Protocol

Select the fundamental cycle c and put:

\omega_0 = \mathbf{1}_{c}

Purpose: to check whether the cycle is harmonious (structural) or attenuated.

IV. Diffusion procedure

Take the obvious step: (I+\Delta t\,\Delta_1)\omega_{k+1}=\omega_k

Options:

• \Delta t = 1
• Max 200 steps
• Stop at:
  \frac{\|\omega_{k+1}-\omega_k\|}{\|\omega_k\|}<10^{-4}

V. Metrics that are fixed

At every step we consider:

1. Flow rate

E_k=\omega_k^\top W_1\omega_k

2. Divergence

D_k=\|\delta_1\omega_k\|_2

3. Whirlwind

C_k=\|d_1\omega_k\|_2

4. Laplacean Energy

L_k=\|\Delta_1\omega_k\|_2

VI. Definition of memory

1. Harmonic part

After convergence: h \approx \omega_{k^\*}

Check: \|\delta_1 h\| < 10^{-6}, \quad \|d_1 h\| < 10^{-6}

2. Spectral verification

Find 10–20 the lowest eigenvalues \lambda_i operator \Delta_1.

Determine: b_1 = \#\{\lambda_i < 10^{-8}\} This is the dimension of memory.

VII. Interpretation of results

Scenario 1: b_1 = 0

The crystal does not have 1-circuit memory.

Any flow is completely broken.

Scenario 2: b_1 > 0

There are b_1 independent stable contours.

These are the structural “loops” of the system.

Scenario 3: Large harmonic energy

\frac{\|h\|_{W_1}}{\|\omega_0\|_{W_1}} \gg 0

→ system tends to fix the contours.

Scenario 4: Rapid attenuation

If E_k falls exponentially fast

→ high spectral gap (large \lambda_2)

→ system assembled.

VIII. Visualization (required)

• E_k time chart (log-scale)
• Graphics D_k and C_k
• Display h on ribs (color and thickness)
• Display the basis of harmonic forms (if b_1>0)

IX. Correctness control

Check:

• \omega_{\mathrm{grad}} + \omega_{\mathrm{curl}} + h = \omega_0
• Orthogonality of component
• numerical stability (the difference does not exceed 10^-8)

X. What is a successful experiment

The experiment is considered completed if:

• Defined by b_1.
• Harmonic basis is calculated.
• It is shown what initial excitations are projected into memory.
• The damping dynamics is recorded.

Next level

If you go deeper, there are two directions:

• Enter Ricci-discretion (Ollivier or Forman) and associate it with attenuation rate.
• Go to the evolution of metrics (a discrete analogue of the Ricci flow).

Then add two extensions to the protocol:

• Discrete Ricci-curve (Ollivier or Forman) and the relationship with the decay rate
• evolution of metrics (discrete analogue of the Ricci flow) and how it affects memory b_1 and harmonics

Below is how “Protocol v0.2” is on top of your Hodge-Diffusion Protocol.

Protocol v0.2

Geometric diagnostics and metric evolution for EQUILIBRIUM-741

XI. Extension 1: Ricci curve on graph and its relationship to attenuation

XI.1. What we measure

For each rib e=(i,j), we consider the curvature \kappa(e). Then we build aggregates:

• average:
  \bar\kappa = \frac{1}{|E|}\sum_{e\in E}\kappa(e)
• Percentage of Negative:
  p_- = \frac{\#\{e:\kappa(e)<0\}}{|E|}
• “voltage map”: top-10% most negative ribs

Interpretation (practical)

• negative \kappa: “dispersion of geodesy”, corridors, discontinuities of consistency, potential bottlenecks of control
• positive \kappa: "exhaustion", high connectivity of local surroundings, good mixing (fast diffusion attenuation)

XI.2. Choosing a default Ricci at the start: Forman (quick and sparse)

In order not to drown in calculations, default:

Forman-Ricci for rib e=(i,j) (without 2-simplexes)

In the simplest version (for a weighted graph), it is calculated according to the local configuration of neighbors. In practice, they take the finished implementation, but the meaning is as follows:

• The weight of the top/edge
• Fines “branching” and “weak” connections
• gives a quick curve map for each edge

Why Forman Default: Linear at |E| and works well at 741 node.

If you want the “right”: Ollivier-Ricci

This is Wasserstein-1 between local distributions \mu_i,\mu_j:

\kappa(i,j)=1-\frac{W_1(\mu_i,\mu_j)}{d(i,j)}

But it's heavier. We leave as a "precision check" mode on the subnet.

XI.3. Ricci’s relationship with diffusion (what exactly we check)

You're already thinking in diffusion:

• E_k=\omega_k^\top W_1\omega_k
• L_k=\|\Delta_1\omega_k\|_2

Add the measurement of “decay rate”:

Global speed (estimate)

\rho_k = \frac{E_{k+1}}{E_k}

(Less - fades faster)

Local speed on ribs

See which ribs hold the amplitude longer |\omega_k(e)|.

