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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.
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# Gauge Theory Goal
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# Gauge Theory Goal: SilverSight as Lattice Gauge Theory
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## Purpose
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## Status: BEAUTIFUL_PROVISIONAL — goal statement, not yet formalized
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Formalize the crossing matrix compression as a gauge theory. The 8-mul bound
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## The Goal
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(4 circulant blocks x 2 eigenvalue products each) is not just an algorithmic
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trick — it reflects an underlying gauge symmetry that diagonalizes the
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interaction.
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## Why Gauge Theory
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Derive the SilverSight model (SU(2) quaternion spins with Baker-Hopf coupling)
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FROM lattice gauge theory first principles, showing that:
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The 2×2 circulant block [[σ,τ],[τ,σ]] has the structure of a gauge field:
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1. The Baker-Hopf coupling IS the gauge connection (not an ansatz)
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2. Frustration IS the Wilson loop holonomy (gauge-invariant observable)
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3. The Baker Λ IS the field strength (curvature)
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4. Ground state degeneracy IS the topological sector count
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5. The AT phases ARE confinement/Higgs/Coulomb phases
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6. The YBE IS the gauge transformation integrability
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7. The NR bracket MC equation IS the Bianchi identity
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| Crossing matrix | Gauge theory analogue |
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## The Correspondence (working backwards from gauge theory)
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|----------------|----------------------|
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| σ (diagonal) | Self-coupling / mass term |
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| τ (off-diagonal) | Pair coupling / interaction |
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| DFT eigenvalues σ±τ | Mass eigenstates |
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| 4 blocks | SU(2) x SU(2) x SU(2) x SU(2) |
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| 8 total muls | Degrees of freedom in the mass basis |
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The Yang-MillsPerformance layer multipliers (cache, memory, sync,
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### Step 1: Gauge Field → Baker-Hopf Coupling
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compression, network) are gauge couplings. The `overheadFactor` for each
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layer is the self-coupling of that gauge field. The `composedThroughput` is
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the gauge product.
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## Specific Goals
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In lattice gauge theory, the gauge field lives on LINKS (not sites).
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The link variable U_ij ∈ SU(2) is the parallel transport from site i to j.
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### 1. Gauge group identification
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U_ij = P exp(∫_i^j A_μ dx^μ) (path-ordered exponential)
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Determine the gauge group G such that the crossing matrix is a connection
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For the SilverSight lattice (Sidon-addressed, all-pairs):
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on a G-bundle over the 8-strand braid space.
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U_ij = exp(J_ij) where J_ij is the connection 1-form
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- **Hypothesis:** G = SU(2)^4 (one SU(2) per circulant block)
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The Baker-Hopf coupling J_ij = log(a_i + a_j) · n̂_ij^Hopf is the
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- **Test:** Does the product of two crossing matrices close under SU(2)^4?
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CONNECTION, and the parallel transport is:
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- **If false:** G = U(2)^4 or a larger group
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U_ij = exp(log(a_i + a_j) · n̂_ij) = (a_i + a_j)^(n̂_ij)
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### 2. DFT as gauge transformation
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This is the quaternion-valued parallel transport, where:
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- The MAGNITUDE (a_i + a_j) is the Baker weight (transcendental)
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- The DIRECTION n̂_ij is the Hopf fibre (geometric)
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- The COMBINATION is the gauge connection
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Prove that the DFT diagonalization [[σ,τ],[τ,σ]] → (σ+τ, σ-τ) is a gauge
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**Derivation goal:** show that the most general SU(2)-valued connection
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transformation to the mass basis.
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on a Sidon-addressed lattice that is (a) address-dependent, (b)
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transcendental (from the PFE/Baker framework), and (c) Hopf-fibre-directed
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IS the Baker-Hopf coupling. No other choice satisfies all three constraints.
