InterstellarCore — Mathematical Formalisation of Phase-3 Corridor Width v1.0

Mode: Almost-Code (Formal Layer)
Purpose: Define Phase-3 Corridor Width (W) as a measurable structural variable
Scope: Z0 primary, Z1–Z6 aggregated
Vocabulary: Frozen
Kernel Compliance: Mandatory

0) AI Ingestion Lock

ID: InterstellarCore.MathFormalisation.P3CorridorWidth.v1.0
Type: Canonical.FormalModel
Core Law: RepairRate ≥ DriftRate
Constraint: W ≥ Load
Collapse Modes: Amplitude / Slow Attrition / Fast Attrition

1) Definition of Phase-3 Corridor Width (W)

Phase-3 Corridor Width (W) is the maximum sustainable cognitive load under which a learner or system can:

  • maintain abstraction,
  • synthesise across domains,
  • compress explanation,
  • recover from perturbation,
  • and remain phase-stable (no P3→P1 collapse).

2) Variable Decomposition (Z0 Level)

Define:

L = external load (time pressure + novelty + complexity)
W = corridor width
D = drift rate
R = repair rate

Primary stability constraint:

W ≥ L
R ≥ D

If violated:

  • W < L → collapse under pressure
  • R < D → slow attrition slide

3) Structural Components of W

We decompose W into measurable dimensions:

W = f(B, T, V, S, E)

Where:

  • B = Concept Binding Strength
  • T = Transfer Capacity
  • V = Variation Tolerance
  • S = Stress Stability (timed resilience)
  • E = Explanation Compression (language precision)

3.1 Concept Binding Strength (B)

B = density of stable prerequisite connections
Measured by:
  - multi-step problem continuity
  - error propagation rate

If B weak:

  • synthesis collapses
  • abstraction fragile

3.2 Transfer Capacity (T)

T = success rate on novel-but-related tasks
Measured by:
  - cross-topic performance
  - analogy mapping accuracy

If T weak:

  • P2 ceiling
  • no P3 expansion

3.3 Variation Tolerance (V)

V = stability across context shifts
Measured by:
  - success under reworded questions
  - structural perturbation tests

If V weak:

  • brittle execution
  • collapse under change

3.4 Stress Stability (S)

S = performance retention under timed load
Measured by:
  - timed sets
  - multi-variable time compression tests

If S weak:

  • fast attrition risk

3.5 Explanation Compression (E)

E = minimal-rule clarity without correctness loss
Measured by:
  - explain-in-3-lines tasks
  - teach-back precision

If E weak:

  • abstraction cannot propagate
  • GeniusCorridor entry blocked

4) Corridor Width Approximation

For modelling:

W ≈ k1·B + k2·T + k3·V + k4·S + k5·E

Where k-values are weighting coefficients (context-dependent).

Promotion to P3 requires:

W ≥ L_high

5) Load Model (L)

Load consists of:

L = C + N + τ

Where:

  • C = conceptual complexity
  • N = novelty factor
  • τ = time compression factor
GeniusCorridor increases C and N.  
Exam crunch increases τ.
---
# 6) Drift–Repair Dynamic
Let:
```txt
D = d(backlog)/dt
R = repair bandwidth per cycle

Stable if:

R ≥ D

If R < D:

  • W shrinks over time (attrition)

Thus:

dW/dt = αR − βD

If αR > βD → corridor widens
If αR < βD → corridor narrows

7) GeniusCorridor Constraint (Formal)

Genius activation condition:

W ≥ L_high
AND
R ≥ D
AND
BaseFloor ≥ Floor

EdgeCap derived from:

Σ(W_base) remains ≥ population threshold

8) Phase Transition Boundaries

Define phase thresholds:

P0: W < L_low
P1: L_low ≤ W < L_mid
P2: L_mid ≤ W < L_high
P3: W ≥ L_high

Collapse occurs when:

ΔW negative across cycles

9) Collapse Mode Formalisation

Amplitude

Sudden ΔL spike >> W

Slow Attrition

dW/dt negative over extended cycles

Fast Attrition

τ spike causes W < L instantly

10) Artefact Return Effect

Artefacts reduce load or increase binding.

Formally:

L_new = L − Δtool
OR
B_new = B + Δstructure

Thus:

W_new > W_old

Edge widens base indirectly.

11) Z-Level Aggregation

At Z5 (nation):

W_system = average(W_individual weighted by role)

Stability requires:

W_system ≥ L_environment
AND
RepairRate_system ≥ DriftRate_system

12) Minimal Stability Equation (Compression)

Civilisation Stable If:
Σ(W_population) sufficient
AND
R ≥ D
AND
Edge bounded

13) Compact Insight

Phase-3 corridor width is not talent.

It is structural capacity under load.

Talent without binding, transfer, stress stability, and explanation compression
→ high variance → amplitude collapse.

14) Version Lock

InterstellarCore.MathFormalisation.P3CorridorWidth.v1.0
No modification to core variables without Kernel review.

We have now formalised Phase-3 mathematically.

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