From dead6528de972caff543a9e2b67cdbc6b3820f5a Mon Sep 17 00:00:00 2001 From: allaun Date: Sun, 5 Jul 2026 14:53:33 -0500 Subject: [PATCH] =?UTF-8?q?docs:=20gauge=20theory=20goal=20=E2=80=94=205?= =?UTF-8?q?=20specific=20goals=20with=20success=20criteria?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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. --- .../docs/specs/GAUGE_THEORY_GOAL.md | 114 ++++++++++++++++++ 1 file changed, 114 insertions(+) create mode 100644 6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md diff --git a/6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md b/6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md new file mode 100644 index 00000000..b3841121 --- /dev/null +++ b/6-Documentation/docs/specs/GAUGE_THEORY_GOAL.md @@ -0,0 +1,114 @@ +# 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