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- **Required:** Show that the DFT matrix F = 1/√2 [[1,1],[1,-1]] is an
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### Step 2: Wilson Loop → Frustration
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element of the gauge group
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- **If true:** The 8-mul cost is the number of mass eigenstates
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- **If false:** The compression is algorithmic, not structural
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### 3. Overhead as gauge coupling
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The Wilson loop around a triangle (i,j,k) is:
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W_ijk = U_ij · U_jk · U_ki = Tr(P exp(∮ A))
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Map each Yang-MillsPerformance layer to a gauge field with coupling
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The loop is TRIVIAL (W = identity) when the connection is flat (no
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constant g_i = overheadFactor(layer_i).
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curvature inside the loop). It is NON-TRIVIAL when there is curvature —
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which is exactly FRUSTRATION.
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- **Prediction:** `composedThroughput = baseRate x ∏(1 - g_i^2)`
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frustrated(i,j,k) ↔ W_ijk ≠ 1 ↔ F_ijk ≠ 0
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- **Current formula:** `composedThroughput = baseRate x ∏(layerMultiplier_i)`
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where `layerMultiplier_i = 1 - overheadFactor_i`
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- **Test:** Does `layerMultiplier_i = 1 - g_i^2` hold for any g_i?
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- **If true:** The performance model is a gauge theory prediction
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- **If false:** The analogy is decorative, not structural
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### 4. Compression bound from gauge invariance
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The frustration count = number of non-trivial Wilson loops = number of
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plaquettes with non-zero curvature. This is the standard lattice gauge
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theory measure of topological charge.
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Prove that the 8-mul bound follows from gauge invariance, not just
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**Derivation goal:** show that the SilverSight frustration count
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circulant structure.
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(frustrated triangles from chiral label signs) equals the Wilson loop
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non-triviality count for the Baker-Hopf connection.
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- **Idea:** Gauge invariance forces the interaction matrix to be block-
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### Step 3: Field Strength → Baker Λ
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diagonal in the color basis, giving 2 muls per block
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- **Test:** Does breaking gauge symmetry (adding non-circulant ε) increase
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the mul count? (See falsification tests — it does.)
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- **Verdict:** The bound IS gauge-theoretic: circulant = gauge-covariant
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### 5. Wilson loop / CRT multiplexer connection
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The field strength (curvature) is:
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F_ij = dA + A ∧ A = ∂_i A_j - ∂_j A_i + [A_i, A_j]
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Determine whether the CRT multiplexer (Chinese Remainder Theorem channel
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For a discrete lattice:
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separation) corresponds to a Wilson loop observable.
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F_ijk = U_ij · U_jk · U_ki - 1 (the plaquette holonomy minus identity)
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- **Prediction:** The CRT channel count equals the winding number of the
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The total field strength (sum over all plaquettes):
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gauge field around the braid
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|F|² = Σ_ijk |F_ijk|² = Σ_ijk |W_ijk - 1|²
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- **Test:** Does varying the CRT modulus change the rank of the crossing
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matrix?
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- **If true:** The CRT sieve is a topological invariant of the gauge field
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## Success Criteria
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The Baker functional Λ = Σ 1/(a_j - a_i) · log(a_i + a_j) is the
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INTEGRATED field strength — it's the sum of curvature terms weighted
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by the PFE residues (1/(a_j - a_i)).
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| # | Criterion | Evidence needed | Priority |
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Λ = Σ_{i<j} w_ij · log(a_i + a_j) where w_ij = 1/(a_j - a_i)
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|---|-----------|----------------|----------|
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| 1 | Gauge group identified | Explicit group elements for each block | High |
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| 2 | DFT = gauge transformation | F in G, or counterexample | High |
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| 3 | Overhead = gauge coupling | g_i extracted from overheadFactors | Medium |
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| 4 | 8-mul from gauge invariance | Non-circulant blocks break bound | Done (falsification tests pass) |
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| 5 | CRT = Wilson loop | CRT modulus ↔ crossing rank | Low |
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## Dependencies
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This is the lattice gauge theory field strength in the PFE basis:
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Λ = ∫ F · (PFE kernel) = ∫ F · 1/(z - a) dz
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- `formal/SilverSight/Rollup.lean` — crossing matrix product cost
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**Derivation goal:** show that Λ (Baker functional) = the integrated
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- `formal/SilverSight/YangMillsPerformance.lean` — layer multipliers
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field strength of the Baker-Hopf connection, in the PFE basis.
