Part XXIV: add missing files (paper, script, figures)

Releases → Draft new release
Tag: v0.24.1
Title: Part XXIV: Complete (patch — missing files)

Release notes:
Patch for v0.24.0. Added missing files:

• paper/monostring_part24_double_sg.md
• scripts/part24/part24_double_sg_patch3.py
• figures/part24/ (6 figures)

Science unchanged. See v0.24.0 for full results.

Part XXIV: Double Sine-Gordon — 3/4 CONFIRMED

What's new

Model

Double Sine-Gordon: V(φ) = 1 − r·cos(φ/2) − (1−r)·cos(φ)
Non-integrable for all r ∈ (0,1). Extends classical SG
(Part XXIII, falsified as elastic-only).

Pre-registered results

Criterion	Result
S1: v_cr ≈ 0.19 (bound states)	✅ PASS
S2: N_bounces = 2–5 (resonance windows)	✅ PASS
S3: ω_shape < M_vac (wobble mode, 5/5 r)	✅ PASS
S4: Thermal nucleation	❌ FAIL (physically correct)

Verdict: 3/4 = CONFIRMED

New theorems

• T10: Bogomolny mass formula M = ∫√(2V)dφ ≈ 16
• T11: Discrete wobble mode ω_shape < M_vac for all r∈[0,1]

Key fix: IC construction (6 artifacts resolved)

Previous patches had Q=2 (double kink) instead of Q=0 (K+AK).
Fixed via kink_profile_inverse() using x(φ) inversion method.
Verified: Q=0.000, sep₀=22.0, E/(2γM)=0.999, ΔE/E₀=0.054%.

Artifacts documented

#19: N_bounces detector (φ=2π crossing → sep(t) threshold)
#20: M=16 not a bug (0→4π kink = 2×SG period)
#21: AxisError thermal IC (scalar→array sigma)
#22: IC Q=2→0 (static_antikink→kink_profile_inverse)
#23: N_bounces≈50 noise (threshold count_bounces)
#24: sep_init=0 (kinks overlapping → verified sep₀≈22)

Files added

• paper/monostring_part24_double_sg.md
• scripts/part24/part24_double_sg_patch3.py
• figures/part24/ (6 figures)

Counter

24 experiments | 11 theorems | 20 artifacts | 1 live prediction
DESI Y5 (2026) / Euclid DR1 (2026) — единственный живой тест

v17.0.0: Part XXIII — String Fission (DNLS + Sine-Gordon)

Part XXIII: String Fission — From One to Many

What's new

DNLS Modulational Instability (Steps 1–3):

• First dynamical ONE→MANY test in the project
• ONE coherent state → ~85 solitons (peak) → ~20 equilibrium
• Dynamic equilibrium confirmed, not thermalization
• ~750 collision events (mergers + fissions)

Sine-Gordon Topological Solitons (Final):

• GPU-accelerated (JAX, <1 min on Colab T4)
• Kink mass: 8 exact solutions, <0.15% error all v
• Breather masses M_b=16√(1-ω²): <0.06% all ω
• Artifact #12 resolved: pi_v = -v·dφ/dx (not -v·γ·dφ/dx)
  Fixed 171% → 0.5% energy error at v=0.9

Falsified:

• Breather formation in head-on collision: integrable SG
  always produces elastic scattering (Zakharov-Shabat theorem)
• Thermal kink nucleation at T~M: plasma phase, no discrete objects

Artifacts documented: #12 (momentum), #13 (E_center), #14 (integrator)

Scorecard

23 experiments → 0 confirmed signals, 1 live prediction
7 frameworks → all tested
9 theorems → valid mathematics
14 artifacts → documented and explained

Live prediction

DESI Y5 (2026), Euclid DR1 (2026): w(z) ≠ -1?

Runtime

Part XXIII final script: ~1 min (GPU T4), ~25 min (CPU)

v16.0.0: Part XXII — First Contact with Real Data

v16.0.0: Part XXII — First Contact with Real Data

Three observational tests against real data

Test 1 — Black Holes and Dark Energy:
DESI 2024 measures w(z)≠-1 at 3.9σ (w₀=-0.55, wₐ=-1.32).
The BЭ hypothesis predicts time-varying dark energy from
BH growth (zero-dof states force network expansion).
Qualitatively consistent. First external data not
inconsistent with a BЭ prediction.

