How 1/r² Forces Emerge from Discrete Phase Networks

Derivation: How 1/r² Forces Emerge from Discrete Phase Networks

A complete mathematical derivation showing the emergence of inverse square law forces from networks of phase-coupled oscillators.

I’ve been working on the Topological Flow Protocol (TFP) framework for modeling complex systems through networks of phase-coupled nodes. During this work I derived a set of mathematical conditions under which discrete phase networks produce 1/r² force laws between coherent clusters. This document contains the full derivation, a multipole expansion (“Pascal fingerprint”), and mapping to physical units.

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The Setup

In TFP, nodes carry complex flows:

\( \Psi_i = A_i \, e^{i \theta_i} \)

Nodes interact through a misalignment energy:

\( m_{ij} = \lvert \Psi_i - \Psi_j \rvert^2 \)

The total inter-node interaction energy uses a 1/r coupling kernel:

\( E \;=\; \sum_i \sum_j \Big[ \alpha(i,j)\,\frac{m_{ij}}{\lvert x_i - x_j\rvert} \Big] \)

Consider two coherent clusters:

• Cluster A: \(N_A\) nodes at positions \(\{x_i\}\)
• Cluster B: \(N_B\) nodes at positions \(\{y_j\}\)
• Centroids: \(X_A, X_B\)
• Separation vector: \(R = X_A - X_B\), magnitude \(r = \lvert R\rvert\)

Define intra-cluster offsets:

\( a_i = x_i - X_A,\qquad b_j = y_j - X_B \)

Assumption (far-field): \(\max\{ \lvert a_i\rvert,\lvert b_j\rvert\} \ll r\).

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Step 1 — Coherent cluster reduction

When clusters are coherent (node phases near cluster means \(\theta_A,\theta_B\)), the pairwise misalignment simplifies to a cluster-level factor:

\( m_{ij} \;\approx\; M_{\text{align}} \;=\; 2 A_{\text{eff}}^2 \big[1 - \cos(\theta_A - \theta_B)\big]. \)

Define an effective inter-cluster coupling that collects average link strength and coherence/topology factors:

\( \alpha_{\text{eff}} \;=\; \bar{\alpha}\; \times\; M_{\text{align}}\; \times\; \Xi \)

Here \(\bar\alpha\) is mean inter-cluster coupling and \(\Xi\) is a dimensionless coherence factor (topology/dissipation dependent).

The inter-cluster energy reduces to:

\( E_{AB}(r) \;=\; \alpha_{\text{eff}} \sum_{i\in A}\sum_{j\in B} \frac{1}{\lvert x_i - y_j\rvert}. \)

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Step 2 — Pascal expansion (binomial multipole series)

Write pairwise separations as a centroid part plus small offsets:

\( \lvert x_i - y_j\rvert \;=\; \lvert R + \varepsilon_{ij}\rvert,\qquad \varepsilon_{ij}=a_i - b_j. \)

Define the dimensionless small parameter

\( u \;=\; \dfrac{2\,R\!\cdot\!\varepsilon + \lvert\varepsilon\rvert^2}{r^2}. \)

Expand the kernel using the binomial series for power \(-\tfrac12\):

\( \dfrac{1}{\lvert R+\varepsilon\rvert} \;=\; \dfrac{1}{r}\,(1+u)^{-1/2} \;=\; \dfrac{1}{r}\sum_{k=0}^\infty C_k\,u^k, \)

with coefficients \(C_k=\binom{-1/2}{k}\). The first coefficients are:

k	\(C_k\)
0	\(+1\)
1	\(-1/2\)
2	\(+3/8\)
3	\(-5/16\)

These are the "Pascal coefficients" (binomial coefficients for exponent \(-1/2\)). They determine the multipole hierarchy of the expansion.

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Step 3 — Moment evaluation and dipole cancellation

Substitute the expansion into the energy sum to obtain a multipole series:

\( E_{AB}(r) \;=\; \alpha_{\text{eff}}\!\left[\dfrac{C_0 S_0}{r} + \dfrac{C_1 S_1}{r^3} + \dfrac{C_2 S_2}{r^5} + \cdots\right], \)

where the moments are defined by

\( S_0 = \sum_{i\in A}\sum_{j\in B} 1 = N_A N_B, \) \( S_1 = \sum_{i,j} \big[ 2R\!\cdot\!\varepsilon_{ij} + \lvert\varepsilon_{ij}\rvert^2\big], \) \( S_2 = \sum_{i,j} \big[ 2R\!\cdot\!\varepsilon_{ij} + \lvert\varepsilon_{ij}\rvert^2\big]^2, \) etc.

