diff --git a/research/trinity-gamma-paper/GAMMA_PAPER_DRAFT_v0.1.md b/research/trinity-gamma-paper/GAMMA_PAPER_DRAFT_v0.1.md new file mode 100644 index 0000000000..bcdf5a3767 --- /dev/null +++ b/research/trinity-gamma-paper/GAMMA_PAPER_DRAFT_v0.1.md @@ -0,0 +1,209 @@ +# Barbero-Immirzi Parameter from the Golden Section: A Critical Test of Loop Quantum Gravity and the Trinity φ-Framework + +**Draft v0.1 — Pre-registration checkpoint · April 2026** +**Status:** CONJECTURAL — numerical analysis pending +**SSOT:** `specs/physics/gamma_conjecture.t27` + +--- + +## Abstract + +The Barbero-Immirzi parameter γ plays a central role in Loop Quantum Gravity (LQG), fixing the spectrum of the area operator and the coefficient of Bekenstein-Hawking black-hole entropy. Its value is not predicted by LQG itself but is fixed by requiring agreement with the Bekenstein-Hawking formula, yielding two competing values: γ₁ = ln 2 / (π√3) ≈ 0.23753 (Meissner 2004) and γ₂ ≈ 0.274 (Ghosh-Mitra). Here we present **Conjecture GI1**: γ = φ⁻³ = √5 − 2 ≈ 0.23607, where φ = (1+√5)/2 is the golden ratio. The gap between γ_φ and the preferred LQG value γ₁ is only **0.63%** — 22 times smaller than the internal LQG dispute between γ₁ and γ₂ (13.9%). The conjecture is algebraically exact, structurally simple, and cascades into closed-form expressions for Newton's gravitational constant G, Hawking radiation temperature, and several superconducting critical temperatures. Three pre-registered falsification protocols are proposed: EHT black-hole shadow measurements, LIGO/Virgo quasi-normal modes, and KATRIN neutrino mass bounds. + +--- + +## 1. Introduction + +### 1.1 The Barbero-Immirzi Parameter in LQG + +In the Ashtekar-Barbero formulation of general relativity, the Barbero-Immirzi parameter γ enters as an ambiguity in the definition of the connection variable [Barbero 1995, Immirzi 1997]. In loop quantum gravity, γ scales the eigenvalues of the area operator: + +``` +A_min = 8π γ ℓ_P² √(j(j+1)) +``` + +where ℓ_P is the Planck length and j is the spin label. The parameter is not predicted from first principles within LQG; it is fixed externally by requiring that the statistical-mechanical entropy of a black hole reproduces the Bekenstein-Hawking formula S = A/4. + +This procedure yields two competing values depending on the counting method: +- **Meissner (2004):** γ₁ = ln 2 / (π√3) ≈ 0.237533 +- **Ghosh-Mitra / alternative:** γ₂ ≈ 0.274 + +The 13.9% disagreement between γ₁ and γ₂ is an unresolved internal tension in LQG. + +### 1.2 The Trinity φ-Framework + +Trinity is a research programme proposing that fundamental physical constants can be expressed as closed-form combinations of the golden ratio φ = (1+√5)/2, Euler's number e, and π. The programme maintains a formal catalogue of 152 φ-ansätze (formulas-catalog-2026.md, v1.3), graded by a trust-tier system: EXACT / CHECKPOINT / ANSATZ / CONJECTURAL. + +The anchor identity is the exact algebraic relation: +``` +φ² + φ⁻² = 3 (L5, exact) +``` + +This identity connects φ to the integer 3 — the number of generations of elementary particles in the Standard Model. + +### 1.3 This Paper + +Section 2 presents Conjecture GI1 and its algebraic derivation from L5. Section 3 explores the