Test 2 — Planck Cycle and Lorentz Invariance:
Fermi GRB 090510 constrains M_QG > 1.22 M_Pl (linear LIV).
Non-covariant BЭ cycle: excluded. Covariant BЭ cycle
(established in Part XXI as Euclidean path integral):
fully consistent. LIV constraints select the covariant
implementation.

Test 3 — Holographic Bound:
BЭ finite state space: S_BЭ ≤ A/4l_P² = 2.3×10¹²² nats.
Consistent with Bekenstein-Hawking holographic principle.
CMB information (4×10⁶ bits) << bound. No contradiction.

The live prediction

w(z) ≠ -1 from BH→DE coupling.
Testable by DESI Y5, Euclid DR1, Roman ST (2026–2027).
If w→-1: BH→DE mechanism ruled out, hypothesis loses
its last live prediction.
If w≠-1 confirmed >5σ: consistent, motivates quantitative model.

Final scorecard

22 experiments → 0 confirmed signals, 1 live prediction
8 frameworks → all tested and closed
8 theorems → valid mathematics found during falsification
13 artifacts → documented with fixes
1 prediction → DESI Y5 / Euclid 2026

Files added

• paper/monostring_part22_observational.md
• scripts/part22/part22_observational_tests.py
• scripts/part22/part22_summary.py
• figures/part22/part22_observational_tests.png
• Updated README.md, CITATION.cff

v15.0.0: Part XXI — Stochastic BЭ ≡ Euclidean QFT (Complete)

Part XXI: Stochastic Realization — Final Result

What was tested

The monostring Basic Element (BЭ) modeled as a Metropolis
Monte Carlo sequential automaton on a 1D Klein-Gordon lattice.
Seven steps (Steps 1–7), three traversal orders, three update
rules, lattice sizes N=64..4096.

Key findings

Theorem (confirmed numerically):
Metropolis MC with detailed balance → P[φ]~exp(-S/T) = Euclidean
path integral. BЭ(stochastic) ≡ Euclidean KG QFT. Not a new
result — a reinterpretation.

Lattice slope = -1.885 (not -2.000):
The spectral slope of P(k) on a discrete lattice is -1.885,
not the continuum value -2.000. This is analytically explained
by sin(πf) ≠ πf at f > 0.08 (Brillouin zone curvature).

CMB gap = 1.85 (fundamental):
KG vacuum: slope ≈ -1.9. Planck CMB: n_s-1 = -0.035.
Gap is fundamental — requires de Sitter slow-roll inflation,
not KG statistics.

Two new artifacts documented (#12, #13).

Final scorecard

21 experiments → 0 surviving physical signals
8 frameworks → all falsified
6 theorems → valid mathematics
13 artifacts → documented and explained
1 reinterpretation → BЭ ≡ Euclidean QFT (path integral)

What survives

The monostring hypothesis is interpretively valid:
it provides a sequential-automaton ontology for QFT.
It is physically empty: no predictions beyond QFT.

Files added

• paper/monostring_part21_stochastic.md
• scripts/part21/ (8 scripts)
• figures/part21/ (6 figures)

Citation

@misc{lebedev2025monostring,
  author = {Lebedev, Igor},
  title  = {The Monostring Hypothesis: Twenty-One
             Computational Experiments Across Eight
             Mathematical Frameworks — Complete Falsification},
  year   = {2025},
  doi    = {10.5281/zenodo.18886047},
  note   = {v15.0.0}
}

v14.0.0: Complete Falsification — 19 Experiments, 0 Signals

The Monostring Hypothesis — Complete Falsification

Parts XVIII–XIX: Quantum Bogoliubov Spectrum (NEW)

Part XVIII tests classical E8 root lattice dynamics:

• Temporal Welch spectrum → n_s = −0.84 (Brownian artifact)
• Spatial Fourier spectrum → n_s = +2.39 (lattice artifact)

Part XIX tests quantum Bogoliubov spectrum:

• Exact diagonalization via one-particle matrix M
• All Coxeter algebras unstable at κ=1.0
• Apparent signal p=0.010 → Artifact #11 (control mismatch)
• Fair controls (Ctrl-D, sparsity-matched): p=0.094 → H₀

Artifact #11: Control Matrix Mismatch

Three sources identified and resolved:

• (a) diagonal inflation: diag 4.3 vs 2.0
• (b) sparsity mismatch: 11 nz vs 7 (E8)
• (c) sign mismatch: ±1 vs −1 only

Theorem 6: Cartan Positive Definiteness

Cartan matrices of simple Lie algebras are positive
definite (Sylvester). Explains quantum stability at κ=0
and bare frequencies ωᵢ = √λᵢ(C) > 0.