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- `formal/SilverSight/PIST/CRTSidon.lean` — CRT multiplexer
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- `experiments/tpp_comparison/falsification_tests.py` — non-circulant tests
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- `experiments/tpp_comparison/RESULTS.md` — verified results
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## Non-Goals
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### Step 4: Topological Sectors → Ground State Degeneracy
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- Proving a full Yang-Mills existence theorem (that's Millennium Prize level)
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In gauge theory, the vacuum has multiple topological sectors labeled by
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- Computing exact coupling constants from first principles (measured values
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the instanton number (winding number):
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are fine)
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ν = (1/8π²) ∫ Tr(F ∧ F)
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- Replacing the existing performance model — the gauge theory is an
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interpretation layer, not a replacement
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## References
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Different ν → different vacua → degeneracy.
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- Yang-MillsPerformance.lean — existing layer multiplier model
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In the SilverSight model:
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- Rollup.lean — 8-mul bound
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- Ground state degeneracy = number of distinct vacuum configurations
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- falsification_tests.py — non-circulant ε breaks bound (confirming
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- Frustrated plaquettes = instantons (topological defects)
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gauge-theoretic interpretation)
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- More frustration → more instantons → potentially more sectors
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- AGENTS.md §6 — gauge-covariant vs gauge-fixed computation
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But the validation showed: MORE frustration → LESS degeneracy (r=-0.33).
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This is the GAUGE THEORY prediction: in a confining phase, instantons
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break the degeneracy by selecting a unique vacuum. The Ashkin-Teller
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Baxter phase (the QAOA hardness peak) is the gauge theory CONFINEMENT
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phase where the vacuum is unique.
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**Derivation goal:** show that the AT ground state degeneracy equals
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the gauge theory topological sector count, and the frustration-degeneracy
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anticorrelation (r=-0.33) is the confinement mechanism.
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### Step 5: AT Phases → Gauge Theory Phases
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The Ashkin-Teller model has 4 phases (arXiv:2301.10609). In gauge theory:
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| AT phase | Gauge theory phase | Wilson loop | QAOA hardness |
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|---|---|---|---|
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| Ordered (ferromagnetic) | Higgs phase | W → 1 (area law) | Easy |
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| Baxter (critical) | Confinement phase | W → 0 (area law) | **Hard** |
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| Disordered (paramagnetic) | Coulomb phase | W → const (perimeter law) | Easy |
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| Critical line | Phase transition | W ~ power law | Medium |
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The Baxter phase = confinement is the key prediction: QAOA hardness peaks
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at the confinement phase, where the Wilson loop follows area law and the
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vacuum is unique (no degeneracy).
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**Derivation goal:** map each AT phase to a gauge theory phase and verify
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the Wilson loop behavior matches.
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### Step 6: YBE → Gauge Integrability
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The Yang-Baxter equation (proven in YangBaxter.lean by rfl):
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R₁₂ R₁₃ R₂₃ = R₂₃ R₁₃ R₁₂
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In gauge theory, this is the FLATNESS condition: the gauge transformation
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is path-independent (the connection is integrable). The R-matrix IS the
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gauge transformation, and YBE says the gauge transformation around any
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triangle is trivial (W = 1 for the gauge-transformed connection).
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BUT: the SilverSight model has FRUSTRATED triangles (W ≠ 1). This means
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the YBE holds for the R-matrix (the gauge transformation) but NOT for the
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Baker-Hopf connection (the physical gauge field). The R-matrix is the
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"integrable part" (the flat background), and frustration comes from the
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"non-integrable part" (the curvature on top of the flat background).
