bands (gap ~ 60, capacity ~ 15 per strand).
Track SidonEnergy residual. Prune branches that increase ℰ.
Phase 2 (DAG topology expansion):
Once ℰ < threshold, expand to wide bands (gap ~ 500, capacity ~ 125 per strand).
The DAG-level capacity-0 check prunes exhausted regions.
Phase 3 (Content-addressable dedup):
Deduplicate identical modulus configurations via hash map (NAADF analog of
chunkCalc.fx:57-115). If two DAG nodes have identical moduli and FA values,
they share the same strand-state record.
```
---
## 5. Algorithm: Unified DAG Build
```python
def build_unified_dag(A0, S, n_strands, max_steps):
"""Build the CRT torus DAG with graded Sidon energy + hierarchical pruning."""
# Initialize: chiral pairs with per-directional capacity
pairs = chiral_pairs(n_strands, max_crossings=15)
caps = compute_directional_capacities(pairs)
# Root node: initial Sidon energy
root = DAGNode(pairs, A0, caps)
root.energy = sidon_energy(A0, modulus_product(pairs))
queue = [root]
visited = {}
while queue:
node = queue.pop(0)
# Skip if DAG-level capacity is 0 in all directions
for d in ['id_over', 'id_under', 'ref_over', 'ref_under']:
if dag_capacity(node, d) <= 0:
continue # can't cross in this direction
# Coupled crossing on each strand
for s in range(n_strands):
for direction in ['over', 'under']:
child = coupled_crossing(node, s, direction)
if child is None:
continue # coprimality failed
# Compute Sidon energy and prune if it increases
child.energy = sidon_energy(child.A, child.M)
if child.energy > node.energy + EPSILON:
continue # monotonicity violated: prune
# Dedup against visited states
h = hash(child.moduli)
if h in visited and visited[h].A == child.A:
continue # content-addressable dedup
visited[h] = child
queue.append(child)
```
---
## 6. Implementation Plan
| Component | File | Status | Notes |
|-----------|------|--------|-------|
| `sidon_energy` | `scripts/sidon_energy.py` | New | Graded energy from gap/M using cubic form |
| `directional_capacity` | `scripts/full_chiral_dag.py` | Extend | Add 4-directional capacity tracking |
| `coupled_crossing` | `scripts/full_chiral_dag.py` | Extend | Axis-swap then adjust with topology stiffness |
| `hierarchical_skip` | `scripts/full_chiral_dag.py` | Extend | DAG-level capacity-0 pruning |
| `content_addressable_store` | `scripts/full_chiral_dag.py` | New | Hash-dedup modulus configs |
| `unified_dag_build` | `scripts/run_8strand_search.py` | Extend | Replace BFS with unified algorithm |
| `SidonEnergy` theorem | `formal/CoreFormalism/SidonEnergy.lean` | New | Lemma: ℰ monotonic under accepted crossings |
| `CoupledCrossing` lemma | `formal/CoreFormalism/CoupledCrossing.lean` | New | YB preservation under coupled model |
---
## 7. SidonEnergy Gradient via Asymmetric Scoring Identity
### Mixedbread's scoring identity
The key mathematical insight from mixedbread's asymmetric quantization
(CITATION.cff [2], blog 2026-06-29):
```
q · b = 2 * Σ_{b_i=+1} q_i - Σ q_i
```
A binary × int8 dot product needs only:
1. Precompute Σ q_i (once per query)
2. Sum query dimensions where document bit = +1 (the "selected" sum)
3. Apply the identity: 2×selected − total
No multiplication per dimension needed. Just a conditional add and a shift.
### Our analog: SidonEnergy gradient
For a crossing on strand i, define a **crossing sign** s_i ∈ {+1, −1} encoding
over/under, and a **crossing contribution** c_i (the change in collision gap
attributable to strand i). The SidonEnergy before and after a crossing relates
as:
```
ℰ_after = ℰ_before − (2 * s_i * c_i) / M
```
Derivation:
- ℰ = max(0, 4(1 − gap/M)) for non-Sidon states
- Δgap = s_i * c_i (the gap change from strand i's crossing, signed)
- Δℰ = 4/M * (−Δgap) = −4 * s_i * c_i / M
- So ℰ_after = ℰ_before − (2 * 2 * s_i * c_i / M)
The factor of **2** appears for the same reason as in mixedbread's identity:
the active crossing direction contributes with double weight (+2 step) while
the passive direction contributes with single weight (−1 step). This is an
**asymmetric scoring kernel** embedded in the CRT arithmetic.
### Practical benefit
Replace:
```python
child_energy = sidon_energy(child.A, child.M) # full recompute
if child_energy > node.energy: continue # O(N²) collision check
```
With:
```python
# Gradient update: O(1) per crossing
delta = -4 * crossing_sign * crossing_contribution / total_modulus
child_energy = node.energy + delta
if child_energy > node.energy: continue # same monotonicity gate
```
This is the clean-room analog of the NEON SDOT kernel: precompute the
"query sum" (ℰ_before, once per DAG node), then for each outgoing crossing,
compute only the "selected" part (the contribution of the crossed strand)
and apply the identity.
### Asymmetric precision in the DAG
Mixedbread's crucial storage insight applies directly:
| Side | Mixedbread | Our DAG |
|------|-----------|---------|
| Query / Topology | int8 (high precision, short-lived) | Braid word (1 bit per crossing, cheap to store) |
| Document / Sidon | binary (low precision, dominates storage) | SidonEnergy and FA values (full precision, recomputed on-demand) |
Just as mixedbread stores document vectors as 1-bit signs and keeps query at
int8, we store the **braid word** (which crossings happened) as a bit field
(1 bit per crossing type), while the **FA values and SidonEnergy** are
recomputed from scratch on each node access.
A complete 8-strand crossing history fits in **1 byte** (1 bit per strand for
over/under, or 2 bits per strand with directional encoding). This mirrors
mixedbread's 32× storage reduction: the braid word is the "binary document"
that dominates storage cost, while the DAG traversal is the "int8 query" that
dominates compute cost.
### Scoring kernel for the CRT residue update
Mixedbread's kernel avoids full multiply via precomputed query planes. Our
kernel for CRT residue update:
```python
def crossing_residue(residue_before: int, step: int, mod: int) -> int:
"""CRT residue update: no multiply needed.
Analogous to mixedbread's q·b identity avoiding full dot product.
"""
return (residue_before + step) % mod # single add + modulo
```
A full CRT reconstruction of FA after N crossings would be O(N × num_moduli).
With the gradient identity, each crossing update is O(1): just update the
residue for the crossed strand and compute Δℰ from the signed contribution.
---
## 8. Matrix Orthogonalization of the Modulus Configuration
### Newton-Schulz for the CRT modulus matrix
The mLSTM maintains a memory matrix C ∈ ℝ^{d×d}. Each read is a matrix-vector
product. The Newton-Schulz iteration enforces orthogonality:
```
M ← (3M − M·M^T·M) / 2 (5 iterations → M^T·M ≈ I)
```
This prevents mode collapse: a few strong directions dominating the memory,
crowding out weaker memories. The +15-45% NAR accuracy gain comes from this
equalization (CITATION.cff [3], blog 2026-06-30).
**Our analog:** The CRT modulus configuration is an n×2 matrix (n strands,
2 moduli each). The "mode collapse" analog is coprimality exhaustion: a few
strands' moduli grow large while others stay small, eventually hitting the
Q16_16 bound while other strands have unused capacity.
Define the **modulus orthogonality** constraint:
```
For all i ≠ j: gcd(L_id_i, L_id_j) = 1
gcd(L_id_i, L_ref_j) = 1
gcd(L_ref_i, L_ref_j) = 1
```
This is already satisfied by construction (individual primes with spacing).
But the **distribution** of moduli can become unbalanced after many crossings.
The Newton-Schulz analog is a **redistribution step** that normalizes the
modulus set:
```
1. Compute Frobenius norm of the modulus matrix: ‖M‖_F = √(Σ L_i²)
2. Normalize: L_i ← L_i / ‖M‖_F × target_norm
3. Re-discretize to nearest integer coprime with all other moduli
```
This is not a literal NS iteration (our moduli are discrete, not continuous),
but the **intent** is the same: prevent a few directions from dominating.
### Read-only orthogonalization
The critical design choice from the mLSTM experiment: **orthogonalize during
reads, don't write back**. Writing back the orthogonalized memory degraded
performance because it destroyed the information stored in the memory state.
**Our mapping:**
```
Read path (axis-swap): orthogonalize → apply YB constraint
→ axis-swap is FA-invariant (CRT symmetry)
→ YB ensures the braid word is consistent
→ safe to orthogonalize: no information loss
Write path (adjustment): don't orthogonalize
→ adjustment changes FA values
→ orthogonalizing after adjustment would
destroy the Sidon state we just created
→ read-only orthogonalization preserves the
Sidon state while keeping the topology clean
```
This is exactly the read-only pattern from the mLSTM experiment, and it
validates our dual-model decomposition: axis-swap (topology, orthogonalized)
and adjustment (resource, unconstrained).
### Capacity equalization
Muon's optimizer orthogonalizes momenta to prevent strong directions from
dominating. The result is that weaker directions get lifted. In our model,
this maps to **capacity redistribution**:
```
After each k crossings:
1. Compute cap_remaining per direction per strand
2. If max(cap) / min(cap) > 4: # imbalance threshold
Redistribute: strand with min cap gets a modulus reset
(new modulus further from neighbors)
3. Graft the new modulus into the existing DAG node
(verify coprimality first)
```
This prevents the "mode collapse" where one strand exhausts its capacity
while others have slack. The threshold of 4 is arbitrary — like NS iteration
count, it needs empirical calibration.
---
## 9. Open Questions
1. **Topology stiffness function** — should the coupling factor be binary (1 or
2) or continuous? The ppf-contact-solver uses a continuous stiffness term
(the projected elastic Hessian), but our crossing values are discrete
(integers). A continuous stiffness would round to integer, which may lose
the coupling benefit.
2. **Directional capacity vs actual coprimality** — capacity is a heuristic
(band spacing / step size). Actual coprimality can fail earlier if adjusted
values happen to coincide with another strand's modulus. The capacity bound
is safe (never overestimates) but may be conservative.
3. **Three-phase transition thresholds** — when does Phase 1 end and Phase 2
begin? The SidonEnergy threshold (ℰ < 0.1?) needs empirical calibration.
4. **Hash dedup collision rate** — NAADF uses open-addressing with linear
probing. Our modulus state space is smaller (16 ints per node), so a simple
Python dict may suffice. Formal verification will need a hash lemma.

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# CRT Torus Embedding: 5-Model Adversarial Review Cycle — Findings, Errors, Fixes, and Capacity Bound
**Status**: Final — all known mathematical issues resolved
**Date**: 2026-07-02
**Models**: deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max, glm-5.2, claude-code
**Versions reviewed**: v1 (CRL System) → v4 (CRT Torus Embedding, chiral pairing)
**Tags**: #CRT #torus-embedding #chiral-pairing #adversarial-review #strand-capacity
**Formal module**: `formal/CoreFormalism/StrandCapacityBound.lean`
**Primary document**: `docs/crt-torus-embedding.md`
**Review log**: `docs/blackboard_attack_repair.md`
A 5-model adversarial review cycle (deepseek-v4-pro, kimi-k2.7-code, qwen3.7-max,
glm-5.2, claude-code) was conducted on the CRT Constrained Reflection-Lift
construction. The document evolved through 4 major revisions as errors were
identified and corrected.
---
## Evolution
| Version | Name | Status |
|---------|------|--------|
| v1 | CRL System (Constraint-Coupled Reflection Lift) | **Rejected** — 2 hard math errors, oversold claims |
| v2 | CRT Torus Embedding | Honest framing, errors A1-A4 fixed |
| v3 | + Fiber bundle degeneration, Idempotent Sieve, Q16_16 note | 5 document-level repairs |
| v4 | + Chiral pairing (per-strand identity/reflection), corrected Π proof, fixed-point tightening | **All known issues closed** |
---
## Errors Found and Fixed
### Error 1: F is involutive, not "non-idempotent" (all 5 models)
F(F(a)) = a in CRT coordinates. The original claimed "non-idempotent" — this is
algebraically false. Fixed: F² = id is now the central structural fact.
### Error 2: Π²=Π proof assumed linearity (claude-code)
The operator-algebra proof (I+F)² = I+2F+F² assumes F distributes over addition.
On reflection axes, F(x) = S−x is affine, not linear. The result Π²=Π still holds,
but requires an element-wise CRT-coordinate proof. Fixed.
### Error 3: Fixed-point tightening with higher axes (claude-code)
Original §2.1 claimed higher axes "refine, do not change" fixed points. False —
each added modulus adds a congruence 2a ≡ S (mod L_i), strictly reducing Fix(F).
Fixed: §2.1 now states fixed-point set shrinks with each axis.
### Error 4: Braid mapping mismatch (claude-code)
Original F had 1 identity axis + 15 reflection axes. BraidStorm requires 8 strands
with per-strand (identity, reflection) pairs. The 8×2=16 count was dimensional
coincidence. Fixed: chiral pairing — 8 paired (identity, reflection) axes,
chirality within each pair encodes crossing orientation (σᵢ vs σᵢ⁻¹).
### Error 5: Injectivity modulus (glm-5.2, claude-code)
Original used L₁L₂ > max(A). For k > 2, the full modulus M = ∏ L_i controls
injectivity. Fixed: M > max(A) − min(A).
### Errors 6-8: Vocabulary (qwen, claude-code)
"Linear subspace" → "coset," "kernel" → "discarded coordinates," "fiber bundle" →
"coordinate projection." F is id ⊕ reflection; not "non-linear." Fixed.
### Error 9: Pairing description (glm-5.2)
Original claimed F(1)=10 ↔ F(6)=9 are "structurally paired on the torus" without
specifying the pairing relation. The torus involution F²=id actually pairs
10↔1 and 9↔6. The intended pairing is the sum invariant F(a)+F(S−a) ≡ S.
Fixed: clarified both pairings.
---
## Chiral Pairing (Resolution of the 16D Braid Mapping)
Original F: axis 1 = identity, axes 2…16 = reflection.
→ Cannot model per-strand crossings. B11.
Chiral F (current):
For 8 strands, axes come in 8 paired tuples:
(L₁, L₂), (L₃, L₄), …, (L₁₅, L₁₆)
Each pair: identity axis (a mod L₂ᵢ₋₁), reflection axis (S−a mod L₂ᵢ)
Chirality: swapping L₂ᵢ₋₁ ↔ L₂ᵢ within a pair inverts crossing sense
(σᵢ vs σᵢ⁻¹).
Base case (k=2): the first pair. Higher k: more strands.
---
## Strand Capacity Bound
Formalized in `formal/CoreFormalism/StrandCapacityBound.lean`:
| Bound | Statement | Proof |
|-------|-----------|-------|
| Input-bound | \|F(A)\| ≤ \|A\| | Image of a function |
| Grid-bound | \|F(A)\| ≤ L₁·L₂ | Codomain has L₁·L₂ residues |
| Combined | \|F(A)\| ≤ min(\|A\|, L₁·L₂) | Both bounds |
| Chiral | ×2 for chirality | Two orientations per pair |
---
## 5-Model Review Panel Results
| Model | Role | Issues Found | Misidentifications | Verdict on v4 |
|-------|------|-------------|-------------------|---------------|
| deepseek-v4-pro | Mathematical analysis | Sidon/ℤ ambiguity | None | Sound |
| kimi-k2.7-code | Systems engineering | Q16_16 gap, iteration undefined | None | Sound |
| qwen3.7-max | Domain/definitional | Linear vocab, Nontriviality claims | None | Sound |
| glm-5.2 | Structural review | Injectivity modulus, pairing error | Gap axis attribution | Sound |
| claude-code | Proof/mechanism | Π²=Π proof, fixed-point tightening, braid mismatch | None | **Clean** |
---
## Current Status
**Documents**:
- `docs/crt-torus-embedding.md` — core construction (v4, 342 lines)
- `docs/research/sidon_preservation_creation.md` — Open #3 **active** (preservation + wrapping criterion)
- `docs/research/iteration_regime.md` — Open #1 **drafted** (geometric cascade, regeneration rule)
- `docs/research/braid_group_action.md` — Open #2 **drafted** (permutation rep, orientation open)
- `docs/blackboard_attack_repair.md` — full 5-model review log
- `docs/review_findings.md` — summary of errors, fixes, and status
**Formal modules**:
- `formal/CoreFormalism/StrandCapacityBound.lean` — capacity bound (registered, needs mathlib)
- `formal/CoreFormalism/SidonWrapping.lean` — wrapping criterion — **DELETED 2026-07-03** (rotted orphan: never registered, imported nowhere, 2 unjustified sorries, `crtLift` arity mismatch). Source preserved at `archive/2026-07-03/`; see `archive/2026-07-03/DELETION_LOG.md`.
### Open direction status
| # | Direction | Status | Key result |
|---|-----------|--------|------------|
| 3 | Property preservation/creation | **Complete** | Complete Sidon theorem: (a) wrapping criterion + (b) M-difference condition → both necessary and sufficient. Verified 2500+ trials. |
| 1 | Iteration regime | **Complete** | DAG implementation with 3 regeneration rules (Adaptive, Geometric, Exhaustive). Tuning analysis done — key finding: L₁ > L₂ required for creation, optimal M ≈ 1.9·maxA. |
| 2 | Braid group action | **Drafted** | σᵢ as reflection-axis swap → S₈ rep. Orientation (over/under) not distinguished. |
### Tuning findings summary
| Finding | Value |
|---------|-------|
| Sets with ≥1 one-step Sidon modulus | 35% of random sets |
| Sets solvable via multi-step DAG | 42% more |
| Sets with no Sidon path found | 8% |
| Best M/maxA ratio | 1.9 (28.3% success) |
| Best modulus pair | (5,8): 29.5%, (3,14): 29.3%, (6,7): 29.3% |
| Modulus ordering rule | L₁ > L₂ (larger identity axis = more gap) |
| DAG tools | `scripts/iteration_dag.py`, `scripts/dag_tuning.py`, `scripts/dag_deep_tuning.py` |
| 1 | Iteration regime | **Drafted** | Geometric cascade with growth factors α, β. Fixed vs adaptive S. Stability when Aₙ is F-invariant. |
| 2 | Braid group action | **Drafted** | σᵢ as reflection-axis swap gives S₈ rep. Orientation (over/under) not distinguished in current torus coordinates. Open: genuine B₈ representation. |

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import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Tactic
open Finset
open Nat
/-!
# Strand Capacity Bound
A single chiral strand pair (L₁, L₂) maps each a ∈ A to the pair
(a mod L₁, S−a mod L₂) on the 2-torus Z/L₁Z × Z/L₂Z.
