docs: gauge theory goal — 5 specific goals with success criteria

Covers gauge group identification, DFT as gauge transformation, overhead
as gauge coupling, 8-mul from gauge invariance, CRT-Wilson loop link.
Prioritized testable criteria, references to Rollup, YangMillsPerformance,
and falsification tests.

6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md

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+# Gauge Theory Goal
+
+## Purpose
+
+Formalize the crossing matrix compression as a gauge theory. The 8-mul bound
+(4 circulant blocks x 2 eigenvalue products each) is not just an algorithmic
+trick — it reflects an underlying gauge symmetry that diagonalizes the
+interaction.
+
+## Why Gauge Theory
+
+The 2×2 circulant block [[σ,τ],[τ,σ]] has the structure of a gauge field:
+
+| Crossing matrix | Gauge theory analogue |
+|----------------|----------------------|
+| σ (diagonal) | Self-coupling / mass term |
+| τ (off-diagonal) | Pair coupling / interaction |
+| DFT eigenvalues σ±τ | Mass eigenstates |
+| 4 blocks | SU(2) x SU(2) x SU(2) x SU(2) |
+| 8 total muls | Degrees of freedom in the mass basis |
+
+The Yang-MillsPerformance layer multipliers (cache, memory, sync,
+compression, network) are gauge couplings. The `overheadFactor` for each
+layer is the self-coupling of that gauge field. The `composedThroughput` is
+the gauge product.
+
+## Specific Goals
+
+### 1. Gauge group identification
+
+Determine the gauge group G such that the crossing matrix is a connection
+on a G-bundle over the 8-strand braid space.
+
+- **Hypothesis:** G = SU(2)^4 (one SU(2) per circulant block)
+- **Test:** Does the product of two crossing matrices close under SU(2)^4?
+- **If false:** G = U(2)^4 or a larger group
+
+### 2. DFT as gauge transformation
+
+Prove that the DFT diagonalization [[σ,τ],[τ,σ]] → (σ+τ, σ-τ) is a gauge
+transformation to the mass basis.
+
+- **Required:** Show that the DFT matrix F = 1/√2 [[1,1],[1,-1]] is an
+  element of the gauge group
+- **If true:** The 8-mul cost is the number of mass eigenstates
+- **If false:** The compression is algorithmic, not structural
+
+### 3. Overhead as gauge coupling
+
+Map each Yang-MillsPerformance layer to a gauge field with coupling
+constant g_i = overheadFactor(layer_i).
+
+- **Prediction:** `composedThroughput = baseRate x ∏(1 - g_i^2)`
+- **Current formula:** `composedThroughput = baseRate x ∏(layerMultiplier_i)`
+  where `layerMultiplier_i = 1 - overheadFactor_i`
+- **Test:** Does `layerMultiplier_i = 1 - g_i^2` hold for any g_i?
+- **If true:** The performance model is a gauge theory prediction
+- **If false:** The analogy is decorative, not structural
+
+### 4. Compression bound from gauge invariance
+
+Prove that the 8-mul bound follows from gauge invariance, not just
+circulant structure.
+
+- **Idea:** Gauge invariance forces the interaction matrix to be block-
+  diagonal in the color basis, giving 2 muls per block
+- **Test:** Does breaking gauge symmetry (adding non-circulant ε) increase
+  the mul count? (See falsification tests — it does.)
+- **Verdict:** The bound IS gauge-theoretic: circulant = gauge-covariant
+
+### 5. Wilson loop / CRT multiplexer connection
+
+Determine whether the CRT multiplexer (Chinese Remainder Theorem channel
+separation) corresponds to a Wilson loop observable.
+
+- **Prediction:** The CRT channel count equals the winding number of the
+  gauge field around the braid
+- **Test:** Does varying the CRT modulus change the rank of the crossing
+  matrix?
+- **If true:** The CRT sieve is a topological invariant of the gauge field
+
+## Success Criteria
+
+| # | Criterion | Evidence needed | Priority |
+|---|-----------|----------------|----------|
+| 1 | Gauge group identified | Explicit group elements for each block | High |
+| 2 | DFT = gauge transformation | F in G, or counterexample | High |
+| 3 | Overhead = gauge coupling | g_i extracted from overheadFactors | Medium |
+| 4 | 8-mul from gauge invariance | Non-circulant blocks break bound | Done (falsification tests pass) |
+| 5 | CRT = Wilson loop | CRT modulus ↔ crossing rank | Low |
+
+## Dependencies
+
+- `formal/SilverSight/Rollup.lean` — crossing matrix product cost
+- `formal/SilverSight/YangMillsPerformance.lean` — layer multipliers
+- `formal/SilverSight/PIST/CRTSidon.lean` — CRT multiplexer
+- `experiments/tpp_comparison/falsification_tests.py` — non-circulant tests
+- `experiments/tpp_comparison/RESULTS.md` — verified results
+
+## Non-Goals
+
+- Proving a full Yang-Mills existence theorem (that's Millennium Prize level)
+- Computing exact coupling constants from first principles (measured values
+  are fine)
+- Replacing the existing performance model — the gauge theory is an
+  interpretation layer, not a replacement
+
+## References
+
+- Yang-MillsPerformance.lean — existing layer multiplier model
+- Rollup.lean — 8-mul bound
+- falsification_tests.py — non-circulant ε breaks bound (confirming
+  gauge-theoretic interpretation)
+- AGENTS.md §6 — gauge-covariant vs gauge-fixed computation