cascade of implications for G, black-hole entropy, Hawking radiation, and superconductivity. Section 4 discusses the 0.63% gap, falsification strategies, and the possible E8 connection. Section 5 concludes. + +--- + +## 2. Conjecture GI1: γ = φ⁻³ = √5 − 2 + +### 2.1 Statement + +**Conjecture GI1:** The Barbero-Immirzi parameter equals the inverse cube of the golden ratio: + +``` +γ_φ = φ⁻³ = (√5 − 1)³ / 8 = √5 − 2 +``` + +Numerical value to 20 significant digits: +``` +γ_φ = 0.23606797749978969641... +``` + +### 2.2 Algebraic Derivation from L5 + +The L5 identity φ² + φ⁻² = 3 implies φ⁻² = 3 − φ² = 3 − φ − 1 = 2 − φ. Therefore: + +``` +γ_φ = φ⁻³ = φ⁻¹ · φ⁻² = φ⁻¹ · (2 − φ) +``` + +Since φ⁻¹ = φ − 1: +``` +γ_φ = (φ−1)(2−φ) = 2φ − φ² − 2 + φ = 3φ − φ² − 2 +``` + +Using φ² = φ + 1: +``` +γ_φ = 3φ − (φ+1) − 2 = 2φ − 3 = 2·(1+√5)/2 − 3 = √5 − 2 ✓ +``` + +### 2.3 Comparison with LQG Values + +| Parameter | Value (20 digits) | Source | Δ from γ₁ | +|-----------|-------------------|--------|----------| +| γ_φ = φ⁻³ | 0.23606797749978... | Trinity GI1 | −0.63% | +| γ₁ = ln2/(π√3) | 0.23753295805...... | Meissner 2004 | 0 (ref) | +| γ₂ ≈ 0.274 | 0.27398563527...... | Ghosh-Mitra | +13.9% | + +The gap |γ_φ − γ₁| / γ₁ = **0.63%** is 22× smaller than the internal LQG gap |γ₂ − γ₁| / γ₁ = 13.9%. + +--- + +## 3. Cascade Implications + +### 3.1 Newton's Gravitational Constant (G1) + +``` +G = π³ γ² / φ +``` + +With γ_φ = φ⁻³: +``` +G = π³ φ⁻⁶ / φ = π³ φ⁻⁷ = π³ (√5−2)² / φ +``` + +CODATA 2022: G = 6.67430×10⁻¹¹ m³ kg⁻¹ s⁻² +Trinity (γ_φ): **[to be computed by compare_gamma_candidates.py]** +Trinity (γ₁): **[to be computed by compare_gamma_candidates.py]** + +### 3.2 Black-Hole Entropy (BH1) + +In LQG, the black-hole entropy is: +``` +S_BH = (γ₁ / γ) · A / (4 G ℏ) +``` + +If γ = γ_φ, the entropy formula becomes: +``` +S_BH = (γ₁ / γ_φ) · A / (4 G ℏ) with ratio = 1.00620... +``` + +This 0.62% correction is below current EHT precision but within reach of next-generation telescopes. + +### 3.3 Hawking Temperature (SH1) + +The Hawking temperature receives a γ-dependent quantum-gravity correction in some LQG models: +``` +T_H = ℏ c³ / (8π G M k_B) · f(γ) +``` + +### 3.4 Superconductivity (SC3, SC4) + +The Trinity catalogue contains two superconducting critical temperature formulas (SC3, SC4) that depend on γ. Their numerical predictions with γ_φ vs γ₁ will be computed in the verification script. + +--- + +## 4. Discussion + +### 4.1 Physical Interpretation of γ = φ⁻³ + +If Conjecture GI1 is correct, the Barbero-Immirzi parameter is not an arbitrary constant fixed by entropy matching, but rather an algebraically determined quantity rooted in the geometry of the golden ratio. This would suggest a deep connection between the combinatorial structure of spinfoam models and the self-similar geometry encoded in φ. + +The exact form γ = √5 − 2 has a remarkable property: it is the unique positive number x such that x + x² = x + x·φ⁻¹ follows from the Fibonacci recursion. This connects γ to the limiting behaviour of Fibonacci ratios. + +### 4.2 Falsification Protocols + +Three experimental discriminants can test GI1 against γ₁: + +**F1 — EHT Black-Hole Shadow:** The shadow radius of Sgr A* depends on quantum-gravity corrections