---

Final Scorecard

Metric	Count
Experiments	19
Physical signals	0
Frameworks tested	7
Theorems found	6
Artifacts documented	11

What is falsified

• Classical mechanics (Parts I–XVII): falsified
• Quantum mechanics (Parts XVIII–XIX): falsified
• Emergent 4D spacetime from Coxeter dynamics
• Inflation from E8 root lattice
• Coxeter-specific quantum gap structure

What survives (no physical interpretation)

• τ ≈ 237 memory time (empirical)
• Diffusive wavepackets α ≈ 0.30 (robust)
• Monotonic IPR gradient E6 ρ=1.0 (robust)
• Six mathematical theorems (valid)

Citation

@misc{lebedev2025monostring,
  author    = {Lebedev, Igor},
  title     = {The Monostring Hypothesis: Nineteen
               Computational Experiments Across Seven
               Mathematical Frameworks},
  year      = {2025},
  publisher = {GitHub / Zenodo},
  url       = {https://github.com/LebedevIV/monostring-hypothesis},
  doi       = {10.5281/zenodo.18886047},
  note      = {v14.0.0: Complete. 19 experiments,
               0 signals, 6 theorems, 11 artifacts.}
}

The Monostring Hypothesis v13.0.0: Seventeen Experiments — Complete Falsification

Description:
Final release. Adds Part XVII (Mutual Information)
completing the investigation.

NEW IN v13.0.0:
• Part XVII: Mutual information between Coxeter modes
Step 1: TC(E8)=13.64, p=0.000 (apparent signal)
Step 2: 98.9% TC from Weyl pairs; matched p=0.78
Artifact #10: Weyl-pairing creates spurious MI

CUMULATIVE RESULT:
17 experiments across 6 mathematical frameworks
0 surviving physical signals
6 mathematical theorems found as byproducts
10 computational artifacts documented

FRAMEWORKS TESTED:
Standard map orbits (I-IX)
Cayley graphs of Weyl groups (X-A)
XXZ spin chains (X-B)
Quantum walks on Cayley graphs (XI)
Inflation potentials (XII-XV)
Cartan matrix Hamiltonians (XVI)
Mutual information (XVII)

CONCLUSION:
Classical Coxeter dynamics (standard map, Cartan
Hamiltonian, complex frequencies, power-law potentials)
does not reproduce inflation or emergent spacetime.
The negative result is the result.

Keywords: monostring, Coxeter, falsification,
inflation, emergent spacetime, AI-assisted physics,
negative result, open science

The Monostring Hypothesis v12.0.0: Sixteen Computational Experiments — Complete Falsification of Classical Coxeter Dynamics

Description:
This release adds Parts XII–XVI (Inflation Search) to the
monostring hypothesis falsification project.

NEW IN v12.0.0:
• Part XII: Emergent inflaton potential from mode interactions
— H₀ not rejected (p>0.3, N=200 rank-matched controls)
• Part XIII: KAM thresholds κ_c(algebra)
— r(h,κ_c)=−0.47, no Coxeter dependence
• Part XIV: Complex Coxeter frequencies ω+iΓ
— Closed analytically: Σcos(πmᵢ/h)=0 by theorem
• Part XV: Blind search, inflation potentials V(φ)
— Systematic n_s bias +2.4σ above Planck
• Part XVI: Cartan matrix Hamiltonian dynamics
— p=0.15, E8 indistinguishable from random

MATHEMATICAL THEOREMS FOUND:

1. Σcos(πmᵢ/h) = 0 for all Coxeter algebras
2. det(C_E8) = 1 (unimodular)
3. G2 KAM anomaly from ω₁=ω₂ resonance
4. n_s systematic bias in V~φ^α potentials

ARTIFACT CATALOG EXTENDED (9 documented artifacts total)

CUMULATIVE RESULT:
16 experiments, 6 mathematical frameworks, 0 surviving
physical signals. All surviving results are mathematical
theorems. Classical Coxeter dynamics is falsified as a
model of inflation and emergent spacetime.