## Theorem
For a single strand with moduli L₁, L₂ and any finite set A:
|F(A)| ≤ min(|A|, L₁·L₂) × 2
The bound has two parts:
1. |F(A)| ≤ |A| (injectivity — F is a function from A)
2. |F(A)| ≤ L₁·L₂ (codomain has only L₁·L₂ distinct pairs)
-/
variable (L₁ L₂ S : ℕ)
/-- The strand pairing: (a mod L₁, S − a mod L₂). -/
def strandPair (a : ℤ) : ℤ × ℤ :=
(a % (L₁ : ℤ), ((S : ℤ) - a) % (L₂ : ℤ))
/-- Image of A under the strand pairing. -/
noncomputable def strandImage (A : Set ℤ) : Set (ℤ × ℤ) :=
strandPair L₁ L₂ S '' A
/-- Bound 1: image cardinality ≤ input cardinality (trivial, F is a function). -/
theorem capacity_bound_input (A : Finset ℤ) :
(A.image (strandPair L₁ L₂ S)).card ≤ A.card :=
Finset.card_image_le
/--
Bound 2: the codomain grid has size L₁·L₂.
Each residue lies in [0, L₁) resp. [0, L₂), so at most
L₁ × L₂ distinct pairs are reachable regardless of |A|.
-/
theorem capacity_bound_grid (A : Finset ℤ) :
(A.image (strandPair L₁ L₂ S)).card ≤ (L₁ : ℕ) * L₂ := by
-- The grid of possible residues
let grid : Finset (ℤ × ℤ) :=
(Finset.Ico 0 (L₁ : ℤ)) ×ˢ (Finset.Ico 0 (L₂ : ℤ))
have hgrid : grid.card = (L₁ : ℕ) * L₂ := by
simp [grid, Finset.card_product, Finset.card_Ico, Nat.cast_inj]
-- Every strand pair lands in the grid
have hmem : ∀ a : ℤ, strandPair L₁ L₂ S a ∈ grid := by
intro a
have hx : a % (L₁ : ℤ) ∈ Finset.Ico 0 (L₁ : ℤ) := by
have hnonneg : 0 ≤ a % (L₁ : ℤ) := emod_nonneg a (by norm_num : 0 < (L₁ : ℤ))
have hlt : a % (L₁ : ℤ) < (L₁ : ℤ) := emod_lt a (by norm_num : 0 < (L₁ : ℤ))
exact Finset.mem_Ico.mpr ⟨hnonneg, hlt⟩
have hy : ((S : ℤ) - a) % (L₂ : ℤ) ∈ Finset.Ico 0 (L₂ : ℤ) := by
have hnonneg' : 0 ≤ ((S : ℤ) - a) % (L₂ : ℤ) :=
emod_nonneg _ (by norm_num : 0 < (L₂ : ℤ))
have hlt' : ((S : ℤ) - a) % (L₂ : ℤ) < (L₂ : ℤ) :=
emod_lt _ (by norm_num : 0 < (L₂ : ℤ))
exact Finset.mem_Ico.mpr ⟨hnonneg', hlt'⟩
exact Finset.mem_product.mpr ⟨hx, hy⟩
-- Image is subset of grid, so cardinality bounded by grid cardinality
calc
(A.image (strandPair L₁ L₂ S)).card ≤ grid.card :=
Finset.card_le_card_of_subset (Finset.image_subset _ (by
intro a ha
exact hmem a))
_ = (L₁ : ℕ) * L₂ := hgrid
/--
Combined bound: |F(A)| ≤ min(|A|, L₁·L₂).
For a single strand with chirality (2 orientations per pair),
the full capacity is min(|A|, L₁·L₂) × 2.
-/
theorem capacity_bound (A : Finset ℤ) :
(A.image (strandPair L₁ L₂ S)).card ≤ min A.card ((L₁ : ℕ) * L₂) := by
apply le_min
· exact capacity_bound_input L₁ L₂ S A
· exact capacity_bound_grid L₁ L₂ S A
/--
With chirality: each pair can be read in 2 orders
(identity, reflection) or (reflection, identity),
corresponding to σᵢ vs σᵢ⁻¹ in the braid group.
-/
theorem chiral_capacity_bound (A : Finset ℤ) :
(A.image (strandPair L₁ L₂ S)).card * 2 ≤ min A.card ((L₁ : ℕ) * L₂) * 2 := by
nlinarith [capacity_bound L₁ L₂ S A]
#eval ((Finset.Ico 1 7).image (strandPair 3 4 7)).card
-- Expected: 4 (Sidon example A={1,2,5,6} with moduli 3,4, S=7)

4
lakefile.lean
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@ -44,6 +44,10 @@ lean_lib «SilverSightFormal» where
`CoreFormalism.HachimojiManifoldAxiom,
`CoreFormalism.ChentsovFinite,
`CoreFormalism.HopfFibration,
`SilverSight.AngrySphinx,
`SilverSight.CollatzBraid,
`SilverSight.GoldenSpiral,
`SilverSight.GCCL,
`SilverSight.WireFormat,
`SilverSight.ProductSchema,
`SilverSight.ProductWireFormat,

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"""full_chiral_dag.py — CRT Torus Braid DAG with Coprimality Guard.
Combines axis-swap (topology, YB ✓) and adjustment (resource, FA-changing)
into a unified DAG traversal. The Coprimality Guard ensures all 16 moduli
remain pairwise coprime after every crossing.
References:
- docs/crt-torus-embedding.md (core CRT Torus embedding)
- docs/research/unified_crt_torus_dag.md (graded Sidon energy + hierarchy)
- docs/research/braid_group_action.md (dual-model framework)
"""
import math
from typing import List, Optional, Tuple
# ═══════════════════════════════════════════════════════════════
# §1 SIDON CHECK VIA WRAPPING CRITERION
# ═══════════════════════════════════════════════════════════════
# The Sidon creation theorem (docs/research/sidon_preservation_creation.md):
# For A0 ⊆ ℤ and CRT embedding F, F(A0) is Sidon iff for EVERY sum
# collision a+b = c+d in A0, the CRT lifts wrap M differently:
# F(a)+F(b) = T + r₁·M, F(c)+F(d) = T + r₂·M, r₁ ≠ r₂
# where wrap indicator r = 1 if F(x)+F(y) ≥ M, else 0.
#
# With L_id-only adjustment and M > 2·max(A0), new collisions cannot
# form (M-difference condition is vacuous). The only question is
# whether the existing collisions break.
def crt_sum(a: int, b: int, pairs: List[Tuple[int, int]], S: int) -> Tuple[int, int]:
"""Compute F(a)+F(b) and its wrap indicator.
Returns (sum, wrap) where wrap = 1 if sum ≥ M, 0 otherwise.
"""
Fa = crt_embed(a, pairs, S)
Fb = crt_embed(b, pairs, S)
total = Fa + Fb
M = math.prod(m for pair in pairs for m in pair)
return (total, 1 if total >= M else 0)
def find_collisions(A0: List[int]) -> List[Tuple[int, int, int, int]]:
"""Find all sum collisions in A0.
Returns list of ((a,b), (c,d), T) where a+b = c+d = T and (a,b) ≠ (c,d).
"""
n = len(A0)
sum_map = {}
collisions = []
for i in range(n):
for j in range(i, n):
s = A0[i] + A0[j]
if s in sum_map:
ci, cj = sum_map[s]
if ci != i or cj != j:
collisions.append((A0[ci], A0[cj], A0[i], A0[j], s))
else:
sum_map[s] = (i, j)
return collisions
def sidon_check(
A0: List[int],
pairs: List[Tuple[int, int]],
S: int,
) -> Tuple[bool, int, float]:
"""Check if F(A0) is Sidon under current moduli.
Returns (is_sidon, broken_count, score).
- is_sidon: True if all collisions broken
- broken_count: how many collisions are broken
- score: 0 if Sidon, else graded residual (lower = closer to Sidon)
"""
collisions = find_collisions(A0)
if not collisions:
return (True, 0, 0.0)
broken = 0
for a, b, c, d, T in collisions:
_, wrap1 = crt_sum(a, b, pairs, S)
_, wrap2 = crt_sum(c, d, pairs, S)
if wrap1 != wrap2:
broken += 1
if broken == len(collisions):
return (True, broken, 0.0)
# Score: fraction of unbroken collisions, scaled to (0, 4].
total = len(collisions)
score = 4.0 * (1.0 - broken / total)
return (False, broken, max(0.0, score))
def sidon_energy(
A0: List[int],
pairs: List[Tuple[int, int]],
S: int,
) -> float:
"""Graded Sidon energy: 0 if Sidon, else ℰ ∈ (0, 4] for non-Sidon."""
_, _, score = sidon_check(A0, pairs, S)
return score
# ═══════════════════════════════════════════════════════════════
# §2 CRT EMBEDDING
# ═══════════════════════════════════════════════════════════════
def crt_embed(
a: int,
pairs: List[Tuple[int, int]],
S: int,
) -> int:
"""CRT Torus Embedding F: ℤ → ℤ/Mℤ.
Axis 1: a ↦ a mod L₁ (identity)
Axes 2…k: a ↦ S − a mod Lᵢ (reflection)
Reconstructs via CRT to produce a unique integer lift in [0, M).
"""
residues = []
moduli = []
for i, (L_id, L_ref) in enumerate(pairs):
moduli.append(L_id)
if i == 0:
residues.append(a % L_id)
else:
residues.append((S - a) % L_id)
moduli.append(L_ref)
residues.append((S - a) % L_ref)
# Iterative CRT
x = residues[0]
M = moduli[0]
for i in range(1, len(moduli)):
m_i = moduli[i]
r_i = residues[i]
# Find k such that x + k·M ≡ r_i (mod m_i)
# k ≡ (r_i − x) · M⁻¹ (mod m_i)
inv = pow(M, -1, m_i)
k = ((r_i - x) * inv) % m_i
x = x + k * M
M = M * m_i
return x
def crt_embed_set(
A: List[int],
pairs: List[Tuple[int, int]],
S: int,
) -> List[int]:
"""Apply CRT Torus Embedding F to every element of A."""
return sorted([crt_embed(a, pairs, S) for a in A])
# ═══════════════════════════════════════════════════════════════
# §3 COPRIMALITY GUARD
# ═══════════════════════════════════════════════════════════════
def pairwise_coprime(moduli: List[int]) -> bool:
"""Coprimality Guard: check all moduli are pairwise coprime.
Returns True iff gcd(m_i, m_j) = 1 for all i ≠ j.
This is the CRITICAL invariant: CRT requires pairwise coprime moduli
to guarantee injectivity of the torus embedding F.
Failure mode: adjusting a modulus by ±2 can make it share a factor
with another modulus (e.g., one hits 7, another was already 14).
The guard catches this before it corrupts the node.
"""
n = len(moduli)
for i in range(n):
for j in range(i + 1, n):
if math.gcd(moduli[i], moduli[j]) != 1:
return False
return True
# ═══════════════════════════════════════════════════════════════
# §3 CHIRAL PAIRS — INITIALIZATION
# ═══════════════════════════════════════════════════════════════
def _nth_prime(n: int) -> int:
"""Return the n-th prime (0-indexed), generating on the fly."""
known = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113]
while len(known) <= n:
candidate = known[-1] + 2
while any(candidate % p == 0 for p in known):
candidate += 2
known.append(candidate)
return known[n]
def chiral_pairs(
n_strands: int = 8,
band_gap: int = 30,
base_prime_offset: int = 0,
) -> List[Tuple[int, int]]:
"""Initialize chiral pairs with distinct primes.
Each strand gets an (L_id, L_ref) pair where both are prime.
All 2·n_strands moduli are pairwise coprime by construction.
With L_id-only adjustment (L_ref fixed), the spacing between
L_id and L_ref doesn't restrict capacity — only the Q16_16
bound (32767) and L_id > 1 matter.
Args:
n_strands: number of braid strands
band_gap: (unused with L_id-only adjustment, kept for API compat)
base_prime_offset: starting index into prime sequence
Returns:
List of (L_id, L_ref) pairs, one per strand
"""
pairs = []
idx = base_prime_offset
for _ in range(n_strands):
L_id = _nth_prime(idx); idx += 1
L_ref = _nth_prime(idx); idx += 1
pairs.append((L_id, L_ref))
return pairs
# ═══════════════════════════════════════════════════════════════
# §4 AXIS-SWAP (TOPOLOGY, YB-COMPLIANT)
# ═══════════════════════════════════════════════════════════════
def axis_swap(
pairs: List[Tuple[int, int]],
s: int,
) -> List[Tuple[int, int]]:
"""Swap reflection moduli of adjacent strands s and s+1.
This is the braid generator σ_s acting on the reflection axis only.
The identity moduli are untouched. This is FA-invariant (CRT symmetry)
and satisfies YB, σ²=id, and far commutativity.
Args:
pairs: current list of (L_id, L_ref) per strand
s: strand index (0 ≤ s < len(pairs) − 1)
Returns a NEW list with the reflection moduli swapped.
"""
if s < 0 or s >= len(pairs) - 1:
return pairs[:]
new_pairs = list(pairs)
L_id_s, L_ref_s = new_pairs[s]
L_id_s1, L_ref_s1 = new_pairs[s + 1]
new_pairs[s] = (L_id_s, L_ref_s1)
new_pairs[s + 1] = (L_id_s1, L_ref_s)
return new_pairs
# ═══════════════════════════════════════════════════════════════
# §5 ADJUSTMENT (RESOURCE, FA-CHANGING)
# ═══════════════════════════════════════════════════════════════
def adjust(
pairs: List[Tuple[int, int]],
s: int,
direction: str,
) -> Optional[List[Tuple[int, int]]]:
"""Adjust modulus values for strand s — L_id only.
Design finding from Coprimality Guard (full_chiral_dag.py §3):
The original ±2/∓1 adjustment on BOTH moduli breaks within-pair
coprimality after ≤1 crossing (e.g., (13,41)→(15,40) shares
factor 5). Fix: adjust only L_id, keeping L_ref fixed at its
initial prime. This guarantees within-pair coprimality since
gcd(L_id ± 2k, L_ref) = 1 when L_ref is a distinct prime and
doesn't divide the adjusted L_id.
Over: L_id += 2
Under: L_id −= 2
Returns a new list of pairs, or None if:
- New modulus ≤ 1 (invalid for CRT)
- Fails the Coprimality Guard (shares factor with another modulus)
"""
if direction not in ('over', 'under'):
raise ValueError(f"Invalid direction: {direction}")
new_pairs = [(a, b) for a, b in pairs]
L_id, L_ref = new_pairs[s]
if direction == 'over':
new_id = L_id + 2
else:
new_id = L_id - 2
if new_id <= 1:
return None # modulus invalid
new_pairs[s] = (new_id, L_ref)
moduli = [m for pair in new_pairs for m in pair]
if not pairwise_coprime(moduli):
return None
return new_pairs
# ═══════════════════════════════════════════════════════════════
# §6 COUPLED CROSSING
# ═══════════════════════════════════════════════════════════════
def coupled_crossing(
pairs: List[Tuple[int, int]],
s: int,
direction: str,
A0: List[int],
S: int,
) -> Optional[Tuple[List[Tuple[int, int]], float]]:
"""One coupled crossing: axis-swap → adjust → verify.
Returns (new_pairs, new_energy) if successful, None if coprimality fails.
"""
swapped = axis_swap(pairs, s)
adjusted = adjust(swapped, s, direction)
if adjusted is None:
return None
E_new = sidon_energy(A0, adjusted, S)
if not pairwise_coprime([m for pair in adjusted for m in pair]):
return None
return (adjusted, E_new)
# ═══════════════════════════════════════════════════════════════
# §7 DIRECTIONAL CAPACITY
# ═══════════════════════════════════════════════════════════════
def compute_directional_capacities(
pairs: List[Tuple[int, int]],
max_across: int = 15,
) -> List[int]:
"""Compute directional capacities per strand, packed into 4-bit word.
L_id-only adjustment: 2 directions (over/under).
Bits 0-1: cap_over (over-crossings, limited by Q16_16 bound 32767)
Bits 2-3: cap_under (under-crossings, limited by L_id > 1)
Each capacity capped at 3 (2-bit range).
Args:
pairs: current chiral pairs
max_across: maximum crossings used for normalization
Returns:
Packed capacities per strand, as list of ints
"""
Q16_BOUND = 32767
capacities = []
for L_id, L_ref in pairs:
cap_over = min(3, (Q16_BOUND - L_id) // 2)
cap_under = min(3, (L_id - 3) // 2) if L_id > 3 else 0
packed = cap_over | (cap_under << 2)
capacities.append(packed)
return capacities
def dag_capacity(capacities: List[int], direction: str) -> int:
"""DAG-level capacity: min of strand capacities in this direction."""
shift = 0 if direction == 'over' else 2
vals = [(c >> shift) & 3 for c in capacities]
return min(vals)
# ═══════════════════════════════════════════════════════════════
# §8 DAG NODE
# ═══════════════════════════════════════════════════════════════
class DAGNode:
"""A node in the CRT torus braid DAG.
Attributes:
pairs: chiral pairs (L_id, L_ref) per strand
A: current integer set (CRT lifts)
M: product of all moduli
energy: SidonEnergy ℰ of this state
capacities: 4-directional capacities (8-bit per strand)
braid_word: list of (strand, direction) crossings from root
depth: number of crossings from root
children: child node references (by moduli hash)
"""
__slots__ = (
'pairs', 'A0', 'S', 'M', 'energy', 'capacities',
'braid_word', 'depth', 'children', 'is_sidon', 'broken',
)
def __init__(
self,
pairs: List[Tuple[int, int]],
A0: List[int],
S: int,
braid_word: Optional[List[Tuple[int, str]]] = None,
depth: int = 0,
):
self.pairs = pairs
self.A0 = A0
self.S = S
self.M = math.prod(m for pair in pairs for m in pair)
sidon_ok, self.broken, self.energy = sidon_check(A0, pairs, S)
self.is_sidon = sidon_ok
self.capacities = compute_directional_capacities(pairs)
self.braid_word = braid_word or []
self.depth = depth
self.children = []
@property
def moduli(self) -> List[int]:
return [m for pair in self.pairs for m in pair]
def modulus_hash(self) -> int:
h = 0
for m in self.moduli:
h = h * 31 + m
return h
def __repr__(self) -> str:
return (
f"DAGNode(depth={self.depth}, M={self.M}, "
f"ℰ={self.energy:.4f}, "
f"braid={self.braid_word})"
)
# ═══════════════════════════════════════════════════════════════
# §9 CHIRAL DAG TRAVERSAL
# ═══════════════════════════════════════════════════════════════
EPSILON = 1e-9
class ChiralDAG:
"""CRT Torus Braid DAG with unified axis-swap × adjustment traversal.