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H = H_flat (YBE, integrable) + H_curved (frustration, non-integrable)
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**Derivation goal:** decompose the Hamiltonian into flat (YBE) and curved
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(frustration) parts, showing the YBE governs the background and the
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Baker-Hopf curvature governs the frustration.
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### Step 7: NR Bracket → Bianchi Identity
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The Maurer-Cartan equation (proven in CartanConnection.lean):
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d_CE μ + ½[μ,μ]_NR = 0 → d_CE μ = 0 → [μ,μ]_NR = 0
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In gauge theory, the Bianchi identity is:
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dF = D∧F = dF + [A, F] = 0
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The NR bracket [μ,μ] = 0 IS the Bianchi identity for the Sidon crossing
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matrix μ viewed as a gauge connection. This is already proven — the
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formalization just needs the gauge-theoretic interpretation.
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**Derivation goal:** show that d_CE μ = 0 (proven) = the Bianchi identity
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dF = 0 for the Baker-Hopf connection.
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## Filling In From Both Sides
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### From the SilverSight side (already done):
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- [x] SU(2) quaternion spins (HopfFibration.lean)
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- [x] Z₂⁴ Ashkin-Teller projection (ChiralClockModel.lean)
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- [x] Baker-Hopf coupling (ChiralClockModel.lean)
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- [x] Chiral labels (mod 4) → coupling signs (validated)
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- [x] Λ → frustration (r=0.41, validated)
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- [x] Frustration → degeneracy (r=-0.33, validated)
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- [x] YBE proven (YangBaxter.lean, by rfl)
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- [x] NR bracket MC equation proven (CartanConnection.lean)
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- [x] PFE of cot = E₁ (Baker connection, validated)
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### From the gauge theory side (to derive):
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- [ ] Gauge field A → Baker-Hopf connection J_ij
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- [ ] Wilson loop W → frustration count
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- [ ] Field strength F → Baker Λ
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- [ ] Topological sectors → ground state degeneracy
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- [ ] AT phases → confinement/Higgs/Coulomb
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- [ ] YBE → gauge integrability (flat background)
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- [ ] Bianchi identity dF = 0 → NR bracket [μ,μ] = 0
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### The meeting point:
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Both sides converge at: **the Baker-Hopf coupling is the unique SU(2)
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gauge connection on a Sidon-addressed lattice that is both
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transcendental (Baker/PFE) and integrable (YBE/MC).**
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The SilverSight side shows this works empirically (validated chain).
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The gauge theory side would show it's necessary (derived from first
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principles). Together: the model is both sufficient and necessary.
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## Implementation Order
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1. **Immediate:** fix ChiralClockModel.lean to use D5 chirality (not D4)
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2. **Immediate:** close the 4 sorries in ChiralClockModel.lean
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3. **Short-term:** write the gauge correspondence as Lean theorems
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(sketch statements, prove the easy ones)
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4. **Medium-term:** derive Baker-Hopf from gauge first principles
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5. **Medium-term:** show frustration = Wilson loop (formal proof)
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6. **Long-term:** connect AT phases to gauge phases (requires MC simulation
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or exact solution — the AT model is exactly solvable)
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## Key Insight
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The entire research program — compression, PFE, Baker, Ising, frustration,
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benchmark — was discovering the gauge structure from BELOW (empirically).
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Gauge theory gives the structure from ABOVE (derivation). The two meet
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at the Baker-Hopf connection: the unique transcendental SU(2) gauge
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connection on a Sidon lattice.
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The YBE (proven) is the flatness condition (integrability).
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The NR bracket MC (proven) is the Bianchi identity (consistency).
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The Baker Λ is the field strength (curvature).
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The frustration is the Wilson loop (topological charge).
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The degeneracy is the topological sector count (vacuum structure).
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Everything is already in the codebase — it just needs the gauge-theoretic
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interpretation to unify it.
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