SilverSight/WORK_LOG.md

# SILVERSIGHT WORK LOG
## Every Step Verifiable with a Calculator. No Code Until Math Passes.

**Rule:** Each step produces a number (or set of numbers). You verify the number
before proceeding. If the number is wrong, STOP. Do not continue. The error must
be found.

---

## PHASE 0: FOUNDATION (COMPLETE — verified)

These are established mathematical results. No code. No proof required. They are
given.

| Step | What | Formula | Status |
|------|------|---------|--------|
| 0.1 | **Fisher metric** | g_p(u,v) = Σᵢ uᵢvᵢ/pᵢ | Given (Chentsov 1972, Amari 1985) |
| 0.2 | **√p embedding** | φ(p) = (√p₁,...,√p₈) ∈ S⁷ | Given (pullback: g_p = 4·φ*g_{S⁷}) |
| 0.3 | **Fisher distance** | d_F(p,q) = 2·arccos(Σᵢ√(pᵢqᵢ)) | Given (Bhattacharyya arc) |
| 0.4 | **Φ-corkscrew** | f(n) = (√n·cos(nψ), √n·sin(nψ)), ψ = 2π/φ² | Given (irrational → injective) |
| 0.5 | **Golden ratio** | φ = (1+√5)/2 = 1.6180339887... | Given |

**Verify 0.5:** Type `(1+sqrt(5))/2` into calculator → 1.6180339887...

---

## PHASE 1: THE THREE GATES (COMPLETE — worksheets written)

Each gate has a worksheet with explicit numerical examples. Every number can be
recomputed by hand or calculator.

### GATE G1: Chaos Game is Contractive

| Substep | Input | Formula | Expected Output | Verify |
|---------|-------|---------|-----------------|--------|
| G1.1 | p = (0.3,0.1,0.15,0.05,0.2,0.08,0.07,0.05) | g_p(u,u) = Σ uᵢ²/pᵢ | 0.07833 | Calculator |
| G1.2 | Same p | φ(p) = (√p₁,...,√p₈) | ‖φ(p)‖₂ = 1.0 | Σ√pᵢ² = 1.0 |
| G1.3 | p, q = (0.2,0.2,0.1,0.1,0.15,0.1,0.1,0.05) | d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ)) | ~0.439 | Calculator |
| G1.4 | C(p) = pair-averaged | d_F(C(p),C(q)) | ~0.100 | Calculator |
| **G1.5** | **Both distances** | **d_F(C(p),C(q)) < d_F(p,q)** | **0.100 < 0.439** | **✓ PASS / ✗ STOP** |
| G1.6 | λ = 1/√2 | Contraction factor | 0.7071 | 1/sqrt(2) |

**Worksheet:** `docs/first_principles/G1_WORKSHEET.md`

### GATE G2: Semantic Collisions Broken

| Substep | Input | Formula | Expected Output | Verify |
|---------|-------|---------|-----------------|--------|
| G2.1 | "a+b=c" | Count 8 byte classes | (0,0.2,0,0.2,0,0,0.6,0) | Count by hand |
| G2.2 | "x+y=z" | Count 8 byte classes | (0,0.2,0,0.2,0,0,0.6,0) | Count by hand |
| G2.3 | "p/q=r" | Count 8 byte classes | (0,0.2,0,0.2,0,0,0.6,0) | Count by hand |
| **G2.4** | **All three vectors** | **F("a+b=c") = F("x+y=z") = F("p/q=r")** | **COLLISION** | **✓ CONFIRMED** |
| G2.5 | Parse trees | τ("a+b=c") = (V,B+,B=,B/) | (0.6,0.2,0.2,0) | Count nodes |
| G2.6 | Parse trees | τ("p/q=r") = (V,B+,B=,B/) | (0.6,0,0.2,0.2) | Count nodes |
| **G2.7** | **τ("a+b=c") vs τ("p/q=r")** | **τ differs** | **COLLISION BROKEN** | **✓ PASS / ✗ STOP** |
| G2.8 | d_F((F,τ₁), (F,τ₂)) | d² = d²_F(F₁,F₂) + d²_F(τ₁,τ₂) | ~1.287 | Calculator |

**Worksheet:** `docs/first_principles/G2_WORKSHEET.md`

### GATE G3: Eigensolid Fixed Point

| Substep | Input | Formula | Expected Output | Verify |
|---------|-------|---------|-----------------|--------|
| G3.1 | p = (0.3,0.1,0.15,0.05,0.2,0.08,0.07,0.05) | C(p)ᵢ = (p_{2k-1}+p_{2k})/2 | (0.2,0.2,0.1,0.1,0.14,0.14,0.06,0.06) | 4 additions, 4 divisions |
| G3.2 | C(p) | C(C(p)) | Same as C(p) | Apply C again |
| **G3.3** | **C(p) vs C(C(p))** | **C∘C = C** | **IDENTICAL** | **✓ PASS / ✗ STOP** |
| G3.4 | (0.4,0.2,0.28,0.12) ∈ Δ₃ | φ(q) = (q₁/2,q₁/2,q₂/2,q₂/2,q₃/2,q₃/2,q₄/2,q₄/2) | = C(p) | Calculator |
| G3.5 | p, q (from G1.3) | d_F(C(p),C(q)) | ~0.100 | Calculator |
| **G3.6** | **Both distances** | **d_F(C(p),C(q)) < d_F(p,q)** | **0.100 < 0.439** | **✓ PASS / ✗ STOP** |
| G3.7 | I_loss(p) | Σₖ sₖ·KL(p_{2k-1}/sₖ, p_{2k}/sₖ ‖ ½,½) | ~0.107 nats | Calculator |