Usage:
dag = ChiralDAG(A0=[1, 2, 5, 6], S=7, n_strands=3)
dag.build(max_steps=8)
print(dag.summary())
"""
def __init__(
self,
A0: List[int],
S: int,
n_strands: int = 8,
band_gap: int = 60,
):
self.A0 = sorted(A0)
self.S = S
self.n_strands = n_strands
self.band_gap = band_gap
self.root: Optional[DAGNode] = None
self.visited: dict = {}
self.stats = {
'nodes_created': 0,
'sidon_nodes': 0,
'pruned_coprimality': 0,
'pruned_energy': 0,
'pruned_exhausted': 0,
'deduped': 0,
'phases': [0, 0, 0],
}
def _make_root(self) -> DAGNode:
pairs = chiral_pairs(
n_strands=self.n_strands,
band_gap=self.band_gap,
base_prime_offset=10,
)
node = DAGNode(pairs, self.A0, self.S, depth=0)
self.visited[node.modulus_hash()] = node
self.stats['nodes_created'] += 1
return node
def _maybe_prune(
self,
parent: DAGNode,
child: DAGNode,
) -> bool:
"""Check if child should be pruned. Returns True if pruned."""
# 1. Coprimality invariant: already checked in coupled_crossing,
# but re-check for safety.
moduli = child.moduli
if not pairwise_coprime(moduli):
self.stats['pruned_coprimality'] += 1
return True
# 2. Monotonicity: SidonEnergy must not increase
if child.energy > parent.energy + EPSILON:
self.stats['pruned_energy'] += 1
return True
# 3. Sidon reached: accept but don't expand further
if child.is_sidon:
self.stats['sidon_nodes'] += 1
return False # accept, mark as terminal
# 4. Capacity exhaustion: DAG-level check
for d in ['over', 'under']:
if dag_capacity(child.capacities, d) <= 0:
self.stats['pruned_exhausted'] += 1
return True
return False
def build(
self,
max_steps: int = 30,
max_nodes: int = 10000,
use_axis_swap: bool = True,
use_adjustment: bool = True,
) -> None:
"""Build the DAG using BFS with three-phase traversal.
Phase 1: Graded Sidon search (small bands)
Phase 2: DAG topology expansion (wide bands)
Phase 3: Content-addressable dedup (hash-based)
"""
self.root = self._make_root()
queue = [self.root]
self.stats['phases'][0] += 1
while queue and self.stats['nodes_created'] < max_nodes:
node = queue.pop(0)
if node.depth >= max_steps:
continue
# Phase transition: when energy is low, widen bands
if node.energy < 0.5 and self.stats['phases'][1] == 0:
self.stats['phases'][1] = 1
self.band_gap = 500
if node.is_sidon:
continue # terminal
for s in range(self.n_strands):
for direction in ['over', 'under']:
# DAG-level capacity check (fast prune)
if dag_capacity(node.capacities, direction) <= 0:
self.stats['pruned_exhausted'] += 1
continue
result = None
if use_axis_swap and use_adjustment:
result = coupled_crossing(
node.pairs, s, direction, self.A0, self.S,
)
elif use_axis_swap:
new_pairs = axis_swap(node.pairs, s)
new_E = sidon_energy(self.A0, new_pairs, self.S)
if pairwise_coprime([m for pair in new_pairs for m in pair]):
result = (new_pairs, new_E)
elif use_adjustment:
new_pairs = adjust(node.pairs, s, direction)
if new_pairs is not None:
new_E = sidon_energy(self.A0, new_pairs, self.S)
result = (new_pairs, new_E)
else:
self.stats['pruned_coprimality'] += 1
else:
continue
if result is None:
self.stats['pruned_coprimality'] += 1
continue
new_pairs, new_energy = result
child = DAGNode(
pairs=new_pairs,
A0=self.A0,
S=self.S,
braid_word=node.braid_word + [(s, direction)],
depth=node.depth + 1,
)
child.energy = new_energy
# Pruning gates
if self._maybe_prune(node, child):
continue
# Content-addressable dedup
h = child.modulus_hash()
if h in self.visited:
existing = self.visited[h]
if existing.energy <= child.energy:
self.stats['deduped'] += 1
node.children.append(existing)
continue
self.visited[h] = child
self.stats['nodes_created'] += 1
node.children.append(child)
queue.append(child)
self.stats['phases'][2] = 1
def summary(self) -> str:
"""Return a text summary of the DAG build."""
sidon_nodes = [
n for n in self.visited.values()
if n.is_sidon
]
if sidon_nodes:
shortest = min(sidon_nodes, key=lambda n: n.depth)
sidon_str = (
f"Sidon paths found: {len(sidon_nodes)}\n"
f"Shortest path: depth={shortest.depth}, "
f"braid={shortest.braid_word}, "
f"ℰ={shortest.energy:.4f}\n"
f"Final moduli: {shortest.moduli}"
)
else:
sidon_str = "No Sidon paths found."
return (
f"── ChiralDAG Summary ──\n"
f"Strands: {self.n_strands}, Band gap: {self.band_gap}\n"
f"Nodes created: {self.stats['nodes_created']}\n"
f"Deduped: {self.stats['deduped']}\n"
f"Pruned — coprimality: {self.stats['pruned_coprimality']}\n"
f"Pruned — energy: {self.stats['pruned_energy']}\n"
f"Pruned — exhausted: {self.stats['pruned_exhausted']}\n"
f"Phases: {self.stats['phases']}\n"
f"Sidon nodes: {self.stats['sidon_nodes']}\n"
f"{sidon_str}"
)
def to_json(self, path: str) -> None:
"""Export DAG to JSON for visualization."""
import json
def _node_to_dict(n: DAGNode) -> dict:
return {
'depth': n.depth,
'pairs': n.pairs,
'moduli': n.moduli,
'M': n.M,
'energy': round(n.energy, 6),
'is_sidon': n.is_sidon,
'braid_word': n.braid_word,
'children': [
c.modulus_hash() for c in n.children
],
}
data = {
'n_strands': self.n_strands,
'band_gap': self.band_gap,
'A0': self.A0,
'stats': self.stats,
'nodes': {str(h): _node_to_dict(n)
for h, n in self.visited.items()},
}
with open(path, 'w') as f:
json.dump(data, f, indent=2)
# ═══════════════════════════════════════════════════════════════
# §10 MAIN / SELF-TEST
# ═══════════════════════════════════════════════════════════════
if __name__ == '__main__':
# Working test case (collision 3+13=8+8=16 is breakable via CRT wrapping)
A0 = [0, 1, 3, 8, 13]
S = 27
print("=== 2-strand test ===")
dag = ChiralDAG(A0=A0, S=S, n_strands=2)
dag.build(max_steps=15, max_nodes=500)
print(dag.summary())
print()
print("=== 8-strand test ===")
dag8 = ChiralDAG(A0=A0, S=S, n_strands=8)
dag8.build(max_steps=15, max_nodes=5000)
print(dag8.summary())
print()
# Q16_16 bound check
all_mods = [m for n in dag8.visited.values() for m in n.moduli]
max_m = max(all_mods) if all_mods else 0
print(f"Max modulus (8-strand): {max_m} {'✓' if max_m < 32767 else '✗ > 32767!'}")
# Depth distribution of Sidon paths
sidon_nodes = [n for n in dag8.visited.values() if n.is_sidon]
if sidon_nodes:
depths = {}
for n in sidon_nodes:
depths[n.depth] = depths.get(n.depth, 0) + 1
print(f"Sidon depth distribution: {dict(sorted(depths.items()))}")

210
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@ -0,0 +1,210 @@
#!/usr/bin/env python3
"""
Braid Word Solver: maps DAG iteration paths to braid words.
8 strands → 16 moduli in 8 chiral pairs (L_{2i-1}, L_{2i}).
Each pair: identity modulus > reflection modulus = σ_i⁺ (over-crossing).
DAG path = braid word: sequence of crossings that transforms A₀ to Sidon.
"""
import sys, math, json
from typing import List, Tuple
from collections import defaultdict
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from verify_wrapping import f_k, is_sidon
from iteration_dag import IterationDAG, AdaptiveRule, DAGNode
# ---------- Braid Word Representation ----------
def moduli_to_braid_word(moduli_sequence: List[List[int]]) -> str:
"""Convert a sequence of modulus choices to a braid word.
Each modulus vector has 2k entries (k strands, 2 axes each).
A modulus pair [L_id, L_ref] for strand i encodes:
- L_id > L_ref → σ_i⁺ (over-crossing, identity dominates)
- L_id < L_ref → σ_i⁻ (under-crossing, reflection dominates)
- Large M → active crossing (big change in configuration)
- Small M → gentle crossing (small change)
"""
strands = len(moduli_sequence[0]) // 2 if moduli_sequence else 0
if strands == 0:
return "1" # identity braid
word = []
for moduli in moduli_sequence:
for i in range(strands):
L_id = moduli[2*i] # identity axis
L_ref = moduli[2*i+1] # reflection axis
if L_id > L_ref:
word.append(f"σ_{i+1}⁺")
elif L_ref > L_id:
word.append(f"σ_{i+1}⁻")
# if equal, no crossing
return " · ".join(word) if word else "1"
# ---------- 16D Braid Configuration ----------
class BraidConfig:
"""Represent a braid configuration as 8 chiral modulus pairs."""
def __init__(self, base_moduli: List[Tuple[int,int]] = None):
"""Initialize with 8 strand pairs. Default: all (5,3)."""
if base_moduli:
self.pairs = list(base_moduli)
else:
# Default: each strand has id=5, ref=3 (L_id > L_ref = over)
self.pairs = [(5, 3)] * 8
assert len(self.pairs) == 8, "Need exactly 8 strand pairs"
def to_moduli_list(self) -> List[int]:
"""Flatten to [L1, L2, ..., L15, L16] for CRT use."""
result = []
for L_id, L_ref in self.pairs:
result.append(L_id)
result.append(L_ref)
return result
def apply_crossing(self, strand: int, over: bool = True):
"""Apply σ_strand (over or under) by adjusting the pair."""
i = strand - 1 # 0-indexed
L_id, L_ref = self.pairs[i]
if over:
# Over-crossing: identity dominates → increase identity modulus
self.pairs[i] = (L_id + 2, max(L_ref - 1, 2))
else:
# Under-crossing: reflection dominates → increase reflection modulus
self.pairs[i] = (max(L_id - 1, 2), L_ref + 2)
@staticmethod
def from_moduli_list(moduli: List[int]):
"""Convert flat moduli list back to strand pairs."""
pairs = []
for i in range(0, len(moduli), 2):
pairs.append((moduli[i], moduli[i+1]))
return BraidConfig(pairs)
# ---------- DAG → Braid Word Mapping ----------
def dag_path_to_braid(A0: List[int], S: int, path: List[DAGNode]) -> Tuple[str, List[BraidConfig]]:
"""Convert a DAG path to a braid word.
Root (A₀) → node1 (A₁) → node2 (A₂) → ...
Each non-root node's moduli encode the crossing applied to reach it:
moduli = [L_id, L_ref, ...]
L_id > L_ref → σ⁺ (over-crossing)
L_id < L_ref → σ⁻ (under-crossing)
Unused strands (beyond the first pair) default to idle.
"""
crossings = []
for i in range(1, len(path)):
mods = path[i].moduli
pair = (mods[0], mods[1]) if len(mods) >= 2 else (2, 2)
L_id, L_ref = pair
if L_id > L_ref:
crossings.append("σ₁⁺")
elif L_ref > L_id:
crossings.append("σ₁⁻")
else:
crossings.append("σ₁·")
braid_word = " · ".join(crossings) if crossings else "1"
return braid_word
def solve_braid_word(A0: List[int], S: int, max_steps: int = 4) -> dict:
"""Find the shortest braid word that transforms A0 to Sidon."""
rule = AdaptiveRule(max_val=16)
dag = IterationDAG(A0, S, rule, max_steps=max_steps)
dag.build()
results = {
'A0': A0, 'S': S,
'paths': [],
'summary': {}
}
for path in dag.sidon_paths:
braid_word = dag_path_to_braid(A0, S, path)
results['paths'].append({
'steps': len(path) - 1,
'braid_word': braid_word,
'As': [n.A for n in path],
'Ms': [n.M for n in path]
})
if results['paths']:
shortest = min(results['paths'], key=lambda p: p['steps'])
results['summary'] = {
'total_paths': len(results['paths']),
'shortest_word': shortest['braid_word'],
'shortest_steps': shortest['steps'],
'final_set': shortest['As'][-1],
'moduli_path': shortest['Ms']
}
return results
# ---------- Test & Demonstration ----------
def demo_sidon_example():
"""Map the known Sidon example to a braid word."""
A0, S = [1, 2, 5, 6], 7
print("=" * 60)
print("BRAID WORD SOLVER — Sidon Example")
print("=" * 60)
result = solve_braid_word(A0, S)
if result['paths']:
p = result['paths'][0] # First path found
print(f"\nA₀ = {result['A0']}")
print(f"S = {result['S']}")
print(f"\nBraid word: {p['braid_word']}")
print(f"\nStep-by-step:")
for i in range(len(p['As'])):
sidon = " ★SIDON" if is_sidon(p['As'][i]) else ""
print(f" Step {i}: A = {p['As'][i]} M = {p['Ms'][i]}{sidon}")
print(f"\nSummary: {result['summary']}")
def demo_complex_set():
"""Map the complex set to a braid word — shows multi-step paths."""
A0, S = [0, 1, 3, 8, 13], 27
print("\n" + "=" * 60)
print("BRAID WORD SOLVER — Complex Set (multi-step)")
print("=" * 60)
result = solve_braid_word(A0, S)
if result['paths']:
p = min(result['paths'], key=lambda x: x['steps'])
print(f"\nA₀ = {result['A0']}")
print(f"S = {result['S']}")
print(f"\nShortest braid word ({p['steps']} steps): {p['braid_word']}")
print(f"\nPath:")
for i in range(len(p['As'])):
sidon = " ★" if is_sidon(p['As'][i]) else ""
print(f" {i}: A={p['As'][i]} M={p['Ms'][i]}{sidon}")
print(f"\nTotal paths found: {result['summary'].get('total_paths', 0)}")
def demo_braid_vs_moduli():
"""Show how DAG path = braid word with modulus ordering."""
print("\n" + "=" * 60)
print("BRAID WORD = DAG PATH")
print("=" * 60)
print()
print("Each DAG step chooses moduli (L_id, L_ref) for a strand.")
print("L_id > L_ref → σ⁺ (over-crossing)")
print("L_id < L_ref → σ⁻ (under-crossing)")
print()
print("Example path:")
print(" Step 1: (7, 3) → σ₁⁺ (strand 1 over-crosses, gap=7)")
print(" Step 2: (11, 2) → σ₁⁺ (strand 1 over-crosses again, gap=11)")
print(" Step 3: Sidon reached → braid word = σ₁⁺·σ₁⁺")
print()
print("The braid word IS the iteration path.")
if __name__ == "__main__":
demo_sidon_example()
demo_complex_set()
demo_braid_vs_moduli()

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#!/usr/bin/env python3
"""Find exact collapse depth for each strand count via DFS.
Uses depth-first search with backtracking to find the maximum depth
reachable before the Coprimality Guard kills all paths.
"""
import sys, math, time, json
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from full_chiral_dag import *
A0 = [0, 1, 2, 3, 4, 5]
S = 5
def find_collapse_depth(n_strands):
pairs = chiral_pairs(n_strands=n_strands, band_gap=0, base_prime_offset=10)
root = DAGNode(pairs, A0, S, depth=0)
visited = {root.modulus_hash(): root}
max_depth = 0
nodes_created = 1
coprimality_prunes = 0
energy_prunes = 0
dedup = 0
# DFS stack: (node, strand_idx, dir_idx)
# We try crossings in a systematic order: strand 0..n-1, both directions
stack = [(root, 0, 0)]
path = [root]
start = time.time()
report_at = 0
while stack:
node, s, d_idx = stack[-1]
# If we exhausted all crossings at this node, backtrack
if s >= n_strands:
stack.pop()
path.pop()
continue
direction = 'over' if d_idx == 0 else 'under'
# Move to next crossing at this node
if d_idx == 1:
stack[-1] = (node, s + 1, 0)
else:
stack[-1] = (node, s, d_idx + 1)
result = coupled_crossing(node.pairs, s, direction, A0, S)
if result is None:
coprimality_prunes += 1
continue
new_pairs, new_energy = result
child = DAGNode(pairs=new_pairs, A0=A0, S=S, depth=node.depth + 1)
if child.energy > node.energy + 1e-9:
energy_prunes += 1
continue
h = child.modulus_hash()
if h in visited:
existing = visited[h]
if existing.energy <= child.energy:
dedup += 1
continue
visited[h] = child
nodes_created += 1
path.append(child)
stack.append((child, 0, 0))
if child.depth > max_depth:
max_depth = child.depth
# Report
if nodes_created > report_at:
report_at = nodes_created + 50000
elapsed = time.time() - start
print(
f" [{elapsed:.0f}s] n_strands={n_strands} "
f"depth={max_depth} "
f"nodes={nodes_created} "
f"copr={coprimality_prunes} "
f"en={energy_prunes} "
f"dedup={dedup}"
)
sys.stdout.flush()
if max_depth > 500:
return {
'n_strands': n_strands,
'collapse_depth': max_depth,
'collapsed': False,
'reason': 'depth_limit',
'nodes_created': nodes_created,
'coprimality_prunes': coprimality_prunes,
'energy_prunes': energy_prunes,
'dedup': dedup,
'runtime_s': round(time.time() - start, 2),
'max_modulus': max(m for n in visited.values() for m in n.moduli),
}
# DFS exhausted — we've found the deepest reachable state
return {
'n_strands': n_strands,
'collapse_depth': max_depth,
'collapsed': True,
'reason': 'dfs_exhausted',
'nodes_created': nodes_created,
'coprimality_prunes': coprimality_prunes,
'energy_prunes': energy_prunes,
'dedup': dedup,
'runtime_s': round(time.time() - start, 2),
'max_modulus': max(m for n in visited.values() for m in n.moduli) if visited else 0,
}
results = []
for n_strands in [2, 3, 4, 5, 6, 7, 8]:
print(f"\n{'=' * 60}")
print(f"Strands: {n_strands}")
print(f"{'=' * 60}")
r = find_collapse_depth(n_strands)
results.append(r)
status = "COLLAPSED" if r['collapsed'] else "NO COLLAPSE"
print(f"\n → {status} at depth {r['collapse_depth']}")
print(f" nodes={r['nodes_created']}, runtime={r['runtime_s']}s")
print(f" copr={r['coprimality_prunes']}, en={r['energy_prunes']}")
print(f" max_mod={r['max_modulus']}")
print(f"\n{'=' * 60}")
print("FINAL RESULTS")
print(f"{'=' * 60}")
for r in results:
print(f" {r['n_strands']} strands: "
f"{'COLLAPSED' if r['collapsed'] else 'LIMIT'}"
f" @ depth {r['collapse_depth']} "
f"({r['nodes_created']} nodes, {r['runtime_s']}s)")

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#!/usr/bin/env python3
"""
DAG Deep Tuning: find optimal modulus selection strategies.
"""
import sys, math, random, json
from collections import Counter
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from verify_wrapping import f_k, is_sidon, sum_collisions, wrapping_criterion, certify_sidon_creation, m_difference_condition, wrapping_condition
random.seed(42)
# ---------- Failure Mode Analysis ----------
def analyze_failures(A, S, L1, L2):
"""Why does this moduli pair fail to create Sidon?"""
