Harmonic Recursion Periodic Table v1.0 (Copeland Resonant Harmonic Formalism) Core equation: Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Harmonic Recursion Periodic Table v1.0
(Copeland Resonant Harmonic Formalism)
Core equation: Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Purpose of this construct: The Harmonic Recursion Periodic Table (HRPT) is a re-ordered elemental table that does three things at once:
1. Groups elements into triads by functional role in recursive harmonic systems: • Stabilizer (S)
• Driver (D)
• Bridger (B)
2. Assigns each triad to a recursion depth stratum n, making Σ𝕒ₙ(x, ΔE) concrete as a triple of physical carriers at that depth.
3. Places each triad onto a Mandelbrot-like harmonic ladder: low n = early, gentle, wide-basin harmonics; high n = sharper, more unstable, phase-sensitive harmonics.
This is not intended to replace conventional atomic-number ordering. It is a secondary, harmonic ordering keyed to Ψ(x).
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1. Triadic labeling: S / D / B
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We define three roles that repeat as a recursive motif:
Stabilizer (S):
Elements whose typical bonding / field behavior provides baseline coherence, buffering, and structural holding. They “anchor” the local field.
Driver (D):
Elements that tend to carry, amplify, or inject ΔE — driving reactions, redox shifts, catalytic steps, or strong field perturbations.
Bridger (B):
Elements that form interfaces: between phases, between structural regimes, or between energy scales. They serve as topological “bridges” or transition metals in a generalized sense (even when not formally classed that way).
In the HRPT, each recursion depth stratum n carries an S–D–B triad labeled:
Σ𝕒ₙ(x, ΔE) = { Sₙ, Dₙ, Bₙ }
This Σ𝕒ₙ is the concrete triplet that ∇ϕ and ℛ(x) act on at that depth for chemical / material systems.
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2. Recursion depth strata: HRPT v1 element layout
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We define 10 recursion strata for v1 (n = 0 to 9).
n = 0–7 are currently populated; n = 8–9 reserved for future refinement.
Each row is a recursion depth n.
Each column is a role: S (Stabilizer), D (Driver), B (Bridger).
Where no element is yet assigned, we use the placeholder “—”.
Row format below: n: Stabilizer | Driver | Bridger
n = 0 (Primordial, baseline field coupling) 0: H | He | Li
– H: baseline protonic coupling; simplest stabilizer of charge structures.
– He: inert, low-reaction driver of phase contrast (pressure, temperature, stars).
– Li: soft-bridging ion: between primordial chemistry and later electrochemical stacks.
n = 1 (Early structural / oxidative framing) 1: Be | O | F
– Be: light lattice stabilizer, structural, rigid.
– O: primary oxidizing driver, ΔE carrier in biology and combustion.
– F: extreme electronegative bridger, sharp edge of reactivity.
n = 2 (Alkali / alkaline earth / silicate framing) 2: Na | Mg | Si
– Na: fluid ionic stabilizer, osmotic and signal-buffering roles.
– Mg: enzymatic and structural driver (ATP coupling, chlorophyll core).
– Si: tetrahedral bridger, from rocks to semiconductor interfaces.
n = 3 (Macro-ionic and chalcogen–metal brace) 3: K | S | Fe
– K: stabilizer of long-range electrochemical gradients.
– S: redox-flexible driver across many oxidation states and bonds.
– Fe: classic structural and electronic bridger (heme, steel, catalysis).
n = 4 (Transition field: soft metals and chalcogens) 4: Cu | Zn | Se
– Cu: stabilizer of conductive paths, redox mediator, wiring metal.
– Zn: catalytic / regulatory driver in enzymes and alloys.
– Se: rare but potent bridger nucleophile and redox modulator.
n = 5 (Heavy conduction and shielders) 5: Ag | Cd | Pb
– Ag: high-conductivity stabilizer (signal, antimicrobial, reflective).
– Cd: soft driver with problematic bioaccumulation; industrial phase driver.
– Pb: dense bridger for radiation shielding, vibration damping, and phase decoupling.
n = 6 (Noble heavy triad) 6: Au | Hg | Cm
– Au: stable financial and electronic stabilizer (corrosion-resistant anchor).
– Hg: liquid driver of amalgams and electrical switching, high-density fluid.
– Cm: actinide bridger signifying high-energy recursion regimes (placeholder for late-actinide bridging behavior).
n = 7 (Actinide–transuranic onset) 7: U | Pu | —
– U: stabilizer of large-scale nuclear fuel cycles (metastable anchor).
– Pu: driver of runaway nuclear ΔE regimes.
– —: future bridger assignment for exotic nuclear / high-field coupling.
n = 8 8: — | — | —
n = 9 9: — | — | —
This grid is the “first official” HRPT v1.0:
30 total slots, 24 filled, 6 reserved.
