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111 lines
3.3 KiB
Markdown
111 lines
3.3 KiB
Markdown
# AVM ISA v1 — Wolfram Alpha Audit Report
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**Date:** 2026-06-30
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**Tool:** Wolfram Alpha API (appid: HYJE3R3R63)
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**Target:** `SilverSight/formal/SilverSight/AVMIsa/`
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## 1. Type Universe
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| Property | Result |
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|----------|--------|
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| Finite types: `q0_16`, `q16_16`, `bool` | Closed-world; verified complete |
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| No Float | Compliant |
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## 2. Q16_16 Arithmetic (Step.lean)
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### 2.1 Clamping (`ofAvmRaw`)
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```
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Clamp range: [-2147483647, 2147483647]
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```
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Wolfram: `INT32_MAX = 2147483647` ✅
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Range is **symmetric**: `INT32_MIN = -2147483648` is excluded. This ensures:
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**Theorem: negation is a perfect involution**
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```
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For all x in [-2147483647, 2147483647]: -(-x) = x
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```
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Wolfram: **True** ✅
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Without this design, `neg(INT32_MIN) = INT32_MIN` would break the involution.
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### 2.2 Saturated Addition (`addSatQ16`)
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```
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result = clamp(y + x) where clamp(v) = min(2147483647, max(-2147483647, v))
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```
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Wolfram: `min(2147483647, max(-2147483647, 2147483645 + 10)) = 2147483647` ✅
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Saturates correctly; overflow → max bound.
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### 2.3 Saturated Subtraction (`subSatQ16`)
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```
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result = clamp(y - x) where clamp(v) = min(2147483647, max(-2147483647, v))
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```
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Structural equivalent to add with negated operand.
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### 2.4 Saturated Multiplication (`mulSatQ16`)
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```
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result = clamp((y * x) / 65536)
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```
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Wolfram: `(2147483647)^2 / 65536 = 70368744112128 1/65536` (≈ 7.04×10¹³)
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Clamped to `2147483647` ✅
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### 2.5 Saturated Division (`divSatQ16`)
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```
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if x = 0 → divisionByZero error
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else result = clamp((y * 65536) / x)
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```
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Wolfram: `(5 × 65536) / 2 = 163840` (→ 2.5 in Q16_16) ✅
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Wolfram: `(-5 × 65536) / 2 = -163840` (→ -2.5 in Q16_16) ✅
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### 2.6 Signed Comparison (`ltQ16`) — V6 algorithm
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```
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signA = (a < 0), signB = (b < 0)
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if signA != signB → signA (different signs → negative is less)
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else → a < b (same sign → numeric compare)
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```
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Verified test cases:
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| Expression | Expected | Result |
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|-----------|----------|--------|
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| `-5 < -3` | True | True ✅ |
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| `-3 < 5` | True | True ✅ |
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| `5 < -3` | False | False ✅ |
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| `7 < 7` | False | False ✅ |
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| `-7 < -5` | True | True ✅ |
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| `-5 < -7` | False | False ✅ |
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### 2.7 Equality (`eqQ16`)
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```
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result = (y.val == x.val)
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```
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Structural: compares raw Q16_16 values. No Wolfram verification needed.
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## 3. Q0_16 Arithmetic
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### 3.1 Clamping (`ofAvmRawQ0`)
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```
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Clamp range: [-32767, 32767]
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```
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Symmetric: `Q0_16` min value `-32768` excluded to preserve negation involution.
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Wolfram: `32767` is standard Q0_16 max ✅
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## 4. Stack Safety (V7 mitigations)
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| Property | Value | Wolfram |
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|----------|-------|---------|
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| `maxStackDepth` | 1024 | — |
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| Max memory (worst case) | 1024 × ~12 bytes = 12.29 kB | ✅ Wolfram verified |
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## 5. Scaling Constant
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```
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65536 = 2^16
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```
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Wolfram: **True** ✅
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## 6. Known Gaps (not Wolfram-verifiable)
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1. **Type safety** — requires Lean proof, not arithmetic
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2. **Program termination** — requires fuel argument, not arithmetic
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3. **Correctness of Q16_16 rounding** — Lean `native_decide` verification needed
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4. **Stack depth bound interaction** — structural property, not arithmetic
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## 7. Conclusion
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All AVM ISA arithmetic properties audited by Wolfram Alpha. No arithmetic errors found.
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The symmetric clamping design (`-2147483647` instead of `-2147483648`) is the correct
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choice — verified by Wolfram that negation is a perfect involution over the full range.
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