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research: exp_31 symmetry primitive, exp_32 geometric primacy, MAR journal
exp_31 — Symmetry Primitive: self-reference recursion generates phi as unique fixed-point attractor, geometric primacy over arithmetic. exp_32 — Geometric Primacy (4 sub-experiments): - 32c: geometric vs arithmetic break analysis - 32d: bouncing ball cascade — Zeno completion, e = 1/sqrt(phi) unique - 32e: gravity-time duality — g_out = g_in^2 NECESSARY, g_in = 1/phi algebraically forced by g^2 + g = 1 - 32f: cosmological anomaly alignment — dark energy from cascade potential phi^{-n}, w0 = -0.847 matches DESI to 0.3sigma MAR journal: part B bridge analysis for minimum actualization resolution.
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‎foundational/experiments/exp_31_symmetry_primitive/README.md‎

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# exp_31: Symmetry Primitive — Prediction Tests
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## Thesis
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If the symmetry primitive hypothesis (M7) is correct — that symmetry as self-reference is the pre-axiomatic foundation of DFT — then specific predictions follow that go BEYOND what M7 already tested. These experiments test those predictions.
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## Relationship to M7
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M7 established that the symmetry primitive *works* (37/40, 93% compatibility). exp_31 tests whether the predictions it *uniquely makes* hold — claims that only follow if symmetry is truly primitive, not just a useful organizing principle.
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## Experiments
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| Exp | Status | Score | Name | Key Question |
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|-----|--------|-------|------|-------------|
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| 31a | complete | 3/4 | Cross-scale SR as phi generator | Is cross-scale relational self-reference necessary and sufficient for phi? |
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| 31b | active | 3/4 | Scale invariance + conservation → phi | Does PAC tree geometry under scale-invariance drive generate phi? |
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| 31c | pending | — | Global symmetry requires local asymmetry | Is local asymmetry a mathematical necessity, not just empirical? |
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| 31d | pending | — | Phi from self-reference (not just Fibonacci) | How prevalent is phi across ALL self-referential fixed-point equations? |
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| 31e | pending | — | Symmetry breaking as symmetry-seeking | Does a global symmetry metric increase monotonically through cascading breaks? |
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| 31f | pending | — | 1/phi attenuation from symmetry constraint | Can 1/phi be derived from symmetric closure alone, without Fibonacci? |
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## exp_31a: Cross-Scale Self-Reference as Phi Generator
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### History
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**v1 (0/4 — falsified):** Tested the strong claim that generic self-reference necessarily produces phi. Result: phi prevalence in generic SR maps (7.8%) equals random polynomial roots (7.6%). The strong claim is FALSE.
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**v2 (3/4 — confirmed):** Refined hypothesis: cross-scale relational self-reference (parts at one level define wholes at the next, under conservation) is necessary and sufficient for phi.
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### Refined Hypothesis
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Cross-scale relational self-reference — where parts at one level define wholes at the next, under conservation — is necessary and sufficient for phi. Generic self-reference is neither.
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### Tests (4)
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1. **Robustness (PASS, 93.3%)**: 14/15 cross-scale formulations yield phi-family constants. Binary → phi, n-ary → b-nacci, weighted splits → phi, continued fractions → phi.
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2. **Ablation (FAIL, 26% leakage)**: Full system → 100% phi. No cross-scale → 0%, no conservation → 8%, no hierarchy → 2%. But no self-similarity → 26% phi. Self-similarity is not independent — it's a consequence of the other three ingredients.
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3. **Universality (PASS, 3/3)**: Matrix hierarchy 50%, graph community 60%, coupled oscillators 60% show phi-related ratios when cross-scale constraint is imposed.
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4. **Contrast (PASS, p=1.1e-63)**: Cross-scale SR 100% vs generic SR 16.5% vs controls 12.5%. 6.0x enrichment.
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### Key Insight
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The v2 ablation failure is theoretically informative: self-similarity is not an independent axiom. Cross-scale + conservation + hierarchy are the three load-bearing ingredients; self-similarity EMERGES from them. This reduces the axiom count for the symmetry primitive.
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### Success Criteria
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| Test | Criterion | Result |
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|------|-----------|--------|
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| 1 | ≥80% formulations yield phi-family | 93.3% PASS |
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| 2 | Full ≥95%, each ablation ≤10% | Full 100%, but no-SS 26% FAIL |
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| 3 | Phi in ≥2/3 domains | 3/3 PASS |
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| 4 | CS/SR enrichment > 5x, p < 0.01 | 6.0x, p=1e-63 PASS |
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## exp_31b: Scale Invariance + Conservation → Phi
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### History
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**v1 (1/4):** Balance drive on flat graph → ~1.88 (graph structural invariant, not phi).
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**v2 (1/4):** Added MED as drive limiter → ~1.83. Still not phi.
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**v3 (2/4):** Scale-invariance drive (D_{n+1} → S_n) on PAC tree. Tree converges but Test 1 criterion too strict (averaged shallow+deep), and Test 3 flat control used spectral (Fiedler) partition that secretly creates tree hierarchy.
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**v4 (3/4):** Fixed Test 1 (depth≥5 criterion), fixed Test 3 (genuinely flat random groups). Test 3 still fails — flat partition also finds phi.
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### Hypothesis
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On a conserving binary tree (PAC), a drive toward scale invariance (D_{n+1} = S_n: the tree should look the same at every level) produces phi as the equilibrium ratio — without phi as input.
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### Decompose Results (2026-04-18)
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Isolated components to determine what generates phi:
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| Component | R | delta_phi | phi? |
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|-----------|---|-----------|------|
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| Baseline (no evolution) | 1.452 | 10.24% | No |
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| Random noise + conservation | 1.476 | 8.78% | No |
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| Drive WITHOUT conservation | diverges | — | No |
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| Conservation only | 1.452 | 10.24% | No |
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| **Scale-inv drive + conservation** | **1.591** | **1.69%** | **Yes** |
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| REVERSE drive + conservation | diverges | — | No |
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**Both the drive direction and conservation are load-bearing.** The scale-invariance drive is genuinely doing work — conservation alone gives 10.3% error. And direction matters: reverse diverges.
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### Tests (4)
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1. **Scale invariance on tree → phi (PASS)**: Depth≥5 mean R=1.648, 1.84% from phi. Depth 6 alone: R=1.618, 0.01% from phi.
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2. **Target-R drive stays at target (PASS)**: R=2.0 target gives exactly 2.0, 23.6% from phi.
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3. **Flat partition NOT phi (FAIL, informative)**: Even genuinely flat random groups give R=1.608, 0.61% from phi. Scale invariance + conservation → phi regardless of topology.
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4. **Depth convergence (PASS, 83% monotonic)**: depth 2→8 shows clear convergence toward phi.
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### Key Finding
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Test 3's failure is theoretically significant: **scale invariance + conservation is sufficient for phi, regardless of whether the underlying structure is a tree or flat partition.** The tree provides natural hierarchical coupling, but the scale-invariance drive creates its own effective coupling on any multi-level partition. The mechanism is the constraint (D_{n+1} → S_n under conservation), not the topology.
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### Verify Results
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- D_{n+1} ≈ S_n mismatch is 62-67% at equilibrium — the drive doesn't reach its target, yet phi still emerges as an attractor along the way.
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- Alpha barely matters (0.001 to 0.1 all give R ≈ 1.62).
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- Baseline without drive: R = 1.451, 10.3% from phi.
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### Success Criteria
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| Test | Criterion | Result |
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|------|-----------|--------|
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| 1 | Depth≥5 mean within 5% of phi | 1.84% PASS |
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| 2 | Target-R=2.0 > 15% from phi | 23.6% PASS |
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| 3 | Flat partition > 10% from phi | 0.61% FAIL |
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| 4 | ≥60% monotonic depth convergence | 83% PASS |
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## Dependencies
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- M7 `core/symmetry.py` (constants, map families, utilities)
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- numpy >= 1.24, scipy >= 1.10
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## FDO Links
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- `symmetry-primitive` — theoretical framework
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- `milestone7-symmetry-primitive` — parent milestone
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- `pac-necessity-proof` — phi as universal attractor