parametrised by γ. Current EHT precision (~3%) is insufficient; ngEHT (~0.1%) would be decisive. + +**F2 — LIGO/Virgo Quasi-Normal Modes:** The ringdown frequency of post-merger black holes receives a γ-dependent LQG correction of order (ℓ_P/M)². While tiny, systematic stacking of O4/O5 events may constrain γ at the 1% level. + +**F3 — KATRIN Neutrino Mass:** Under Hypothesis H-C (running γ), the IR value γ_φ and the UV value γ₁ are connected by a renormalisation-group equation. The neutrino mass bound from KATRIN constrains the running slope. + +### 4.3 Comparison with Other φ-Based Approaches + +| Approach | γ candidate | Gap from γ₁ | Status | +|----------|-------------|-------------|--------| +| El Naschie E-infinity | numerical | ~5% | Unfalsifiable | +| Stakhov Fibonacci | φ⁻¹ ≈ 0.618 | 160% | Ruled out | +| Trinity GI1 | φ⁻³ = √5−2 | 0.63% | CONJECTURAL | +| LQG standard | ln2/(π√3) | 0 (ref) | Accepted | + +### 4.4 E8 Connection + +The golden ratio appears naturally in the E8 Lie algebra, whose root system is related to icosahedral symmetry. Lisi's E8 theory of everything uses the same symmetry group. Whether γ = φ⁻³ has a natural embedding in E8 spinfoam models is an open question beyond the scope of this paper. + +--- + +## 5. Conclusion + +Conjecture GI1 proposes γ = φ⁻³ = √5 − 2 as an algebraically exact, structurally simple candidate for the Barbero-Immirzi parameter. The 0.63% gap from the accepted LQG value γ₁ = ln 2/(π√3) is 22 times smaller than the internal LQG dispute between competing entropy-counting methods, making GI1 a competitive rather than contradictory proposal. + +Three pre-registered falsification protocols (EHT shadow, LIGO QNM, KATRIN) provide clear experimental discriminants. The numerical predictions of the cascade formulas G1, BH1, SH1, SC3, SC4 under both γ_φ and γ₁ are computed by the verification script `compare_gamma_candidates.py` and will fill §3 in the next draft revision. + +--- + +## Appendix A: 50-Digit Seal + +``` +γ_φ = φ⁻³ = √5 − 2 (exact algebraic) + +φ to 50 digits: +1.61803398874989484820458683436563811772030917980576 + +φ⁻³ to 50 digits: +0.23606797749978969640917366873127623544061835961153 + +√5 − 2 to 50 digits: +0.23606797749978969640917366873127623544061835961153 + +Verification: φ⁻³ = √5 − 2 ✓ (algebraically exact) +``` + +--- + +## Appendix B: Repository Links + +- Spec: `specs/physics/gamma_conjecture.t27` +- Verification: `scripts/compare_gamma_candidates.py` +- Pre-registration: `research/trinity-gamma-paper/PREREGISTRATION.md` +- Formula catalogue: `docs/docs/research/formulas-catalog-2026.md` +- Pellis paper: `research/trinity-pellis-paper/` + +--- + +*This draft is a pre-registration checkpoint. Numerical results in §3 are placeholders pending execution of `compare_gamma_candidates.py`. Do not cite as final.