Jonathan's attempt mathematical stab to visualize a possible dimension progression. How do dimensions take aspects of the previous dimension and apply them forward. Note that dimensions 1 and 2 are exceptions. This is telling in that 2D is euclidean and may establish the prinicpal foundation of {2,3}, and 1D is collapsed. These are then carried into the 3D unfolding and beyond and regress back to 1 with a flatening curve. The curve itself is interesting as it is left leaning and tappers off, it is triangular when graphed in a graph (see below). It is an interesting curiousity and likely just a game of numerology rather than REAL mathematics. But it does start the process of finding a connector to each dimension. the plastic number is; p = 1.32471795724474602596... and falls relatively close in the 5th dimension (though could be coincidence or a product of using Fibonnaci sequencing). Its not a confirmation of any sort; more a curiousity at this junction. If this formula is meaningful in any way, we will likely find this more relavent. In the interim, the formula is treated as an attempt, NOT as fact or prediction. FORMAT Fibonacci indexing: F0=1, F1=1, F2=2, F3=3, F4=5, F5=8, F6=13, F7=21, F8=34, F9=55, ... d = dimension General formula (d >= 3): a_d = 1 + ( F_{d+1} ^ (1 / F_{d-1}) ) / F_{d-1} Special cases: d=1: 1 d=2: 1.5 Table: d | root order | arg | denom | expression | approx --|------------|-----|-------|-----------------------------|-------- 1 | - | - | - | 1 | 1.000 2 | - | - | - | 1.5 | 1.500 3 | 2 | 5 | 2 | 1 + sqrt(5)/2 | 1.618 4 | 3 | 8 | 3 | 1 + cbrt(8)/3 | 1.667 5 | 5 | 13 | 5 | 1 + 13^(1/5)/5 | ~1.334 6 | 8 | 21 | 8 | 1 + 21^(1/8)/8 | ~1.186 7 | 13 | 34 | 13 | 1 + 34^(1/13)/13 | ~1.101 8 | 21 | 55 | 21 | 1 + 55^(1/21)/21 | ~1.059 1.70 ┼ • (peak d=4) 1.65 ┼ ↗ ↘ 1.60 ┼ ↗ ↘ 1.55 ┼ ↗ ↘ 1.50 ┼ ↗ ↘ 1.45 ┼ ↗ ↘ 1.40 ┼ ↗ ↘ 1.35 ┼ ↗ ↘ ← near ρ at d=5 1.30 ┼ ↘ 1.25 ┼ ↘ 1.20 ┼ ↘ 1.10 ┼ ↘ 1.05 ┼ ↘ 1.00 ┼───────────────────────────────────────────────→ ∞ 1 2 3 4 5 6 7 8 9 10 15 20 d ================================================================================ FIBONACCI PAIR TABLE — DIMENSIONAL PROGRESSION ================================================================================ Added: 2026-03-13 Source: Derived from the Fibonacci indexing above and confirmed against TLT-002 Extended 2D lattice test results (11 materials tested) Each dimension is characterized by a Fibonacci pair {F_{d-1}, F_d}. The SUM of each pair equals the first element of the NEXT dimension's pair, forming a chain. Pair Dimension Sum Bridge to next Role of each number ----------------------------------------------------------------------- {1,1} 1D 2 → 2 in {2,3} 1: one direction, 1: standing wave {2,3} 2D 5 → 5 in {3,5} 2: sublattices, 3: directional components (N=3) {3,5} 3D 8 → 8 in {5,8} 3: ?, 5: directional components (N=5)? {5,8} 4D 13 → 13 in {8,13} (untested) {8,13} 5D 21 → ... (untested) Key: - The SECOND number in each pair determines the GEOMETRY of that dimension (confirmed for 2D: "3" → N=3 → hexagonal, 9/9 materials) - The FIRST number determines the SUBLATTICE or structural multiplicity (confirmed for 2D: "2" → 2 sublattices in all honeycomb structures) - The SUM is the BRIDGE NUMBER — it exists in the current dimension but does not produce a clean periodic structure (confirmed for 2D: "5" → borophene coordination 5, polymorphic, no clean N-wave match) ================================================================================ DIMENSIONAL UNIDIRECTIONALITY — OBSERVATION FROM 2D LATTICE TESTS ================================================================================ Added: 2026-03-13 Source: TLT-002 Extended results (buckled honeycomb materials: Si, Ge, Sn) Status: Observation with testable prediction, NOT confirmed QUESTION: Do dimensional influences propagate unidirectionally (lower → higher only) or bidirectionally (higher dimensions backward-influence lower)? OBSERVATION FROM THE DATA: The 2D lattice tests show a clear asymmetry in dimensional influence: AT THE UPPER FREQUENCY RANGE of 2D (heavier elements): Silicene (Si): buckling = 0.44 A, bond angle 116° — 3D influence emerging Germanene (Ge): buckling = 0.65 A, bond angle 113° — stronger 3D influence Stanene (Sn): buckling = 0.85 A, bond angle 109.5° — tetrahedral (3D) → 3D geometry is EMERGING in 2D space at the upper energy/mass range. → The heavier the element, the more the 2D structure lifts into 3D. → Stanene's bond angle (109.5°) is the diamond cubic angle — a 3D structure manifesting in a nominally 2D material. AT THE LOWER FREQUENCY RANGE of 2D (lighter elements): Graphene (C): perfectly flat, bond angle exactly 120° — pure 2D hBN (B, N): perfectly flat, bond angle exactly 120° — pure 2D → NO 3D influence at the lower end. Pure Euclidean 2D. → The lighter elements are fully described by the {2,3} framework with no contamination from higher-dimensional geometry. CONCLUSION: Dimensional influence is UNIDIRECTIONAL. Higher-dimensional patterns emerge at the upper frequency boundary of the current dimension but do NOT reach back down to influence the lower frequency range. This is consistent with the theory's claim that time (and therefore information) is unidirectional — frame X+1 cannot influence frame X. The pattern: - Lower frequencies in dimension D → pure D-dimensional geometry - Upper frequencies in dimension D → (D+1)-dimensional patterns emerging - The transition is continuous, not discrete (no clean break) - The bridge number (sum of Fibonacci pair) marks the transitional zone TESTABLE PREDICTION: If this pattern holds, then in 3D space ({3,5}): - Lower-mass/lower-frequency 3D structures should exhibit pure 3D geometry with no 4D influence (e.g., simple BCC/FCC metals at low atomic number) - Upper-mass/upper-frequency 3D structures should begin to exhibit characteristics associated with {5,8} / 4D-like behavior - 4D patterns should NOT backward-influence low-frequency 3D structures Additionally, working backward: - In 1D ({1,1}), the upper frequency range should show 2D-like emergence — and indeed, 1D standing waves (N=2 stripes) are the starting point from which 2D lattices emerge via the addition of the third directional component This prediction is testable with the 3D extension of the N-wave interference framework, and with systematic analysis of 3D crystal structures across the periodic table sorted by atomic mass/Compton frequency. ================================================================================