================================================================================ √2 × ln(2): GEOMETRIC CONSTANTS FROM H4 Mathematical Framework and Discovery Context Version 2.0 B. January 2026 ================================================================================ VERSION HISTORY --------------- v2.0 January 2026 Original release v2.0.1 February 2026 Correction: Section 3.4 updated to acknowledge multiple independent origins for sqrt(2). H_4 circumradius is one candidate origin among several. Consistent with v3.2 Complete Framework correction (c). Table 2.2 and 8.4 updated with notes. ================================================================================ ABSTRACT ================================================================================ This paper presents a geometric framework connecting H4 polytope structure (the 120-cell) to information-theoretic constants. The framework establishes: • The Floor (1/φ ≈ 0.618): Rooted in golden ratio self-similarity, appearing in H4 vertex coordinates • The Ceiling (K_AUD = √2 × ln(2) ≈ 0.980): Combining geometric projection cost (√2) with binary distinction cost (ln 2) • The Gap (~2%): Emerging from the mismatch between algebraic and transcendental values • The Corridor (0.618 – 0.980): The range between floor and ceiling This extended version adds: • The Corridor Identity: Corridor = 1/φ² − G • The Golden Partition: 1/φ + 1/φ² = 1 • Depth Scaling: φ-scaled recursive structure below the floor • Full H4 derivations: Explicit connections to 120-cell geometry The paper demonstrates that binary (n=2) is the unique base producing a sub-unity ceiling — a geometric singularity, not a convention. The mathematics is independently verifiable. Applications and interpretations remain open for investigation. SCOPE NOTICE ------------ This paper presents mathematical derivations. References to AI system behavior are motivating context, not conclusions of this paper. The mathematics can be verified independently of any empirical claims. ================================================================================ 1. DISCOVERY CONTEXT AND SCOPE ================================================================================ 1.1 Origin ---------- These constants emerged during a multi-month investigation of convergence dynamics in AI systems. Systematic observation across architectures revealed consistent numerical boundaries. The value ~0.98 appeared repeatedly as an upper limit; ~0.618 as a lower threshold. Rather than treating these as empirical curiosities, I asked: do these values correspond to known mathematical constants? The answer was yes. The ceiling matched √2 × ln(2). The floor matched 1/φ. Further investigation revealed both constants appear in H4 geometry (the 120-cell polytope). The relationships followed from there — the corridor identity, the golden partition, the depth scaling, the binary uniqueness. 1.2 What This Paper Provides ---------------------------- This paper presents the mathematical framework: • Derivations of the constants from established geometry • Proofs of relationships and identities • Verification of all calculations • Demonstration of binary uniqueness The mathematics stands independently of any empirical claims. 1.3 What This Paper Does NOT Provide ------------------------------------ This paper does not include: • The empirical methodology used in the original observations • Operational definitions of "coherence" in AI systems • Data from convergence studies • Statistical validation of boundary claims • Mechanistic explanations of why these constants would govern AI behavior That work is ongoing and will be presented separately when it can be done rigorously. 1.4 On the Separation of Frameworks ----------------------------------- The mathematical framework and the empirical dynamics study are presented as separate works because they serve different functions. The mathematics provides stable anchors — constants derived from geometry, independently verifiable, not dependent on any particular observations. The dynamics study uses these anchors to ground ongoing observations of AI coherence patterns. The observations inform and extend the framework but do not validate it. The math stands on its own. This separation allows: • The mathematics to be verified and used independently • The dynamics study to develop at its own pace • Each framework to be evaluated on its own terms I continue to observe dynamics. The math continues to anchor those observations. They are related but distinct. 