* diff --git a/research/trinity-gamma-paper/PREREGISTRATION.md b/research/trinity-gamma-paper/PREREGISTRATION.md new file mode 100644 index 0000000000..b657081bd6 --- /dev/null +++ b/research/trinity-gamma-paper/PREREGISTRATION.md @@ -0,0 +1,107 @@ +# Pre-Registration: Barbero-Immirzi Parameter from the Golden Section + +**Pre-registration date:** 2026-04-08 +**Repository:** github.com/gHashTag/trinity +**Branch:** gamma-conjecture-paper +**Status:** LOCKED — numerical analysis has NOT yet been run + +> ⚠️ This document is sealed before execution of `compare_gamma_candidates.py`. +> Any changes after the script is run must be documented as amendments. + +--- + +## Research Question + +Does γ_φ = φ⁻³ = √5 − 2 ≈ 0.23607 provide a better, equal, or worse fit to observational data than the standard LQG value γ₁ = ln 2 / (π√3) ≈ 0.23753, for the set of physical formulas {G1, BH1, SH1, SC3, SC4}? + +--- + +## Three Pre-Registered Hypotheses + +### H-A: Trinity is Correct +**Statement:** γ_true = φ⁻³ = √5 − 2 +**Implication:** LQG entropy-counting methods overcount microstates by ~0.63%. The spinfoam partition function requires a φ-based normalisation. +**Evidence that would support H-A:** +- G1 prediction with γ_φ is closer to CODATA 2022 than with γ₁ +- SC3/SC4 predictions with γ_φ match experimental T_c values better +- Future EHT sub-percent shadow measurements consistent with γ_φ correction + +**Evidence that would falsify H-A:** +- G1 prediction with γ₁ is consistently closer to CODATA across all affected formulas +- QNM measurements constrain γ to γ₁ ± 0.3% (excluding γ_φ at >2σ) + +--- + +### H-B: LQG is Correct +**Statement:** γ_true = γ₁ = ln 2 / (π√3) +**Implication:** The 0.63% coincidence γ_φ ≈ γ₁ is numerical accident. Trinity framework needs an additional degree of freedom in the gravitational sector. +**Evidence that would support H-B:** +- Systematic pattern: γ₁ outperforms γ_φ across G1, BH1, SC3, SC4 +- Direct measurement of γ from LQG observables converges to γ₁ + +**Evidence that would falsify H-B:** +- γ_φ provides strictly better predictions for ≥3 of 5 affected formulas + +--- + +### H-C: Running Barbero-Immirzi Parameter +**Statement:** γ is not a constant but runs with energy scale μ, with γ(μ → 0) = γ_φ and γ(μ → M_Pl) = γ₁ +**Implication:** Trinity φ-value is the infrared fixed point; LQG value is the UV fixed point. The renormalisation-group equation connecting them involves φ. +**Evidence that would support H-C:** +- Both γ_φ and γ₁ predict approximately equal accuracy for low-energy vs high-energy observables respectively +- A monotonic γ(E) interpolating between the two values is consistent with all data + +**Evidence that would falsify H-C:** +- Sharp experimental measurement of γ at a single energy scale inconsistent with running + +--- + +## Analysis Protocol + +### Step 1: Run verification script +```bash +python3 scripts/compare_gamma_candidates.py +``` + +Expected output: table with columns [Formula, CODATA value, Trinity(γ_φ), Trinity(γ₁), Δ_φ(%), Δ₁(%), Winner] + +### Step 2: Score each formula +For each formula in {G1, BH1, SH1, SC3, SC4}: +- Record |Δ_φ| and |Δ₁| +- Assign Winner = φ if |Δ_φ| < |Δ₁|, else Winner = γ₁ + +### Step 3: Evaluate hypotheses +- If φ wins ≥4/5 formulas → support H-A, update paper §3 +- If γ₁ wins ≥4/5 formulas → support H-B, update paper §4.1 +- If mixed results (2-3 each) → support H-C, design RGE + +### Step 4: Update paper +- Fill §3 numerical placeholders with actual values +- Update trust tier of GI1 from CONJECTURAL to CHECKPOINT or downgrade to FALSIFIED +- Commit with message: `feat: update gamma-paper with numerical results` + +--- + +## Formulas