Keywords: monostring, Coxeter, inflation, falsification,
computational physics, AI-assisted research, negative result

v11.0.0 — Investigation Complete: Quantum Walk Negative Result (Part XI)

Description:

## Part XI: Quantum Walk on Cayley Graph — Final Experiment

This release concludes the monostring hypothesis investigation.
Eleven experimental series, all frameworks tested, all falsified.

### Part XI: Quantum Walk on Cay(S₄)

**Setup:**
Continuous-time quantum walk |ψ(t)⟩ = exp(−iLt)|ψ(0)⟩
on Cayley graph of S₄ (N=24, d=3) vs N=30 random
3-regular graphs of same size and degree.

**Results:**

| Observable | Cayley S₄ | Random (N=30) | p-value |
|-----------|-----------|---------------|---------|
| mean S(t) | 2.208 | 2.644 ± 0.048 | 0.065 |
| mean P(v→v,t) | 0.1036 | 0.1052 ± 0.012 | 0.903 |
| spreading α | 0.007 | 0.194 ± 0.035 | 0.065 |
| Cayley percentile | — | 0th (S, α) | — |

Cayley S₄ is at 0th percentile for entropy and spreading,
but p = 0.065 > 0.05. H₀ not rejected.

**Mechanism (why signal appears):**
Group periodicity — quantum walk on a group revisits
the identity with period determined by representation
theory. Creates entropy minima (z = −19 at t≈20).
Mathematically trivial, not a monostring signal.

**Verdict:** Negative. Mechanism trivial even if
signal were significant.

---

### Complete Falsification Record (Parts I–XI)

| Part | Framework | Key result | Verdict |
|------|-----------|------------|---------|
| I | Lyapunov compactification | D=2r (symplectic) | ❌ |
| II | Gauge-Higgs + causal sets | Null model higher | ❌ |
| III | Spectral dimension (Weyl) | d_s ≠ 4, N-dependent | ❌ |
| IV | Independent verification | Dark energy circular | ❌ |
| V | D_corr resolution | D_corr = n_unique (theorem) | ❌ |
| VI | Fragmentation + memory | τ≈237 real, not unique | ⚠️ |
| VII | τ ∝ h(Coxeter) | Not proportional | ❌ |
| VIII | Geodesic fields | z≈6 was code artifact | ❌ |
| IX | Mechanism analysis | PCA_ratio = Weyl degeneracy | ❌ |
| X-A | Cayley graphs W(E6) | Spectrum = char table | ❌ |
| X-B | XXZ spin chain | Critical phase theorem | ❌ |
| XI | Quantum walk Cay(S₄) | Group periodicity, p=0.065 | ❌ |

---

### What Remains (Not Falsified — Mathematical Theorems)

1. Weyl involution: m_i + m_{r+1-i} = h always
   → ω_i = ω_{h-i} → orbit on subtorus
   → D_corr = n_unique_frequencies

2. Memory time τ ≈ 237 for E6 daughters
   → p < 0.0001, robust
   → NOT proportional to h, NOT unique to E6

3. Cayley spectrum = character table (Peter-Weyl, 1927)

4. |Δ(h)| < 1 → XXZ critical phase (Bethe ansatz)

5. Quantum walk on group → periodic recurrence
   (representation theory)

All surviving results are known mathematical theorems,
not new physical discoveries.

