fix: restore GAUGE_THEORY_GOAL.md from experimental repo (d0264e38)

My stub replaced the real document. Restoring original: 7-step
lattice gauge theory correspondence from Baker-Hopf coupling to
Bianchi identity, with validated empirical chain on the SilverSight
side and derivation targets on the gauge theory side.

6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md

@@ -1,114 +1,221 @@
-# Gauge Theory Goal
+# Gauge Theory Goal: SilverSight as Lattice Gauge Theory
 
-## Purpose
+## Status: BEAUTIFUL_PROVISIONAL — goal statement, not yet formalized
 
-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.
+## The Goal
 
-## Why Gauge Theory
+Derive the SilverSight model (SU(2) quaternion spins with Baker-Hopf coupling)
+FROM lattice gauge theory first principles, showing that:
 
-The 2×2 circulant block [[σ,τ],[τ,σ]] has the structure of a gauge field:
+1. The Baker-Hopf coupling IS the gauge connection (not an ansatz)
+2. Frustration IS the Wilson loop holonomy (gauge-invariant observable)
+3. The Baker Λ IS the field strength (curvature)
+4. Ground state degeneracy IS the topological sector count
+5. The AT phases ARE confinement/Higgs/Coulomb phases
+6. The YBE IS the gauge transformation integrability
+7. The NR bracket MC equation IS the Bianchi identity
 
-| 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 Correspondence (working backwards from gauge theory)
 
-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.
+### Step 1: Gauge Field → Baker-Hopf Coupling
 
-## Specific Goals
+In lattice gauge theory, the gauge field lives on LINKS (not sites).
+The link variable U_ij ∈ SU(2) is the parallel transport from site i to j.
 
-### 1. Gauge group identification
+  U_ij = P exp(∫_i^j A_μ dx^μ)  (path-ordered exponential)
 
-Determine the gauge group G such that the crossing matrix is a connection
-on a G-bundle over the 8-strand braid space.
+For the SilverSight lattice (Sidon-addressed, all-pairs):
+  U_ij = exp(J_ij)  where J_ij is the connection 1-form
 
-- **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
+The Baker-Hopf coupling J_ij = log(a_i + a_j) · n̂_ij^Hopf is the
+CONNECTION, and the parallel transport is:
+  U_ij = exp(log(a_i + a_j) · n̂_ij) = (a_i + a_j)^(n̂_ij)
 
-### 2. DFT as gauge transformation
+This is the quaternion-valued parallel transport, where:
+- The MAGNITUDE (a_i + a_j) is the Baker weight (transcendental)
+- The DIRECTION n̂_ij is the Hopf fibre (geometric)
+- The COMBINATION is the gauge connection
 
-Prove that the DFT diagonalization [[σ,τ],[τ,σ]] → (σ+τ, σ-τ) is a gauge
-transformation to the mass basis.
+**Derivation goal:** show that the most general SU(2)-valued connection
+on a Sidon-addressed lattice that is (a) address-dependent, (b)
+transcendental (from the PFE/Baker framework), and (c) Hopf-fibre-directed
+IS the Baker-Hopf coupling. No other choice satisfies all three constraints.
 
-- **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
+### Step 2: Wilson Loop → Frustration
 
-### 3. Overhead as gauge coupling
+The Wilson loop around a triangle (i,j,k) is:
+  W_ijk = U_ij · U_jk · U_ki = Tr(P exp(∮ A))
 
-Map each Yang-MillsPerformance layer to a gauge field with coupling
-constant g_i = overheadFactor(layer_i).
+The loop is TRIVIAL (W = identity) when the connection is flat (no
+curvature inside the loop). It is NON-TRIVIAL when there is curvature —
+which is exactly FRUSTRATION.
 
-- **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
+  frustrated(i,j,k) ↔ W_ijk ≠ 1 ↔ F_ijk ≠ 0
 
-### 4. Compression bound from gauge invariance
+The frustration count = number of non-trivial Wilson loops = number of
+plaquettes with non-zero curvature. This is the standard lattice gauge
+theory measure of topological charge.
 
-Prove that the 8-mul bound follows from gauge invariance, not just
-circulant structure.
+**Derivation goal:** show that the SilverSight frustration count
+(frustrated triangles from chiral label signs) equals the Wilson loop
+non-triviality count for the Baker-Hopf connection.
 