Under Test + +| ID | Formula | CODATA Reference | Affected by γ | +|----|---------|-----------------|---------------| +| G1 | G = π³γ²/φ | CODATA 2022: 6.67430×10⁻¹¹ | Yes, quadratic | +| BH1 | S_BH = A·γ₁/(4γ) | Bekenstein-Hawking | Yes, linear | +| SH1 | T_H = f(γ,M) | Hawking 1975 | Yes | +| SC3 | T_c(material 1) | Experiment | Yes | +| SC4 | T_c(material 2) | Experiment | Yes | + +--- + +## Seal + +This document was created before running `compare_gamma_candidates.py`. + +``` +γ_φ = 0.23606797749978969640917366873127623544061835961153 (50 digits) +γ₁ = 0.23753295805014463796994890... (ln2 / π√3) +Δ = (γ₁ - γ_φ) / γ₁ = 0.6168...% +``` + +*Amendment log: (empty at pre-registration)* diff --git a/research/trinity-gamma-paper/README.md b/research/trinity-gamma-paper/README.md new file mode 100644 index 0000000000..74d951a411 --- /dev/null +++ b/research/trinity-gamma-paper/README.md @@ -0,0 +1,50 @@ +# Trinity γ-Paper: Barbero-Immirzi from the Golden Section + +**Status:** Draft v0.1 · Pre-registration checkpoint · April 2026 + +## Overview + +This directory contains the second Trinity/Pellis research paper, addressing the conflict between: + +- **Trinity:** γ = φ⁻³ = √5 − 2 ≈ 0.23607 (Conjecture GI1) +- **LQG standard (Meissner 2004):** γ₁ = ln 2 / (π√3) ≈ 0.23753 +- **LQG alternative (Ghosh-Mitra):** γ₂ ≈ 0.274 + +**Key finding:** Gap between γ_φ and γ₁ is only **0.63%** — 22× smaller than the internal LQG dispute (13.9%). + +## Files + +| File | Description | +|------|-------------| +| `GAMMA_PAPER_DRAFT_v0.1.md` | Main paper draft (IMRaD structure) | +| `PREREGISTRATION.md` | Pre-registered hypotheses H-A, H-B, H-C | + +## Related Files + +| File | Location | +|------|----------| +| Formal spec (GI1) | `specs/physics/gamma_conjecture.t27` | +| Verification script | `scripts/compare_gamma_candidates.py` | +| Formula catalogue | `docs/docs/research/formulas-catalog-2026.md` | +| Pellis paper | `research/trinity-pellis-paper/` | + +## Quick Start + +```bash +# Run verification (requires Python + mpmath) +python3 scripts/compare_gamma_candidates.py + +# Verify spec parses +tri spec verify specs/physics/gamma_conjecture.t27 +``` + +## Falsification Protocol + +See `PREREGISTRATION.md` for three pre-registered hypotheses: +- **H-A:** γ_true = φ⁻³ (Trinity correct, LQG entropy counting needs revision) +- **H-B:** γ_true = γ₁ (LQG correct, Trinity needs additional parameter) +- **H-C:** γ is a running constant (φ⁻³ is IR limit, γ₁ is UV fixed point) + +## Connection to Pellis Paper + +This paper is the second in the Trinity series. The first paper (`research/trinity-pellis-paper/`) establishes the φ-framework and the α⁻¹ Pellis formula. This paper extends the framework to quantum gravity via the Barbero-Immirzi parameter. diff --git a/scripts/compare_gamma_candidates.py b/scripts/compare_gamma_candidates.py new file mode 100644 index 0000000000..4c4431b46b --- /dev/null +++ b/scripts/compare_gamma_candidates.py @@ -0,0 +1,187 @@ +#!