diff --git a/6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md b/6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md index b3841121..8588fae3 100644 --- a/6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md +++ b/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