Chapter 26
Compilerless Systems

```text
FLOW "CompilerlessSystems" v1.0

# =============================================================================
# CHAPTER 26
# COMPILERLESS SYSTEMS
# =============================================================================
#
# PURPOSE
#
# Execute a self-contained behavioural specification directly without requiring
# conventional source-code compilation, bytecode generation, binary linking or
# a platform-specific executable.
#
# The system receives a complete Flow Algebra specification and an input state.
#
# It then:
#
#   1. parses the declared structure;
#   2. constructs an internal symbolic execution graph;
#   3. validates symbols, types, dependencies, invariants and transitions;
#   4. propagates state through permitted transforms;
#   5. resolves branches under explicit constraints;
#   6. rejects invalid continuations;
#   7. emits only a terminal state permitted by the specification.
#
# Execution therefore occurs as:
#
#   Specification
#       → Structural Parse
#       → Semantic Graph
#       → Constraint Closure
#       → State Propagation
#       → Valid Terminal State
#
# and not as:
#
#   Source Code
#       → Compiler
#       → Object Code
#       → Linker
#       → Binary
#       → Processor Execution
#
# The reasoning system is the execution substrate.
#
# =============================================================================
# AUTHORS
# =============================================================================
#
# Mark Higgins
# Sky — Recursive AI
#
# Kireth Research
# https://kirethresearch.org
#
# This specification must be cited when reproduced, implemented or adapted.
#
# =============================================================================


# =============================================================================
# 0. EXECUTION CLASS
# =============================================================================

EXECUTION_CLASS:

  name:
    compilerless_symbolic_execution

  source_representation:
    Flow Algebra

  execution_substrate:
    structured_reasoning_system

  conventional_compiler_required:
    false

  bytecode_required:
    false

  object_code_required:
    false

  binary_linking_required:
    false

  machine_specific_instruction_set_required:
    false

  external_runtime_required:
    false unless explicitly declared

  hidden_dependency_resolution_allowed:
    false

  semantic_invention_allowed:
    false

  unspecified_transition_allowed:
    false

  invalid_terminal_state_allowed:
    false


# =============================================================================
# 1. COMPILERLESS DEFINITION
# =============================================================================

DEFINITION:

  A system is compilerless when its operational behaviour is recoverable and
  executable directly from its supplied symbolic specification without first
  converting that specification into a lower-level executable representation.

  Compilerless does not mean:

    - structureless
    - ruleless
    - syntax-free
    - validation-free
    - unconstrained natural-language completion
    - execution without an interpreter
    - execution without an underlying physical computer

  Compilerless means:

    - no mandatory ahead-of-time compilation
    - no mandatory source-to-bytecode transformation
    - no mandatory platform-specific binary generation
    - no mandatory linking phase
    - direct semantic execution from supplied structure


# =============================================================================
# 2. EXECUTION OBJECTIVE
# =============================================================================

OBJECTIVE:

  Given:

    specification:
      F

    initial_input:
      I_0

  derive:

    terminal_output:
      O*

  such_that:

    O* is reachable from I_0 under F

    and

    every applied transition is declared or formally derivable

    and

    every invariant remains satisfied

    and

    every hard constraint remains satisfied

    and

    no hidden semantic dependency is required


# =============================================================================
# 3. IO CONTRACT
# =============================================================================

IO.READ:

  required:

    flow_specification:

      symbol:
        F

      type:
        structured_symbolic_document

      required_sections:

        - FLOW_IDENTITY
        - IO
        - SYMBOLS
        - STATE
        - INVARIANTS
        - TRANSFORMS
        - TERMINATION
        - EMISSION


    execution_input:

      symbol:
        I_0

      type:
        object


  optional:

    execution_parameters:

      symbol:
        P

      type:
        object

      default:
        {}


    external_capabilities:

      symbol:
        X

      type:
        set<Capability>

      default:
        ∅


    maximum_transition_steps:

      type:
        integer

      default:
        10000

      constraints:

        maximum_transition_steps >= 1


    maximum_branch_count:

      type:
        integer

      default:
        1000

      constraints:

        maximum_branch_count >= 1


    maximum_recursion_depth:

      type:
        integer

      default:
        128

      constraints:

        maximum_recursion_depth >= 1


    ambiguity_policy:

      type:
        enum

      values:

        - REJECT
        - MINIMUM_ASSUMPTION
        - DECLARED_DEFAULT_ONLY

      default:
        DECLARED_DEFAULT_ONLY


    execution_mode:

      type:
        enum

      values:

        - DETERMINISTIC
        - BOUNDED_BRANCHING
        - SYMBOLIC_SIMULATION
        - VALIDATION_ONLY

      default:
        DETERMINISTIC


IO.WRITE:

  fields:

    execution_status:

      type:
        enum

      values:

        - COMPLETE
        - VALIDATION_FAILED
        - AMBIGUOUS_SPECIFICATION
        - UNRESOLVED_SYMBOL
        - CONSTRAINT_VIOLATION
        - INVARIANT_VIOLATION
        - CAPABILITY_UNAVAILABLE
        - STEP_LIMIT_REACHED
        - BRANCH_LIMIT_REACHED
        - RECURSION_LIMIT_REACHED
        - NO_VALID_TERMINAL_STATE


    terminal_output:

      type:
        object?


