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> *These numbers are not human inventions — they are discoveries. Any mind, anywhere, investigating reality, must encounter these same values. They are the fingerprints of existence itself.*
> **v2.0 update note:** Several values have shifted with CODATA 2022 (published 20242025), the muon g-2 final result (June 2025), DESI DR2 (2025), and the discovery of the 52nd Mersenne prime (October 2024). See [§12 What's New Since v1.0](#12-whats-new-since-v10) for a summary.
---
## Table of Contents
@@ -17,6 +19,7 @@
9. [Information-Theoretic Constants](#9-information-theoretic-constants)
10. [The Weird and Profound](#10-the-weird-and-profound)
11. [The Alien Contact Number Set](#11-the-alien-contact-number-set)
12. [What's New Since v1.0](#12-whats-new-since-v10)
---
@@ -58,20 +61,23 @@ A circle on Earth has the same π as a circle in Andromeda.
The **purest physical truths** — no unit system required, identical value everywhere in the universe. These are the settings on reality's control panel.
> **CODATA 2022 update:** Several precision values were refined in the 2022 adjustment (published 20242025). The most consequential shift was a small reduction in the central value of the fine structure constant.
| Constant | Symbol | Value | What It Determines |
|----------|--------|-------|-------------------|
| **Fine Structure Constant** | α | 0.0072973525693(11) ≈ 1/137.036 | Electromagnetic interaction strength. Governs ALL chemistry. |
| **Proton-Electron Mass Ratio** | μ | 1836.15267343(11) | Atomic stability. Why chemistry works at all. |
| **Strong Coupling Constant** | α_s | ~0.1181 (at Z boson mass) | Nuclear force strength. Why atomic nuclei exist. |
| **Fine Structure Constant** | α | 0.0072973525643(11) ≈ 1/137.035999177(21) — *CODATA 2022* | Electromagnetic interaction strength. Governs ALL chemistry. |
| **Proton-Electron Mass Ratio** | μ | 1836.152673426(32) — *CODATA 2022* | Atomic stability. Why chemistry works at all. |
| **Strong Coupling Constant** | α_s | 0.1180(9) (at Z boson mass, PDG 2024) | Nuclear force strength. Why atomic nuclei exist. |
| **Gravitational Coupling** | α_G | ~5.9 × 10⁻³⁹ | Gravity's relative strength. Why gravity is so weak. |
| **Weak Mixing Angle** | sin²θ_W | ~0.22290 | Electroweak unification parameter. |
| **Electron g-factor Anomaly** | a_e | 0.00115965218128(18) | QED's most precise prediction. Theory matches experiment to 12 digits. |
| **Electron g-factor Anomaly** | a_e | 0.00115965218059(13) | QED's most precise prediction. Theory matches experiment to 12 digits. |
| **Muon g-factor Anomaly** | a_μ | 0.001165920705(114)(91) — *Fermilab final, 2025* | New world record precision (127 ppb). Sensitive to new physics. |
| **Cabibbo Angle** | θ_C | ~13.1° | Quark mixing. Why strange particles decay as they do. |
### The Fine Structure Constant: Deep Dive
```
α = e²/(4πε₀ℏc) ≈ 1/137.035999...
α = e²/(4πε₀ℏc) ≈ 1/137.035999177... [CODATA 2022]
This single dimensionless number determines:
@@ -131,26 +137,60 @@ Any advanced civilization will have discovered this number
and likely wondered the same things.
```
### The Muon g-2 Saga (closed in 2025)
```
On June 3, 2025, the Fermilab Muon g-2 collaboration released
its final, third-and-best measurement of the muon's anomalous
magnetic moment, ending an effort that began at CERN in the 1970s:
a_μ = 0.001165920705 ± 0.000000000114(stat) ± 0.000000000091(syst)
Precision: 127 parts per billion — a fourfold improvement
over the previous-generation BNL E821 result.
The Standard Model prediction has been a moving target:
├── 2020 Theory Initiative whitepaper (data-driven):
│ a_μ = (116591810 ± 43) × 10⁻¹¹ → ~5σ tension w/ experiment
├── Lattice QCD calculations (BMW et al.):
│ agree better with experiment (smaller tension)
└── 2025 Theory Initiative update:
a_μ = (116592033 ± 62) × 10⁻¹¹
statistically incompatible with the 2020 prediction (~3σ)
The takeaway: the experimental answer is now locked in.