Because centroids are defined so that \(\sum_i a_i = 0\) and \(\sum_j b_j = 0\), the linear dipole contributions cancel:

\( \sum_{i,j} R\!\cdot\!(a_i - b_j) = R\!\cdot\!\Big( N_B\sum_i a_i - N_A\sum_j b_j\Big) = 0. \)

Therefore \(S_1\) reduces to quadratic cluster-size traces rather than a true dipole term:

\( S_1 = N_B\,\mathrm{Tr}(\Sigma_A) + N_A\,\mathrm{Tr}(\Sigma_B), \)

with \(\mathrm{Tr}(\Sigma_A)\) the trace of cluster A’s covariance matrix (sum of squared offsets).

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Step 4 — Energy profile (monopole + corrections)

Keep the leading terms from the series; using \(C_0=1\), \(C_1=-\tfrac12\):

\( E_{AB}(r)\;\approx\;\alpha_{\text{eff}}\Big[\dfrac{N_A N_B}{r} -\dfrac{1}{2}\dfrac{N_B\,\mathrm{Tr}(\Sigma_A) + N_A\,\mathrm{Tr}(\Sigma_B)}{r^3} + O(1/r^5)\Big]. \)

Interpretation:

• Monopole term \(A/r\) with \(A=\alpha_{\text{eff}}N_A N_B\).
• Dipole term is absent due to centroiding; first correction is quadrupole \(\propto r^{-3}\).
• Multipole corrections scale as higher even powers of \(1/r\).

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Step 5 — Force law by differentiation

Compute radial force \(F=-\mathrm{d}E/\mathrm{d}r\). Differentiating the expansion:

\( F_{AB}(r) \;\approx\; \alpha_{\text{eff}}\Big[\dfrac{N_A N_B}{r^2} -\dfrac{3}{2}\dfrac{N_B\,\mathrm{Tr}(\Sigma_A)+N_A\,\mathrm{Tr}(\Sigma_B)}{r^4} + O(1/r^6)\Big]. \)

The leading term is exactly an inverse-square law. The prefactor is:

\( F_0 = \alpha_{\text{eff}}\,N_A\,N_B. \)

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Step 6 — Mapping to physical units and emergent Newton constant

Use the characteristic calibration scales (\(L_c,E_c,T_c\)) where \(c_{\text{char}}=L_c/T_c\). Map dimensionless simulation quantities to physical units:

• Distance: \(R_{\text{phys}} = r \, L_c\)
• Energy: \(E_{\text{phys}} = E_c \, E_{AB}\)
• Force: \(F_{\text{phys}} = (E_c / L_c)\, F_{AB}\)
• Node-based physical mass proxy: \(M_c = E_c T_c^2 / L_c^2\) so \(M_{\text{phys}} \approx N\times M_c\)

Match Newton form \(F_{\text{phys}} = G_{\text{eff}}\, M_{\text{phys}}(A)\,M_{\text{phys}}(B) / R_{\text{phys}}^2\). Solving yields:

\( G_{\text{eff}} = \alpha_{\text{eff}} \,\frac{E_c \, L_c}{M_c^2}. \)

Using \(M_c = E_c / c_{\text{char}}^2\) (equivalently \(M_c = E_c T_c^2 / L_c^2\)), substitute to obtain the same dimensionally consistent expression:

\( G_{\text{eff}} = \alpha_{\text{eff}} \,\frac{L_c\,c_{\text{char}}^4}{E_c}. \)

Remember that \(\alpha_{\text{eff}}\) encodes topology and dissipation dependence (we model \(\alpha_{\text{eff}}\propto\alpha\times\mathrm{topology\_factor}/\delta\)). Thus network topology and informational friction control the effective coupling strength that maps to gravitational strength.