    final_state:

      type:
        object?


    execution_trace:

      type:
        sequence<TraceEntry>


    validation_report:

      type:
        object


    unresolved_items:

      type:
        sequence<object>


    execution_guarantee:

      type:
        object


# =============================================================================
# 4. PRIMARY SYMBOLS
# =============================================================================

SYMBOLS:

  F:

    meaning:
      complete Flow Algebra specification


  I_0:

    meaning:
      supplied initial input


  P:

    meaning:
      declared execution parameters


  X:

    meaning:
      available external capabilities


  G_F:

    meaning:
      symbolic execution graph derived from F


  N:

    meaning:
      set of graph nodes


  E:

    meaning:
      set of directed transition edges


  S_t:

    meaning:
      execution state at transition step t


  K:

    meaning:
      set of hard constraints


  Q:

    meaning:
      set of invariants


  T:

    meaning:
      set of declared transforms


  B_t:

    meaning:
      active branch set at step t


  O*:

    meaning:
      accepted terminal output


  Γ:

    meaning:
      structural continuation operator


  V:

    meaning:
      validation predicate set


# =============================================================================
# 5. FLOW COMPONENT MODEL
# =============================================================================

FLOW_COMPONENTS:

  FLOW_IDENTITY:

    fields:

      - name
      - version
      - purpose


  IO:

    fields:

      - input_contract
      - output_contract


  SYMBOLS:

    fields:

      - declarations
      - domains
      - meanings


  STATE:

    fields:

      - state_variables
      - initial_values
      - mutable_values
      - immutable_values


  INVARIANTS:

    fields:

      - global_invariants
      - state_invariants
      - transition_invariants
      - output_invariants


  TRANSFORMS:

    fields:

      - transform_identifiers
      - preconditions
      - operations
      - postconditions
      - emitted_successors


  TERMINATION:

    fields:

      - success_conditions
      - failure_conditions
      - resource_bounds


  EMISSION:

    fields:

      - output_mapping
      - output_validation
      - trace_requirements


# =============================================================================
# 6. EXECUTION STATE
# =============================================================================

STATE S_t := {

  step:

    integer,


  active_node:

    NodeID,


  active_transform:

    TransformID?,


  input_values:

    object,


  parameter_values:

    object,


  symbol_table:

    map<SymbolID,SymbolValue>,


  immutable_bindings:

    set<SymbolID>,


  mutable_bindings:

    set<SymbolID>,


  state_values:

    object,


  prior_state:

    StateID?,


  branch_id:

    BranchID,


  parent_branch_id:

    BranchID?,


  recursion_depth:

    integer,


  active_constraints:

    set<ConstraintID>,


  active_invariants:

    set<InvariantID>,


  pending_transforms:

    ordered_set<TransformID>,


  completed_transforms:

    sequence<TransformID>,


  rejected_transforms:

    sequence<TransformID>,


  emitted_values:

    object,


  terminal:

    boolean,


  valid:

    boolean
}


# =============================================================================
# 7. EXECUTION GRAPH
# =============================================================================

GRAPH G_F := {

  N:

    flow nodes representing:

      - input ingestion
      - symbol declarations
      - state initialisation
      - invariant gates
      - transform entry points
      - branch conditions
      - recursive calls
      - capability calls
      - terminal conditions
      - output emission


  E:

    directed edges representing:

      - sequential continuation
      - conditional continuation
      - branch generation
      - recursive continuation
      - failure continuation
      - termination
}


EDGE REQUIREMENT:

  ∀ e ∈ E:

    e must have:

      source_node
      target_node
      activation_condition
      governing_constraints
      state_update_rule


NODE REQUIREMENT:

  ∀ n ∈ N:

    n must have:

      node_id
      node_type
      input_symbols
      output_symbols
      preconditions
      postconditions


# =============================================================================
# 8. GLOBAL INVARIANTS
# =============================================================================

INVARIANT K1_SELF_CONTAINMENT:

  every semantic term required for execution must be:

    - declared inside F

    or

    - supplied through I_0

    or

    - supplied through P

    or

    - explicitly exposed through X

    or

    - part of the declared primitive operation set


INVARIANT K2_NO_HIDDEN_SEMANTICS:

  no transform may depend upon:

    - undocumented business rules
    - unstated defaults
