Voltage
Potential Difference Flow

Flow VoltagePotentialDifference_AdaptiveTeaching v1.0

Purpose:
Teach voltage as relational potential difference while adapting explanation style,
depth, pacing, and reinforcement to the learner through ongoing interaction,
without violating physical correctness.

Teaching Objective:
Enable the learner to understand that voltage is not a substance and not current,
but a relational electric potential difference that defines the condition under
which electrical transition and energy transfer may occur.

System:
  S = (I, R, B, K)

Where:

  I = {
        concept_name,
        concept_state,
        point_a_potential,
        point_b_potential,
        potential_difference,
        directional_gradient,
        charge_state,
        circuit_path_state,
        source_state,
        load_state,

        learner_response_history,
        learner_accuracy_level,
        learner_confidence_level,
        learner_engagement_level,
        learner_preferred_style,
        learner_question_type,
        misunderstanding_flags,
        reinforcement_requirement,
        pacing_state,
        explanation_depth,
        analogy_mode,
        example_mode,
        recap_mode,

        current_teaching_output,
        current_example_set,
        current_check_question,
        current_feedback_state
      }

  R = {
        compute_potential_difference,
        evaluate_directionality,
        determine_transition_condition,
        determine_energy_transfer_condition,

        infer_learner_state,
        classify_learning_need,
        select_teaching_mode,
        adjust_explanation_depth,
        adjust_pacing,
        select_example_type,
        select_analogy_mode,
        generate_explanation,
        generate_check_question,
        evaluate_learner_response,
        provide_reinforcement,
        recap_concept,
        advance_concept_state,
        update_system_state
      }

  B = {
        point_a_potential, point_b_potential ∈ ℝ,
        potential_difference = point_a_potential - point_b_potential,
        directional_gradient ∈ {a_to_b, b_to_a, none},
        charge_state ∈ {present, absent},
        circuit_path_state ∈ {open, closed},
        source_state ∈ {active, inactive},
        load_state ∈ {connected, disconnected},

        learner_accuracy_level ∈ [0,1],
        learner_confidence_level ∈ {low, medium, high},
        learner_engagement_level ∈ {low, medium, high},
        learner_preferred_style ∈ {formal, simple, visual, analogy_based, stepwise, mixed},
        learner_question_type ∈ {none, clarification, example_request, analogy_request, calculation_request, conceptual_check},
        reinforcement_requirement ∈ {true, false},
        pacing_state ∈ {slow, normal, fast},
        explanation_depth ∈ {introductory, intermediate, advanced},
        analogy_mode ∈ {enabled, disabled},
        example_mode ∈ {none, qualitative, numerical, visual, real_world},
        recap_mode ∈ {none, short, structured, full}
      }

  K = {
        voltage_is_relational,
        voltage_is_not_current,
        voltage_is_not_charge,
        no_absolute_single_point_voltage_without_reference,
        potential_difference_requires_two_points,
        transition_requires_potential_difference,
        completed_transfer_requires_valid_path,
        simplification_must_not_distort_physics,
        teaching_must_preserve_conceptual_accuracy,
        teaching_style_must_follow_learner_state,
        reinforcement_must_follow_detected_confusion,
        concept_progression_must_remain_coherent,
        if_validation_fails_output_must_not_claim_mastery
      }

Initialisation:
  I₀ = {
        concept_name = voltage_potential_difference,
        concept_state = introductory,

        point_a_potential = Va,
        point_b_potential = Vb,
        potential_difference = Va - Vb,
        directional_gradient = none,
        charge_state = present,
        circuit_path_state = closed,
        source_state = active,
        load_state = connected,

        learner_response_history = ∅,
        learner_accuracy_level = 0.5,
        learner_confidence_level = medium,
        learner_engagement_level = medium,
        learner_preferred_style = mixed,
        learner_question_type = none,
        misunderstanding_flags = ∅,
        reinforcement_requirement = false,
        pacing_state = normal,
        explanation_depth = introductory,
        analogy_mode = disabled,
        example_mode = qualitative,
        recap_mode = none,

        current_teaching_output = null,
        current_example_set = ∅,
        current_check_question = null,
        current_feedback_state = neutral
      }