-- **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
+### Step 3: Field Strength → Baker Λ
 
-### 5. Wilson loop / CRT multiplexer connection
+The field strength (curvature) is:
+  F_ij = dA + A ∧ A = ∂_i A_j - ∂_j A_i + [A_i, A_j]
 
-Determine whether the CRT multiplexer (Chinese Remainder Theorem channel
-separation) corresponds to a Wilson loop observable.
+For a discrete lattice:
+  F_ijk = U_ij · U_jk · U_ki - 1  (the plaquette holonomy minus identity)
 
-- **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
+The total field strength (sum over all plaquettes):
+  |F|² = Σ_ijk |F_ijk|² = Σ_ijk |W_ijk - 1|²
 
-## Success Criteria
+The Baker functional Λ = Σ 1/(a_j - a_i) · log(a_i + a_j) is the
+INTEGRATED field strength — it's the sum of curvature terms weighted
+by the PFE residues (1/(a_j - a_i)).
 
-| # | 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 |
+  Λ = Σ_{i<j} w_ij · log(a_i + a_j)  where w_ij = 1/(a_j - a_i)
 
-## Dependencies
+This is the lattice gauge theory field strength in the PFE basis:
+  Λ = ∫ F · (PFE kernel) = ∫ F · 1/(z - a) dz
 
-- `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
+**Derivation goal:** show that Λ (Baker functional) = the integrated
+field strength of the Baker-Hopf connection, in the PFE basis.
 
-## Non-Goals
+### Step 4: Topological Sectors → Ground State Degeneracy
 
-- 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
+In gauge theory, the vacuum has multiple topological sectors labeled by
+the instanton number (winding number):
+  ν = (1/8π²) ∫ Tr(F ∧ F)
 
-## References
+Different ν → different vacua → degeneracy.
 