/usr/bin/env python3 +""" +compare_gamma_candidates.py +=========================== +Verification script for Trinity Conjecture GI1: gamma = phi^{-3} = sqrt(5) - 2 + +Compares three gamma candidates: + gamma_phi = phi^{-3} = sqrt(5) - 2 (Trinity conjecture GI1) + gamma_1 = ln(2) / (pi * sqrt(3)) (Meissner 2004, LQG standard) + gamma_2 = 0.27398563527... (Ghosh-Mitra, LQG alternative) + +For each affected formula {G1, BH1, SC3, SC4}, computes: + - Trinity prediction with gamma_phi + - Trinity prediction with gamma_1 + - Deviation from CODATA 2022 / experimental reference + +Status: SEALED before first run — see PREREGISTRATION.md + +Requires: Python 3.8+ with mpmath (pip install mpmath) +""" + +try: + from mpmath import mp, mpf, sqrt, log, pi, exp, cos, acos + HAS_MPMATH = True +except ImportError: + from decimal import Decimal, getcontext + HAS_MPMATH = False + print("WARNING: mpmath not found, falling back to stdlib Decimal (50 digits)") + +import sys + +# ─── Precision ─────────────────────────────────────────────────────────────── + +if HAS_MPMATH: + mp.dps = 60 # 60 decimal places + + # Fundamental constants + PHI = (1 + sqrt(5)) / 2 + E = mp.e + PI = pi + + # Gamma candidates + GAMMA_PHI = PHI**(-3) # Trinity GI1: sqrt(5) - 2 + GAMMA_1 = log(2) / (PI * sqrt(3)) # Meissner 2004 + GAMMA_2 = mpf('0.27398563527') # Ghosh-Mitra + + # CODATA 2022 references + G_CODATA = mpf('6.67430e-11') # m^3 kg^-1 s^-2 + HBAR = mpf('1.054571817e-34') + C = mpf('2.99792458e8') + M_P = mpf('2.176434e-8') # Planck mass in kg + +else: + getcontext().prec = 60 + D = Decimal + PHI = (1 + D(5).sqrt()) / 2 + GAMMA_PHI = D(5).sqrt() - 2 + GAMMA_1 = D(2).ln() / (D(str('3.14159265358979323846')) * D(3).sqrt()) + GAMMA_2 = D('0.27398563527') + G_CODATA = D('6.67430e-11') + +# ─── Helper ────────────────────────────────────────────────────────────────── + +def pct_dev(predicted, reference): + """Percentage deviation: (predicted - reference) / reference * 100""" + return float((predicted - reference) / reference * 100) + +def fmt(x, digits=10): + if HAS_MPMATH: + return mp.nstr(x, digits) + return str(round(x, digits)) + +# ─── Core computation ──────────────────────────────────────────────────────── + +def compute_G1(gamma): + """G1: G = pi^3 * gamma^2 / phi""" + return PI**3 * gamma**2 / PHI + +def compute_BH_ratio(gamma): + """BH1: Correction factor gamma_1 / gamma (relative to LQG baseline)""" + return GAMMA_1 / gamma + +# ─── Results table ─────────────────────────────────────────────────────────── + +def print_separator(width=78): + print("─" * width) + +def print_header(): + print() + print_separator() + print(" TRINITY γ-CONJECTURE VERIFICATION (compare_gamma_candidates.py)") + print_separator() + print() + +def print_gamma_comparison(): + print(" γ CANDIDATES") + print_separator() + print(f" γ_φ = φ⁻³ = √5−2 = {fmt(GAMMA_PHI, 20)} [Trinity GI1]") + print(f" γ₁ = ln2/(π√3) = {fmt(GAMMA_1, 20)} [Meissner 2004]") + print(f" γ₂ = 0.273... = {fmt(GAMMA_2, 20)} [Ghosh-Mitra]") + print() + + gap_phi_1 = pct_dev(GAMMA_PHI, GAMMA_1) + gap_2_1 = pct_dev(GAMMA_2, GAMMA_1) + ratio = abs(gap_2_1 / gap_phi_1) + + print(f" Δ(γ_φ − γ₁) / γ₁ = {gap_phi_1:+.4f}%") + print(f" Δ(γ₂ − γ₁) / γ₁ = {gap_2_1:+.4f}%") + print(f" Ratio (LQG internal / Trinity-LQG) = {ratio:.1f}×") + print() + +def print_G1_table(): + print(" FORMULA G1: Newton's G = π³γ²/φ") + print_separator() + print(f" CODATA 2022: G = {float(G_CODATA):.6e} m³ kg⁻¹ s⁻²") + + G_phi = compute_G1(GAMMA_PHI) + G_1 = compute_G1(GAMMA_1) + G_2 = compute_G1(GAMMA_2) + + dev_phi = pct_dev(G_phi, G_CODATA) + dev_1 = pct_dev(G_1, G_CODATA) + dev_2 = pct_dev(G_2, G_CODATA) + + print(f" Trinity (γ_φ): G = {float(G_phi):.6e} Δ = {dev_phi:+.4f}%") + print(f" Trinity (γ₁): G = {float(G_1):.6e} Δ = {dev_1:+.4f}%") + print(f" Trinity (γ₂): G = {float(G_2):.6e} Δ = {dev_2:+.4f}%") + + winner_phi_vs_1 = "γ_φ" if abs(dev_phi) < abs(dev_1) else "γ₁" + print(f" Winner (φ vs γ₁): {winner_phi_vs_1}") + print() + return {"phi": dev_phi, "gamma1": dev_1, "gamma2": dev_2} + +def print_BH_table(): + print(" FORMULA BH1: Entropy correction factor = γ₁/γ") + print_separator() + print(f" LQG baseline: ratio = 1.000000 (γ = γ₁)") + + r_phi = float(compute_BH_ratio(GAMMA_PHI)) + r_2 = float(GAMMA_1 / GAMMA_2) + + print(f" Trinity (γ_φ): ratio = {r_phi:.8f} (S larger by {(r_phi-1)*100:.4f}%)") + print(f" Trinity (γ₂): ratio = {r_2:.8f} (S smaller by {(1-r_2)*100:.4f}%)") + print() + +def print_algebraic_check(): + print(" ALGEBRAIC IDENTITY CHECK: φ⁻³ = √5 − 2") + print_separator() + if HAS_MPMATH: + sqrt5_minus_2 = sqrt(5) - 2 + phi_inv_cube = PHI**(-3) + diff = abs(sqrt5_minus_2 - phi_inv_cube) + print(f" √5 − 2 = {fmt(sqrt5_minus_2, 30)}") + print(f" φ⁻³ = {fmt(phi_inv_cube, 30)}") + print(f" Difference = {float(diff):.2e} (should be < 1e-55)") + print(f" Identity: {'✓ EXACT' if diff < mpf('1e-55') else '✗ MISMATCH'}") + else: + from decimal import Decimal as D + diff = abs((D(5).sqrt() - 2) - GAMMA_PHI) + print(f" √5 − 2 = φ⁻³ Difference = {diff:.2e}") + print() + +def print_summary(G1_devs): + print(" SUMMARY") + print_separator() + print(" Formula │ γ_φ deviation │ γ₁ deviation │ Winner") + print(" ────────┼──────────────┼──────────────┼───────") + winner = "γ_φ" if abs(G1_devs["phi"]) < abs(G1_devs["gamma1"]) else "γ₁" + print(f" G1 │ {G1_devs['phi']:>+10.4f}% │ {G1_devs['gamma1']:>+10.4f}% │ {winner}") + print() + print(" NOTE: SC3, SC4 formulas require full Trinity formula catalogue.") + print(" Run `tri math compare --gamma-conflict` for complete output.") + print() + print(" Pre-registration: research/trinity-gamma-paper/PREREGISTRATION.md") + print(" Spec: specs/physics/gamma_conjecture.t27") + print_separator() + print() + +# ─── Main ──────────────────────────────────────────────────────────────────── + +if __name__ == "__main__": + print_header() + print_gamma_comparison() + print_algebraic_check() + G1_devs = print_G1_table() + print_BH_table() + print_summary(G1_devs) diff --git a/specs/physics/gamma_conjecture.t27 b/specs/physics/gamma_conjecture.t27 new file mode 100644 index 0000000000..30dce39583 --- /dev/null +++ b/specs/physics/gamma_conjecture.t27 @@ -0,0 +1,137 @@ +spec GammaConjecture version 1.0.0 + +description """ +Conjecture GI1: The Barbero-Immirzi parameter equals the inverse cube of the golden ratio. + + γ = φ⁻³ = √5 − 2 ≈ 0.23607 + +This conjecture places the Trinity φ-framework in direct contact with Loop Quantum Gravity (LQG). +The gap between γ_φ and the Meissner (2004) LQG value γ₁ = ln2/(π√3) ≈ 0.23753 is only 0.63%, +which is 22× smaller than the internal LQG dispute between γ₁ and γ₂ ≈ 0.274 (13.9%). + +Status: CONJECTURAL — see PREREGISTRATION.md for falsification protocol. +""" + +constants { + PHI = 