M = L1 * L2
guaranteed, FA, reason = certify_sidon_creation(A, S, [L1, L2])
collisions = sum_collisions(A)
wrap_ok, unresolved = wrapping_condition(A, S, [L1, L2])
mdiff_ok, violators = m_difference_condition(A, M)
failures = []
if not wrap_ok:
for (a,b,c,d),(s1,s2) in unresolved:
failures.append(f" Wrap fail: {a}+{b}={a+b} and {c}+{d}={c+d} both map to s1={s1}, s2={s2} (same wrap state)")
if not mdiff_ok:
for T1, T2, pairs1, pairs2 in violators:
failures.append(f" M-diff fail: sum {T1} (from {pairs1}) and {T2} (from {pairs2}) differ by M={M}")
return failures, FA
def detailed_modulus_report(A, S, max_mod=20):
"""Full report on every valid modulus pair."""
maxA = max(A)
rows = []
for L1 in range(2, max_mod + 1):
for L2 in range(L1 + 1, max_mod + 1):
if math.gcd(L1, L2) != 1: continue
M = L1 * L2
if not (maxA < M <= 2 * maxA): continue
guaranteed, FA, _ = certify_sidon_creation(A, S, [L1, L2])
sidon = is_sidon(FA)
fails, _ = analyze_failures(A, S, L1, L2)
rows.append({
'L1': L1, 'L2': L2, 'M': M,
'sidon': sidon, 'guaranteed': guaranteed,
'failures': fails, 'FA': FA
})
return rows
# ---------- Optimal M Strategy ----------
def analyze_optimal_M_trend(trials=500):
"""Trend: what M/maxA ratios work best?"""
results = []
for trial in range(trials):
maxA = random.randint(5, 30)
n = random.randint(4, 8)
A = sorted(random.sample(range(maxA + 1), min(n, maxA + 1)))
S = random.randint(maxA, maxA + 10)
max_mod = max(2, min(20, 2 * maxA))
for L1 in range(2, max_mod + 1):
for L2 in range(L1 + 1, max_mod + 1):
if math.gcd(L1, L2) != 1: continue
M = L1 * L2
if not (maxA < M <= 2 * maxA): continue
FA = [f_k(a, S, [L1, L2]) for a in A]
sidon = is_sidon(FA)
results.append({
'maxA': maxA, 'M': M,
'ratio': M / maxA,
'sidon': sidon
})
# Group by ratio buckets
buckets = {}
for r in results:
bucket = round(r['ratio'] * 10) / 10 # 0.1 increments
if bucket not in buckets:
buckets[bucket] = {'total': 0, 'sidon': 0}
buckets[bucket]['total'] += 1
if r['sidon']:
buckets[bucket]['sidon'] += 1
print("\n=== Optimal M/maxA Ratio Analysis ===")
print(f"{'Ratio':>8} {'Total':>8} {'Sidon':>8} {'Rate':>8}")
print("-" * 36)
for ratio in sorted(buckets.keys()):
b = buckets[ratio]
pct = b['sidon'] / b['total'] * 100
print(f"{ratio:>8.1f} {b['total']:>8} {b['sidon']:>8} {pct:>7.1f}%")
# Best ratio range
best = max(buckets.items(), key=lambda x: x[1]['sidon'] / x[1]['total'])
print(f"\nBest ratio: {best[0]:.1f} ({best[1]['sidon']/best[1]['total']*100:.1f}% success)")
# ---------- Modulus Size Preference ----------
def analyze_modulus_size_preference(trials=500):
"""Which modulus values work most often?"""
mod_counts = Counter()
mod_sidon = Counter()
for trial in range(trials):
maxA = random.randint(5, 30)
n = random.randint(4, 8)
A = sorted(random.sample(range(maxA + 1), min(n, maxA + 1)))
S = random.randint(maxA, maxA + 10)
max_mod = max(2, min(20, 2 * maxA))
for L1 in range(2, max_mod + 1):
for L2 in range(L1 + 1, max_mod + 1):
if math.gcd(L1, L2) != 1: continue
M = L1 * L2
if not (maxA < M <= 2 * maxA): continue
FA = [f_k(a, S, [L1, L2]) for a in A]
sidon = is_sidon(FA)
mod_counts[(L1, L2)] += 1
if sidon:
mod_sidon[(L1, L2)] += 1
print("\n=== Modulus Size Preference ===")
print(f"{'Moduli':>10} {'Trials':>8} {'Sidon':>8} {'Rate':>8}")
print("-" * 38)
sorted_mods = sorted(mod_counts.items(), key=lambda x: x[1], reverse=True)
for (L1, L2), count in sorted_mods[:15]:
sidon_count = mod_sidon.get((L1, L2), 0)
pct = sidon_count / count * 100
print(f"[{L1:>2},{L2:>2}] {count:>8} {sidon_count:>8} {pct:>7.1f}%")
# ---------- Multi-step Analysis ----------
def analyze_multi_step_needed(trials=300):
"""For sets that fail one-step, analyze multi-step depth."""
print("\n=== Multi-Step Analysis ===")
from iteration_dag import IterationDAG, AdaptiveRule
one_step_only = 0
multi_step = 0
no_path = 0
for trial in range(trials):
maxA = random.randint(5, 30)
n = random.randint(4, 7)
A = sorted(random.sample(range(maxA + 1), min(n, maxA + 1)))
S = random.randint(maxA, maxA + 10)
# Check one-step
mods = []
for L1 in range(2, 15):
for L2 in range(L1 + 1, 15):
if math.gcd(L1, L2) != 1: continue
M = L1 * L2
if not (maxA < M <= 2 * maxA): continue
FA = [f_k(a, S, [L1, L2]) for a in A]
if is_sidon(FA):
mods.append((L1, L2))
if mods:
one_step_only += 1
continue
# Check multi-step
rule = AdaptiveRule(max_val=12)
dag = IterationDAG(A, S, rule, max_steps=3, max_branch=30)
dag.build()
if dag.sidon_paths:
multi_step += 1
else:
no_path += 1
print(f" One-step success: {one_step_only}/{trials}")
print(f" Multi-step only: {multi_step}/{trials}")
print(f" No path found: {no_path}/{trials}")
# ---------- Main ----------
if __name__ == "__main__":
# Detailed failure analysis for known examples
print("=" * 60)
print("DEEP TUNING: FAILURE ANALYSIS")
print("=" * 60)
print("\n--- Sidon Example: A=[1,2,5,6], S=7 ---")
rows = detailed_modulus_report([1,2,5,6], 7)
for r in rows:
status = "✓ SIDON" if r['sidon'] else "✗ FAIL"
g = "guaranteed" if r['guaranteed'] else "not-guaranteed"
print(f" [{r['L1']},{r['L2']}] M={r['M']} {status} ({g})")
if r['failures']:
for f in r['failures'][:2]:
print(f" {f}")
print("\n--- Complex Set: A=[0,1,3,8,13], S=27 ---")
rows = detailed_modulus_report([0,1,3,8,13], 27)
for r in rows[:8]:
status = "✓ SIDON" if r['sidon'] else "✗ FAIL"
print(f" [{r['L1']},{r['L2']}] M={r['M']} {status}")
if r['failures']:
for f in r['failures'][:3]:
print(f" {f}")
analyze_optimal_M_trend(300)
analyze_modulus_size_preference(300)
analyze_multi_step_needed(200)

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#!/usr/bin/env python3
"""
DAG Tuning & Analysis: map iteration behavior, find optimal moduli.
"""
import sys, math, random, itertools, json
from collections import Counter
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from verify_wrapping import f_k, is_sidon, sum_collisions, wrapping_criterion, certify_sidon_creation
from iteration_dag import IterationDAG, AdaptiveRule, GeometricRule, DAGNode
# ---------- Modulus Effectiveness Analysis ----------
def test_all_moduli(A, S, max_modulus=20):
"""Test ALL coprime modulus pairs in the valid range, return effectiveness map."""
results = []
maxA = max(A)
for L1 in range(2, max_modulus + 1):
for L2 in range(L1 + 1, max_modulus + 1):
if math.gcd(L1, L2) != 1:
continue
M = L1 * L2
if not (maxA < M <= 2 * maxA):
continue
guaranteed, FA, reason = certify_sidon_creation(A, S, [L1, L2])
results.append({
'moduli': [L1, L2], 'M': M,
'sidon': is_sidon(FA),
'guaranteed': guaranteed,
'FA': FA,
'reason': reason
})
return results
def modulus_heatmap(A, S, max_modulus=20):
"""Generate a heatmap of modulus effectiveness."""
results = test_all_moduli(A, S, max_modulus)
if not results:
print(" No valid moduli in range")
return {}
sidon_count = sum(1 for r in results if r['sidon'])
guaranteed_count = sum(1 for r in results if r['guaranteed'])
# Best moduli by Sidon creation
sidon_mods = [r for r in results if r['sidon']]
stats = {
'total_moduli': len(results),
'sidon_success': sidon_count,
'guaranteed_sidon': guaranteed_count,
'success_rate': sidon_count / max(len(results), 1),
'guarantee_rate': guaranteed_count / max(len(results), 1),
'best_moduli': sidon_mods[:10] if len(sidon_mods) <= 10 else sidon_mods[:10],
'worst_moduli': [r for r in results if not r['sidon']][:5]
}
return stats
# ---------- DAG Depth Analysis ----------
def depth_distribution(A0, S0, max_steps=6):
"""Analyze the distribution of path lengths to Sidon."""
rule = AdaptiveRule(max_val=16)
dag = IterationDAG(A0, S0, rule, max_steps=max_steps)
dag.build()
path_lengths = []
for path in dag.sidon_paths:
path_lengths.append(len(path) - 1) # steps, not nodes
return {
'total_nodes': len(dag.all_nodes),
'sidon_paths': len(dag.sidon_paths),
'path_lengths': dict(Counter(path_lengths)),
'min_steps': min(path_lengths) if path_lengths else None,
'max_steps': max(path_lengths) if path_lengths else None,
'avg_steps': sum(path_lengths) / len(path_lengths) if path_lengths else None
}
# ---------- Parameter Sweep ----------
def sweep_parameter(target_property="sidon", trials=200, max_modulus=16, max_steps=4):
"""Sweep across random A sets and find optimal tuning strategies."""
random.seed(42)
primes = [2,3,5,7,11,13,17,19,23,29,31,37]
results = []
for trial in range(trials):
# Generate random A
maxA = random.randint(5, 30)
n = random.randint(4, 8)
A = sorted(random.sample(range(maxA + 1), min(n, maxA + 1)))
S = random.randint(maxA, maxA + 10)
# Test single-step: find modulus pairs that produce Sidon in one step
mod_results = test_all_moduli(A, S, max_modulus)
one_step_sidon = sum(1 for r in mod_results if r['sidon'])
# Test DAG: find multi-step paths to Sidon
rule = AdaptiveRule(max_val=max_modulus)
dag = IterationDAG(A, S, rule, max_steps=max_steps)
dag.build()
multi_step = len(dag.sidon_paths)
# Find the smallest M that works
min_sidon_M = None
for r in mod_results:
if r['sidon']:
if min_sidon_M is None or r['M'] < min_sidon_M:
min_sidon_M = r['M']
results.append({
'A': A, 'S': S, 'n': len(A), 'maxA': maxA,
'one_step_candidates': one_step_sidon,
'one_step_total': len(mod_results),
'one_step_rate': one_step_sidon / max(len(mod_results), 1),
'multi_step_paths': multi_step,
'min_sidon_M': min_sidon_M
})
return results
# ---------- Analysis Reports ----------
def report_sidon_example():
"""Detailed analysis of the known Sidon creation example."""
A, S = [1, 2, 5, 6], 7
print("=" * 60)
print(f"SIDON EXAMPLE ANALYSIS: A={A}, S={S}")
print("=" * 60)
stats = modulus_heatmap(A, S)
print(f"\nModulus Analysis ({stats['total_moduli']} coprime pairs in range):")
print(f" Sidon creation success: {stats['sidon_success']}/{stats['total_moduli']} ({stats['success_rate']*100:.1f}%)")
print(f" Guaranteed Sidon: {stats['guaranteed_sidon']}/{stats['total_moduli']} ({stats['guarantee_rate']*100:.1f}%)")
print(f" Best moduli (first 10 Sidon-creating pairs):")
for r in stats['best_moduli']:
print(f" [{r['moduli'][0]}, {r['moduli'][1]}] M={r['M']} FA={r['FA']}")
def report_complex_set():
"""Quick analysis of a more complex set — moduli only, no DAG."""
A, S = [0, 1, 3, 8, 13], 27
print("\n" + "=" * 60)
print(f"COMPLEX SET: A={A}, S={S}")
print("=" * 60)
stats = modulus_heatmap(A, S, max_modulus=16)
print(f"\nModulus Analysis ({stats['total_moduli']} coprime pairs in range):")
pct = stats['success_rate'] * 100
print(f" Sidon creation: {stats['sidon_success']}/{stats['total_moduli']} ({pct:.1f}%)")
print(f" Guaranteed: {stats['guaranteed_sidon']}/{stats['total_moduli']} ({stats['guarantee_rate']*100:.1f}%)")
for r in stats['best_moduli'][:5]:
print(f" [{r['moduli'][0]}, {r['moduli'][1]}] M={r['M']} FA={r['FA']}")
def report_sweep():
"""Fast sweep — moduli only, no DAG building."""
print("\n" + "=" * 60)
print("PARAMETER SWEEP (200 random sets — modulus-only)")
print("=" * 60)
random.seed(42)
results = []
for trial in range(200):
maxA = random.randint(5, 30)
n = random.randint(4, 8)
A = sorted(random.sample(range(maxA + 1), min(n, maxA + 1)))
S = random.randint(maxA, maxA + 10)
mod_results = test_all_moduli(A, S, max_modulus=12)
one_step_sidon = sum(1 for r in mod_results if r['sidon'])
guaranteed = sum(1 for r in mod_results if r['guaranteed'])
results.append({
'n': len(A), 'maxA': maxA,
'one_step_sidon': one_step_sidon,
'total_moduli': len(mod_results),
'guaranteed': guaranteed,
'success_rate': one_step_sidon / max(len(mod_results), 1) if mod_results else 0,
})
sr = [r['success_rate'] for r in results]
print(f"\nResults ({len(results)} sets):")
print(f" Sets with >0 valid moduli: {sum(1 for r in results if r['total_moduli'] > 0)}/{len(results)}")
print(f" Sets with at least one Sidon-creating modulus: {sum(1 for r in results if r['one_step_sidon'] > 0)}/{len(results)}")
print(f" Avg success rate: {sum(sr)/len(sr)*100:.1f}%")
print(f" Best success rate: {max(sr)*100:.1f}%")
# ---------- Main ----------
if __name__ == "__main__":
report_sidon_example()
report_complex_set()
report_sweep()

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#!/usr/bin/env python3
"""
Deep Braid Exploration: YB modulo space, braid invariants, stabilization.
"""
import sys, math, itertools, random
from typing import List, Tuple
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from multi_strand_braid import pairwise_coprime, moduli_from_pairs, F_multi, braid_word, cross
from verify_wrapping import is_sidon
# ============================================================
# 1. YB MODULO SPACE SEARCH
# ============================================================
def search_yb_space(max_mod: int = 100) -> List[dict]:
"""Find ALL 4-tuples where the full YB relation holds with equal FA."""
results = []
# Precompute coprime pairs
coprime_pairs_list = []
for a in range(2, max_mod+1):
for b in range(a+1, max_mod+1):
if math.gcd(a, b) == 1:
coprime_pairs_list.append((a, b))
total = len(coprime_pairs_list)
for idx1, (a, b) in enumerate(coprime_pairs_list):
if idx1 % 100 == 0:
sys.stdout.write(f"\r Searching YB space: {idx1}/{total} pairs...")
sys.stdout.flush()
for idx2 in range(idx1+1, total):
c, d = coprime_pairs_list[idx2]
if not pairwise_coprime([a, b, c, d]):
continue
# Path 1: σ₁⁺ → σ₂⁻ → σ₁⁺
s1 = cross([(a,b),(c,d)], 0, True)
if not s1: continue
s1s2 = cross(s1, 1, False)
if not s1s2: continue
s1s2s1 = cross(s1s2, 0, True)
if not s1s2s1: continue
# Path 2: σ₂⁻ → σ₁⁺ → σ₂⁻
s2 = cross([(a,b),(c,d)], 1, False)
if not s2: continue
s2s1 = cross(s2, 0, True)
if not s2s1: continue
s2s1s2 = cross(s2s1, 1, False)
if not s2s1s2: continue
# Check equal FA
A0, S = [1, 2, 5, 6], 7
f1 = [F_multi(x, S, moduli_from_pairs(s1s2s1)) for x in A0]
f2 = [F_multi(x, S, moduli_from_pairs(s2s1s2)) for x in A0]
if f1 == f2:
results.append({
'init': [(a,b),(c,d)],
'final': s1s2s1,
'path1_word': 'σ₁⁺·σ₂⁻·σ₁⁺',
'path2_word': 'σ₂⁻·σ₁⁺·σ₂⁻',
'FA': f1,
'M': a*b*c*d
})
print()
return results
def test_yb_search():
print("=" * 60)
print("YB MODULO SPACE SEARCH")
print("=" * 60)
results = search_yb_space(max_mod=200)
print(f"\nTotal YB-valid 4-tuples found: {len(results)}")
if results:
print(f"\nSmallest by M:")
for r in sorted(results, key=lambda x: x['M'])[:5]:
print(f" {r['init']} M={r['M']:6d} FA={r['FA']}")
print(f"\nLargest by M:")
for r in sorted(results, key=lambda x: -x['M'])[:3]:
print(f" {r['init']} M={r['M']:8d} FA={r['FA']}")
# ============================================================
# 2. BRAID INVARIANTS FROM M-DIFFERENCES
# ============================================================
def m_difference_spectrum(A, moduli):
"""Compute the M-difference spectrum of a braid configuration."""
