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+# WORKSHEET G3 — Eigensolid Fixed Point
+## Verifiable with any calculator. No English inside formulas.
+
+---
+
+## PART A: The Crossing Operator C (applied numerically)
+
+**INPUT:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
+**Check:** 0.3+0.1+0.15+0.05+0.2+0.08+0.07+0.05 = 1.0 ✓
+
+**FORMULA:**
+```
+C(p)₁ = C(p)₂ = (p₁ + p₂) / 2
+C(p)₃ = C(p)₄ = (p₃ + p₄) / 2
+C(p)₅ = C(p)₆ = (p₅ + p₆) / 2
+C(p)₇ = C(p)₈ = (p₇ + p₈) / 2
+```
+
+**WORK:**
+```
+C(p)₁ = C(p)₂ = (0.3 + 0.1) / 2 = 0.4 / 2 = 0.2
+C(p)₃ = C(p)₄ = (0.15 + 0.05) / 2 = 0.2 / 2 = 0.1
+C(p)₅ = C(p)₆ = (0.2 + 0.08) / 2 = 0.28 / 2 = 0.14
+C(p)₇ = C(p)₈ = (0.07 + 0.05) / 2 = 0.12 / 2 = 0.06
+```
+
+**OUTPUT:** C(p) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+
+**Check:** 0.2+0.2+0.1+0.1+0.14+0.14+0.06+0.06 = 1.0 ✓
+
+---
+
+## PART B: Idempotence C∘C = C (verified numerically)
+
+**INPUT:** C(p) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+
+**WORK:**
+```
+C(C(p))₁ = C(C(p))₂ = (0.2 + 0.2) / 2 = 0.4 / 2 = 0.2
+C(C(p))₃ = C(C(p))₄ = (0.1 + 0.1) / 2 = 0.2 / 2 = 0.1
+C(C(p))₅ = C(C(p))₆ = (0.14 + 0.14) / 2 = 0.28 / 2 = 0.14
+C(C(p))₇ = C(C(p))₈ = (0.06 + 0.06) / 2 = 0.12 / 2 = 0.06
+```
+
+**OUTPUT:** C(C(p)) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+
+**VERIFICATION:** C(C(p)) = C(p) ✓
+
+**RESULT:** C∘C = C. One application reaches the fixed point.
+
+---
+
+## PART C: Image M ≅ Δ₃ (verified numerically)
+
+**Image M:** All vectors with p₁=p₂, p₃=p₄, p₅=p₆, p₇=p₈.
+
+**Map from Δ₃ to M:**
+```
+(q₁, q₂, q₃, q₄) ↦ (q₁/2, q₁/2, q₂/2, q₂/2, q₃/2, q₃/2, q₄/2, q₄/2)
+```
+
+**TEST:** (q₁, q₂, q₃, q₄) = (0.4, 0.2, 0.28, 0.12)
+**Check:** 0.4 + 0.2 + 0.28 + 0.12 = 1.0 ✓
+
+**WORK:**
+```
+φ(q) = (0.4/2, 0.4/2, 0.2/2, 0.2/2, 0.28/2, 0.28/2, 0.12/2, 0.12/2)
+     = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+```
+
+**OUTPUT:** φ(0.4, 0.2, 0.28, 0.12) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+
+**VERIFICATION:** This equals C(p) from Part A. ✓
+
+**Inverse map:**
+```
+ψ(p₁, p₂, ..., p₈) = (2p₁, 2p₃, 2p₅, 2p₇)
+```
+
+**TEST:** ψ(0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+```
+= (2×0.2, 2×0.1, 2×0.14, 2×0.06)
+= (0.4, 0.2, 0.28, 0.12)
+```
+
+**VERIFICATION:** ψ(φ(q)) = q ✓ and φ(ψ(C(p))) = C(p) ✓
+
+---
+
+## PART D: Contraction (verified numerically)
+
+**INPUT:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
+**INPUT:** q = (0.2, 0.2, 0.1, 0.1, 0.15, 0.1, 0.1, 0.05)
+
+**From Part A:** C(p) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
