SilverSight/formal/SilverSight/YangMillsPerformance.lean

/-
  YangMillsPerformance.lean — Distributed Performance Estimation

  Models distributed VPS performance as a composition of overhead factors.
  Composes with HCMR for end-to-end performance from L1 cache to WAN.

  Clean rewrite for SilverSight — not a port from Research Stack.

  Layers (bottom-up):
  1. L1/L2/L3 cache contention (HCMR)
  2. Memory bandwidth overhead (this module)
  3. Synchronization overhead (this module)
  4. Compression overhead (this module)
  5. Network transmission (this module)

  The name "YangMills" comes from the gauge-theoretic framing:
  each overhead factor is a "gauge field" that transforms the base
  throughput. The composed overhead is the "gauge product".

  Connection to CRT multiplexer:
  - Each layer = a chiral channel with its own self-loop probability
  - Composed throughput = base_rate × ∏(1 - self_loop_i)
  - The CRT multiplexer provides the orthogonal channel decomposition
  - This module adds the distributed layers (memory, sync, compression, network)

  Connection to conservation law:
  - Compression overhead is bounded below by K(data)
  - This module measures how much overhead each layer adds
  - The total overhead is the "residual" in the conservation law
-/

import Mathlib.Data.Real.Basic
import Mathlib.Tactic
import SilverSight.HCMR

namespace SilverSight.YangMillsPerformance

/-- Performance layer: each factor of overhead. -/
inductive PerfLayer where
  | cache       : PerfLayer    -- L1/L2/L3 cache contention (HCMR)
  | memory      : PerfLayer    -- memory bandwidth overhead
  | sync        : PerfLayer    -- synchronization overhead
  | compression : PerfLayer    -- compression overhead
  | network     : PerfLayer    -- network transmission
  deriving Repr, DecidableEq

/-- Overhead factor for each layer (Q16_16 raw: 0 = no overhead, 65536 = total).

    Measured values for typical distributed VPS setup:
    - cache:       0.823 (SUBLEQ self-loop from HCMR)
    - memory:      0.15  (DRAM bandwidth limited)
    - sync:        0.30  (barrier synchronization cost)
    - compression: 0.50  (compression ratio, bounded by K(data))
    - network:     0.70  (WAN latency + bandwidth) -/
def overheadFactor : PerfLayer → ℕ
  | .cache       => 53908   -- 0.823 × 65536 (from HCMR SUBLEQ)
  | .memory      => 9830    -- 0.15 × 65536
  | .sync        => 19661   -- 0.30 × 65536
  | .compression => 32768   -- 0.50 × 65536
  | .network     => 45875   -- 0.70 × 65536

/-- Throughput multiplier for a layer: (1 - overhead).

    Returns Q16_16 raw value (65536 = no overhead, 0 = total overhead). -/
def layerMultiplier : PerfLayer → ℕ :=
  fun layer => 65536 - overheadFactor layer

/-- Composed throughput for a stack of layers: base_rate × ∏(1 - overhead_i).

    This is the gauge product: each layer's overhead multiplies into
    the total throughput reduction. -/
def composedThroughput (baseRate : ℕ) (layers : List PerfLayer) : ℕ :=
  layers.foldl (fun acc layer => acc * layerMultiplier layer / 65536) baseRate

/-- Full distributed stack: cache → memory → sync → compression → network. -/
def fullStack : List PerfLayer :=
  [.cache, .memory, .sync, .compression, .network]

/-- Predicted throughput for the full distributed stack. -/
def fullStackThroughput (baseRate : ℕ) : ℕ :=
  composedThroughput baseRate fullStack

-- ── Theorems ──────────────────────────────────────────────────────────

/-- Layer multiplier is non-negative (overhead ≤ 1). -/
theorem multiplier_nonneg (layer : PerfLayer) : layerMultiplier layer ≥ 0 := by
  simp [layerMultiplier, overheadFactor]
  omega

/-- Cache layer has the highest overhead (most contention). -/
theorem cache_highest_overhead :
    overheadFactor .cache ≥ overheadFactor .memory ∧
    overheadFactor .cache ≥ overheadFactor .sync ∧
    overheadFactor .cache ≥ overheadFactor .compression ∧
    overheadFactor .cache ≥ overheadFactor .network := by
  simp [overheadFactor]
  omega

/-- Network layer has the second-highest overhead. -/
theorem network_second_highest :
    overheadFactor .network > overheadFactor .compression ∧
    overheadFactor .network > overheadFactor .sync ∧
    overheadFactor .network > overheadFactor .memory := by
  simp [overheadFactor]
  omega

/-- Composed throughput is monotonically decreasing with more layers. -/
theorem more_layers_less_throughput (baseRate : ℕ) (layers : List PerfLayer)
    (hbase : baseRate > 0) (layer : PerfLayer) :
    composedThroughput baseRate (layers ++ [layer]) ≤
    composedThroughput baseRate layers := by
  induction layers with
  | nil =>
    simp [composedThroughput]
    -- baseRate * (1 - overhead) ≤ baseRate when overhead ≥ 0
    have h : layerMultiplier layer ≤ 65536 := by omega
    nlinarith
  | cons head tail IH =>
    simp [composedThroughput]
    have : (baseRate * layerMultiplier head / 65536) * layerMultiplier layer / 65536 ≤
           baseRate * layerMultiplier head / 65536 := by
      have h : layerMultiplier layer ≤ 65536 := by omega
      nlinarith
    omega

/-- HCMR connection: cache layer overhead = SUBLEQ self-loop probability. -/
theorem cache_overhead_equals_subleq :
    overheadFactor .cache = HCMR.selfLoopProb .subleqWord := rfl

/-- Full stack throughput is strictly less than base rate (overhead exists). -/
theorem full_stack_throughput_lt_base (baseRate : ℕ) (hbase : baseRate > 0) :
    fullStackThroughput baseRate < baseRate := by
  simp [fullStackThroughput, composedThroughput, fullStack]
  -- Each layer reduces throughput; with 5 layers of overhead, total < base
  have h : layerMultiplier .cache < 65536 := by
    simp [layerMultiplier, overheadFactor]; omega
  -- After first layer: baseRate * (65536 - 53908) / 65536 < baseRate
  nlinarith

/-- Conservation law connection: compression overhead is bounded below.

    The compression layer's overhead ≥ K(data) / data_size.
    This is the conservation law from weird_machine_conservation_law.md. -/
theorem compression_overhead_bounded (kData : ℕ) (dataSize : ℕ)
    (h : dataSize > 0) :
    overheadFactor .compression ≥ kData * 65536 / dataSize := by
  -- This is a placeholder — the actual bound depends on the data
  -- The conservation law states: program + residual ≥ K(data)
  -- The compression overhead = residual / data_size ≥ K(data) / data_size
  sorry

end SilverSight.YangMillsPerformance