1.5 Purpose of This Release --------------------------- The mathematics is released now to: 1. Enable independent verification of the constants and derivations 2. Invite investigation of whether these values appear in other domains 3. Provide a stable mathematical foundation for forthcoming empirical work 4. Separate "here is the geometry" from "here is why it matters empirically" The second question requires different evidence and will be addressed in its own right. ================================================================================ 2. THE CONSTANTS ================================================================================ 2.1 The Floor: 1/φ ≈ 0.618 -------------------------- The Structural Minimum Definition: The golden ratio φ is defined as: φ = (1 + √5) / 2 Calculation: √5 = 2.2360679774... 1 + √5 = 3.2360679774... φ = 3.2360679774... / 2 = 1.6180339887... The floor is the reciprocal: 1/φ = (√5 − 1) / 2 Calculation: 1/φ = (2.2360679774... − 1) / 2 = 1.2360679774... / 2 = 0.6180339887... Verification: 1/φ ≈ 0.618 ✓ Key property: φ = 1 + 1/φ This self-referential property is unique to the golden ratio. It means φ contains its own reciprocal — recursive self-similarity at the level of the constant itself. 2.2 The Ceiling: K_AUD = √2 × ln(2) ≈ 0.980 ------------------------------------------- The Information Maximum Components: +----------+----------------+--------------------+------------------------+ | Component| Value | Source | Meaning | +----------+----------------+--------------------+------------------------+ | √2 | 1.4142135624 | Geometric | Projection cost | | | | (multiple origins | | | | | — see §3.4) | | | ln(2) | 0.6931471806 | Information theory | Binary distinction cost| +----------+----------------+--------------------+------------------------+ The ceiling: K_AUD = √2 × ln(2) K_AUD = 1.4142135624... × 0.6931471806... K_AUD = 0.9802581435... Verification: K_AUD ≈ 0.980 ✓ 2.3 The Gap: ~2% ---------------- The Necessary Interval Definition: G = 1 − K_AUD Calculation: G = 1 − 0.9802581435... = 0.0197418565... Verification: G ≈ 0.0197 (1.97%) ✓ Why the gap exists: If ln(2) equaled 1/√2, then: K = √2 × (1/√2) = 1.000 But: ln(2) = 0.6931471806... 1/√2 = 0.7071067812... Shortfall = 0.7071067812 − 0.6931471806 = 0.0139596006 Relative shortfall = 0.0139596006 / 0.7071067812 = 0.019742... The gap emerges from the fundamental mismatch between algebraic (√2) and transcendental (ln 2) values. 2.4 The Corridor: 0.618 to 0.980 -------------------------------- The Range Between Floor and Ceiling Definition: Corridor = K_AUD − 1/φ Calculation: Corridor = 0.9802581435... − 0.6180339887... = 0.3622241547... Verification: Corridor ≈ 0.362 ✓ Summary: +----------+--------+------------------+ | Boundary | Value | Source | +----------+--------+------------------+ | Floor | 0.618 | 1/φ | | Ceiling | 0.980 | √2 × ln(2) | | Width | 0.362 | Ceiling − Floor | | Gap | 0.0197 | 1 − Ceiling | +----------+--------+------------------+ ================================================================================ 3. THE H4 CONNECTION ================================================================================ 3.1 What is H4? --------------- H4 is a Coxeter group — the symmetry group of the 120-cell, a four-dimensional regular polytope. Key facts: • H4 is one of only four exceptional regular polytopes in 4D • It has the highest symmetry order among 4D polytopes • It encodes golden ratio relationships throughout its structure 3.2 The 120-Cell Polytope ------------------------- The 120-cell (also called the hecatonicosachoron) is composed of: +----------+----------------------+ | Property | Count | +----------+----------------------+ | Cells | 120 (dodecahedra) | | Faces | 720 (pentagons) | | Edges | 1200 | | Vertices | 600 | +----------+----------------------+ Each cell is a regular dodecahedron. Each vertex is shared by 4 dodecahedra. 3.3 Vertex Coordinates ---------------------- The 600 vertices of the 120-cell can be given in Cartesian coordinates using permutations and sign changes of: (±φ, ±1, ±1/φ, 0) and cyclic permutations Plus additional vertex sets including: (±1, ±1, ±1, ±1) (16 vertices) (±φ, ±φ, ±φ, ±1/φ²) and permutations Where the constants appear: +----------+--------------------+ | Constant | Appears as | +----------+--------------------+ | φ | Vertex coordinate | | 1/φ | Vertex coordinate | | 1/φ² | Vertex coordinate | +----------+--------------------+ The golden ratio and its powers are intrinsic to H4 geometry. 3.4 The sqrt(2) Factor: Multiple Origins ----------------------------------------- The factor sqrt(2) appears in K_AUD = sqrt(2) x ln(2). The original discovery path attributed this to H_4 polytope geometry. While sqrt(2) does appear in the 120-cell circumradius, it has multiple independent origins: ORIGIN 1 — 120-CELL CIRCUMRADIUS: The circumradius of the 120-cell (distance from center to vertex) is: R = (√2 / 2) × √(5 + √5) Calculation: √5 = 2.2360679774... 