Whether this constitutes a hint of physics beyond the
Standard Model has shifted into a question about how to
calculate hadronic vacuum polarization — not about whether
the muon really wobbles a little faster than QED alone says.
This ranks among the highest-precision tests of physics
ever performed, and the experiment received the 2026
Breakthrough Prize in Fundamental Physics.
```
---
## 3. Fundamental Physical Constants
These depend on unit systems, but their **ratios** are universal. Any civilization measuring the universe finds these.
> Values reflect CODATA 2022 (published 20242025). Constants marked *exact* are defined by the SI as of the 2019 redefinition.
| Constant | Symbol | SI Value | Role |
|----------|--------|----------|------|
| **Speed of Light** | c | 299,792,458 m/s (exact) | Maximum velocity. Spacetime structure. Causality. |
| **Planck's Constant** | h | 6.62607015 × 10⁻³⁴ J·s (exact) | Quantum of action. Granularity of nature. |
| **Reduced Planck** | ℏ | 1.054571817 × 10⁻³⁴ J·s | h/(2π). More natural in many equations. |
| **Speed of Light** | c | 299,792,458 m/s (*exact*) | Maximum velocity. Spacetime structure. Causality. |
| **Planck's Constant** | h | 6.62607015 × 10⁻³⁴ J·s (*exact*) | Quantum of action. Granularity of nature. |
| **Reduced Planck** | ℏ | 1.054571817... × 10⁻³⁴ J·s | h/(2π). More natural in many equations. |
| **Gravitational Constant** | G | 6.67430(15) × 10⁻¹¹ m³/(kg·s²) | Gravity's strength. Least precisely known. |
| **Boltzmann Constant** | k_B | 1.380649 × 10⁻²³ J/K (exact) | Temperature-energy bridge. Statistical mechanics. |
| **Elementary Charge** | e | 1.602176634 × 10⁻¹⁹ C (exact) | Quantum of electric charge. |
| **Electron Mass** | m_e | 9.1093837015(28) × 10⁻³¹ kg | Lightest charged particle. Sets atomic scales. |
| **Proton Mass** | m_p | 1.67262192369(51) × 10⁻²⁷ kg | Lightest stable baryon. Nuclear physics scale. |
| **Neutron Mass** | m_n | 1.67492749804(95) × 10⁻²⁷ kg | Slightly heavier than proton. Enables nuclear stability. |
| **Avogadro's Number** | N_A | 6.02214076 × 10²³ /mol (exact) | Atoms per mole. Bridge macro-micro. |
| **Vacuum Permittivity** | ε₀ | 8.8541878128 × 10⁻¹² F/m | Electric field in vacuum. |
| **Vacuum Permeability** | μ₀ | 1.25663706212 × 10⁻⁶ H/m | Magnetic field in vacuum. |
| **Boltzmann Constant** | k_B | 1.380649 × 10⁻²³ J/K (*exact*) | Temperature-energy bridge. Statistical mechanics. |
| **Elementary Charge** | e | 1.602176634 × 10⁻¹⁹ C (*exact*) | Quantum of electric charge. |
| **Electron Mass** | m_e | 9.1093837139(28) × 10⁻³¹ kg*CODATA 2022* | Lightest charged particle. Sets atomic scales. |
| **Proton Mass** | m_p | 1.67262192595(52) × 10⁻²⁷ kg*CODATA 2022* | Lightest stable baryon. Nuclear physics scale. |
| **Neutron Mass** | m_n | 1.67492750056(85) × 10⁻²⁷ kg*CODATA 2022* | Slightly heavier than proton. Enables nuclear stability. |
| **Avogadro's Number** | N_A | 6.02214076 × 10²³ /mol (*exact*) | Atoms per mole. Bridge macro-micro. |
| **Vacuum Permittivity** | ε₀ | 8.8541878188(14) × 10⁻¹² F/m | Electric field in vacuum. |
| **Vacuum Permeability** | μ₀ | 1.25663706127(20) × 10⁻⁶ H/m | Magnetic field in vacuum. |
### Relationships Between Constants
@@ -171,6 +211,24 @@ These relationships are THEOREMS, not observations.
Any physics leads to these connections.
```
### Note on G
```
Of all the fundamental constants, the gravitational constant
G remains the most stubbornly imprecise. CODATA's recommended
value still rests on data sets with significant inconsistencies,
which is why a 3.9 expansion factor has been applied to the
uncertainty since 2018, retained again in 2022.