XI.4. Hypotheses/expected regularities (check by experiment)

• The more p_- and the lower \bar\kappa, the slower the mixing (worse the attenuation): \rho_k is closer to 1.
• Ribs with the most negative \kappa are more likely to fall into:
• “Bottlenecks” (Bottlenecks)
• Memory contours h
• With positive curvature, “loops” are less stable: often b_1 decreases with increased connectivity (but depends on the addition/removal of triangles and how you update the weight).

XII. Extension 2: The evolution of metrics is a discrete analogue of the Richie stream

Idea: metric/weights are not fixed. We give them an evolution that "cures" geometry, improves connectivity and controllability, and then look at how b_1, the spectrum of \Delta_1 and harmonics change.

XII.1. What is “metrics” in a discrete world?

We have two presentations:

• cost w_e (length/cost of rib)
• conductivity c_e (link strength), often c_e=\exp(-w_e/\tau)

Ricci flow is more logical to do on the “lengths” of w_e, but it is computationally convenient to update c_e.

XII.2. Default update scheme (flow on weights through curvature)

Fix the curvature on the rib \kappa_e (Forman or Ollivier). We set a “goal” \kappa^\* (usually 0 Or a little positive).

Length update (exponentially stable)

w_e^{(t+1)} = w_e^{(t)} \,\exp\Big(-\eta\big(\kappa_e^{(t)}-\kappa^\*\big)\Big)

• if \kappa_e is below target (too negative) → Multiplier >1 or <1 depending on the selected sign; we choose to “treat” negative zones:
  default: negative curvature → bond gain → length reduction
  Then we take:
  w_e^{(t+1)} = w_e^{(t)} \,\exp\Big(+\eta\big(\kappa_e^{(t)}-\kappa^\*\big)\Big)
  (if \kappa Negative, \kappa-\kappa^\*<0, Length is reduced)

Then we recalculate the conductivity:

c_e^{(t+1)}=\exp(-w_e^{(t+1)}/\tau)

and update W_1.

Default parameters:

• \kappa^\*=0
• \eta = 0.05 (small step)
• 20–50 iterations

XII.3. What we keep “invariant”

So that evolution does not “break” the graph:

• preserve the topology of the ribs E (do not add / remove, only weights)
• Limiting the weight:
  w_e \in [w_{\min}, w_{\max}]
  e.g. [0.1, 10] on a relative scale

XII.4. Built-in cycle experiment (Ricci Flow + Hodge Diffusion)

One external metric flow step and one internal diffusion start:

For t=0..T:

• by current weights W_1^t) we think \kappa^t) on the ribs
• Update to W^{(t+1)and W_1^{(t+1)}
• (optional) reassembly W_2^(t+1)
• reassembling \Delta_1^QQQQQ(t+1)QQQQQ
• Make one of the modes:
• (a) spectrally: find b_1^(t+1)-=\dim\ker(\Delta_1)
• (b) Dynamic: run Algorithm 3 and get h^(t+1)

XIII. Metrics of observation during the flow of metrics

At each external step t we fix:

• Ricci-statistics: \bar\kappa^{(t)}, p_-^{(t)}
• Spectrum 1-Laplasian: a few of the smallest \lambda_i^{(t)}
• Memory: b_1^{(t)} = \#\{\lambda_i^{(t)}<\epsilon\}
• Diffusion rate: \rho^t) (energy decline)
• Harmonic stability: correlation/angle between bases \ker(\Delta_1^t) ) and \ker(\Delta_1^{(t+1)})

XIV. Expected effects and how to read them

XIV.1. If the flow “treats” negative zones

• \bar\kappa growing, p_- falling
• diffusion accelerates (energy drops faster)
• harmonic h becomes "cleaner" (less gradient/vortex impurities when decomposed)

XIV.2. What can happen to b_1

Important: b_1 is the topological value of the complex (if not to change E and triangles), ideally should be stable.

But in computational practice, “almost zero” eigenvalues can:

• stick to zero (if weights create a strong regularization)
• or vice versa to approach zero (if the metric enhances the separation of substructures)

Therefore, we distinguish:

• Strict Memory: \lambda<10^{-10}
• Soft Memory: \lambda\in[10^{-10},10^{-6almost loops)

XV. Recommended run plan (minimum)

• Baseline: without Ricci-evolution — measure \bar\kappa, p_-, b_1, \rho
• Flow-20: 20 Iterations of the flow - see trends
• Flow-50: 50 iterations - sustainability or degradation
• In each mode we drive 3 Type \omega_0 (A/B/C from the previous protocol)

XVI. The resulting artifact (what should happen)

• Curvature map for ribs (where voltage)
• Graphs \bar\kappa^{(t)}, p_-^{(t)}
• Spectral graphs \lambda_i^jjjjjjj(t)jj nil
• Set of harmonics (memory base) before/after flow
• Comparison of diffusion damping rate before/after

I will take the next step in the form of a “Unified EQUILIBRIUM-741 Pipeline”: pseudocode at the “run it and get a report” level (build → triangles → operators → hodge → diffusion → ricci → flow → compare), with clearly defined parameters and logging points.