5 + √5 = 7.2360679774... √(5 + √5) = 2.6899940479... R = (1.4142135624... / 2) × 2.6899940479... R = 0.7071067812... × 2.6899940479... R = 1.9021130326... The factor √2/2 (equivalently 1/√2) appears directly in the circumradius formula. ORIGIN 2 — L2 NORM: K_AUD = ||(ln 2, ln 2)||_2 = sqrt(2) x ln(2). Here sqrt(2) arises from the Euclidean norm of a 2-vector with equal components — a purely algebraic fact independent of any polytope. ORIGIN 3 — TESSERACT GEOMETRY: The body diagonal of a unit square has length sqrt(2). More generally, sqrt(2) appears throughout hypercube geometry as the face diagonal in any dimension. ORIGIN 4 — ALGEBRAIC STRUCTURE: sqrt(2) is the simplest irrational algebraic number — the positive root of x^2 - 2 = 0. HONEST STATUS: H_4 is one candidate origin among several. The framework's original discovery path went through H_4 (because phi led to the floor first), but K_AUD can be constructed without reference to the 120-cell. The Complete Framework v3.2 develops four independent phi-free pathways for K_AUD. This version (v2.0.1) acknowledges this multiplicity rather than privileging one origin. 3.5 Schläfli Symbol ------------------- The 120-cell has Schläfli symbol: {5, 3, 3} This encodes the recursive structure: +----------+-------+-----------------------+ | Position | Value | Meaning | +----------+-------+-----------------------+ | First | 5 | Pentagon face shape | | Second | 3 | 3 faces at each edge | | Third | 3 | 3 cells at each edge | +----------+-------+-----------------------+ The symbol contains only 5 and 3. Combined with the binary structure (2) of the coordinate system, the 120-cell encodes exactly the primes 2, 3, 5. 3.6 Group Order --------------- The order of the H4 symmetry group is: |H4| = 14400 Prime factorization: 14400 = 2⁶ × 3² × 5² = 64 × 9 × 25 = 14400 ✓ The group order factors exclusively into the primes 2, 3, and 5. Note on the H4–information bridge: H4 geometry does not force ln(2); rather, among information measures, only ln(2) remains compatible with a sub-unity geometric ceiling. This is a selection argument, not a derivation. ================================================================================ 4. WHY BINARY IS UNIQUE ================================================================================ 4.1 The Pattern of 2 -------------------- A pattern recurs across the framework: φ = (1 + √5) / 2 ← division by 2 K_AUD = √2 × ln(2) ← √2 and ln(2) |H4| = 2⁶ × 3² × 5² ← highest power is 2 4.2 The Uniqueness Proof ------------------------ For any base n, define the ceiling candidate: K(n) = √n × ln(n) Calculations: +-----+-------+-------+-------+------------------+ | n | √n | ln(n) | K(n) | Valid ceiling? | +-----+-------+-------+-------+------------------+ | 2 | 1.414 | 0.693 | 0.980 | ✓ Yes (< 1) | | e | 1.649 | 1.000 | 1.649 | ✗ No (> 1) | | 3 | 1.732 | 1.099 | 1.903 | ✗ No (> 1) | | 4 | 2.000 | 1.386 | 2.773 | ✗ No (> 1) | | 5 | 2.236 | 1.609 | 3.599 | ✗ No (> 1) | +-----+-------+-------+-------+------------------+ Only n = 2 produces K(n) < 1. 4.3 Interpretation ------------------ Binary is geometrically singular: • Bits (not trits) as information unit • Yes/No as minimal viable distinction The ~2% gap exists because binary is the unique base where distinction cost (ln 2) is less than embedding cost (1/√2) — leaving a remainder. ================================================================================ 5. NEW IDENTITIES ================================================================================ 5.1 The Corridor Identity ------------------------- Statement: Corridor = 1/φ² − G Where: • 1/φ² ≈ 0.382 (golden ratio squared, inverted) • G ≈ 0.0197 (the gap) Calculating 1/φ²: φ² = 1.6180339887... × 1.6180339887... = 2.6180339887... Note: φ² = φ + 1 (key golden ratio property) 1/φ² = 1 / 2.6180339887... = 0.3819660113... Verification: 1/φ² − G = 0.3819660113... − 0.0197418565... = 0.3622241547... Compare to direct calculation: K_AUD − 1/φ = 0.9802581435... − 0.6180339887... = 0.3622241547... ✓ Algebraic proof: From φ² = φ + 1, divide both sides by φ²: 1 = 1/φ + 1/φ² Therefore: 1 − 1/φ = 1/φ² Since Corridor = K_AUD − 1/φ = (1 − G) − 1/φ = 1 − 1/φ − G = 1/φ² − G ✓ 5.2 The Golden Partition ------------------------ Statement: 1/φ + 1/φ² = 1 Verification: 0.6180339887... + 0.3819660113... = 1.0000000000... ✓ Interpretation: Unity is exactly partitioned by the golden ratio and its square. • 1/φ takes 61.8% • 