**Worksheet:** `docs/first_principles/G3_WORKSHEET.md`

---

## PHASE 2: FEATURE EXTRACTION (NEXT — math → numbers)

**Prerequisite:** Gates G1, G2, G3 all PASS.

| Step | Input | Formula | Output | Verify |
|------|-------|---------|--------|--------|
| 2.1 | String E (e.g. "a+b=c") | Count 8 byte classes → fᵢ | f = (0,1,0,1,0,0,3,0) | Count by hand |
| 2.2 | Frequency vector f | pᵢ = fᵢ / Σⱼfⱼ | p = (0,0.2,0,0.2,0,0,0.6,0) | Calculator: 1/5=0.2 |
| 2.3 | Probability vector p | Check: Σpᵢ = 1, pᵢ ≥ 0 | 1.0, all ≥ 0 | Calculator sum |
| **2.4** | **p** | **‖φ(p)‖₂ = 1.0** | **Verify: (√0.2)²×2 + (√0.6)² = 1.0** | **✓ PASS / ✗ STOP** |
| 2.5 | Same E | Parse tree → count 4 node types → τ | τ = (0.6,0.2,0.2,0) | Count nodes by hand |
| 2.6 | Combined Φ(E) = (p, τ) | Check: p ∈ Δ₇, τ ∈ Δ₃ | Both sum to 1 | Calculator |
| 2.7 | Reference set {E₁,...,Eₖ} | For each: compute Φ(Eᵢ), store | Database of (p,τ) pairs | N/A |

**No code yet.** Just the arithmetic of counting and normalizing.

---

## PHASE 3: THE CHAOS GAME (math → convergence)

**Prerequisite:** Phase 2 complete. Reference points exist.

| Step | Input | Formula | Output | Verify |
|------|-------|---------|--------|--------|
| 3.1 | Target Φ(E) = (p,τ) | Pick random reference Φ(Eᵢ) | Starting point | N/A |
| 3.2 | Current point x = (p,τ) | Pick random reference Φ(Eⱼ) | Random selection | N/A |
| 3.3 | x, target | w(x) = ½·x + ½·Φ(Eⱼ) | New point | Calculator: average |
| 3.4 | Iterate n times | x_{k+1} = ½·x_k + ½·Φ(E_{random}) | Sequence x₁, x₂, ... | Each step: calculator |
| **3.5** | **Sequence x₁,...,x₁₀₀** | **Distance between consecutive points** | **Shrinks by ~½ each time** | **✓ PASS / ✗ STOP** |
| 3.6 | Limit point x* | Check: d_F(x*, w(x*)) < ε for all w | Converged | Calculator |

**The contraction factor is ½ = 0.5.** After 20 iterations, error < 2^{-20} ≈ 10^{-6}.
Type `0.5^20` into calculator → 9.5367×10^{-7}.

---

## PHASE 4: THE EIGENSOLID COMPRESSOR (math → one step)

**Prerequisite:** Phase 3 complete. Convergence point x* exists.

| Step | Input | Formula | Output | Verify |
|------|-------|---------|--------|--------|
| 4.1 | Converged point x* = (p*, τ*) | Apply C: average each pair in p* | C(p*) | 4 additions, 4 divisions |
| 4.2 | C(p*) | Apply C again | C(C(p*)) | Same as C(p*) |
| **4.3** | **C(p*) vs C(C(p*))** | **Identical?** | **✓ PASS / ✗ STOP** |
| 4.4 | C(p*) | 4 pair values → q ∈ Δ₃ | (2·C(p*)₁, 2·C(p*)₃, 2·C(p*)₅, 2·C(p*)₇) | Calculator |
| 4.5 | q ∈ Δ₃ | Check: Σqᵢ = 1 | 1.0 | Calculator |
| 4.6 | Original p*, compressed q | KL divergence: Σᵢ p*ᵢ·ln(p*ᵢ/C(p*)ᵢ) | Information loss (nats) | Calculator |

**Total operations for full compression:**
- Chaos game: ~20 iterations × (1 random pick + 8 multiplications + 8 additions)
- Eigensolid: 4 additions + 4 divisions
- **Under 500 arithmetic operations total.**