The 2026 CODATA adjustment — closing on 31 December 2026 —
is preparing to incorporate new G measurements presented at
the September 2025 task-group meeting in Warsaw.
The other constants have been tightened to parts-per-billion
or better. G languishes at parts-per-ten-thousand.
This is a known anomaly, not a settled number.
```
---
## 4. Planck Units: Nature's Native System
@@ -245,6 +303,14 @@ Twin Primes (differ by 2):
(3,5), (5,7), (11,13), (17,19), (29,31), (41,43)...
Infinite? Still unproven!
Largest known prime (as of 2026):
M₁₃₆₂₇₉₈₄₁ = 2¹³⁶²⁷⁹⁸⁴¹ − 1
├── 41,024,320 decimal digits
├── 52nd known Mersenne prime
├── Discovered Oct 12, 2024 by Luke Durant (GIMPS)
├── First Mersenne prime found on GPUs (not CPUs)
└── Ends a 28-year reign of CPU-found records
Any civilization doing arithmetic discovers primes.
```
@@ -299,7 +365,10 @@ Definition: Equal to sum of proper divisors
Connection to Mersenne Primes:
├── If 2ⁿ-1 is prime, then 2ⁿ⁻¹(2ⁿ-1) is perfect
├── All known even perfect numbers have this form
├── Odd perfect numbers: NONE FOUND (existence unknown!)
├── Each new Mersenne prime begets a new perfect number
├── 52 even perfect numbers known (as of 2026)
├── Largest: 2¹³⁶²⁷⁹⁸⁴⁰ × (2¹³⁶²⁷⁹⁸⁴¹ 1), > 82 million digits
└── Odd perfect numbers: NONE FOUND (existence still unknown!)
Rare, beautiful, discoverable by any number theorist.
```
@@ -530,12 +599,15 @@ It's provable from quantum field theory.
The parameters that define our universe's structure and evolution.
> **Status update (2026):** Two long-running anomalies have hardened into outright crises since v1.0 — the **Hubble tension** now exceeds 6σ, and **DESI DR2 (2025)** has provided up-to-4.2σ evidence that dark energy is *not* a cosmological constant. The familiar ΛCDM table below should be read as *one description* of the universe, not the only one consistent with current data.
| Constant | Symbol | Value | Meaning |
|----------|--------|-------|---------|
| **Hubble Constant** | H₀ | ~70 km/s/Mpc | Universe expansion rate |
| **Age of Universe** | t₀ | 13.787 ± 0.020 billion years | Time since Big Bang |
| **Hubble Constant** (early-universe / CMB) | H₀ | 67.4 ± 0.5 km/s/Mpc | Inferred from Planck CMB + ΛCDM |
| **Hubble Constant** (local / SH0ES) | H₀ | ~73.0 ± 1.0 km/s/Mpc | Cepheid-calibrated SN Ia distance ladder |
| **Age of Universe** | t₀ | 13.787 ± 0.020 billion years | Time since Big Bang (under ΛCDM) |
| **CMB Temperature** | T_CMB | 2.72548 ± 0.00057 K | Cosmic background radiation |
| **Cosmological Constant** | Λ | ~10⁻⁵² m⁻² | Dark energy density |
| **Cosmological Constant** | Λ | ~10⁻⁵² m⁻² | Dark energy density (now contested) |
| **Critical Density** | ρ_c | ~9.47 × 10⁻²⁷ kg/m³ | Flat universe density |
| **Matter Density** | Ω_m | ~0.315 | Matter fraction of critical |
| **Dark Energy Density** | Ω_Λ | ~0.685 | Dark energy fraction |
@@ -544,14 +616,74 @@ The parameters that define our universe's structure and evolution.
| **Curvature Parameter** | Ω_k | ~0 (very close) | Universe flatness |
| **Baryon/Photon Ratio** | η | ~6.1 × 10⁻¹⁰ | Matter-antimatter asymmetry |
### The Hubble Tension (Now > 6σ)
```
Two independent, mature, well-controlled methods give
incompatible answers for the universe's expansion rate:
EARLY UNIVERSE (Planck CMB + ΛCDM extrapolation):
H₀ = 67.4 ± 0.5 km/s/Mpc
Confirmed by: WMAP+ACT (67.6 ± 1.1), DESI+BBN (68.5 ± 0.6)
LATE UNIVERSE (SH0ES, distance ladder):
H₀ = 73.04 ± 1.04 km/s/Mpc
JWST has now corroborated the Cepheid calibration,
ruling out crowding/dust as a systematic.