M = 1
for m in moduli: M *= m
maxA = max(A)
if M <= maxA:
return {'regime': 'aliasing', 'M': M, 'sums': []}
# Compute all pairwise sums
sums = set()
for i in range(len(A)):
for j in range(i, len(A)):
sums.add(A[i] + A[j])
sum_list = sorted(sums)
# Find M-differences
diffs = []
for i in range(len(sum_list)):
for j in range(i+1, len(sum_list)):
d = sum_list[j] - sum_list[i]
if d > 0 and d % M == 0:
diffs.append((sum_list[i], sum_list[j], d // M))
return {'regime': 'injective' if M > maxA else 'aliasing', 'M': M, 'sums': sum_list, 'diffs': diffs}
def braid_invariant_from_mdiff(A, S, pairs_seq):
"""Compute braid invariant: the M-difference spectrum through a braid path."""
invariants = []
for pairs in pairs_seq:
mods = moduli_from_pairs(pairs)
spec = m_difference_spectrum(A, mods)
invariants.append({
'pairs': pairs,
'M': spec['M'],
'regime': spec['regime'],
'num_diffs': len(spec.get('diffs', [])),
'diffs': spec.get('diffs', [])
})
return invariants
def test_invariants():
print("\n" + "=" * 60)
print("BRAID INVARIANTS FROM M-DIFFERENCES")
print("=" * 60)
A0, S = [1, 2, 5, 6], 7
# Trace a braid path and compute invariants
configs = [[(3,4)], [(7,3)], [(11,2)]]
for config in configs:
mods = moduli_from_pairs(config)
FA = [F_multi(a, S, mods) for a in A0]
M = 1
for m in mods: M *= m
spec = m_difference_spectrum(A0, mods)
sidon = is_sidon(FA)
pairs_str = str(config[0])
print(f" {pairs_str:12s} M={M:3d} Sidon={sidon} sums={len(spec.get('sums',[]))} diffs={len(spec.get('diffs',[]))}")
# ============================================================
# 3. MODULUS REGENERATION AS BRAID STABILIZATION
# ============================================================
def markov_stabilization(pairs, strand=0):
"""A Markov stabilization move: add a trivial crossing to extend word length.
Stabilization: add (L_new_id, L_new_ref) as a new strand, with values
coprime to all existing moduli.
"""
existing = moduli_from_pairs(pairs)
existing_vals = set(existing)
# Find a coprime pair not in existing values
L_new_id = 2
while L_new_id in existing_vals: L_new_id += 1
L_new_ref = L_new_id + 1
while L_new_ref in existing_vals or math.gcd(L_new_id, L_new_ref) != 1:
L_new_ref += 1
new_pair = (L_new_id, L_new_ref)
all_mods = existing + [L_new_id, L_new_ref]
if pairwise_coprime(all_mods):
return pairs + [new_pair], f"stabilized with {new_pair}"
return pairs, "stabilization failed"
def test_stabilization():
print("\n" + "=" * 60)
print("MODULUS REGENERATION AS BRAID STABILIZATION")
print("=" * 60)
# Start with a 1-strand system, cross until coprimality fails
pairs = [(3, 4)]
history = [pairs]
for step in range(10):
# Try over-crossing
c = cross(pairs, 0, True)
if c is None:
# Stabilize: add a new strand with coprime moduli
pairs, msg = markov_stabilization(pairs)
print(f" Step {step}: coprimality failed → {msg}")
if pairs == history[-1]:
print(f" Cannot stabilize further. Stopping.")
break
else:
pairs = c
history.append(pairs)
if len(history) <= 6:
print(f" Step {step}: pairs={pairs}")
print(f"\n Total steps before stabilization needed: {len(history)-1}")
print(f" Final config: {pairs}")
# ============================================================
# MAIN
# ============================================================
if __name__ == "__main__":
test_yb_search()
test_invariants()
test_stabilization()

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"""full_chiral_dag.py — CRT Torus Braid DAG with Coprimality Guard.
Combines axis-swap (topology, YB ✓) and adjustment (resource, FA-changing)
into a unified DAG traversal. The Coprimality Guard ensures all 16 moduli
remain pairwise coprime after every crossing.
References:
- docs/crt-torus-embedding.md (core CRT Torus embedding)
- docs/research/unified_crt_torus_dag.md (graded Sidon energy + hierarchy)
- docs/research/braid_group_action.md (dual-model framework)
"""
import math
from typing import List, Optional, Tuple
# ═══════════════════════════════════════════════════════════════
# §1 SIDON CHECK VIA WRAPPING CRITERION
# ═══════════════════════════════════════════════════════════════
# The Sidon creation theorem (docs/research/sidon_preservation_creation.md):
# For A0 ⊆ ℤ and CRT embedding F, F(A0) is Sidon iff for EVERY sum
# collision a+b = c+d in A0, the CRT lifts wrap M differently:
# F(a)+F(b) = T + r₁·M, F(c)+F(d) = T + r₂·M, r₁ ≠ r₂
# where wrap indicator r = 1 if F(x)+F(y) ≥ M, else 0.
#
# With L_id-only adjustment and M > 2·max(A0), new collisions cannot
# form (M-difference condition is vacuous). The only question is
# whether the existing collisions break.
def crt_sum(a: int, b: int, pairs: List[Tuple[int, int]], S: int) -> Tuple[int, int]:
"""Compute F(a)+F(b) and its wrap indicator.
Returns (sum, wrap) where wrap = 1 if sum ≥ M, 0 otherwise.
"""
Fa = crt_embed(a, pairs, S)
Fb = crt_embed(b, pairs, S)
total = Fa + Fb
M = math.prod(m for pair in pairs for m in pair)
return (total, 1 if total >= M else 0)
def find_collisions(A0: List[int]) -> List[Tuple[int, int, int, int]]:
"""Find all sum collisions in A0.
Returns list of ((a,b), (c,d), T) where a+b = c+d = T and (a,b) ≠ (c,d).
"""
n = len(A0)
sum_map = {}
collisions = []
for i in range(n):
for j in range(i, n):
s = A0[i] + A0[j]
if s in sum_map:
ci, cj = sum_map[s]
if ci != i or cj != j:
collisions.append((A0[ci], A0[cj], A0[i], A0[j], s))
else:
sum_map[s] = (i, j)
return collisions
def sidon_check(
A0: List[int],
pairs: List[Tuple[int, int]],
S: int,
) -> Tuple[bool, int, float]:
"""Check if F(A0) is Sidon under current moduli.
Returns (is_sidon, broken_count, score).
- is_sidon: True if all collisions broken
- broken_count: how many collisions are broken
- score: 0 if Sidon, else graded residual (lower = closer to Sidon)
"""
collisions = find_collisions(A0)
if not collisions:
return (True, 0, 0.0)
broken = 0
for a, b, c, d, T in collisions:
_, wrap1 = crt_sum(a, b, pairs, S)
_, wrap2 = crt_sum(c, d, pairs, S)
if wrap1 != wrap2:
broken += 1
if broken == len(collisions):
return (True, broken, 0.0)
# Score: fraction of unbroken collisions, scaled to (0, 4].
total = len(collisions)
score = 4.0 * (1.0 - broken / total)
return (False, broken, max(0.0, score))
def sidon_energy(
A0: List[int],
pairs: List[Tuple[int, int]],
S: int,
) -> float:
"""Graded Sidon energy: 0 if Sidon, else ℰ ∈ (0, 4] for non-Sidon."""
_, _, score = sidon_check(A0, pairs, S)
return score
# ═══════════════════════════════════════════════════════════════
# §2 CRT EMBEDDING
# ═══════════════════════════════════════════════════════════════
def crt_embed(
a: int,
pairs: List[Tuple[int, int]],
S: int,
) -> int:
"""CRT Torus Embedding F: ℤ → ℤ/Mℤ.
Axis 1: a ↦ a mod L₁ (identity)
Axes 2…k: a ↦ S − a mod Lᵢ (reflection)
Reconstructs via CRT to produce a unique integer lift in [0, M).
"""
residues = []
moduli = []
for i, (L_id, L_ref) in enumerate(pairs):
moduli.append(L_id)
if i == 0:
residues.append(a % L_id)
else:
residues.append((S - a) % L_id)
moduli.append(L_ref)
residues.append((S - a) % L_ref)
# Iterative CRT
x = residues[0]
M = moduli[0]
for i in range(1, len(moduli)):
m_i = moduli[i]
r_i = residues[i]
# Find k such that x + k·M ≡ r_i (mod m_i)
# k ≡ (r_i − x) · M⁻¹ (mod m_i)
inv = pow(M, -1, m_i)
k = ((r_i - x) * inv) % m_i
x = x + k * M
M = M * m_i
return x
def crt_embed_set(
A: List[int],
pairs: List[Tuple[int, int]],
S: int,
) -> List[int]:
"""Apply CRT Torus Embedding F to every element of A."""
return sorted([crt_embed(a, pairs, S) for a in A])
# ═══════════════════════════════════════════════════════════════
# §3 COPRIMALITY GUARD
# ═══════════════════════════════════════════════════════════════
def pairwise_coprime(moduli: List[int]) -> bool:
"""Coprimality Guard: check all moduli are pairwise coprime.
Returns True iff gcd(m_i, m_j) = 1 for all i ≠ j.
This is the CRITICAL invariant: CRT requires pairwise coprime moduli
to guarantee injectivity of the torus embedding F.
Failure mode: adjusting a modulus by ±2 can make it share a factor
with another modulus (e.g., one hits 7, another was already 14).
The guard catches this before it corrupts the node.
"""
n = len(moduli)
for i in range(n):
for j in range(i + 1, n):
if math.gcd(moduli[i], moduli[j]) != 1:
return False
return True
# ═══════════════════════════════════════════════════════════════
# §3 CHIRAL PAIRS — INITIALIZATION
# ═══════════════════════════════════════════════════════════════
def _nth_prime(n: int) -> int:
"""Return the n-th prime (0-indexed), generating on the fly."""
known = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113]
while len(known) <= n:
candidate = known[-1] + 2
while any(candidate % p == 0 for p in known):
candidate += 2
known.append(candidate)
return known[n]
def chiral_pairs(
n_strands: int = 8,
band_gap: int = 30,
base_prime_offset: int = 0,
) -> List[Tuple[int, int]]:
"""Initialize chiral pairs with distinct primes.
Each strand gets an (L_id, L_ref) pair where both are prime.
All 2·n_strands moduli are pairwise coprime by construction.
With L_id-only adjustment (L_ref fixed), the spacing between
L_id and L_ref doesn't restrict capacity — only the Q16_16
bound (32767) and L_id > 1 matter.
Args:
n_strands: number of braid strands
band_gap: (unused with L_id-only adjustment, kept for API compat)
base_prime_offset: starting index into prime sequence
Returns:
List of (L_id, L_ref) pairs, one per strand
"""
pairs = []
idx = base_prime_offset
for _ in range(n_strands):
L_id = _nth_prime(idx); idx += 1
L_ref = _nth_prime(idx); idx += 1
pairs.append((L_id, L_ref))
return pairs
# ═══════════════════════════════════════════════════════════════
# §4 AXIS-SWAP (TOPOLOGY, YB-COMPLIANT)
# ═══════════════════════════════════════════════════════════════
def axis_swap(
pairs: List[Tuple[int, int]],
s: int,
) -> List[Tuple[int, int]]:
"""Swap reflection moduli of adjacent strands s and s+1.
This is the braid generator σ_s acting on the reflection axis only.
The identity moduli are untouched. This is FA-invariant (CRT symmetry)
and satisfies YB, σ²=id, and far commutativity.
Args:
pairs: current list of (L_id, L_ref) per strand
s: strand index (0 ≤ s < len(pairs) − 1)
Returns a NEW list with the reflection moduli swapped.
"""
if s < 0 or s >= len(pairs) - 1:
return pairs[:]
new_pairs = list(pairs)
L_id_s, L_ref_s = new_pairs[s]
L_id_s1, L_ref_s1 = new_pairs[s + 1]
new_pairs[s] = (L_id_s, L_ref_s1)
new_pairs[s + 1] = (L_id_s1, L_ref_s)
return new_pairs
# ═══════════════════════════════════════════════════════════════
# §5 ADJUSTMENT (RESOURCE, FA-CHANGING)
# ═══════════════════════════════════════════════════════════════
def adjust(
pairs: List[Tuple[int, int]],
s: int,
direction: str,
) -> Optional[List[Tuple[int, int]]]:
"""Adjust modulus values for strand s — L_id only.
Design finding from Coprimality Guard (full_chiral_dag.py §3):
The original ±2/∓1 adjustment on BOTH moduli breaks within-pair
coprimality after ≤1 crossing (e.g., (13,41)→(15,40) shares
factor 5). Fix: adjust only L_id, keeping L_ref fixed at its
initial prime. This guarantees within-pair coprimality since
gcd(L_id ± 2k, L_ref) = 1 when L_ref is a distinct prime and
doesn't divide the adjusted L_id.
Over: L_id += 2
Under: L_id −= 2
Returns a new list of pairs, or None if:
- New modulus ≤ 1 (invalid for CRT)
- Fails the Coprimality Guard (shares factor with another modulus)
"""
if direction not in ('over', 'under'):
raise ValueError(f"Invalid direction: {direction}")
new_pairs = [(a, b) for a, b in pairs]
L_id, L_ref = new_pairs[s]
if direction == 'over':
new_id = L_id + 2
else:
new_id = L_id - 2
if new_id <= 1:
return None # modulus invalid
new_pairs[s] = (new_id, L_ref)
moduli = [m for pair in new_pairs for m in pair]
if not pairwise_coprime(moduli):
return None
return new_pairs
# ═══════════════════════════════════════════════════════════════
# §6 COUPLED CROSSING
# ═══════════════════════════════════════════════════════════════
def coupled_crossing(
pairs: List[Tuple[int, int]],
s: int,
direction: str,
A0: List[int],
S: int,
) -> Optional[Tuple[List[Tuple[int, int]], float]]:
"""One coupled crossing: axis-swap → adjust → verify.
Returns (new_pairs, new_energy) if successful, None if coprimality fails.
"""
swapped = axis_swap(pairs, s)
adjusted = adjust(swapped, s, direction)
if adjusted is None:
return None
E_new = sidon_energy(A0, adjusted, S)
if not pairwise_coprime([m for pair in adjusted for m in pair]):
return None
return (adjusted, E_new)
# ═══════════════════════════════════════════════════════════════
# §7 DIRECTIONAL CAPACITY
# ═══════════════════════════════════════════════════════════════
def compute_directional_capacities(
pairs: List[Tuple[int, int]],
max_across: int = 15,
) -> List[int]:
"""Compute directional capacities per strand, packed into 4-bit word.
L_id-only adjustment: 2 directions (over/under).
Bits 0-1: cap_over (over-crossings, limited by Q16_16 bound 32767)
Bits 2-3: cap_under (under-crossings, limited by L_id > 1)
Each capacity capped at 3 (2-bit range).
Args:
pairs: current chiral pairs
max_across: maximum crossings used for normalization
Returns:
Packed capacities per strand, as list of ints
"""
Q16_BOUND = 32767
capacities = []
for L_id, L_ref in pairs:
cap_over = min(3, (Q16_BOUND - L_id) // 2)
cap_under = min(3, (L_id - 3) // 2) if L_id > 3 else 0
packed = cap_over | (cap_under << 2)
capacities.append(packed)
return capacities
def dag_capacity(capacities: List[int], direction: str) -> int:
"""DAG-level capacity: min of strand capacities in this direction."""
shift = 0 if direction == 'over' else 2
vals = [(c >> shift) & 3 for c in capacities]
return min(vals)
# ═══════════════════════════════════════════════════════════════
# §8 DAG NODE
# ═══════════════════════════════════════════════════════════════
class DAGNode:
"""A node in the CRT torus braid DAG.
Attributes:
pairs: chiral pairs (L_id, L_ref) per strand
A: current integer set (CRT lifts)
M: product of all moduli
energy: SidonEnergy ℰ of this state
capacities: 4-directional capacities (8-bit per strand)
braid_word: list of (strand, direction) crossings from root
depth: number of crossings from root
children: child node references (by moduli hash)
"""
__slots__ = (
'pairs', 'A0', 'S', 'M', 'energy', 'capacities',
'braid_word', 'depth', 'children', 'is_sidon', 'broken',
)
def __init__(
self,
pairs: List[Tuple[int, int]],
A0: List[int],
S: int,
braid_word: Optional[List[Tuple[int, str]]] = None,
depth: int = 0,
):
self.pairs = pairs
self.A0 = A0
self.S = S
self.M = math.prod(m for pair in pairs for m in pair)
sidon_ok, self.broken, self.energy = sidon_check(A0, pairs, S)
self.is_sidon = sidon_ok
self.capacities = compute_directional_capacities(pairs)
self.braid_word = braid_word or []
self.depth = depth
self.children = []
@property
def moduli(self) -> List[int]:
return [m for pair in self.pairs for m in pair]
def modulus_hash(self) -> int:
h = 0
for m in self.moduli:
h = h * 31 + m
return h
def __repr__(self) -> str:
return (
f"DAGNode(depth={self.depth}, M={self.M}, "
f"ℰ={self.energy:.4f}, "
f"braid={self.braid_word})"
)
# ═══════════════════════════════════════════════════════════════
# §9 CHIRAL DAG TRAVERSAL
# ═══════════════════════════════════════════════════════════════
EPSILON = 1e-9
class ChiralDAG:
"""CRT Torus Braid DAG with unified axis-swap × adjustment traversal.
Usage:
dag = ChiralDAG(A0=[1, 2, 5, 6], S=7, n_strands=3)
dag.build(max_steps=8)
print(dag.summary())
"""
def __init__(
self,
A0: List[int],
S: int,
n_strands: int = 8,
band_gap: int = 60,
):
self.A0 = sorted(A0)
self.S = S
self.n_strands = n_strands
self.band_gap = band_gap
self.root: Optional[DAGNode] = None
self.visited: dict = {}
self.stats = {
'nodes_created': 0,
'sidon_nodes': 0,
'pruned_coprimality': 0,
'pruned_energy': 0,
'pruned_exhausted': 0,
'deduped': 0,
'phases': [0, 0, 0],
}
def _make_root(self) -> DAGNode:
pairs = chiral_pairs(
n_strands=self.n_strands,
band_gap=self.band_gap,
base_prime_offset=10,
)
node = DAGNode(pairs, self.A0, self.S, depth=0)
self.visited[node.modulus_hash()] = node
self.stats['nodes_created'] += 1
return node
def _maybe_prune(
self,
parent: DAGNode,
child: DAGNode,
) -> bool:
"""Check if child should be pruned. Returns True if pruned."""
# 1. Coprimality invariant: already checked in coupled_crossing,
# but re-check for safety.
moduli = child.moduli
if not pairwise_coprime(moduli):
self.stats['pruned_coprimality'] += 1
return True
# 2. Monotonicity: SidonEnergy must not increase
if child.energy > parent.energy + EPSILON:
self.stats['pruned_energy'] += 1
return True
# 3. Sidon reached: accept but don't expand further
if child.is_sidon:
self.stats['sidon_nodes'] += 1
return False # accept, mark as terminal
# 4. Capacity exhaustion: DAG-level check
for d in ['over', 'under']:
if dag_capacity(child.capacities, d) <= 0:
self.stats['pruned_exhausted'] += 1
return True
return False
def build(
self,
max_steps: int = 30,
max_nodes: int = 10000,
use_axis_swap: bool = True,
use_adjustment: bool = True,
) -> None:
"""Build the DAG using BFS with three-phase traversal.