---

## PHASE 5: THE Φ-CORKSCREW ENCODING (math → spiral index)

**Prerequisite:** Phase 4 complete. Compressed q ∈ Δ₃ exists.

| Step | Input | Formula | Output | Verify |
|------|-------|---------|--------|--------|
| 5.1 | q = (q₁,q₂,q₃,q₄) | c₀ = q₁, c₁ = q₂, c₂ = q₃, c₃ = q₄ | 4 coefficients | Direct |
| 5.2 | (c₀,c₁,c₂,c₃) | Pad to 8 with zeros: (c₀,c₁,c₂,c₃,0,0,0,0) | 8 coefficients | N/A |
| 5.3 | 8 coefficients | Normalize: cᵢ' = cᵢ / Σⱼcⱼ | (c₀',c₁',c₂',c₃',0,0,0,0) | Calculator |
| 5.4 | Normalized coefficients | Phinary packing: n = Σᵢ₌₀⁷ floor(cᵢ' × 256) × 8ⁱ | Integer n | Calculator |
| 5.5 | Integer n | Spiral: f(n) = (√n·cos(nψ), √n·sin(nψ)) | (x,y) ∈ ℝ² | Calculator |
| **5.6** | **Two different q, q'** | **f(n) ≠ f(n')** | **Different (x,y)** | **✓ PASS / ✗ STOP** |

**Verify injectivity (conceptual):** ψ/2π = 1/φ² ≈ 0.381966. This is irrational.
If f(m) = f(n), then (m-n)ψ ∈ 2πℤ, so ψ/2π = k/(m-n) ∈ ℚ. Contradiction.

---

## PHASE 6: FULL PIPELINE VERIFICATION (end-to-end)

**Prerequisite:** Phases 2-5 all PASS individually.

| Step | Input | Operation | Expected | Verify |
|------|-------|-----------|----------|--------|
| 6.1 | "a+b=c" | Phase 2 → p, τ | p=(0,0.2,0,0.2,0,0,0.6,0), τ=(0.6,0.2,0.2,0) | Count by hand |
| 6.2 | (p,τ) | Phase 3 → x* | Converged point in Δ₇×Δ₃ | Check distance shrink |
| 6.3 | x* | Phase 4 → q | q ∈ Δ₃, C(C(q)) = C(q) | One-step idempotent |
| 6.4 | q | Phase 5 → n | Spiral index n ∈ ℕ | Integer result |
| **6.5** | **"x+y=z"** | **Same pipeline** | **Same n (semantically equivalent)** | **✓ PASS** |
| **6.6** | **"p/q=r"** | **Same pipeline** | **Different n (different operation)** | **✓ PASS / ✗ STOP** |

**If 6.6 produces same n as 6.5:** The parse-tree feature is not distinguishing
division from addition. STOP. Go back to Phase 2 and add operator-type features.

---

## PHASE 7: ONLY NOW CODE (all math verified)

**Prerequisite:** Phases 1-6 all PASS. Every number verified by calculator.

| Step | What | Test | Pass Criteria |
|------|------|------|---------------|
| 7.1 | Implement byte-count (2.1-2.2) | Input: "a+b=c" | Output matches hand-count |
| 7.2 | Implement parse-tree (2.5) | Input: "a+b=c" | Output matches hand-count |
| 7.3 | Implement chaos game (3.1-3.5) | Run 100 iterations | Final distance < 10^{-6} |
| 7.4 | Implement eigensolid (4.1-4.4) | Input: any p ∈ Δ₇ | C(C(p)) == C(p) |
| 7.5 | Implement Φ-corkscrew (5.1-5.5) | Input: q ∈ Δ₃ | Output: integer n |
| 7.6 | Full pipeline (6.1-6.6) | "a+b=c", "p/q=r" | Different spiral indices |
| **7.7** | **Cross-check all outputs** | **Same inputs as worksheets** | **Numbers match EXACTLY** | **✓ ALL PASS / ✗ STOP** |

---

## THE STOP CONDITIONS

**STOP and do not proceed if:**

1. Any worksheet number does not match calculator output
2. d_F(C(p), C(q)) ≥ d_F(p,q) in G1.5 or G3.6
3. τ("a+b=c") = τ("p/q=r") in G2.7
4. f(n) = f(m) for n ≠ m in 5.6
5. "a+b=c" and "p/q=r" produce same spiral index in 6.6
6. Code output does not match hand-computed worksheet number in 7.7

**There are no other reasons to stop.** Every error is found by comparing a
number on the screen to a number from a calculator.

---

## PROGRESS TRACKING

| Phase | Status | Date | Verified By |
|-------|--------|------|-------------|
| 0: Foundation | ✅ COMPLETE | 2026-06-23 | Given theorems |
| 1: Three Gates | ✅ WORKSHEETS WRITTEN | 2026-06-23 | G1, G2, G3 worksheets |
| 2: Feature Extraction | ⬜ NOT STARTED | — | — |
| 3: Chaos Game | ⬜ NOT STARTED | — | — |
| 4: Eigensolid | ⬜ NOT STARTED | — | — |
| 5: Φ-Corkscrew | ⬜ NOT STARTED | — | — |
| 6: Full Pipeline | ⬜ NOT STARTED | — | — |
| 7: Code | ⬜ NOT STARTED | — | — |

**Current milestone:** Complete numerical verification of all three gate
worksheets. Every number must match calculator output before proceeding to
Phase 2.