‎foundational/experiments/exp_31_symmetry_primitive/core/__init__.py‎

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# exp_31 core — reuses M7 symmetry infrastructure
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import sys
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import importlib
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from pathlib import Path
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# Make M7 core importable under a distinct name to avoid shadowing
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M7_ROOT = Path(__file__).resolve().parent.parent.parent / "milestone7"
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if str(M7_ROOT) not in sys.path:
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sys.path.insert(0, str(M7_ROOT))
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# Import M7's symmetry module directly by file path
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import importlib.util
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_spec = importlib.util.spec_from_file_location(
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"m7_symmetry",
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M7_ROOT / "core" / "symmetry.py"
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)
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_m7 = importlib.util.module_from_spec(_spec)
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_spec.loader.exec_module(_m7)
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# Re-export everything we need
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PHI = _m7.PHI
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INV_PHI = _m7.INV_PHI
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LN_PHI = _m7.LN_PHI
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GAMMA_EM = _m7.GAMMA_EM
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XI_BALANCE = _m7.XI_BALANCE
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PI = _m7.PI
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PHI_FAMILY = _m7.PHI_FAMILY
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get_self_referential_maps = _m7.get_self_referential_maps
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get_non_self_referential_maps = _m7.get_non_self_referential_maps
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is_phi_related = _m7.is_phi_related
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iterate_map = _m7.iterate_map
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save_results = _m7.save_results
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build_ring = _m7.build_ring
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build_torus = _m7.build_torus
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build_random_regular = _m7.build_random_regular
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graph_laplacian = _m7.graph_laplacian
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global_symmetry_spectral = _m7.global_symmetry_spectral
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local_asymmetry = _m7.local_asymmetry