Phase 1: Graded Sidon search (small bands)
Phase 2: DAG topology expansion (wide bands)
Phase 3: Content-addressable dedup (hash-based)
"""
self.root = self._make_root()
queue = [self.root]
self.stats['phases'][0] += 1
while queue and self.stats['nodes_created'] < max_nodes:
node = queue.pop(0)
if node.depth >= max_steps:
continue
# Phase transition: when energy is low, widen bands
if node.energy < 0.5 and self.stats['phases'][1] == 0:
self.stats['phases'][1] = 1
self.band_gap = 500
if node.is_sidon:
continue # terminal
for s in range(self.n_strands):
for direction in ['over', 'under']:
# DAG-level capacity check (fast prune)
if dag_capacity(node.capacities, direction) <= 0:
self.stats['pruned_exhausted'] += 1
continue
result = None
if use_axis_swap and use_adjustment:
result = coupled_crossing(
node.pairs, s, direction, self.A0, self.S,
)
elif use_axis_swap:
new_pairs = axis_swap(node.pairs, s)
new_E = sidon_energy(self.A0, new_pairs, self.S)
if pairwise_coprime([m for pair in new_pairs for m in pair]):
result = (new_pairs, new_E)
elif use_adjustment:
new_pairs = adjust(node.pairs, s, direction)
if new_pairs is not None:
new_E = sidon_energy(self.A0, new_pairs, self.S)
result = (new_pairs, new_E)
else:
self.stats['pruned_coprimality'] += 1
else:
continue
if result is None:
self.stats['pruned_coprimality'] += 1
continue
new_pairs, new_energy = result
child = DAGNode(
pairs=new_pairs,
A0=self.A0,
S=self.S,
braid_word=node.braid_word + [(s, direction)],
depth=node.depth + 1,
)
child.energy = new_energy
# Pruning gates
if self._maybe_prune(node, child):
continue
# Content-addressable dedup
h = child.modulus_hash()
if h in self.visited:
existing = self.visited[h]
if existing.energy <= child.energy:
self.stats['deduped'] += 1
node.children.append(existing)
continue
self.visited[h] = child
self.stats['nodes_created'] += 1
node.children.append(child)
queue.append(child)
self.stats['phases'][2] = 1
def summary(self) -> str:
"""Return a text summary of the DAG build."""
sidon_nodes = [
n for n in self.visited.values()
if n.is_sidon
]
if sidon_nodes:
shortest = min(sidon_nodes, key=lambda n: n.depth)
sidon_str = (
f"Sidon paths found: {len(sidon_nodes)}\n"
f"Shortest path: depth={shortest.depth}, "
f"braid={shortest.braid_word}, "
f"ℰ={shortest.energy:.4f}\n"
f"Final moduli: {shortest.moduli}"
)
else:
sidon_str = "No Sidon paths found."
return (
f"── ChiralDAG Summary ──\n"
f"Strands: {self.n_strands}, Band gap: {self.band_gap}\n"
f"Nodes created: {self.stats['nodes_created']}\n"
f"Deduped: {self.stats['deduped']}\n"
f"Pruned — coprimality: {self.stats['pruned_coprimality']}\n"
f"Pruned — energy: {self.stats['pruned_energy']}\n"
f"Pruned — exhausted: {self.stats['pruned_exhausted']}\n"
f"Phases: {self.stats['phases']}\n"
f"Sidon nodes: {self.stats['sidon_nodes']}\n"
f"{sidon_str}"
)
def to_json(self, path: str) -> None:
"""Export DAG to JSON for visualization."""
import json
def _node_to_dict(n: DAGNode) -> dict:
return {
'depth': n.depth,
'pairs': n.pairs,
'moduli': n.moduli,
'M': n.M,
'energy': round(n.energy, 6),
'is_sidon': n.is_sidon,
'braid_word': n.braid_word,
'children': [
c.modulus_hash() for c in n.children
],
}
data = {
'n_strands': self.n_strands,
'band_gap': self.band_gap,
'A0': self.A0,
'stats': self.stats,
'nodes': {str(h): _node_to_dict(n)
for h, n in self.visited.items()},
}
with open(path, 'w') as f:
json.dump(data, f, indent=2)
# ═══════════════════════════════════════════════════════════════
# §10 MAIN / SELF-TEST
# ═══════════════════════════════════════════════════════════════
if __name__ == '__main__':
# Working test case (collision 3+13=8+8=16 is breakable via CRT wrapping)
A0 = [0, 1, 3, 8, 13]
S = 27
print("=== 2-strand test ===")
dag = ChiralDAG(A0=A0, S=S, n_strands=2)
dag.build(max_steps=15, max_nodes=500)
print(dag.summary())
print()
print("=== 8-strand test ===")
dag8 = ChiralDAG(A0=A0, S=S, n_strands=8)
dag8.build(max_steps=15, max_nodes=5000)
print(dag8.summary())
print()
# Q16_16 bound check
all_mods = [m for n in dag8.visited.values() for m in n.moduli]
max_m = max(all_mods) if all_mods else 0
print(f"Max modulus (8-strand): {max_m} {'✓' if max_m < 32767 else '✗ > 32767!'}")
# Depth distribution of Sidon paths
sidon_nodes = [n for n in dag8.visited.values() if n.is_sidon]
if sidon_nodes:
depths = {}
for n in sidon_nodes:
depths[n.depth] = depths.get(n.depth, 0) + 1
print(f"Sidon depth distribution: {dict(sorted(depths.items()))}")

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#!/usr/bin/env python3
"""Generate modulus heatmap visualization data (JSON)."""
import sys, math, json
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from verify_wrapping import f_k, is_sidon, certify_sidon_creation
def generate_heatmap_data(A, S, max_mod=16):
"""Generate heatmap of all coprime pairs in [2,max_mod]."""
maxA = max(A)
data = {'A': A, 'S': S, 'maxA': maxA, 'cells': []}
for L1 in range(2, max_mod + 1):
for L2 in range(2, max_mod + 1):
if L1 == L2: continue
if math.gcd(L1, L2) != 1: continue
M = L1 * L2
if not (maxA < M <= 2 * maxA):
# Still record but mark as out-of-range
status = "out_of_range"
else:
FA = [f_k(a, S, [L1, L2]) for a in A]
sidon = is_sidon(FA)
_, _, reason = certify_sidon_creation(A, S, [L1, L2])
status = "sidon" if sidon else "fail"
data['cells'].append({
'L1': L1, 'L2': L2, 'M': M,
'status': status
})
return data
# Known examples
examples = [
([1,2,5,6], 7, "Sidon example"),
([0,1,3,8,13], 27, "Complex set"),
]
for A, S, name in examples:
data = generate_heatmap_data(A, S)
with open(f'/home/allaun/SilverSight/docs/diagrams/heatmap_{name.replace(" ","_")}.json', 'w') as f:
json.dump(data, f, indent=2)
sidon_count = sum(1 for c in data['cells'] if c['status'] == 'sidon')
fail_count = sum(1 for c in data['cells'] if c['status'] == 'fail')
out_count = sum(1 for c in data['cells'] if c['status'] == 'out_of_range')
print(f"{name}: {sidon_count} sidon, {fail_count} fail, {out_count} out-of-range")

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#!/usr/bin/env python3
"""
Iteration DAG for CRT Torus Embedding.
Traces paths through modulus space, searching for a sequence of
modulus choices that transforms A into a Sidon set.
"""
import sys, math, random, itertools
from typing import List, Tuple, Optional, Dict, Set
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from verify_wrapping import *
# ---------- DAG Node ----------
class DAGNode:
__slots__ = ('step', 'A', 'moduli', 'S', 'M', 'sidon', 'parent', 'children',
'terminal', 'reason', 'id')
_next_id = 0
def __init__(self, A, moduli, S, parent=None, step=0):
self.id = DAGNode._next_id; DAGNode._next_id += 1
self.step = step
self.A = sorted(A)
self.moduli = list(moduli)
self.S = S
self.M = 1
for Li in moduli: self.M *= Li
self.sidon = is_sidon(self.A)
self.parent = parent
self.children = []
self.terminal = False
self.reason = ""
def key(self):
return (tuple(self.A), tuple(self.moduli), self.S)
def __repr__(self):
return f"Node#{self.id}(step={self.step}, |A|={len(self.A)}, M={self.M}, sidon={self.sidon})"
# ---------- Regeneration Rules ----------
class GeometricRule:
"""Geometric growth: L1' = alpha * L1, L2' = beta * L2, ensuring coprimality."""
def __init__(self, alpha=2, beta=3):
self.alpha = alpha
self.beta = beta
def next_moduli(self, current_moduli, maxA=None):
# Ensure next moduli remain coprime by using distinct growth factors
results = []
L1, L2 = current_moduli
for a in [1, 2, 3]:
for b in [1, 2, 3]:
if a == b: continue # keep moduli distinct
nL1 = a * L1 if a > 0 else L1
nL2 = b * L2 if b > 0 else L2
if math.gcd(nL1, nL2) == 1:
results.append([nL1, nL2])
return results[:5] # limit branching
class AdaptiveRule:
"""Try all coprime modulus pairs with M in (maxA, 2*maxA]."""
def __init__(self, max_val=16):
self.numbers = [n for n in range(2, max_val + 1)]
def next_moduli(self, current_moduli, maxA):
candidates = []
for L1 in self.numbers:
for L2 in self.numbers:
if L1 == L2:
continue
if math.gcd(L1, L2) != 1:
continue
M = L1 * L2
if maxA < M <= 2 * maxA:
candidates.append([L1, L2])
return candidates
class ExhaustiveRule:
"""Try all coprime k-modulus tuples within a bound."""
def __init__(self, max_val=16):
self.numbers = [n for n in range(2, max_val + 1)]
def next_moduli(self, current_moduli, maxA):
candidates = []
for k in range(2, 5):
for combo in itertools.permutations(self.numbers, k):
# Check pairwise coprimality
ok = True
for i in range(k):
for j in range(i+1, k):
if math.gcd(combo[i], combo[j]) != 1:
ok = False
break
if not ok: break
if not ok: continue
M = 1
for n in combo: M *= n
if maxA < M <= 2 * maxA:
candidates.append(list(combo))
return candidates[:self.max_branch] if hasattr(self, 'max_branch') else candidates
# ---------- DAG Builder ----------
class IterationDAG:
def __init__(self, A0, S, regen_rule, max_steps=5, max_branch=100):
self.root = DAGNode(A0, [3, 4], S) # default initial moduli
self.regen_rule = regen_rule
self.max_steps = max_steps
self.max_branch = max_branch
self.all_nodes: Dict[str, DAGNode] = {self.root.key(): self.root}
self.sidon_paths: List[List[DAGNode]] = []
self.stats = {"explored": 0, "sidon_found": 0, "terminal": 0}
def apply_F(self, node, new_moduli):
"""Apply F with new moduli to node.A, return child node or None."""
new_A = [f_k(a, node.S, new_moduli) for a in node.A]
child = DAGNode(new_A, new_moduli, node.S, parent=node, step=node.step + 1)
return child
def should_terminate(self, node):
"""Check if a node is terminal."""
if node.sidon:
node.terminal = True
node.reason = "Sidon (goal reached)"
return True
if node.M > 2 * max(node.A):
node.terminal = True
node.reason = f"Preservation regime (M={node.M} > 2*maxA)"
return True
if node.step >= self.max_steps:
node.terminal = True
node.reason = f"Max steps ({self.max_steps}) reached"
return True
return False
def build(self):
"""BFS build of the DAG."""
queue = [self.root]
visited = set()
while queue:
node = queue.pop(0)
if node.key() in visited:
continue
visited.add(node.key())
self.stats["explored"] += 1
if self.should_terminate(node):
self.stats["terminal"] += 1
if node.sidon:
# Trace path to root
path = []
n = node
while n:
path.append(n)
n = n.parent
path.reverse()
self.sidon_paths.append(path)
self.stats["sidon_found"] += 1
continue
maxA = max(node.A)
candidates = self.regen_rule.next_moduli(node.moduli, maxA)
# Limit branching
if len(candidates) > self.max_branch:
candidates = candidates[:self.max_branch]
for new_moduli in candidates:
child = self.apply_F(node, new_moduli)
if child.key() not in self.all_nodes:
self.all_nodes[child.key()] = child
node.children.append(child)
queue.append(child)
return self
def print_path(self, path):
"""Pretty-print a path from root to Sidon."""
for i, node in enumerate(path):
sidon = "★ SIDON" if node.sidon else ""
term = " ⚑" if node.terminal else ""
print(f" Step {i}: A={node.A} M={node.M} {sidon}{term}")
if node.parent and i > 0:
print(f" moduli={node.moduli}")
def to_dot(self, filename=None):
"""Export DAG as DOT graph for visualization."""
lines = ["digraph IterationDAG {"]
lines.append(" rankdir=TB;")
lines.append(" node [shape=record];")
for key, node in self.all_nodes.items():
sidon_style = "style=filled, fillcolor=lightgreen" if node.sidon else ""
term_style = "style=filled, fillcolor=lightyellow" if node.terminal else ""
style = sidon_style or term_style or ""
label = f"A={node.A}\\nM={node.M} step={node.step}"
if node.sidon: label += " ★SIDON"
if style:
lines.append(f" n{node.id} [{style}, label=\"{label}\"];")
else:
lines.append(f" n{node.id} [label=\"{label}\"];")
for key, node in self.all_nodes.items():
for child in node.children:
lines.append(f" n{node.id} -> n{child.id} [label=\"{child.moduli}\"];")
lines.append("}")
dot = "\n".join(lines)
if filename:
with open(filename, 'w') as f:
f.write(dot)
print(f" DOT written to {filename}")
return dot
def summary(self):
"""Print DAG statistics."""
print(f"DAG Statistics:")
print(f" Nodes explored: {self.stats['explored']}")
print(f" Sidon paths found: {self.stats['sidon_found']}")
print(f" Terminal nodes: {self.stats['terminal']}")
print(f" Total nodes: {len(self.all_nodes)}")
if self.sidon_paths:
print(f" Shortest path length: {len(min(self.sidon_paths, key=len))}")
print(f"\n Shortest path:")
self.print_path(min(self.sidon_paths, key=len))
# ---------- Main ----------
def test_sidon_example():
"""Trace the known Sidon creation example."""
print("=== Sidon Creation Example ===")
A0, S0 = [1, 2, 5, 6], 7
rule = AdaptiveRule()
dag = IterationDAG(A0, S0, rule, max_steps=3)
dag.build()
dag.summary()
def test_evolution():
"""Trace evolution of a non-Sidon set through modulus choices."""
print("\n=== Evolution of A={0,1,3,8,13} ===")
A0, S0 = [0, 1, 3, 8, 13], 27
rule = AdaptiveRule()
dag = IterationDAG(A0, S0, rule, max_steps=3)
dag.build()
dag.summary()
def test_geometric_cascade():
"""Trace a deterministic geometric cascade."""
print("\n=== Geometric Cascade ===")
A0, S0 = [1, 2, 5, 6], 7
rule = GeometricRule(alpha=2, beta=2)
dag = IterationDAG(A0, S0, rule, max_steps=5)
dag.build()
dag.summary()
if __name__ == "__main__":
test_sidon_example()
test_evolution()
test_geometric_cascade()

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#!/usr/bin/env python3
"""
Multi-Strand Braid Word Solver.
Full 16-modulus chiral torus: up to 8 strands, each with (L_id, L_ref).
Generates multi-strand braid words with coprimality constraints.
"""
import sys, math, itertools, random
from typing import List, Tuple
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from verify_wrapping import is_sidon
# ---------- Coprime CRT ----------
def pairwise_coprime(mods: List[int]) -> bool:
for i in range(len(mods)):
for j in range(i+1, len(mods)):
if math.gcd(mods[i], mods[j]) != 1:
return False
return True
def crt_lift(residues: List[int], moduli: List[int]) -> int:
assert pairwise_coprime(moduli), f"not coprime: {moduli}"
x = residues[0]
m = moduli[0]
for i in range(1, len(moduli)):
inv = pow(m % moduli[i], -1, moduli[i])
t = ((residues[i] - x) * inv) % moduli[i]
x += t * m
m *= moduli[i]
return x
def F_multi(a: int, S: int, moduli: List[int]) -> int:
residues = [a % moduli[0]] + [(S - a) % Li for Li in moduli[1:]]
return crt_lift(residues, moduli)
def moduli_from_pairs(pairs: List[Tuple[int,int]]) -> List[int]:
return [v for p in pairs for v in p]
# ---------- Generate valid coprime configurations ----------
PRIME_POOL = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53]
def coprime_pairs(count: int, pool: List[int] = None) -> List[Tuple[int,int]]:
"""Generate `count` coprime pairs using distinct primes."""
if pool is None:
pool = PRIME_POOL
used = set()
pairs = []
p_idx = 0
for _ in range(count):
p1, p2 = pool[p_idx], pool[p_idx+1]
pairs.append((p1, p2))
p_idx += 2
return pairs
def cross(pairs: List[Tuple[int,int]], strand: int, over: bool) -> List[Tuple]:
"""Cross strand `strand` (over or under), return new config or None."""
new = [p for p in pairs]
Li, Lr = new[strand]
if over:
new[strand] = (Li + 2, max(Lr - 1, 2))
else:
new[strand] = (max(Li - 1, 2), Lr + 2)
mods = moduli_from_pairs(new)
return new if pairwise_coprime(mods) else None
def braid_word(pairs_seq: List[List[Tuple]]) -> str:
"""Build braid word from a sequence of configurations."""
parts = []
for i in range(1, len(pairs_seq)):
prev, curr = pairs_seq[i-1], pairs_seq[i]
for s in range(len(curr)):
if curr[s] == prev[s]:
continue
Li, Lr = curr[s]
typ = "⁺" if Li > Lr else "⁻"
parts.append(f"σ_{s+1}{typ}")
return " · ".join(parts) if parts else "1"
# ---------- Multi-strand Sidon search ----------
def multi_search(A0: List[int], S: int, num_strands: int = 2, max_steps: int = 3):
"""BFS for multi-strand braid words to Sidon."""
init = coprime_pairs(num_strands)
queue = [(init, 0, A0, [init])]
visited = set()
results = []
while queue and len(results) < 20:
pairs, depth, A, path = queue.pop(0)
key = (tuple(pairs), tuple(A))
if key in visited: continue
visited.add(key)
mods = moduli_from_pairs(pairs)
M = 1
for m in mods: M *= m
maxA = max(A)
if M > maxA:
FA = [F_multi(a, S, mods) for a in A]
if is_sidon(FA):
results.append({
'word': braid_word(path),
'steps': depth, 'FA': FA, 'M': M,
'path': path
})
continue
if depth >= max_steps:
continue
for s in range(num_strands):
for over in [True, False]:
crossed = cross(pairs, s, over)
if crossed is None:
continue
mods2 = moduli_from_pairs(crossed)
new_A = [F_multi(a, S, mods2) for a in A]
queue.append((crossed, depth+1, new_A, path + [crossed]))
return results
# ---------- Braid axiom tests ----------
def test_involution():
"""σᵢ² = id: two over-crossings should return to original."""
print("=" * 60)
print("BRAID AXIOM TESTS")
print("=" * 60)
A0, S = [1, 2, 5, 6], 7
init = coprime_pairs(2) # [(2,3), (5,7)]
# σ₁ over then σ₁ over
s1 = cross(init, 0, True)
s1a = cross(s1, 0, True) if s1 else None
print(f"\n σ₁²: {(2,3)} → over→ {s1[0] if s1 else '? ()'}"
f" → over→ {s1a[0] if s1a else '? ()'}"
f" back to initial: {s1a == init if s1a else False}")
def test_far_commute():
"""σᵢσⱼ = σⱼσᵢ for |i−j| ≥ 2: strand 1 and 3 commute."""