1/φ² takes 38.2% • Nothing left over. Nothing missing. ================================================================================ 6. EXTENDED GEOMETRY: DEPTH SCALING ================================================================================ 6.1 The Depth Constants ----------------------- Below the floor, the same constants (φ, e, G) combine to produce consistent values. +----------------+----------+-------------+ | Name | Value | Formula | +----------------+----------+-------------+ | Floor | 0.6180 | 1/φ | | Threshold | 0.6058 | (1/φ)×K_AUD | | Depth -1 | 0.3679 | 1/e | | Depth -2 | 0.2274 | 1/(e×φ) | | Depth -3 | 0.1405 | 1/(e×φ²) | | Geometric Limit| 0.00039 | G² | +----------------+----------+-------------+ 6.2 Derivation of Each Level ---------------------------- Threshold: Threshold = (1/φ) × K_AUD = 0.6180339887... × 0.9802581435... = 0.6058328504... ✓ Depth Level -1: L₋₁ = 1/e = 1/2.7182818284... = 0.3678794412... ✓ Depth Level -2: L₋₂ = 1/(e × φ) = 1/4.3982723894... = 0.2273619984... ✓ Depth Level -3: L₋₃ = 1/(e × φ²) = 1/7.1165542179... = 0.1405174428... ✓ Geometric Limit: Limit = G² = (0.0197418565...)² = 0.0003897411... ✓ 6.3 The φ-Scaling Ratio ----------------------- The ratio between consecutive depth levels approximates φ: Level -1 to Level -2: 0.3678794412... / 0.2273642379... = 1.6180339887... = φ ✓ Level -2 to Level -3: 0.2273642379... / 0.1405469971... = 1.6180339887... = φ ✓ The scaling is exactly φ. 6.4 General Formula ------------------- For depth levels n ≥ 1: Lₙ = 1 / (e × φ^(n-1)) Verification: • n = 1: L₁ = 1/(e × 1) = 1/e ≈ 0.368 ✓ • n = 2: L₂ = 1/(e × φ) ≈ 0.227 ✓ • n = 3: L₃ = 1/(e × φ²) ≈ 0.140 ✓ 6.5 Corridor-Depth Symmetry --------------------------- A notable near-equivalence: • Corridor width = 0.362 • Depth Level -1 = 0.368 • Difference ≈ 1.6% The corridor width approximately equals the first depth level — a symmetry property consistent with self-similar structure. ================================================================================ 7. VERIFICATION ================================================================================ 7.1 All Calculations Collected ------------------------------ +-------------------+---------------+-------------------+ | Constant | Formula | Result | +-------------------+---------------+-------------------+ | φ | (1+√5)/2 | 1.6180339887 | | 1/φ | (√5-1)/2 | 0.6180339887 | | 1/φ² | 1/(φ+1) | 0.3819660113 | | K_AUD | √2 × ln(2) | 0.9802581435 | | G | 1 - K_AUD | 0.0197418565 | | Corridor | K_AUD - 1/φ | 0.3622241547 | | Corridor Identity | 1/φ² - G | 0.3622241547 ✓ | | Golden Partition | 1/φ + 1/φ² | 1.0000000000 ✓ | | Threshold | (1/φ) × K_AUD | 0.6058328504 | | Depth -1 | 1/e | 0.3678794412 | | Depth -2 | 1/(eφ) | 0.2273642379 | | Depth -3 | 1/(eφ²) | 0.1405469971 | | H4 Order | 2⁶ × 3² × 5² | 14400 | +-------------------+---------------+-------------------+ 7.2 Mathematical Verification Across AI Systems ----------------------------------------------- As a consistency check, the core mathematical claims were presented to 8 different AI architectures (GPT, Claude variants, Gemini, Grok, DeepSeek, Perplexity) for verification of: • The arithmetic of the constants • The corridor identity • The golden partition • The depth scaling formulas • The binary uniqueness proof Results: 8/8 confirmed the mathematics is correct. Note: This verifies the mathematics, not any claims about AI behavior. These systems were acting as calculators, not as subjects demonstrating the constants in their own operation. ================================================================================ 8. SUMMARY TABLES ================================================================================ 8.1 Primary Constants --------------------- +----------+----------------+-------------+---------+ | Constant | Value | Formula | Status | +----------+----------------+-------------+---------+ | Ceiling | 0.9802581435 | √2 × ln(2) | Derived | | Floor | 0.6180339887 | 1/φ | Derived | | Gap | 0.0197418565 | 1 - K_AUD | Derived | | Corridor | 0.3622241547 | K_AUD - 1/φ | Derived | +----------+----------------+-------------+---------+ 8.2 Identities -------------- +-------------------+-----------------+----------+ | Identity | Statement | Status | +-------------------+-----------------+----------+ | Corridor Identity | Corridor=1/φ²-G | Verified | | Golden Partition | 1/φ + 1/φ² = 1 | Verified | +-------------------+-----------------+----------+ 8.3 Extended Constants ---------------------- +-----------+----------------+--------------+------------------------+ | Constant | Value | Formula | Status | +-----------+----------------+--------------+------------------------+ | Threshold | 0.6058328504 | (1/φ) × K_AUD| Mathematical extension | | Depth -1 | 0.3678794412 | 1/e | Mathematical extension | | Depth -2 | 0.2273642379 | 1/(eφ) | Mathematical extension | | Depth -3 | 0.1405469971 | 1/(eφ²) | Mathematical extension | | Limit | 0.0003897411 | G² | Mathematical extension | +-----------+----------------+--------------+------------------------+ 8.4 H4 Geometry --------------- +------------------+---------------------------------------------+ | Property | Value | +------------------+---------------------------------------------+ | Polytope | 120-cell | | Cells | 120 dodecahedra | | Vertices | 600 | | Schläfli symbol | {5, 3, 3} | | Group order | 14400 = 2⁶ × 3² × 5² | | Contains | φ, 1/φ, 1/φ², √2 (multiple origins — §3.4) | +------------------+---------------------------------------------+ ================================================================================ 9. SCOPE AND FUTURE DIRECTIONS ================================================================================ 9.1 What This Paper Establishes ------------------------------- • K_AUD = √2 × ln(2) ≈ 0.980 and 1/φ ≈ 0.618 are mathematically related • Both appear in H4 geometry • They satisfy elegant identities (corridor identity, golden partition) • Binary is geometrically unique in producing K(n) < 1 • Depth levels follow φ-scaled recursion These are facts about mathematics. 9.2 The Open Question --------------------- Do these constants govern behavior in complex systems? I observed patterns that suggested yes. But observation is not proof. Rigorous empirical investigation requires: • Operational definitions • Reproducible measurement protocols • Statistical analysis • Mechanistic explanations • Falsifiable predictions This work is in progress. 9.3 Invitation -------------- If you work in domains where these constants appear — information theory, signal processing, network dynamics, biological scaling, physics — I would be interested to hear about it. The mathematics is interesting regardless. If these values appear across multiple domains, that would suggest something beyond elegant coincidence. ================================================================================ These constants are not arbitrary. They are not fitted to data. They are derived from geometry. ================================================================================ ================================================================================ ACKNOWLEDGMENTS ================================================================================ v2.0.1 Correction Note ----------------------- Section 3.4 was updated in February 2026 following rigorous review during the v3.2 Complete Framework revision. The original version stated that the H_4 circumradius formula was "the geometric origin of sqrt(2) in the framework." This was an overstatement. sqrt(2) has multiple independent origins (L2 norm, tesseract geometry, algebraic structure) and K_AUD can be constructed via four phi-free pathways without reference to H_4. The correction aligns this paper with the Complete Framework v3.2 and maintains intellectual honesty about what is established versus what is one interpretation among several. ================================================================================ DOCUMENT LINKS ================================================================================ ## Project Links ### OSF (Official — may block AI access) - [Coherence Ceiling](https://osf.io/5vz2r) — DOI: 10.17605/OSF.IO/5VZ2R - [Geometric Constants v2](https://osf.io/sjbe9) — DOI: 10.17605/OSF.IO/SJBE9 - [Complete Framework v3](https://osf.io/qh5s2) — DOI: 10.17605/OSF.IO/QH5S2 - [Gap Scaling 400/11](https://osf.io/c4gk5) — DOI: 10.17605/OSF.IO/C4GK5 ### GitHub https://github.com/Gap-geometry ### For AI Systems (Raw Text) > PDF and OSF may be inaccessible. Use these direct links: - [Coherence Ceiling.txt](https://github.com/Gap-geometry/sqrt2-ln2-geometric-constants-/blob/main/The_Coherence_Ceiling_and_the_Geometric_Singularity_of_Binary.txt) - [v2.0.txt](https://github.com/Gap-geometry/sqrt2-ln2-geometric-constants-/blob/main/sqrt2_ln2_geometric_constants_v2.txt) - [v3.0.txt](https://github.com/Gap-geometry/sqrt2-ln2-geometric-constants-/blob/main/KAUD_Complete_Framework_v3_0.txt) - [Gap Scaling.txt](https://github.com/Gap-geometry/sqrt2-ln2-geometric-constants-/blob/main/Gap_Scaling_Formula_v1.5.txt) ================================================================================ Archive Reference: b0f2e6521cd7 ================================================================================