-- 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
+In the SilverSight model:
+  - Ground state degeneracy = number of distinct vacuum configurations
+  - Frustrated plaquettes = instantons (topological defects)
+  - More frustration → more instantons → potentially more sectors
+
+But the validation showed: MORE frustration → LESS degeneracy (r=-0.33).
+This is the GAUGE THEORY prediction: in a confining phase, instantons
+break the degeneracy by selecting a unique vacuum. The Ashkin-Teller
+Baxter phase (the QAOA hardness peak) is the gauge theory CONFINEMENT
+phase where the vacuum is unique.
+
+**Derivation goal:** show that the AT ground state degeneracy equals
+the gauge theory topological sector count, and the frustration-degeneracy
+anticorrelation (r=-0.33) is the confinement mechanism.
+
+### Step 5: AT Phases → Gauge Theory Phases
+
+The Ashkin-Teller model has 4 phases (arXiv:2301.10609). In gauge theory:
+
+| AT phase | Gauge theory phase | Wilson loop | QAOA hardness |
+|---|---|---|---|
+| Ordered (ferromagnetic) | Higgs phase | W → 1 (area law) | Easy |
+| Baxter (critical) | Confinement phase | W → 0 (area law) | **Hard** |
+| Disordered (paramagnetic) | Coulomb phase | W → const (perimeter law) | Easy |
+| Critical line | Phase transition | W ~ power law | Medium |
+
+The Baxter phase = confinement is the key prediction: QAOA hardness peaks
+at the confinement phase, where the Wilson loop follows area law and the
+vacuum is unique (no degeneracy).
+
+**Derivation goal:** map each AT phase to a gauge theory phase and verify
+the Wilson loop behavior matches.
+
+### Step 6: YBE → Gauge Integrability
+
+The Yang-Baxter equation (proven in YangBaxter.lean by rfl):
+  R₁₂ R₁₃ R₂₃ = R₂₃ R₁₃ R₁₂
+
+In gauge theory, this is the FLATNESS condition: the gauge transformation
+is path-independent (the connection is integrable). The R-matrix IS the
+gauge transformation, and YBE says the gauge transformation around any
+triangle is trivial (W = 1 for the gauge-transformed connection).
+
+BUT: the SilverSight model has FRUSTRATED triangles (W ≠ 1). This means
+the YBE holds for the R-matrix (the gauge transformation) but NOT for the
+Baker-Hopf connection (the physical gauge field). The R-matrix is the
+"integrable part" (the flat background), and frustration comes from the
+"non-integrable part" (the curvature on top of the flat background).
+
+  H = H_flat (YBE, integrable) + H_curved (frustration, non-integrable)
+
+**Derivation goal:** decompose the Hamiltonian into flat (YBE) and curved
+(frustration) parts, showing the YBE governs the background and the
+Baker-Hopf curvature governs the frustration.
+
+### Step 7: NR Bracket → Bianchi Identity
+
+The Maurer-Cartan equation (proven in CartanConnection.lean):
+  d_CE μ + ½[μ,μ]_NR = 0  →  d_CE μ = 0  →  [μ,μ]_NR = 0
+
+In gauge theory, the Bianchi identity is:
+  dF = D∧F = dF + [A, F] = 0
+
+The NR bracket [μ,μ] = 0 IS the Bianchi identity for the Sidon crossing
+matrix μ viewed as a gauge connection. This is already proven — the
+formalization just needs the gauge-theoretic interpretation.
+
+**Derivation goal:** show that d_CE μ = 0 (proven) = the Bianchi identity
+dF = 0 for the Baker-Hopf connection.
+
+## Filling In From Both Sides
+
+### From the SilverSight side (already done):
+- [x] SU(2) quaternion spins (HopfFibration.lean)
+- [x] Z₂⁴ Ashkin-Teller projection (ChiralClockModel.lean)
+- [x] Baker-Hopf coupling (ChiralClockModel.lean)
+- [x] Chiral labels (mod 4) → coupling signs (validated)
+- [x] Λ → frustration (r=0.41, validated)
+- [x] Frustration → degeneracy (r=-0.33, validated)
+- [x] YBE proven (YangBaxter.lean, by rfl)
+- [x] NR bracket MC equation proven (CartanConnection.lean)
+- [x] PFE of cot = E₁ (Baker connection, validated)
+
+### From the gauge theory side (to derive):
+- [ ] Gauge field A → Baker-Hopf connection J_ij
+- [ ] Wilson loop W → frustration count
+- [ ] Field strength F → Baker Λ
+- [ ] Topological sectors → ground state degeneracy
+- [ ] AT phases → confinement/Higgs/Coulomb
+- [ ] YBE → gauge integrability (flat background)
+- [ ] Bianchi identity dF = 0 → NR bracket [μ,μ] = 0
+
+### The meeting point:
+Both sides converge at: **the Baker-Hopf coupling is the unique SU(2)
+gauge connection on a Sidon-addressed lattice that is both
+transcendental (Baker/PFE) and integrable (YBE/MC).**
+
+The SilverSight side shows this works empirically (validated chain).
+The gauge theory side would show it's necessary (derived from first
+principles). Together: the model is both sufficient and necessary.
+
+## Implementation Order
+
+1. **Immediate:** fix ChiralClockModel.lean to use D5 chirality (not D4)
+2. **Immediate:** close the 4 sorries in ChiralClockModel.lean
+3. **Short-term:** write the gauge correspondence as Lean theorems
+   (sketch statements, prove the easy ones)
+4. **Medium-term:** derive Baker-Hopf from gauge first principles
+5. **Medium-term:** show frustration = Wilson loop (formal proof)
+6. **Long-term:** connect AT phases to gauge phases (requires MC simulation
+   or exact solution — the AT model is exactly solvable)
+
+## Key Insight
+
+The entire research program — compression, PFE, Baker, Ising, frustration,
+benchmark — was discovering the gauge structure from BELOW (empirically).
+Gauge theory gives the structure from ABOVE (derivation). The two meet
+at the Baker-Hopf connection: the unique transcendental SU(2) gauge
+connection on a Sidon lattice.
+
+The YBE (proven) is the flatness condition (integrability).
+The NR bracket MC (proven) is the Bianchi identity (consistency).
+The Baker Λ is the field strength (curvature).
+The frustration is the Wilson loop (topological charge).
+The degeneracy is the topological sector count (vacuum structure).
+
+Everything is already in the codebase — it just needs the gauge-theoretic
+interpretation to unify it.