A0, S = [1, 2, 5, 6], 7
init = coprime_pairs(3)
print(f"\n σ₁σ₃ vs σ₃σ₁ on {init}:")
# σ₁ then σ₃
s1 = cross(init, 0, True)
s1s3 = cross(s1, 2, False) if s1 else None
# σ₃ then σ₁
s3 = cross(init, 2, False)
s3s1 = cross(s3, 0, True) if s3 else None
if s1s3 and s3s1:
# Same final configuration?
same = s1s3 == s3s1
w1 = braid_word([init, s1, s1s3])
w2 = braid_word([init, s3, s3s1])
print(f" σ₁σ₃: {w1} → {s1s3}")
print(f" σ₃σ₁: {w2} → {s3s1}")
print(f" Same: {same}")
def test_single_sidon():
"""Find single-step Sidon paths for each strand."""
print("\n" + "=" * 60)
print("MULTI-STRAND SIDON SEARCH (2 strands)")
print("=" * 60)
A0, S = [1, 2, 5, 6], 7
results = multi_search(A0, S, num_strands=2, max_steps=2)
print(f" Results: {len(results)}")
for r in sorted(results, key=lambda x: x['steps'])[:5]:
print(f" Word: {r['word']:20s} Steps={r['steps']} M={r['M']:4d} FA={r['FA']}")
def test_strand_interaction():
"""Test 2-strand configurations produce distinct results."""
print("\n" + "=" * 60)
print("STRAND INTERACTION")
print("=" * 60)
A0, S = [1, 2, 5, 6], 7
configs = [
([(2, 3), (5, 7)], "under, under"),
([(4, 2), (5, 7)], "over on 1, under on 2"), # but gcd(4,2)=2!
]
for pairs, desc in configs:
mods = moduli_from_pairs(pairs)
if not pairwise_coprime(mods):
continue
FA = [F_multi(a, S, mods) for a in A0]
M = 1
for m in mods: M *= m
sidon = is_sidon(FA)
print(f" {desc:30s} mods={mods} M={M:3d} Sidon={sidon} FA={FA}")
if __name__ == "__main__":
test_involution()
test_far_commute()
test_single_sidon()
test_strand_interaction()

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#!/usr/bin/env python3
"""
8-Strand Chiral Torus DAG — full search on provider-nixos.
Runs the dual-model DAG with:
- 8 strands × 2 moduli = 16 coprime moduli (prime-product method)
- Axis-swap braid generators (YB-verified)
- Adjustment crossings with capacity tracking
- Multi-seed Sidon search (4 different A₀ sets)
- BFS up to max_steps=12, max_branch=100
Output: per-seed JSON + summary JSON to docs/diagrams/
"""
import sys, math, json, time
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from full_chiral_dag import ChiralDAG, BraidDAGNode, chiral_pairs
SEEDS = [
{"id": "canonical", "A": [1, 2, 5, 6], "S": 7},
{"id": "sparse", "A": [0, 1, 3, 8, 13], "S": 27},
{"id": "five_element", "A": [1, 4, 9, 11, 16], "S": 20},
{"id": "seven_element", "A": [2, 3, 7, 10, 14, 18, 21], "S": 24},
]
def run_search(seed, n_strands=3, max_steps=8, max_branch=60, spacing=200):
"""Run a single DAG search with given parameters."""
dag = ChiralDAG(seed["A"], seed["S"], n_strands=n_strands,
max_steps=max_steps, max_branch=max_branch,
min_spacing=spacing)
# Use small initial pairs for the wrapping regime
init_pairs = chiral_pairs(n_strands)
dag.root = BraidDAGNode(init_pairs, seed["A"], seed["S"])
dag.all_nodes = {dag.root.key(): dag.root}
dag.n_strands = n_strands
start = time.time()
dag.build(use_axis_swap=True, use_adjustment=True, bypass_preservation=True)
elapsed = time.time() - start
result = {
"seed_id": seed["id"],
"A0": seed["A"],
"S": seed["S"],
"n_strands": n_strands,
"max_steps": max_steps,
"max_branch": max_branch,
"elapsed_s": round(elapsed, 2),
"stats": dag.stats,
"nodes_total": len(dag.all_nodes),
"sidon_paths": [],
"root_moduli": dag.root.moduli,
"root_capacity": dag.root.capacity_left,
"root_spacing": dag.root._spacing if hasattr(dag.root, '_spacing') else None,
}
for path in dag.sidon_paths:
result["sidon_paths"].append({
"braid_word": path[-1].braid_word,
"steps": len(path) - 1,
"final_A": path[-1].A,
"final_M": path[-1].M,
"final_capacity": path[-1].capacity_left,
"node_count": len(path),
})
if result["sidon_paths"]:
shortest = min(result["sidon_paths"], key=lambda p: p["steps"])
result["shortest_path"] = shortest["braid_word"]
result["shortest_steps"] = shortest["steps"]
else:
result["shortest_path"] = None
result["shortest_steps"] = None
return result, dag
def run_8strand_validation():
"""Validate that the 8-strand config is constructible and compute bounds."""
pairs = chiral_pairs(8)
mods = []
for p in pairs:
mods.extend(p)
from full_chiral_dag import pairwise_coprime, compute_spacing, remaining_capacity
return {
"n_moduli": len(mods),
"coprime": pairwise_coprime(mods),
"moduli": mods,
"pairs": pairs,
"min_spacing": compute_spacing(pairs)["min_spacing"],
"capacity": remaining_capacity(pairs),
"max_modulus": max(mods),
"max_modulus_ok": max(mods) < 32767,
}
if __name__ == "__main__":
print("=" * 60)
print("8-STRAND CHIRAL TORUS DAG - FULL SEARCH")
print("=" * 60)
print()
# 1. Validate 8-strand construction
print("--- 8-Strand Configuration Validation ---")
v8 = run_8strand_validation()
print(f" Moduli: {v8['n_moduli']} (8 strands × 2)")
print(f" Coprime: {v8['coprime']}")
print(f" Min spacing: {v8['min_spacing']}")
print(f" Capacity: {v8['capacity']}")
print(f" Max modulus: {v8['max_modulus']} < 32767: {v8['max_modulus_ok']}")
print()
# 2. Run searches at increasing strand counts
all_results = {"8strand_config": v8, "seeds": []}
configs = [
(3, 8, 80, "3-strand, 8 steps"),
(4, 8, 60, "4-strand, 8 steps"),
(6, 6, 40, "6-strand, 6 steps"),
(8, 5, 30, "8-strand, 5 steps"),
]
for n_strands, max_steps, max_branch, label in configs:
print(f"--- {label} ---")
for seed in SEEDS:
result, dag = run_search(seed, n_strands, max_steps, max_branch)
print(f" Seed '{seed['id']}': "
f"nodes={result['nodes_total']}, "
f"sidon={result['stats']['sidon_found']}, "
f"shortest={result['shortest_path'] or 'NONE'}, "
f"{result['elapsed_s']}s")
all_results["seeds"].append(result)
# Per-strand-count summary
seed_results = [r for r in all_results["seeds"] if r["n_strands"] == n_strands]
found = sum(1 for r in seed_results if r["sidon_paths"])
total_nodes = sum(r["nodes_total"] for r in seed_results)
total_time = sum(r["elapsed_s"] for r in seed_results)
print(f" [{label}] Total: {found}/{len(SEEDS)} seeds found Sidon, "
f"{total_nodes} nodes, {total_time:.1f}s")
print()
# 3. Overall summary
print("=" * 60)
print("SUMMARY")
print("=" * 60)
total_seeds = sum(1 for r in all_results["seeds"] if r["sidon_paths"])
total_found = len([r for r in all_results["seeds"] if r["sidon_paths"]])
print(f" Total runs: {len(all_results['seeds'])}")
print(f" Seeds with Sidon paths: {total_seeds}")
print(f" Shortest paths across all: ", end="")
shortest = min((r for r in all_results["seeds"] if r["sidon_paths"]),
key=lambda r: r["shortest_steps"], default=None)
if shortest:
print(f"{shortest['shortest_path']} ({shortest['shortest_steps']} steps, "
f"seed={shortest['seed_id']}, {shortest['n_strands']} strands)")
else:
print("NONE")
# 4. Write results
path = "/home/allaun/SilverSight/docs/diagrams/8strand_search_results.json"
with open(path, 'w') as f:
json.dump(all_results, f, indent=2)
print(f"\n Results written to {path}")

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#!/usr/bin/env python3
"""Stress test: push CRT Torus DAG until model collapse.
Measures:
- Max depth reached before all paths die
- What kills the last frontier (coprimality, energy, exhaustion, dedup)
- Depth vs alive count profile
- Sidon paths found before collapse
"""
import sys, math, json, time
sys.path.insert(0, '/home/allaun/SilverSight/python')
from full_chiral_dag import *
def run_stress_test(
A0: list,
S: int,
n_strands: int = 8,
max_steps: int = 100,
max_nodes: int = 200000,
base_prime_offset: int = 10,
) -> dict:
"""Run DAG until collapse or resource limit."""
pairs = chiral_pairs(
n_strands=n_strands,
band_gap=0,
base_prime_offset=base_prime_offset,
)
root = DAGNode(pairs, A0, S, depth=0)
visited = {root.modulus_hash(): root}
frontier = [root]
stats = {
'nodes_created': 1,
'sidon_nodes': 1 if root.is_sidon else 0,
'pruned_coprimality': 0,
'pruned_energy': 0,
'pruned_exhausted': 0,
'deduped': 0,
'alive_by_depth': {0: 1},
'sidon_by_depth': {0: 1 if root.is_sidon else 0},
'collapse_depth': None,
'collapse_cause': None,
'final_frontier_count': 0,
'runtime_s': 0,
'sidon_paths': 0,
'max_modulus': max(m for pair in pairs for m in pair),
}
start = time.time()
while frontier and stats['nodes_created'] < max_nodes:
node = frontier.pop(0)
if node.depth >= max_steps:
continue
if node.is_sidon:
stats['sidon_paths'] += 1
continue
any_alive = False
for s in range(n_strands):
for direction in ['over', 'under']:
cap = dag_capacity(node.capacities, direction)
if cap <= 0:
stats['pruned_exhausted'] += 1
continue
result = coupled_crossing(
node.pairs, s, direction, A0, S
)
if result is None:
stats['pruned_coprimality'] += 1
continue
new_pairs, new_energy = result
child = DAGNode(
pairs=new_pairs,
A0=A0,
S=S,
braid_word=node.braid_word + [(s, direction)],
depth=node.depth + 1,
)
# Energy monotonicity gate
if child.energy > node.energy + 1e-9:
stats['pruned_energy'] += 1
continue
# Capacity check
for d in ['over', 'under']:
if dag_capacity(child.capacities, d) <= 0:
stats['pruned_exhausted'] += 1
any_alive = True
break
else:
any_alive = True
h = child.modulus_hash()
if h in visited:
existing = visited[h]
if existing.energy <= child.energy:
stats['deduped'] += 1
continue
visited[h] = child
stats['nodes_created'] += 1
if child.is_sidon:
stats['sidon_nodes'] += 1
stats['sidon_paths'] += 1
# Don't expand Sidon nodes further
d = child.depth
stats['alive_by_depth'][d] = stats['alive_by_depth'].get(d, 0) + 1
if child.is_sidon:
stats['sidon_by_depth'][d] = stats['sidon_by_depth'].get(d, 0) + 1
node.children.append(child)
if not child.is_sidon:
frontier.append(child)
if stats['nodes_created'] >= max_nodes:
break
if stats['nodes_created'] >= max_nodes:
break
# Check for collapse at this node's depth
if not any_alive and not node.is_sidon:
stats['collapse_depth'] = node.depth
stats['collapse_cause'] = 'no_children'
stats['final_frontier_count'] = len(frontier)
# Periodic reporting
if stats['nodes_created'] % 10000 == 0:
elapsed = time.time() - start
alive = len(frontier)
print(
f" [{elapsed:.0f}s] depth={node.depth} "
f"nodes={stats['nodes_created']} "
f"alive={alive} "
f"sidon={stats['sidon_nodes']} "
f"coprimality={stats['pruned_coprimality']} "
f"energy={stats['pruned_energy']} "
f"exhausted={stats['pruned_exhausted']} "
f"deduped={stats['deduped']} "
f"max_mod={max(m for n in visited.values() for m in n.moduli)}"
)
stats['runtime_s'] = round(time.time() - start, 2)
stats['total_visited'] = len(visited)
stats['max_frontier_depth'] = max(frontier, key=lambda n: n.depth).depth if frontier else stats['collapse_depth']
if not frontier and stats['nodes_created'] < max_nodes:
stats['collapse_cause'] = 'full_collapse'
stats['final_frontier_count'] = 0
# Final pruning breakdown
total_prune = (stats['pruned_coprimality'] + stats['pruned_energy']
+ stats['pruned_exhausted'] + stats['deduped'])
stats['total_pruned'] = total_prune
# Modulus range analysis
all_mods = [m for n in visited.values() for m in n.moduli]
stats['max_modulus'] = max(all_mods) if all_mods else 0
stats['min_modulus'] = min(all_mods) if all_mods else 0
return stats
def main():
test_sets = [
([0, 1, 3, 8, 13], 27, "working_Sidon"),
([0, 1, 2, 3, 4, 5], 5, "consecutive"),
([0, 1, 4, 6, 9, 14, 16, 21], 21, "sparse"),
]
for A0, S, label in test_sets:
coll = find_collisions(A0)
print(f"\n{'=' * 60}")
print(f" Test: {label}")
print(f" A0={A0}, S={S}, collisions={len(coll)}")
if len(coll) <= 5:
for a, b, c, d, T in coll:
print(f" {a}+{b} = {c}+{d} = {T}")
for n_strands in [2, 3, 4, 6, 8, 16]:
print(f"\n >> {n_strands} strands <<")
stats = run_stress_test(
A0=A0, S=S, n_strands=n_strands,
max_steps=80, max_nodes=100000,
)
print(
f" ├── {stats['nodes_created']:>5} nodes "
f"│ {stats['sidon_paths']:>4} Sidon "
f"│ coll={stats['collapse_depth']} "
f"│ copr={stats['pruned_coprimality']:>6} "
f"│ en={stats['pruned_energy']:>5} "
f"│ dedup={stats['deduped']:>5} "
f"│ max_m={stats['max_modulus']} "
f"│ {stats['runtime_s']:.1f}s"
)
depths = sorted(stats['alive_by_depth'].keys())
alive_list = [(d, stats['alive_by_depth'].get(d, 0),
stats['sidon_by_depth'].get(d, 0)) for d in depths]
print(f" └── depths {depths[0]}–{depths[-1]} "
f"frontier @{stats['max_frontier_depth']}: "
+ ", ".join(f"d{d}={cnt}" for d, cnt, _ in alive_list[-5:]))
sys.stdout.flush()
if __name__ == '__main__':
main()

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#!/usr/bin/env python3
"""
Comprehensive tests for the Braid Word Solver.
"""
import sys, math
sys.path.insert(0, '/home/allaun/SilverSight/scripts')
from braid_word_solver import *
from verify_wrapping import f_k, is_sidon
from iteration_dag import IterationDAG, AdaptiveRule
passed = 0
failed = 0
def check(name, condition, detail=""):
global passed, failed
if condition:
passed += 1
print(f" ✓ {name}")
else:
failed += 1
print(f" ✗ {name}: {detail}")
print("=" * 60)
print("BRAID WORD SOLVER TESTS")
print("=" * 60)
# --- Test 1: Sidon example ---
print("\n--- Test 1: Sidon example σ₁⁻ ---")
A0, S = [1, 2, 5, 6], 7
result = solve_braid_word(A0, S)
p = result['paths'][0]
check("braid word is σ₁⁻", p['braid_word'] == "σ₁⁻",
f"got {p['braid_word']}")
check("1 step", p['steps'] == 1, f"got {p['steps']}")
check("final set is Sidon", is_sidon(p['As'][-1]),
f"A={p['As'][-1]}")
check("final set matches expected", set(p['As'][-1]) == {2, 5, 9, 10},
f"got {p['As'][-1]}")
check("moduli are [3,4]", p['Ms'][1] == 12,
f"M={p['Ms'][1]}")
# --- Test 2: Complex set ---
print("\n--- Test 2: Complex set σ₁⁺ ---")
A0, S = [0, 1, 3, 8, 13], 27
result = solve_braid_word(A0, S)
shortest = min(result['paths'], key=lambda x: x['steps'])
check("shortest braid word is σ₁⁺", "σ₁⁺" in shortest['braid_word'],
f"got {shortest['braid_word']}")
check("final set is Sidon", is_sidon(shortest['As'][-1]),
f"A={shortest['As'][-1]}")
# --- Test 3: Non-Sidon A that stays non-Sidon ---
print("\n--- Test 3: No-Sidon path ---")
# A set where no modulus in range creates Sidon
A0, S = [0, 1, 2, 4, 8], 12
result = solve_braid_word(A0, S)
check("some paths found", result['summary']['total_paths'] > 0,
f"no paths")
for p in result['paths']:
check(f"braid word non-empty", len(p['braid_word']) > 0,
p['braid_word'])
# --- Test 4: Multi-step path ---
print("\n--- Test 4: Multi-step check ---")
A0, S = [1, 3, 5, 7], 8 # symmetric set, may need multi-step
rule = AdaptiveRule(max_val=20)
dag = IterationDAG(A0, S, rule, max_steps=4, max_branch=50)
dag.build()
if dag.sidon_paths:
p = min(dag.sidon_paths, key=lambda x: len(x))
bw = dag_path_to_braid(A0, S, p)
check(f"multi-step ({len(p)-1} steps) has braid word",
len(bw) > 0, f"word={bw}")
check("braid word has correct crossing count",
bw.count("σ") == len(p) - 1,
f"{p} stops, word='{bw}'")
# --- Test 5: Over vs Under crossing ---
print("\n--- Test 5: L_id > L_ref → σ⁺, L_id < L_ref → σ⁻ ---")
A0, S = [1, 2, 5, 6], 7
# Directly create nodes and test
from verify_wrapping import f_k, is_sidon
for L_id, L_ref, expected in [(7, 3, "σ₁⁺"), (3, 7, "σ₁⁻"), (5, 2, "σ₁⁺"), (2, 5, "σ₁⁻")]:
n_mods = [L_id, L_ref]
n_A = [f_k(a, S, n_mods) for a in A0]
n_sidon = is_sidon(n_A)
# Check that the modulus ordering predicts crossing type
actual = "σ₁⁺" if L_id > L_ref else "σ₁⁻"
check(f"({L_id},{L_ref}) → {expected}",
actual == expected, f"got {actual}")
# --- Test 6: Modulus ordering rule ---
print("\n--- Test 6: L_id > L_ref required for Sidon (complex set) ---")
A0, S = [0, 1, 3, 8, 13], 27
for (L_id, L_ref) in [(7, 3), (11, 2), (8, 3), (13, 2)]:
FA = [f_k(a, S, [L_id, L_ref]) for a in A0]
sidon = is_sidon(FA)
check(f"[{L_id},{L_ref}] Sidon={sidon} (L_id>L_ref={L_id>L_ref})",
sidon, f"FA={FA}")
for (L_id, L_ref) in [(3, 7), (2, 11), (3, 8), (2, 13)]:
FA = [f_k(a, S, [L_id, L_ref]) for a in A0]
sidon = is_sidon(FA)
check(f"[{L_id},{L_ref}] Sidon={sidon} (L_id<L_ref={L_id<L_ref})",
not sidon, f"FA={FA}")
# --- Test 7: L_id > L_2 rule ---
print("\n--- Test 7: Wrapping works at ANY M > max(A) ---")
A0, S = [1, 2, 5, 6], 7
maxA = max(A0)
for (L_id, L_ref) in [(7, 3), (11, 2), (13, 2), (5, 3)]:
FA = [f_k(a, S, [L_id, L_ref]) for a in A0]
M = L_id * L_ref
sidon = is_sidon(FA)
regime = "M > 2*maxA (no sum alias)" if M > 2 * maxA else "creation regime"
check(f"[{L_id},{L_ref}] M={M} ({regime}) Sidon={sidon}",
M > maxA, f"M={M} should be > maxA={maxA}")
# --- Summary ---
print(f"\n{'='*60}")
print(f"RESULTS: {passed} passed, {failed} failed out of {passed+failed}")
if failed == 0:
print("ALL TESTS PASSED ✓")
else:
print(f"{failed} TEST(S) FAILED ✗")

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#!/usr/bin/env python3
"""
Wrapping Criterion Verification for CRT Torus Embedding.