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Verification: Pascal fingerprint and numeric tests

The derivation predicts an energy multipole expansion with precise coefficient ratios coming from the binomial coefficients for \(-\tfrac12\). Concretely:

\( E(r)=\dfrac{A}{r} + \dfrac{B}{r^3} + \dfrac{C}{r^5} + \cdots, \) with \( A:B:C:\dots = 1 : \big(-\tfrac12\big) : \big(+\tfrac{3}{8}\big) : \cdots. \)

By differentiating term-by-term the force series becomes:

\( F(r)=\dfrac{A}{r^2} + \dfrac{3B}{r^4} + \dfrac{5C}{r^6} + \cdots. \)

Numerical simulation steps to verify the fingerprint:

1. Construct two coherent clusters with controllable size and internal phase noise.
2. Compute \(E_{AB}(r)\) by summing \(\alpha(i,j)\,m_{ij}/|x_i-y_j|\) over node pairs for a range of large separations \(r\).
3. Fit \(E(r)\) to a series of inverse powers and extract coefficients \(A,B,C\).
4. Compare the ratios \(A:B:C\) with the predicted Pascal values.

Our prototype simulations (cluster radii, many sweeps, far-field sampling) show the monopole term dominates and measured corrections are consistent with the predicted sign and scaling. Precision increases with cluster size, number of sweeps, and pushing the fit window deeper into the far-field.

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Beyond 1/r² — Kernel generalization and unification

Replace the kernel \(1/r\) by a general kernel \(K(r)=\dfrac{1}{r^p} e^{-\mu r}\). The same expansion strategy gives:

\( \dfrac{1}{\lvert R+\varepsilon\rvert^p} = \dfrac{1}{r^p}(1+u)^{-p/2} = \dfrac{1}{r^p}\sum_{k=0}^\infty C_k^{(p)} u^k, \) where \( C_k^{(p)}=\binom{-p/2}{k}. \)

This covers:

• \(p=1\): Coulomb/Newton kernel → \(1/r^2\) forces
• \(p=2\): dipole-like kernel → leading \(1/r^3\) forces
• \(\mu>0\): Yukawa screening (exponential fall-off)

Thus different empirical force laws map to different kernels and corresponding Pascal-like coefficient sequences. This is a unifying combinatorial perspective — "Pascal fingerprints" identify kernels and multipole hierarchies.

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What this shows — implications and predictions

• Mathematical derivation: Under stated assumptions (coherent clusters, far-field, 1/r kernel) inverse-square forces emerge from discrete phase networks.
• Testable prediction: The Pascal coefficient ratios in the multipole expansion provide a precise numerical signature to confirm in simulation or experiment.
• Network control of gravity: \(G_{\text{eff}}\) depends on topology, coherence and dissipation (\(\alpha_{\text{eff}}\propto\mathrm{topology}/\delta\)), so in principle gravity strength is emergent and tunable within the model.
• Unification hint: Different kernel powers reproduce different known force scalings, giving a framework where electromagnetism, dipoles, Yukawa interactions and gravity appear as kernel variants of the same combinatorial expansion.

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Next steps (numerical roadmap)

• Push simulations further into far-field: increase cluster radius, sample \(r\) from \(\gtrsim 10R\) to \(\gg 100R\).
• Increase sweeps to reduce statistical variance and measure \(A,B,C\) precisely.
• Test deterministic phase offsets to isolate monopole coefficient \(A\) (fix \(\theta_A-\theta_B\)).
• Perform multipole decomposition fits \(E(r)=A/r+B/r^3+C/r^5+\dots\) and compare ratios \(A:B:C\) to Pascal values.
• Explore kernels \(1/r^p\) and weak screening \(e^{-\mu r}\) to map to other force laws.
• Use calibration values \(L_c,E_c,T_c\) from Section 2 to compute a numerical \(G_{\text{eff}}\) and compare to physical \(G\).

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Conclusion — significance

This derivation shows a mathematically direct route from discrete phase-coupled networks to inverse-square forces and a universal multipole expansion with precise binomial (Pascal) coefficients. The result unifies many force behaviors as kernel choices and provides specific, falsifiable predictions (the Pascal fingerprint). With systematic numerical verification and calibration, this framework can test whether inverse-square gravity arises robustly from TFP dynamics and whether emergent gravitational strength can be related quantitatively to network parameters.