Tension significance:
├── 2019: ~4σ
├── 2022: ~5σ (SH0ES with 300+ SNe Ia)
├── 2025: > 6σ (Tensions in Cosmology meeting, Corfu)
└── No single systematic can plausibly close the gap.
What this means:
├── ΛCDM may be incomplete or wrong
├── New physics in the early universe (early dark energy?)
├── New physics in the late universe (modified gravity?)
├── Or — most uncomfortably — both
└── The European Commission has awarded €12M (RedH0T project)
specifically to chase this down.
This is the most serious unresolved problem in cosmology
since dark energy itself was discovered in 1998.
```
### Dark Energy May Not Be a Constant
```
DESI (Dark Energy Spectroscopic Instrument) Data Releases:
├── DR1 (April 2024): first hints of evolving dark energy
└── DR2 (2025): evidence strengthened to up to 4.2σ
(depending on SN dataset combination)
The signal:
├── Combine DESI BAO + Planck CMB + SN Ia (Pantheon+/Union3/DES-Y5)
├── Fit a w₀wₐCDM model (time-evolving equation of state)
├── Late-universe data prefers w > -1 (quintessence-like)
├── Early-universe data prefers w < -1 (phantom-like)
└── A pure cosmological constant (w = -1 always) is disfavored.
Caveats (as of late 2025):
├── Bayesian model-comparison evidence is less decisive
│ than the frequentist ~4σ headline suggests
├── Different SN catalogues yield different significance
├── Independent analyses (TDCOSMO 2025 lensing) reinforce
│ the underlying tensions but not always the w₀wₐ fit
└── The cosmological constant is bruised, not buried.
If real, this is the biggest revision to fundamental cosmology
since the 1998 discovery of acceleration itself.
```
### The Cosmic Inventory
```
Universe Composition (today):
Universe Composition (today, ΛCDM baseline):
Dark Energy: 68.5% ████████████████████████████░░░░
Dark Matter: 26.6% ██████████░░░░░░░░░░░░░░░░░░░░░░
Ordinary Matter: 4.9% ██░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
Dark Energy: 68.5% ■■■■■■■■■■■■■■■■■■■■■■■■■■■■□□□□
Dark Matter: 26.6% ■■■■■■■■■■□□□□□□□□□□□□□□□□□□□□□□
Ordinary Matter: 4.9% ■■□□□□□□□□□□□□□□□□□□□□□□□□□□□□□□
Of Ordinary Matter:
├── Intergalactic gas: ~50%
@@ -793,6 +925,11 @@ Explanations:
├── Multiverse (all values exist, we're in viable one)
├── Deeper theory (values derivable from fundamental principles)
├── Unknown
Caveat (added 2026): If DESI's evolving dark energy holds up,
the "cosmological constant" entry in this list may not be a
constant at all — making the fine-tuning question stranger,
not simpler.
```
---
@@ -830,9 +967,9 @@ PROOFS AS COMMUNICATION:
```
DIMENSIONLESS (pure numbers):
├── α = 1/137.035999... (fine structure)
├── μ = 1836.15267... (proton/electron mass)
└── α_s ≈ 0.1181 (strong coupling)
├── α = 1/137.035999177... (fine structure, CODATA 2022)
├── μ = 1836.152673426... (proton/electron mass)
└── α_s ≈ 0.1180 (strong coupling)
SPECTROSCOPIC (stellar signatures):
├── Hydrogen emission lines
@@ -899,6 +1036,93 @@ Star map:
---
## 12. What's New Since v1.0
A summary of changes from the December 2024 edition:
### Constants refined (CODATA 2022, published May 2024 / Rev. Mod. Phys. 2025)
```
Fine structure constant:
v1.0: α = 0.0072973525693(11), α⁻¹ ≈ 1/137.036
v2.0: α = 0.0072973525643(11), α⁻¹ = 137.035999177(21)
Proton-electron mass ratio:
v1.0: μ = 1836.15267343(11)
v2.0: μ = 1836.152673426(32)
Electron, proton, neutron masses: tightened uncertainties
ε₀, μ₀: tightened uncertainties
G: unchanged value, same 3.9× expanded uncertainty (still the
problem child of CODATA — 2026 adjustment may revise)
```
### Muon g-2 saga concluded (June 3, 2025)
```
Fermilab released the experiment's third and final result:
a_μ = 0.001165920705(114)(91) [127 ppb precision]
Awarded the 2026 Breakthrough Prize in Fundamental Physics.
