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Article 13 — Symmetry Budget & Collapse Rate Law (ρ, D(t)) (Almost-Code Canonical) v1.0

Collapse risk and collapse speed rise when symmetry breaking exceeds a group’s symmetry budget.

Summary (Canonical)

Summary (Canonical)

Every group/system has a Symmetry Budget: how much structural change (choice/symmetry breaking) it can absorb per unit time while staying reliable.
If symmetry breaking is injected faster than the budget allows, phase shear accumulates, binds delete, and collapse speed becomes measurable.

Core ratio:ρ(t)=Sinj(t)Scap(G,t)ρ(t)=Scap​(G,t)Sinj​(t)​

  • Stable if ρ<1ρ<1
  • Failure regime if ρ≥1ρ≥1

Collapse-speed proxy:D(t)=k⋅(max⁡(0,ρ(t)−1))α, α>1D(t)=k⋅(max(0,ρ(t)−1))α, α>1

1) First Principles

1.1 Why “budget”

Systems do not fail because change exists.
They fail because change arrives faster than repair and re-stabilisation capacity.

A budget model makes this computable:

  • “How much change is too much?”
  • “How fast will collapse accelerate if we exceed it?”

1.2 Choice must be weighted by disruption

Not all choices are equal.
We must weight each change by magnitude (ΔS).

2) Definitions (Locked)

2) Definitions (Locked)

2.1 Injection (symmetry breaks arriving)

In a time window tt:

  • C(t)C(t) = count of meaningful choice events
  • ΔSi∈[0,1]ΔSi​∈[0,1] = symmetry break magnitude of event i

Sinj(t)=∑i=1C(t)ΔSiSinj​(t)=i=1∑C(t)​ΔSi​

2.2 Capacity (symmetry breaks absorbable)

For group GG:

  • NN = active agents/roles
  • BB = redundancy/bind strength
  • LL = load/tempo (time pressure)
  • RR = role mix (A,V,O,Op)

Scap(G,t)=Sbase(N,B)⋅g(L)⋅m(R)Scap​(G,t)=Sbase​(N,B)⋅g(L)⋅m(R)

Where:

  • g(L)g(L) decreases with load/tempo
  • m(R)m(R) increases with Oracle gating and decreases with Architect flooding in execution lanes

2.3 Overload ratio (core trigger)

ρ(t)=Sinj(t)Scap(G,t)ρ(t)=Scap​(G,t)Sinj​(t)​

3) The Symmetry Budget Law (Locked)

3.1 Stability

ρ(t)<1⇒stable bandρ(t)<1⇒stable band

3.2 Shear accumulation

ρ(t)≈1⇒shear accumulates (drift risk)ρ(t)≈1⇒shear accumulates (drift risk)

3.3 Failure regime

ρ(t)≥1⇒failure regime; bind deletion acceleratesρ(t)≥1⇒failure regime; bind deletion accelerates

This is the symmetry-space equivalent of Rate Dominance R=D˙/G˙R=D˙/G˙.

4) Collapse Rate / Lattice Destruction Rate

Let:

  • D(t)D(t) = lattice destruction rate (bind deletion / reliability loss rate)
  • k>0k>0 = fragility coefficient (lane-specific)
  • α>1α>1 = cascade exponent (nonlinear amplification)

D(t)=k⋅(max⁡(0,ρ(t)−1))αD(t)=k⋅(max(0,ρ(t)−1))α

Interpretation:

  • slightly above threshold → manageable damage
  • far above threshold → fast cascade

5) Phase Coupling (P0–P3)

Use moving average ρ‾ρ​ over window WW:

  • P3: ρ‾≤0.7ρ​≤0.7
  • P2: 0.7<ρ‾<1.00.7<ρ​<1.0
  • P1: 1.0≤ρ‾≤1.31.0≤ρ​≤1.3
  • P0: ρ‾>1.3ρ​>1.3 or repeated spikes >1.6>1.6

(Bands are tunable; the ratio structure is locked.)

(Bands are tunable; the ratio structure is locked.)

) AVOO Interpretation (Why this is role-true)

  • Architect activity increases SinjSinj​ unless sandboxed.
  • Operator lanes require low SinjSinj​ to preserve throughput.
  • Oracle gating increases effective ScapScap​ (filters unsafe breaks).
  • Visionary sets direction but can raise injection if cadence is too fast.

So collapse occurs when:

  • boundary exploration leaks into interior execution under high tempo
  • policy churn floods Operator lanes
  • exceptions multiply and branch complexity rises

7) System Optimisation (Control Levers)

To reduce ρρ:

7.1 Reduce injection SinjSinj​

  • freeze changes (reduce C)
  • cap disruption (reduce ΔS)
  • remove exceptions and forks
  • revert to stable SOP corridors

7.2 Increase capacity ScapScap​

  • strengthen binds/redundancy (increase B)
  • add backups and modularity
  • slow tempo (reduce L)
  • improve Oracle gating (increase O_w)
  • sandbox Architect activity (reduce effective A injection into interior)

This is truncation + stitching in symmetry space.This is truncation + stitching in symmetry space.

8) Failure Mode Trace (Required)

Change/choices accelerate → ΔS accumulates → (\rho) crosses 1 → shear accumulates → repair latency rises → bind deletion accelerates (D(t)↑) → P2→P1 drift → shock arrives → P0 collapse.
Repair: truncation to drop (\rho) below 1; stitching to raise capacity and restore redundancy.

Almost-Code Spec Block (Copyable)

CivOS.SymmetryBudgetRateLaw.v1.0

Inputs:
  Choice events i=1..C(t)
  ΔS_i ∈ [0,1]  # symmetry break magnitude per event
Injection:
  S_inj(t) = Σ ΔS_i
Capacity:
  S_cap(G,t) = S_base(N,B) * g(L) * m(R)
    N := active agents/roles
    B := redundancy/bind strength
    L := load/tempo (g'(L) < 0)
    R := role mix (A,V,O,Op)
Overload Ratio:
  ρ(t) = S_inj(t) / S_cap(G,t)
Law:
  Stable iff ρ(t) < 1
  Failure regime iff ρ(t) >= 1
  Shear accumulates as ρ approaches/exceeds 1
Destruction Rate:
  D(t) = k * (max(0, ρ(t)-1))^α , α>1

FAQ (Short)

Q1: Why not just count number of changes?
Because disruption size matters. We weight by ΔS.

Q2: What’s the simplest intervention?
Truncate: freeze changes and remove exceptions until ρ<1.

Q3: Can systems die from too little change?
Yes—ossification removes boundary corridors. But that is a different drift mode; this article handles overload-driven collapse speed.

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