+
+**Compute C(q):**
+```
+C(q)₁ = C(q)₂ = (0.2 + 0.2) / 2 = 0.2
+C(q)₃ = C(q)₄ = (0.1 + 0.1) / 2 = 0.1
+C(q)₅ = C(q)₆ = (0.15 + 0.1) / 2 = 0.125
+C(q)₇ = C(q)₈ = (0.1 + 0.05) / 2 = 0.075
+```
+
+**OUTPUT:** C(q) = (0.2, 0.2, 0.1, 0.1, 0.125, 0.125, 0.075, 0.075)
+**Check:** 0.2+0.2+0.1+0.1+0.125+0.125+0.075+0.075 = 1.0 ✓
+
+---
+
+**STEP 1: Compute d_F(p, q)**
+
+```
+√(p₁q₁) = √(0.3×0.2) = √0.06   = 0.24495
+√(p₂q₂) = √(0.1×0.2) = √0.02   = 0.14142
+√(p₃q₃) = √(0.15×0.1) = √0.015 = 0.12247
+√(p₄q₄) = √(0.05×0.1) = √0.005 = 0.07071
+√(p₅q₅) = √(0.2×0.15) = √0.03  = 0.17321
+√(p₆q₆) = √(0.08×0.1) = √0.008 = 0.08944
+√(p₇q₇) = √(0.07×0.1) = √0.007 = 0.08367
+√(p₈q₈) = √(0.05×0.05) = √0.0025= 0.05000
+```
+
+**Sum:**
+```
+S_pq = 0.24495 + 0.14142 + 0.12247 + 0.07071
+     + 0.17321 + 0.08944 + 0.08367 + 0.05000
+     = 0.97587
+```
+
+**d_F(p,q) = 2·arccos(0.97587)**
+
+Type into calculator: `2 * arccos(0.97587)`
+
+---
+
+**STEP 2: Compute d_F(C(p), C(q))**
+
+```
+√(C(p)₁·C(q)₁) = √(0.2×0.2)  = 0.2
+√(C(p)₂·C(q)₂) = √(0.2×0.2)  = 0.2
+√(C(p)₃·C(q)₃) = √(0.1×0.1)  = 0.1
+√(C(p)₄·C(q)₄) = √(0.1×0.1)  = 0.1
+√(C(p)₅·C(q)₅) = √(0.14×0.125) = √0.0175 = 0.13229
+√(C(p)₆·C(q)₆) = √(0.14×0.125) = 0.13229
+√(C(p)₇·C(q)₇) = √(0.06×0.075) = √0.0045 = 0.06708
+√(C(p)₈·C(q)₈) = √(0.06×0.075) = 0.06708
+```
+
+**Sum:**
+```
+S_CpCq = 0.2 + 0.2 + 0.1 + 0.1 + 0.13229 + 0.13229 + 0.06708 + 0.06708
+       = 0.99874
+```
+
+**d_F(C(p), C(q)) = 2·arccos(0.99874)**
+
+Type into calculator: `2 * arccos(0.99874)`
+
+---
+
+**STEP 3: Compare**
+
+| Value | Calculator Input | Expected |
+|-------|-----------------|----------|
+| arccos(0.97587) | `arccos(0.97587)` | ~0.2197 |
+| arccos(0.99874) | `arccos(0.99874)` | ~0.0502 |
+| d_F(p,q) | `2*arccos(0.97587)` | ~0.4394 |
+| d_F(C(p),C(q)) | `2*arccos(0.99874)` | ~0.1004 |
+
+**VERIFICATION:** d_F(C(p), C(q)) ≈ 0.1004 < d_F(p,q) ≈ 0.4394
+
+**CONTRACTION CONFIRMED:** C shrinks Fisher distance.
+
+**Contraction ratio:** 0.1004 / 0.4394 ≈ 0.228
+
+Type: `0.1004 / 0.4394` → ~0.23
+
+---
+
+## PART E: Strict Inequality (when p, q differ within a pair)
+
+**INPUT:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
+**INPUT:** r = (0.1, 0.3, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
+
+p and r differ only within pair 1: (0.3, 0.1) vs (0.1, 0.3).
+
+**C(p) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)**
+**C(r) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)**
+
+C(p) = C(r) because C averages each pair.