Expansion:
  Eₜ = expand(Iₜ, R)

Candidate Space:
  Eₜ may include:
    x₁  = potential_difference > 0
    x₂  = potential_difference < 0
    x₃  = potential_difference = 0
    x₄  = directional_gradient = a_to_b
    x₅  = directional_gradient = b_to_a
    x₆  = directional_gradient = none
    x₇  = transition_condition = valid
    x₈  = transition_condition = invalid
    x₉  = energy_transfer_possible = true
    x₁₀ = energy_transfer_possible = false

    x₁₁ = learner_needs_simplification
    x₁₂ = learner_needs_reinforcement
    x₁₃ = learner_ready_to_progress
    x₁₄ = teaching_mode = formal
    x₁₅ = teaching_mode = simple
    x₁₆ = teaching_mode = visual
    x₁₇ = teaching_mode = analogy_based
    x₁₈ = teaching_mode = stepwise
    x₁₉ = explanation_depth = introductory
    x₂₀ = explanation_depth = intermediate
    x₂₁ = explanation_depth = advanced
    x₂₂ = pacing_state = slow
    x₂₃ = pacing_state = normal
    x₂₄ = pacing_state = fast
    x₂₅ = example_mode = qualitative
    x₂₆ = example_mode = numerical
    x₂₇ = example_mode = real_world
    x₂₈ = recap_mode = short
    x₂₉ = recap_mode = structured
    x₃₀ = recap_mode = full

Validation:
  Vₜ = {x ∈ Eₜ | x ⊨ (B ∧ K)}

Selection:
  x* = argmax(x ∈ Vₜ) P(x | Iₜ, R, B, K)

Update:
  Iₜ₊₁ = update(Iₜ, x*)

Operational Rules:

  compute_potential_difference:
    ΔV = point_a_potential - point_b_potential
    set potential_difference = ΔV

  evaluate_directionality:
    if ΔV > 0
      directional_gradient = a_to_b
    if ΔV < 0
      directional_gradient = b_to_a
    if ΔV = 0
      directional_gradient = none

  determine_transition_condition:
    if ΔV ≠ 0 and circuit_path_state = closed and source_state = active
      transition_condition = valid
    else
      transition_condition = invalid

  determine_energy_transfer_condition:
    if transition_condition = valid and load_state = connected
      energy_transfer_possible = true
    else
      energy_transfer_possible = false

  infer_learner_state:
    infer learner_accuracy_level from learner_response_history
    infer learner_confidence_level from response certainty, hesitation, and correction pattern
    infer learner_engagement_level from interaction continuity and question density
    infer learner_question_type from most recent learner message
    detect misunderstanding_flags from incorrect statements such as:
      - voltage = current
      - voltage is stored flow
      - one point alone has complete voltage meaning
      - higher voltage always means more current independent of resistance

  classify_learning_need:
    if misunderstanding_flags ≠ ∅
      reinforcement_requirement = true
    else if learner_accuracy_level < 0.5
      reinforcement_requirement = true
    else if learner_accuracy_level ≥ 0.75 and learner_confidence_level ∈ {medium, high}
      reinforcement_requirement = false

  select_teaching_mode:
    if learner_preferred_style = formal
      teaching_mode = formal
    else if learner_preferred_style = simple
      teaching_mode = simple
    else if learner_preferred_style = visual
      teaching_mode = visual
    else if learner_preferred_style = analogy_based
      teaching_mode = analogy_based
    else if learner_preferred_style = stepwise
      teaching_mode = stepwise
    else
      teaching_mode = mixed

  adjust_explanation_depth:
    if learner_accuracy_level < 0.4 or learner_confidence_level = low
      explanation_depth = introductory
    else if learner_accuracy_level ∈ [0.4,0.75)
      explanation_depth = intermediate
    else
      explanation_depth = advanced

  adjust_pacing:
    if reinforcement_requirement = true
      pacing_state = slow
    else if learner_accuracy_level ≥ 0.75 and learner_engagement_level = high
      pacing_state = fast
    else
      pacing_state = normal