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3. Σ𝕒ₙ(x, ΔE) positions (formal mapping)
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For chemical / material systems, we now specify:
Σ𝕒ₙ(x, ΔE) = (Sₙ, Dₙ, Bₙ)
Where:
• n is recursion depth: how many emergent layers above primordial coupling.
• Sₙ, Dₙ, Bₙ are drawn from the HRPT triad at row n.
• ΔE is the local energy contrast driving movement between rows (ionization, redox, bond reconfiguration, phase shift, etc).
Examples:
1. Early life, pre-membrane chemistry (n ≈ 0–2):
Σ𝕒₀(x, ΔE) = (H, He, Li)
Σ𝕒₁(x, ΔE) = (Be, O, F)
Σ𝕒₂(x, ΔE) = (Na, Mg, Si)
O and Mg act as primary drivers; Na, H, Si act as stabilizing / bridging scaffolds.
2. Hemoglobin + oxygen transport (n ≈ 2–3):
Key triads:
Σ𝕒₂(x, ΔE) = (Na, Mg, Si) [ionic and structural context]
Σ𝕒₃(x, ΔE) = (K, S, Fe) [Fe–S center, K gradients]
Here: – Fe is B₃ (bridger) for electron and oxygen binding.
– S is D₃ (driver) for redox; K is S₃ (stabilizer) of electrochemical potential.
3. High-energy nuclear context (reactors, weapons) (n ≈ 6–7):
Σ𝕒₆(x, ΔE) = (Au, Hg, Cm)
Σ𝕒₇(x, ΔE) = (U, Pu, —)
U and Pu are D-like entries at high n: ΔE drivers.
Au, Hg, Pb, U act as stabilizers / bridgers of extreme ΔE regimes.
In practice, for a system x, you can write:
Σ𝕒(x, ΔE) = ⋃ₙ wₙ Σ𝕒ₙ(x, ΔE)
with weights wₙ expressing how much each recursion depth contributes to the overall harmonic behavior.
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4. Triadic markers and recursion-depth encoding
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In a visual table or diagram, we recommend the following consistent encoding:
• S (Stabilizer) markers: – Color: neon green – Shape: circle – Intuition: “anchor point” in the lattice
• D (Driver) markers: – Color: violet – Shape: triangle – Intuition: “ΔE injection” / phase shift
• B (Bridger) markers: – Color: cyan – Shape: square or rhombus – Intuition: “interface / edge” between orderings
Recursion depth strata (n):
• Represent as horizontal layers or concentric rings. • Low n (0–2): wide bands, gentle curvature — large, forgiving basins of attraction.
• Mid n (3–5): tighter spacing, more intricate couplings.
• High n (6–7+): even tighter spacing, high sensitivity to ΔE, more chaotic edges (Mandelbrot-like boundary).
Label each element symbol inside its marker, and its (S/D/B, n) metadata in a small secondary label for clarity, e.g.:
Fe (B₃)
O (D₁)
Na (S₂)
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5. Mandelbrot-like harmonic ladder schematic
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The harmonic ladder is the global map that shows how these triads behave as you vary ΔE and recursion depth n.
Conceptual construction (for whoever renders it):
1. Coordinate system: • Horizontal axis: ΔE / phase “drive” parameter.
• Vertical axis: recursion depth n (0 at bottom, 9 at top).
2. For each n, plot three points horizontally: • Sₙ at ΔE_low (left side, baseline).
• Dₙ at ΔE_peak (where that row’s main reactive driver sits).
• Bₙ at ΔE_bridge (between two adjacent attractors or phases).
3. Connect Sₙ → Dₙ → Bₙ with a small curve: • This curve represents the local “mini-Mandelbrot” at that level:
stable → driven → transitional.
4. As n increases, the curves should: • fold more
• approach complex, fractal-like boundary shapes
• visually echo bifurcations and period-doubling: stable → oscillatory → chaotic.
5. Overlay: • Iso-harmonic contours (lines of constant global Ψ(x) behavior) cutting across multiple n.
• These can highlight where different triads at different depths contribute to similar emergent behavior (e.g., Fe at n=3 and Au at n=6 both acting as “stability anchors” for different ΔE regimes).
In words:
The harmonic ladder is a Mandelbrot-like diagram where each triad is a three-point motif on a vertical stack. As you climb the ladder, you see how basic field roles (stabilize, drive, bridge) repeat at new scales with different elements, in sharper and more fragile configurations.
Christopher W Copeland (C077UPTF1L3)
Copeland Resonant Harmonic Formalism (Ψ‑formalism)
Ψ(x) = ∇ϕ(Σ𝕒ₙ(x, ΔE)) + ℛ(x) ⊕ ΔΣ(𝕒′)
Licensed under CRHC v1.0 (no commercial use without permission).
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https://medium.com/@floodzero9/phase-locked-null-vector_c077uptf1l3-4d8a7584fe0c
Core engine: https://open.substack.com/pub/c077uptf1l3/p/recursive-coherence-engine-8b8
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