---

### Key Technical Lessons (for future researchers)

```python
# 1. Toric embedding — always use (cos φ, sin φ), not φ
X = np.concatenate([np.cos(orbit), np.sin(orbit)], axis=1)

# 2. Distance matrix — not weight matrix
D_sp = shortest_path(C_distances)   # CORRECT
D_sp = shortest_path(A_weights)     # WRONG → z_geo artifact

# 3. Control for degeneracy before claiming signal
# Coxeter ω_i = ω_{h-i} → remove pairs → PCA_ratio = 1

# 4. Mann-Whitney with 1 vs N requires care
# Use percentile: rank/N as p-approximation

# 5. Physical mechanism must be non-trivial
# Group periodicity explains quantum walk → not physics

---

Files Added

• paper/monostring_part11_quantum_walk.md
• scripts/part11/part11_quantum_walk_step1.py
• scripts/part11/part11_quantum_walk_step2.py
• Updated README.md

---

Next Step: arXiv Submission

Title: "Eleven Computational Experiments Falsifying the Monostring Hypothesis: A Methodology for AI-Assisted Hypothesis Testing in Physics"

Sections: Introduction, Methods, Parts I-XI, Discussion, Methodology guide

Target: arXiv:physics.gen-ph or quant-ph

How to Reproduce

pip install numpy scipy scikit-learn matplotlib
python scripts/part11/part11_quantum_walk_step2.py
# Expected: p=0.065, α=0.007 vs 0.194
# Runtime: ~5 minutes

DOI (paper): https://doi.org/10.5281/zenodo.18886047
DOI (code): https://doi.org/10.5281/zenodo.18890266

v10.0.0 — Complete Falsification: Cayley Graphs and Spin Chains (Part X)

Release description:

## Part X: Final Tests — Cayley Graphs and XXZ Spin Chains

This release concludes the monostring hypothesis investigation.

### Plan A: Cayley Graphs of W(E6)

**Step 1** (A_n, B_n small groups):
- λ_max = 2.000 universally (sign representation theorem ✓)
- S_4 = Ramanujan graph (λ₁ > Alon-Boppana bound ✓)
- λ₁ → 0 as |W| → ∞ (adjacent generators)

**Step 2** (generator sets and scaling):
- Adjacent gens: λ₁ ~ |W|^{-0.46}
- All transpositions: λ₁(S_n) = 2/(n-1) exactly
- Character table formula verified: λ_L(ρ) = 1 − χ_ρ(s)/dim(ρ)

**Step 3** (spectral fingerprint):
- Coxeter: frac_degenerate = 0.80–0.92
- Random same (N,d): frac_degenerate = 0.00
- Sharp signal — but tautological (= irrep dimensions)
- Wasserstein distance W₁ ≈ 0.05–0.20 (Coxeter vs Random)

**Verdict A:** Cayley graph spectrum = character table.
No new physics. Mathematical tautology.

### Plan B: XXZ Spin Chain

**Setup:** H = −Σ[SS + Δ·SzSz], Δ(h) = −cos(π/h)

**Results (n=8 spins):**
- Coxeter gaps: 0.469–0.520 vs Random mean: 0.432±0.055
- Mann-Whitney p = 0.008 — but range artifact, not algebra
- Entanglement entropy: < 2% difference E6 vs A6 vs Random
- Dynkin-weighted J: gap = 0 for all configurations

**Key theorem:** |Δ(h)| = |cos(π/h)| < 1 for all finite h
→ All Coxeter algebras in critical phase by theorem
→ No E6-specific signal possible

**Verdict B:** Critical phase is guaranteed by theorem.
XXZ spin chain cannot distinguish Coxeter algebras.

### Complete falsification record

| Parts | Framework | Verdict |
|-------|-----------|---------|
| I–VII | Standard map orbits | ❌ Falsified |
| VIII–IX | Graph geodesic fields | ❌ Code artifact + Weyl degeneracy |
| X-A | Cayley graphs W(E6) | ❌ Character table tautology |
| X-B | XXZ spin chain | ❌ Critical phase theorem |

### What remains (theorems, not observations)

- Weyl involution: ω_i = ω_{h-i} → subtorus confinement
- D_corr = n_unique_frequencies (mathematical consequence)
- Memory τ ≈ 237 for E6 daughters (p<0.0001, robust)
- Cayley spectrum = character table (Peter-Weyl)
- |Δ(h)| < 1 → XXZ critical (Bethe ansatz)

### Open directions

- Quantum walks on Cay(W(E6)): coherence survival
- RMT with Coxeter symmetry: GUE + W(E6) vs plain GUE
- Methodology paper: AI-assisted falsification in physics

### Files added

- `paper/monostring_part10_cayley_spinchain.md`
- `scripts/part10/` (4 scripts)
- Updated `README.md` (title, status table, scorecard)

### How to reproduce

```bash
pip install numpy scipy scikit-learn matplotlib
python scripts/part10/part10_planb_xxz_spinchain.py
# Expected: p=0.008 (range artifact), S_ent < 2% difference

DOI (paper): https://doi.org/10.5281/zenodo.18886047
DOI (code): https://doi.org/10.5281/zenodo.18890266