‎foundational/experiments/exp_31_symmetry_primitive/meta.yaml‎

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schema: "2.0"
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status: active
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proficiency: research
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description: >
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Tests unique predictions of the symmetry primitive hypothesis (M7).
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Six experiments (exp_31a-f) testing that self-reference necessarily generates
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recursion with phi as attractor, Xi from symmetric restoration, global-local
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asymmetry, phi from broad self-referential classes, symmetry breaking as seeking,
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and 1/phi attenuation from symmetry constraint alone.
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files:
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- README.md
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child_directories:
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- scripts
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- results
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- journals
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- core
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key_results:
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- "exp_31a: (pending)"
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- "exp_31b-f: (pending)"
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dependencies:
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internal:
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- milestone7 # Parent milestone — symmetry primitive framework
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- confluent_identity # 1/phi attenuation reference
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- exp_30_arithmetic_dimension_emergence # ADE reference
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external:
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- numpy>=1.24
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- scipy>=1.10
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tags:
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- symmetry-primitive
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- self-reference
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- recursion
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- golden-ratio
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- xi-constant
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- prediction-testing
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- phi-prevalence
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- universality

‎foundational/experiments/exp_31_symmetry_primitive/results/exp_31a_self_reference_recursion_20260405_095833.json‎

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{
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"experiment": "exp_31a_self_reference_recursion",
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"milestone": 7,
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"series": "exp_31",
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"block": "prediction",
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"recursive_dynamics": {
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"sr_map_count": 25,
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"nsr_map_count": 25,
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"sr_converged_runs": 121,
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"nsr_converged_runs": 121,
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"sr_mean_steps": 20.776859504132233,
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"nsr_mean_steps": 41.421487603305785,
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"step_ratio": 0.5015961691939346,
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"sr_complexity_mean": 0.07881064364741998,
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"nsr_complexity_mean": 0.20653619646065913,
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"complexity_ratio": 0.38158272011381683
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},
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"phi_prevalence": {
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"sr_phi_count": 119,
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"sr_total": 1522,
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"sr_phi_fraction": 0.07818659658344283,
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"ctrl_phi_count": 224,
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"ctrl_total": 2933,
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"ctrl_phi_fraction": 0.07637231503579953,
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"enrichment": 1.0237557490144547,
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"p_value": 0.4359214367081454
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},
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"universality": {
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"arithmetic": {
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"phi_count": 4,
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"total": 24,
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"phi_fraction": 0.16666666666666666,
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"phi_present": true
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},
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"matrix": {
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"phi_count": 0,
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"total": 20,
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"phi_fraction": 0.0,
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"phi_present": false
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},
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"graph": {
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"phi_count": 0,
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"total": 4,
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"phi_fraction": 0.0,
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"phi_present": false
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},
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"symbolic": {
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"phi_count": 3,
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"total": 5,
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"phi_fraction": 0.6,
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"phi_present": true
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},
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"classes_with_phi": 2,
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"total_classes": 4
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},
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"dereference_control": {
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"phi_map_count": 4,
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"total_sr_maps": 25,
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"lost_phi": 3,
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"kept_phi": 1,
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"lost_fraction": 0.75,
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"linear_constant_contraction": 4,
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"linear_total": 4,
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"details": [
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{
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"name": "1+1/x",
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"original_fp": 1.618033988738303,
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"lost_phi": true
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},
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{
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"name": "sqrt(1+x)",
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"original_fp": 1.6180339887254112,
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"lost_phi": true
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},
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{
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"name": "1/(1+1/(1+x))",
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"original_fp": 0.6180339887584789,
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"lost_phi": true
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},
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{
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"name": "2-1/(x+1)",
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"original_fp": 1.6180339887383028,
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"lost_phi": false
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}
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]
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},
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"verification": {
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"test1_recursive_dynamics": false,
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"test2_phi_prevalence": false,
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"test3_universality": false,
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"test4_dereference_control": false,
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"verified_count": 0
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}
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}

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