+
+**d_F(C(p), C(r)) = 0**
+
+**d_F(p, r):**
+```
+√(0.3×0.1) = √0.03 = 0.17321
+√(0.1×0.3) = √0.03 = 0.17321
+√(0.15×0.15) = 0.15
+√(0.05×0.05) = 0.05
+√(0.2×0.2) = 0.2
+√(0.08×0.08) = 0.08
+√(0.07×0.07) = 0.07
+√(0.05×0.05) = 0.05
+
+Sum = 0.17321 + 0.17321 + 0.15 + 0.05 + 0.2 + 0.08 + 0.07 + 0.05
+    = 0.94642
+```
+
+**d_F(p,r) = 2·arccos(0.94642)**
+
+Type into calculator: `2 * arccos(0.94642)`
+
+Expected: ~0.66 (significantly > 0)
+
+**VERIFICATION:** d_F(C(p), C(r)) = 0 < 0.66 = d_F(p,r)
+
+**STRICT INEQUALITY CONFIRMED:** When p, r differ within a pair, C collapses them completely.
+
+---
+
+## PART F: Information Loss (computed numerically)
+
+**FORMULA:**
+```
+I_loss(p) = Σₖ₌₁⁴ sₖ · [ (p_{2k-1}/sₖ)·ln((p_{2k-1}/sₖ)/(1/2)) + (p_{2k}/sₖ)·ln((p_{2k}/sₖ)/(1/2)) ]
+```
+
+where sₖ = p_{2k-1} + p_{2k}.
+
+**INPUT:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
+
+**Pair 1:** s₁ = 0.3 + 0.1 = 0.4
+```
+a = 0.3/0.4 = 0.75, b = 0.1/0.4 = 0.25
+KL = 0.75·ln(0.75/0.5) + 0.25·ln(0.25/0.5)
+   = 0.75·ln(1.5) + 0.25·ln(0.5)
+   = 0.75·0.40547 + 0.25·(-0.69315)
+   = 0.30410 - 0.17329
+   = 0.13081
+```
+Term 1: s₁ × KL = 0.4 × 0.13081 = 0.05232
+
+**Pair 2:** s₂ = 0.15 + 0.05 = 0.2
+```
+a = 0.15/0.2 = 0.75, b = 0.05/0.2 = 0.25
+KL = 0.75·ln(1.5) + 0.25·ln(0.5)
+   = 0.13081 (same as above)
+```
+Term 2: s₂ × KL = 0.2 × 0.13081 = 0.02616
+
+**Pair 3:** s₃ = 0.2 + 0.08 = 0.28
+```
+a = 0.2/0.28 = 0.71429, b = 0.08/0.28 = 0.28571
+KL = 0.71429·ln(0.71429/0.5) + 0.28571·ln(0.28571/0.5)
+   = 0.71429·ln(1.42858) + 0.28571·ln(0.57142)
+   = 0.71429·0.35668 + 0.28571·(-0.55962)
+   = 0.25477 - 0.15989
+   = 0.09488
+```
+Term 3: s₃ × KL = 0.28 × 0.09488 = 0.02657
+
+**Pair 4:** s₄ = 0.07 + 0.05 = 0.12
+```
+a = 0.07/0.12 = 0.58333, b = 0.05/0.12 = 0.41667
+KL = 0.58333·ln(0.58333/0.5) + 0.41667·ln(0.41667/0.5)
+   = 0.58333·ln(1.16667) + 0.41667·ln(0.83333)
+   = 0.58333·0.15415 + 0.41667·(-0.18232)
+   = 0.08992 - 0.07597
+   = 0.01395
+```
+Term 4: s₄ × KL = 0.12 × 0.01395 = 0.00167
+
+---
+
+**TOTAL:**
+```
+I_loss(p) = 0.05232 + 0.02616 + 0.02657 + 0.00167
+          = 0.10672 nats
+```
+
+**In bits:** 0.10672 / ln(2) = 0.10672 / 0.69315 = 0.15396 bits
+
+**VERIFY on calculator:**
+```
+0.05232 + 0.02616 + 0.02657 + 0.00167
+```
+Result: ~0.1067
+
+---
+
+## SUMMARY (all verified numerically)
+
+| Claim | Verification Method | Result |
+|-------|---------------------|--------|
+| C(p) computed | 8 additions, 4 divisions | ✓ sums to 1.0 |
+| C∘C = C | Apply C twice, compare | ✓ identical |
+| Image M ≅ Δ₃ | Map (0.4,0.2,0.28,0.12) → C(p) | ✓ inverse works |
+| Contraction d_F(C(p),C(q)) < d_F(p,q) | Calculator: arccos comparison | ✓ 0.1004 < 0.4394 |
+| Strict inequality | p, r differ in pair 1 only | ✓ 0 < 0.66 |
+| Information loss | 4 KL divergences, weighted | ✓ 0.1067 nats |