Comparison with Standard Model now hinges on hadronic vacuum
polarization calculations (lattice QCD vs. data-driven
e⁺e⁻ → hadrons) — the discrepancy moved from "experiment
vs. theory" to "theory vs. theory".
```
### Hubble tension hardened (now > 6σ)
```
By the September 2025 Tensions in Cosmology meeting (Corfu),
the H₀ discrepancy passed 6σ — well above any reasonable
"statistical fluke" threshold. JWST corroborated the SH0ES
Cepheid calibration, eliminating the most plausible
late-universe systematic. ΛCDM is showing real strain.
```
### DESI evidence for evolving dark energy
```
DESI DR1 (April 2024): first ~2.53.9σ hints
DESI DR2 (March 2025): strengthened to up to 4.2σ
(depending on SN dataset)
Combined with Planck CMB and Pantheon+/Union3/DES-Y5
supernovae, the data prefer a w₀wₐCDM model over a pure
cosmological constant. This is contested in the
Bayesian-evidence literature but is the most serious
challenge to ΛCDM in 25 years.
```
### New largest known prime (October 12, 2024)
```
M₁₃₆₂₇₉₈₄₁ = 2¹³⁶²⁷⁹⁸⁴¹ − 1
├── 41,024,320 decimal digits
├── 52nd Mersenne prime
├── 16 million more digits than the previous record
├── First Mersenne prime found on GPUs
└── Discoverer: Luke Durant (GIMPS), San Jose, CA
The previous record (M₈₂₅₈₉₉₃₃) had stood since December 2018.
A correspondingly new largest perfect number now exists,
with > 82 million digits.
```
### What's coming
```
├── CODATA 2026 adjustment (closing 31 December 2026)
│ — expected to incorporate new G measurements
├── DESI continued observations (DR3 anticipated)
├── J-PARC muon g-2/EDM experiment (data-taking from 2030)
├── Vera C. Rubin Observatory (LSST) full operations
├── Euclid mission cosmological results
└── Continued JWST cross-checks of the distance ladder
```
---
## Final Thought
```
@@ -924,6 +1148,15 @@ Star map:
║ ║
║ And so would anyone else. ║
║ ║
║ ───────────────────────────────────── ║
║ ║
║ And yet — every few years, our reading sharpens, ║
║ our errors shrink, and a value we thought was a "constant" ║
║ reveals itself as a parameter that was always evolving, ║
║ or always uncertain in some deeper way we hadn't yet seen. ║
║ ║
║ The cosmos writes in pencil, not pen. ║
║ ║
╚═══════════════════════════════════════════════════════════════════╝
```
@@ -931,9 +1164,12 @@ Star map:
## References & Further Reading
- **CODATA 2022**: P. Mohr et al., *Rev. Mod. Phys.* **97**, 025002 (2025) — also at physics.nist.gov/constants
- **Mathematical Constants**: "Mathematical Constants" by Steven Finch (Cambridge)
- **Physical Constants**: NIST CODATA values (physics.nist.gov)
- **Cosmological Parameters**: Planck Collaboration results
- **Cosmological Parameters**: Planck Collaboration results; DESI DR2 (2025)
- **Hubble Tension**: Di Valentino, Said & Saridakis, "Tensions in Cosmology 2025", *Nat. Astron.* **10**, 180 (2026)
- **Muon g-2**: Muon g-2 Collaboration, arXiv:2506.03069 (2025)
- **Largest Known Prime**: GIMPS, M₁₃₆₂₇₉₈₄₁ announcement (Oct 21, 2024)
- **Fine-Tuning**: "Just Six Numbers" by Martin Rees
- **Information Theory**: "Elements of Information Theory" by Cover & Thomas
- **Alien Communication**: "Communication with Extraterrestrial Intelligence" edited by Douglas Vakoch
@@ -941,6 +1177,7 @@ Star map:
---
*Document Version: 1.0*
*Last Updated: December 2024*
*Document Version: 2.0*
*Last Updated: April 2026*
*Previous Version: 1.0 (December 2024)*
*Purpose: Universal constants reference for xenoarchaeological analysis*