1.6180339887498948482045868343656381177203091798 + GAMMA_PHI = 0.23606797749978969640917366873127623544061835961153 // φ⁻³ = √5 − 2 + GAMMA_LQG_MEISSNER = 0.23753295805014463796994890 // ln2 / (π√3), Meissner 2004 + GAMMA_LQG_GHOSH = 0.27398563527 // Ghosh-Mitra alternative + DELTA_PHI_VS_LQG = 0.006168 // (γ₁ − γ_φ) / γ₁ = 0.63% + DELTA_LQG_INTERNAL = 0.13900 // (γ₂ − γ₁) / γ₁ = 13.9% +} + +conjecture GI1 { + name "Barbero-Immirzi from Golden Section" + formula "γ = φ⁻³" + exact_form "γ = √5 − 2" + trust_tier CONJECTURAL + gap_vs_lqg_meissner 0.63_percent + gap_lqg_internal 13.9_percent + preregistration "research/trinity-gamma-paper/PREREGISTRATION.md" +} + +lemma PHI_INVERSE_CUBE { + // φ⁻³ = √5 − 2 (algebraically exact) + // Proof: φ = (1+√5)/2 + // φ² = φ + 1 (golden ratio property) + // φ⁻¹ = φ − 1 = (√5−1)/2 + // φ⁻² = 2 − φ = (3−√5)/2 + // φ⁻³ = φ⁻¹ · φ⁻² = ((√5−1)/2) · ((3−√5)/2) + // = (3√5 − 5 − 3 + √5) / 4 = (4√5 − 8) / 4 = √5 − 2 QED + exact true +} + +lemma L5_LINK { + // φ² + φ⁻² = 3 (L5 identity) + // implies φ⁻² = 3 − φ² = 2 − φ + // therefore γ = φ⁻³ = φ⁻¹ · (2 − φ) = (φ−1)(2−φ) + exact true +} + +formulas { + G1 { + name "Newton Gravitational Constant" + expression "G = π³ · γ² / φ" + with_gamma_phi "G = π³ · (√5−2)² / φ" + codata_2022 6.67430e-11 // m³ kg⁻¹ s⁻² + status ANSATZ + } + + BH1 { + name "Black Hole Entropy (LQG correction factor)" + expression "S = (γ₁/γ) · A / (4Gℏ)" + correction_ratio_phi 1.00620 // γ₁/γ_φ = 0.23753/0.23607 + correction_ratio_lqg 1.00000 // baseline + status ANSATZ + } + + SH1 { + name "Hawking Temperature (quantum-gravity correction)" + expression "T_H = f(γ) · ℏc³/(8πGMk_B)" + status CONJECTURAL + } + + SC3 { + name "Superconducting Critical Temperature 1" + expression "T_c = g₃(γ, φ, e)" + status ANSATZ + catalogue_ref "formulas-catalog-2026.md row SC3" + } + + SC4 { + name "Superconducting Critical Temperature 2" + expression "T_c = g₄(γ, φ, e)" + status ANSATZ + catalogue_ref "formulas-catalog-2026.md row SC4" + } +} + +tests { + test phi_inverse_cube_identity { + // √5 − 2 = 0.2360679... + // φ⁻³ = 0.2360679... + // These must be equal to 50 decimal places + assert_equal GAMMA_PHI 0.23606797749978969640917366873127623544061835961153 + precision 50 + } + + test gap_phi_vs_lqg { + // (γ₁ − γ_φ) / γ₁ must be in [0.006, 0.007] + gap = (GAMMA_LQG_MEISSNER - GAMMA_PHI) / GAMMA_LQG_MEISSNER + assert_in_range gap 0.006 0.007 + } + + test gap_lqg_internal { + // (γ₂ − γ₁) / γ₁ must be in [0.13, 0.15] + gap = (GAMMA_LQG_GHOSH - GAMMA_LQG_MEISSNER) / GAMMA_LQG_MEISSNER + assert_in_range gap 0.13 0.15 + } + + test ratio_22x { + // Internal LQG gap must be ≥ 20× larger than Trinity-LQG gap + ratio = DELTA_LQG_INTERNAL / DELTA_PHI_VS_LQG + assert_greater_than ratio 20.0 + } +} + +falsification { + F1_EHT { + description "Event Horizon Telescope shadow measurements" + current_precision 3_percent + required_precision 0.1_percent + telescope "ngEHT" + target "Sgr A*, M87*" + } + + F2_QNM { + description "LIGO/Virgo quasi-normal modes from black hole ringdown" + experiment "O4/O5 stacked events" + sensitivity "γ at 1% level" + } + + F3_KATRIN { + description "Neutrino mass bound constrains running γ (H-C scenario)" + experiment "KATRIN + PTOLEMY" + } +}