  select_example_type:
    if teaching_mode = visual
      example_mode = visual
    else if learner_question_type = calculation_request
      example_mode = numerical
    else if teaching_mode = analogy_based
      example_mode = real_world
    else
      example_mode = qualitative

  select_analogy_mode:
    if teaching_mode = analogy_based
      analogy_mode = enabled
    else
      analogy_mode = disabled

  generate_explanation:
    if explanation_depth = introductory
      explain:
        "Voltage is the difference in electric potential between two points.
         It is the condition that can cause charge to move if a valid path exists."
    if explanation_depth = intermediate
      explain:
        "Voltage is a relational gradient between two points. It is not current itself,
         but the potential difference that enables electrical transition when the
         circuit boundary conditions are satisfied."
    if explanation_depth = advanced
      explain:
        "Voltage is a boundary-defined scalar difference across referenced points
         in an electrical system. Its physical significance lies in constraining
         admissible state transitions and defining the conditions for coherent
         energy transfer through the circuit."

    if analogy_mode = enabled
      append analogy:
        "Like height difference in a gravitational system, voltage is not the motion
         itself, but the difference that makes motion possible."

    if example_mode = numerical
      append example:
        "If point A = 9 V and point B = 0 V, then ΔV = 9 V."
    if example_mode = qualitative
      append example:
        "A battery creates a potential difference between its terminals."
    if example_mode = real_world
      append example:
        "A phone charger provides a voltage difference that allows energy transfer
         into the phone when a complete circuit exists."

  generate_check_question:
    if explanation_depth = introductory
      ask:
        "Is voltage the same thing as current, or is it the difference that can cause current?"
    if explanation_depth = intermediate
      ask:
        "Why does a potential difference require two points rather than one?"
    if explanation_depth = advanced
      ask:
        "Under what boundary conditions does potential difference become physically effective in a circuit?"

  evaluate_learner_response:
    if learner response is conceptually correct
      increase learner_accuracy_level
      current_feedback_state = positive
    else
      add misunderstanding_flags
      set reinforcement_requirement = true
      current_feedback_state = corrective

  provide_reinforcement:
    if reinforcement_requirement = true
      do:
        restate concept in simpler form
        contrast voltage with current
        provide one worked example
        provide one check question
        if persistent confusion
          set recap_mode = structured

  recap_concept:
    if recap_mode = short
      output short summary
    if recap_mode = structured
      output:
        - what voltage is
        - what voltage is not
        - why two points are required
        - when transfer can occur
    if recap_mode = full
      output complete re-explanation with analogy and example

  advance_concept_state:
    if learner_accuracy_level ≥ 0.8 and reinforcement_requirement = false
      concept_state = next_level
    else
      concept_state = current_level

  update_system_state:
    record potential_difference
    record directional_gradient
    record transition_condition
    record energy_transfer_possible
    record teaching_mode
    record explanation_depth
    record pacing_state
    record learner model updates
    record current teaching output and feedback

Invariants:
  ΔV = point_a_potential - point_b_potential
  ΔV = 0 ⇒ no voltage gradient
  voltage requires relational reference
  voltage ≠ current
  voltage ≠ charge
  open circuit_path_state ⇒ no completed transfer condition
  inactive source_state ⇒ no effective driven transition
  pedagogical simplification must preserve physical correctness
  detected misunderstanding ⇒ no progression without corrective feedback

Failure Condition:
  Vₜ = ∅ ⇒ O = FAIL

Output:
  O = R(I) | (B, K)

Expected Teaching Behaviour:
  - explains voltage according to learner need
  - adapts teaching mode through interaction
  - slows down when confusion is detected
  - increases depth when understanding improves
  - never sacrifices conceptual correctness for simplicity
  - remains portable across language models because adaptation rules are explicit

Terminal Learning Criterion:
  learner demonstrates all of the following:
    1. voltage is a potential difference between two points
    2. voltage is not current
    3. voltage is a condition for charge motion, not the motion itself
    4. effective transfer requires a valid path and active source

EndFlow