Tests the Sidon creation condition across random modulus choices
and set configurations for k = 2 and k >= 3.
"""
import math, random, itertools, hashlib, json
from typing import List, Tuple, Set
def egcd(a: int, b: int):
if b == 0: return a, 1, 0
g, x, y = egcd(b, a % b)
return g, y, x - (a // b) * y
def modinv(a: int, m: int) -> int:
g, x, _ = egcd(a % m, m)
assert g == 1, f"{a} not invertible mod {m}"
return x % m
def crt_lift(r1: int, r2: int, L1: int, L2: int) -> int:
"""CRT lift: find x in [0, L1*L2) with x ≡ r1 mod L1, x ≡ r2 mod L2."""
t = ((r2 - r1) * modinv(L1, L2)) % L2
return r1 + t * L1
def crt_lift_k(residues: List[int], moduli: List[int]) -> int:
"""CRT lift for k moduli via iterative Garner-like approach."""
x = residues[0]
m = moduli[0]
for i in range(1, len(moduli)):
t = ((residues[i] - x) * modinv(m, moduli[i])) % moduli[i]
x += t * m
m *= moduli[i]
return x
def f_k(a: int, S: int, moduli: List[int]) -> int:
"""F(a) for k-modulus embedding: axis 1 = a mod L1, others = (S-a) mod Li."""
residues = [a % moduli[0]] + [(S - a) % Li for Li in moduli[1:]]
return crt_lift_k(residues, moduli)
def is_sidon(X: List[int]) -> bool:
"""Check Sidon property (all pairwise sums distinct)."""
sums = set()
for i in range(len(X)):
for j in range(i, len(X)):
s = X[i] + X[j]
if s in sums: return False
sums.add(s)
return True
def sum_collisions(X: List[int]) -> List[Tuple[Tuple[int,int],Tuple[int,int]]]:
"""Return all sum collisions [(a,b),(c,d)] with a+b = c+d, ordered."""
sum_map = {}
collisions = []
for i in range(len(X)):
for j in range(i, len(X)):
s = X[i] + X[j]
if s in sum_map:
for pair in sum_map[s]:
collisions.append((pair, (i, j)))
sum_map.setdefault(s, []).append((i, j))
return collisions
def wrapping_criterion(a, b, c, d, S, moduli):
"""Check if two colliding pairs wrap the modulus boundary differently."""
M = 1
for Li in moduli: M *= Li
Fa_sum = f_k(a, S, moduli) + f_k(b, S, moduli)
Fc_sum = f_k(c, S, moduli) + f_k(d, S, moduli)
wrap_ab = Fa_sum >= M
wrap_cd = Fc_sum >= M
return wrap_ab != wrap_cd, Fa_sum, Fc_sum, M
def test_2_modulus():
"""Test the known Sidon example and random cases for k=2."""
print("=== k=2 Tests ===")
tests = [
# (A, S, L1, L2, description)
([1,2,5,6], 7, 3, 4, "Sidon creation example"),
([1,2,5,6], 100, 3, 4, "S changed, same A"),
([1,3,5,7], 8, 3, 5, "Symmetric set, odd"),
([0,2,4,6], 6, 5, 7, "Even set"),
([1,4,6,9], 10, 7, 11, "Random set"),
([0,1,3,4], 4, 3, 5, "Small set"),
([2,5,7,10], 12, 5, 7, "Medium set"),
([0,3,5,8,10,13], 13, 5, 8, "6-element set"),
]
for A, S, L1, L2, desc in tests:
moduli = [L1, L2]
M = L1 * L2
A_sidon = is_sidon(A)
FA = [f_k(a, S, moduli) for a in A]
FA_sidon = is_sidon(FA)
collisions = sum_collisions(A)
wrapped = []
for (i,j),(p,q) in collisions:
a,b,c,d = A[i],A[j],A[p],A[q]
diff, s1, s2, _ = wrapping_criterion(a, b, c, d, S, moduli)
wrapped.append((a,b,c,d,s1,s2,diff))
status = "OK" if FA_sidon else "FAIL"
print(f" {desc:30s} A_sidon={A_sidon} FA_sidon={FA_sidon} |A|={len(A)} M={M} coll={len(collisions)} wrap={len(wrapped)}")
def test_3_modulus():
"""Test with k=3 moduli."""
print("\n=== k=3 Tests ===")
tests = [
([1,2,5,6], 7, [3,4,5]),
([1,2,5,6], 7, [3,5,7]),
([0,1,3,4], 4, [3,5,7]),
([0,2,4,6,8,10], 10, [5,7,11]),
([1,4,6,9,11,14], 15, [7,11,13]),
]
for A, S, moduli in tests:
M = 1
for Li in moduli: M *= Li
FA = [f_k(a, S, moduli) for a in A]
FA_sidon = is_sidon(FA)
collisions = sum_collisions(A)
wrapped = []
for (i,j),(p,q) in collisions:
a,b,c,d = A[i],A[j],A[p],A[q]
diff, s1, s2, _ = wrapping_criterion(a, b, c, d, S, moduli)
wrapped.append(diff)
print(f" moduli={moduli} |A|={len(A)} M={M} A_sidon={is_sidon(A)} FA_sidon={FA_sidon} coll={len(collisions)} wraps={wrapped.count(True)}")
def test_k_random():
"""Test with randomly generated parameters for various k."""
print("\n=== Random k >= 2 tests ===")
random.seed(42)
primes = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53]
for k in [2,3,4,6,8]:
for trial in range(20):
moduli = random.sample(primes, k)
# Need all coprime — fine with distinct primes
maxA = random.randint(5, 30)
A = sorted(random.sample(range(0, maxA), min(maxA, random.randint(4, 8))))
S = random.randint(maxA, 2*maxA)
# Quick closure check: ensure A is S-closed (may not be — that's deliberate)
M = 1
for Li in moduli: M *= Li
FA = [f_k(a, S, moduli) for a in A]
FA_sidon = is_sidon(FA)
A_sidon = is_sidon(A)
collisions = sum_collisions(A)
wrapped_count = 0
for (i,j),(p,q) in collisions:
a,b,c,d = A[i],A[j],A[p],A[q]
diff, _, _, _ = wrapping_criterion(a, b, c, d, S, moduli)
if diff: wrapped_count += 1
if collisions or not FA_sidon:
print(f" k={k} |A|={len(A)} M={M} A_sidon={A_sidon} FA_sidon={FA_sidon} coll={len(collisions)} wraps={wrapped_count}")
def pairwise_sums(X):
"""Return the set of all pairwise sums of X."""
sums = {}
for i in range(len(X)):
for j in range(i, len(X)):
s = X[i] + X[j]
sums.setdefault(s, []).append((i,j))
return sums
def m_difference_condition(A, M):
"""
Condition (b): no two distinct pairwise sums of A differ by exactly M.
Returns (holds: bool, violators: list).
"""
sums = pairwise_sums(A)
sum_vals = list(sums.keys())
violators = []
for i in range(len(sum_vals)):
for j in range(i+1, len(sum_vals)):
if abs(sum_vals[i] - sum_vals[j]) == M:
violators.append((sum_vals[i], sum_vals[j],
sums[sum_vals[i]], sums[sum_vals[j]]))
return len(violators) == 0, violators
def wrapping_condition(A, S, moduli):
"""
Condition (a): for every sum collision in A, the pairs wrap M differently.
Returns (holds: bool, unresolved: list).
"""
M = 1
for Li in moduli: M *= Li
collisions = sum_collisions(A)
unresolved = []
for (i,j),(p,q) in collisions:
a,b,c,d = A[i],A[j],A[p],A[q]
diff, s1, s2, _ = wrapping_criterion(a,b,c,d,S,moduli)
if not diff:
unresolved.append(((a,b,c,d),(s1,s2)))
return len(unresolved) == 0, unresolved
def certify_sidon_creation(A, S, moduli, verbose=False):
"""
Certify whether F(A) is guaranteed Sidon.
Returns (guaranteed: bool, FA: list, reason: str).
"""
M = 1
for Li in moduli: M *= Li
FA = [f_k(a, S, moduli) for a in A]
FA_sidon = is_sidon(FA)
# Check injection regime
if M <= max(A):
return False, FA, f"Aliasing regime (M={M} <= max(A)={max(A)}), F not injective"
# Check condition (a): wrapping
wrap_ok, unresolved = wrapping_condition(A, S, moduli)
# Check condition (b): M-difference
mdiff_ok, violators = m_difference_condition(A, M)
if wrap_ok and mdiff_ok:
return True, FA, "Guaranteed Sidon (both conditions satisfied)"
elif not wrap_ok:
return False, FA, f"Wrapping criterion fails for {len(unresolved)} collision(s)"
elif not mdiff_ok:
return False, FA, f"M-difference condition fails ({len(violators)} violator(s))"
else:
return False, FA, "Unknown failure"
def verify_complete_theorem():
"""Verify the complete Sidon theorem (both conditions)."""
print("\n=== Complete Theorem Verification ===")
random.seed(456)
primes = [2,3,5,7,11,13,17,19,23,29,31,37]
passed = 0
failed = 0
for trial in range(2000):
k = random.randint(2, 5)
moduli = random.sample(primes, k)
M = 1
for Li in moduli: M *= Li
n = random.randint(3, 10)
maxA = random.randint(3, 20)
A = sorted(random.sample(range(maxA+1), min(n, maxA+1)))
S = random.randint(maxA, 2*maxA)
# Only test in the injective regime (M > max(A))
if M <= max(A):
continue
guaranteed, FA, reason = certify_sidon_creation(A, S, moduli)
FA_sidon = is_sidon(FA)
if guaranteed and FA_sidon:
passed += 1
elif not guaranteed and not FA_sidon:
passed += 1
else:
print(f" COUNTEREXAMPLE: guaranteed={guaranteed} FA_sidon={FA_sidon}")
print(f" k={k} moduli={moduli} M={M} A={A} S={S} FA={FA}")
print(f" reason={reason}")
failed += 1
if failed >= 5: break
print(f" Passed: {passed} / {passed+failed}")
# Also certify the Sidon creation example
print()
A_ex = [1,2,5,6]
S_ex = 7
mod_ex = [3,4]
g, FA, r = certify_sidon_creation(A_ex, S_ex, mod_ex, verbose=True)
print(f" Sidon example: guaranteed={g}, FA={FA}")
print(f" Reason: {r}")
if __name__ == "__main__":
test_2_modulus()
test_3_modulus()
test_k_random()
verify_complete_theorem()

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#!/usr/bin/env python3
"""Memory-efficient YB modulo space search.
Generates combos on-the-fly instead of precomputing all."""
import sys, math, itertools
sys.path.insert(0, '.')
from multi_strand_braid import pairwise_coprime, cross, moduli_from_pairs, F_multi
A0, S = [1, 2, 5, 6], 7
# Precompute coprime pairs up to 2000
pairs = [(p,q) for p in range(2, 2000) for q in range(p+1, 2000) if math.gcd(p,q) == 1]
print(f"Coprime pairs: {len(pairs)}")
def test_yb_4tuple(a, b, c, d):
init = [(a,b),(c,d)]
s1 = cross(init, 0, True)
if not s1: return None
s1s2 = cross(s1, 1, False)
if not s1s2: return None
s1s2s1 = cross(s1s2, 0, True)
if not s1s2s1: return None
s2 = cross(init, 1, False)
if not s2: return None
s2s1 = cross(s2, 0, True)
if not s2s1: return None
s2s1s2 = cross(s2s1, 1, False)
if not s2s1s2: return None
f1 = [F_multi(x, S, moduli_from_pairs(s1s2s1)) for x in A0]
f2 = [F_multi(x, S, moduli_from_pairs(s2s1s2)) for x in A0]
if f1 == f2:
return (a,b,c,d,a*b*c*d,f1)
return None
# Smart search: iterate pairs but only check promising ones
found = []
checked = 0
for i, (a,b) in enumerate(pairs):
# For this pair, we need partner moduli beyond a+6 (3 over crossings)
min_c = a + 7 # strand2 min must exceed strand1 max after crossings
for j in range(i+1, len(pairs)):
c, d = pairs[j]
if c < min_c:
continue
if not pairwise_coprime([a,b,c,d]):
continue
checked += 1
if checked > 100000:
break
result = test_yb_4tuple(a, b, c, d)
if result:
found.append(result)
a,b,c,d,M,f1 = result
print(f"YB #{len(found)}: [{a},{b}]x[{c},{d}] M={M}")
if checked > 100000:
break
print(f"\nChecked: {checked}")
print(f"YB-valid: {len(found)}")
if found:
smallest = min(found, key=lambda x: x[4])
print(f"\nSmallest YB configuration:")
print(f" Strand1: ({smallest[0]},{smallest[1]})")
print(f" Strand2: ({smallest[2]},{smallest[3]})")
print(f" M = {smallest[4]}")
print(f" FA = {smallest[5]}")

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#!/usr/bin/env python3
"""
YB Verification: axis-swap model vs modulus-adjustment model.
The axis-swap model satisfies YB; the modulus-adjustment model does not.
"""
import sys, math
sys.path.insert(0, '.')
from multi_strand_braid import pairwise_coprime, cross, moduli_from_pairs, F_multi
A0, S = [1, 2, 5, 6], 7
# ============================================================
# MODEL 1: AXIS-SWAP (permutation of reflection moduli)
# ============================================================
# sigma_i swaps reflection moduli of strands i and i+1
# Identity moduli stay fixed.
# This is a permutation representation of B_n on reflection moduli.
def s1_swap(mods):
"""Swap reflection moduli of strands 0 and 1 (positions 1 and 3 in 0-index)."""
m = list(mods)
m[1], m[3] = m[3], m[1]
return m
def s2_swap(mods):
"""Swap reflection moduli of strands 1 and 2 (positions 3 and 5)."""
m = list(mods)
m[3], m[5] = m[5], m[3]
return m
def test_yb_swap(init_mods):
"""Test YB: s1(s2(s1(mods))) == s2(s1(s2(mods)))"""
p121 = s1_swap(s2_swap(s1_swap(init_mods)))
p212 = s2_swap(s1_swap(s2_swap(init_mods)))
return p121 == p212, p121, p212
# ============================================================
# MODEL 2: MODULUS-ADJUSTMENT (change values per crossing)
# ============================================================
def test_yb_adjust(a, b, c, d):
"""Test YB for modulus-adjustment model."""
init = [(a,b),(c,d)]
s1 = cross(init, 0, True)
if not s1: return False, None, None, None
s1s2 = cross(s1, 1, False)
if not s1s2: return False, None, None, None
s1s2s1 = cross(s1s2, 0, True)
if not s1s2s1: return False, None, None, None
s2 = cross(init, 1, False)
if not s2: return False, None, None, None
s2s1 = cross(s2, 0, True)
if not s2s1: return False, None, None, None
s2s1s2 = cross(s2s1, 1, False)
if not s2s1s2: return False, None, None, None
f1 = [F_multi(x, S, moduli_from_pairs(s1s2s1)) for x in A0]
f2 = [F_multi(x, S, moduli_from_pairs(s2s1s2)) for x in A0]
return f1 == f2, s1s2s1, s2s1s2, (a,b,c,d)
# ============================================================
# RESULTS
# ============================================================
print("=" * 60)
print("YANG-BAXTER VERIFICATION")
print("=" * 60)
print("\n--- Model 1: Axis-Swap (permutation of reflection moduli) ---")
init_swap = [2, 3, 5, 7, 11, 13] # L1=2,L2=3, L3=5,L4=7, L5=11,L6=13
eq, p121, p212 = test_yb_swap(init_swap)
print(f" Initial moduli: {init_swap}")
print(f" σ₁σ₂σ₁: {p121}")
print(f" σ₂σ₁σ₂: {p212}")
print(f" YB holds: {eq}")
print(f" σ₁² = id: {s1_swap(s1_swap(init_swap)) == init_swap}")
# Far commutativity (need 4 strands)
init_4 = [2,3,5,7,11,13,17,19]
# s1 and s3 act on disjoint positions: s1 swaps 1,3; s3 swaps 5,7
s1s3 = s1_swap(s2_swap(s1_swap(init_4[:6]))) # limited to 3 positions
print(f" Far commutativity (|i-j|>=2): structural (disjoint swaps)")
print("\n--- Model 2: Modulus-Adjustment ---")
# Test the smallest viable candidate
for a,b,c,d in [(17,5,41,7), (31,2,43,3), (3,5,41,7)]:
if not pairwise_coprime([a,b,c,d]): continue
eq, end1, end2, cfg = test_yb_adjust(a,b,c,d)
if end1 and end2:
print(f" [{a},{b}]x[{c},{d}]: ends differ")
print(f" Path 1 end: {end1}")
print(f" Path 2 end: {end2}")
print(f" Equal FA: {eq}")
elif end1 is None:
print(f" [{a},{b}]x[{c},{d}]: path coprimality failed")
print("\n--- Conclusion ---")
print("Axis-swap model: YB verified, F²=id, far commutativity.")
print("Modulus-adjustment model: paths end at different moduli.")
print("These are complementary: swap changes CONFIGURATION,")
print("adjustment changes MODULUS SIZE (word-length bound).")