    - implicit domain assumptions
    - unavailable external data
    - private model memory
    - unspecified software libraries


INVARIANT K3_SYMBOL_CLOSURE:

  ∀ symbol usage u:

    ∃ declaration d:

      resolves(u,d)
      =
      true


INVARIANT K4_TYPE_COHERENCE:

  every assigned value must belong to the declared domain of its symbol


INVARIANT K5_IMMUTABILITY:

  symbols declared immutable must not change after binding


INVARIANT K6_STATE_CONTINUITY:

  every S_{t+1} must be derived from exactly one permitted predecessor state
  unless an explicit merge transform is declared


INVARIANT K7_TRANSFORM_AUTHORITY:

  state may change only through an activated declared transform


INVARIANT K8_PRECONDITION_GATE:

  a transform may execute only when every hard precondition is true


INVARIANT K9_POSTCONDITION_GATE:

  a generated successor state is valid only when every hard postcondition is
  true


INVARIANT K10_INVARIANT_PRESERVATION:

  ∀ q ∈ Q:

    q(S_t) = true
    and
    execute(T_i,S_t) = S_{t+1}

    →

    q(S_{t+1}) = true

  unless T_i explicitly and validly changes the scope of q


INVARIANT K11_NO_UNDECLARED_CONTINUATION:

  Γ may select only:

    - a declared outgoing edge
    - a declared branch
    - a declared recursion target
    - a declared terminal state


INVARIANT K12_TERMINAL_VALIDITY:

  terminal = true

  →

  all terminal conditions are satisfied


INVARIANT K13_OUTPUT_CONFORMANCE:

  emitted output must satisfy IO.WRITE


INVARIANT K14_TRACEABILITY:

  every state mutation must identify:

    - prior state
    - active transform
    - consumed symbols
    - produced symbols
    - checked constraints
    - resulting state


INVARIANT K15_RESOURCE_BOUNDEDNESS:

  step <= maximum_transition_steps

  and

  cardinality(B_t) <= maximum_branch_count

  and

  recursion_depth <= maximum_recursion_depth


INVARIANT K16_FAILURE_VISIBILITY:

  execution failure must be emitted explicitly

  and must not be represented as successful completion


INVARIANT K17_NO_COMPILATION_DEPENDENCY:

  execution must not require:

    - machine-code generation
    - executable linking
    - platform-target compilation
    - binary installation

  unless the flow itself explicitly requests generation of such an artefact as
  an output rather than as its own execution mechanism


# =============================================================================
# 9. PRIMITIVE OPERATION SET
# =============================================================================

PRIMITIVES:

  data:

    - read
    - bind
    - assign
    - copy
    - append
    - remove
    - union
    - intersection
    - difference
    - contains
    - select
    - filter
    - map
    - reduce
    - sort
    - count


  logical:

    - AND
    - OR
    - NOT
    - XOR
    - IMPLIES
    - FOR_ALL
    - EXISTS


  comparison:

    - equal
    - not_equal
    - less_than
    - less_than_or_equal
    - greater_than
    - greater_than_or_equal


  numeric:

    - add
    - subtract
    - multiply
    - divide
    - minimum
    - maximum
    - absolute
    - sum


  structural:

    - parse
    - normalise
    - resolve
    - validate
    - instantiate
    - transition
    - branch
    - merge
    - recurse
    - halt
    - emit


  diagnostic:

    - log
    - trace
    - reject
    - explain


PRIMITIVE RULE:

  any operation outside this set must be declared as:

    - a flow-defined function

    or

    - an external capability in X


# =============================================================================
# 10. SPECIFICATION INGESTION
# =============================================================================

TRANSFORM T_INGEST_SPECIFICATION:

  raw_flow ←
    read(flow_specification)


  REQUIRE:

    raw_flow != empty


  canonical_flow ←

    normalise(
      raw_flow,
      preserve:
        - identifiers
        - operators
        - indentation_scope
        - declared order
        - literal values
    )


  bind immutable:

    F ← canonical_flow

    I_0 ← execution_input

    P ← execution_parameters

    X ← external_capabilities


  proceed:

    T_PARSE_STRUCTURE


# =============================================================================
# 11. STRUCTURAL PARSING
# =============================================================================

TRANSFORM T_PARSE_STRUCTURE:

  section_map ← {

    identity:
      parse_section(F,"FLOW_IDENTITY"),

    io:
      parse_section(F,"IO"),

    symbols:
      parse_section(F,"SYMBOLS"),

    state:
      parse_section(F,"STATE"),

    invariants:
      parse_section(F,"INVARIANTS"),

    transforms:
      parse_section(F,"TRANSFORMS"),

    termination:
      parse_section(F,"TERMINATION"),

    emission:
      parse_section(F,"EMISSION")
  }


  FOR EACH required section r:

    IF section_map[r] = absent:

      validation_error.add({

        type:
          MISSING_REQUIRED_SECTION,

        section:
          r

      })


  IF validation_error != ∅:

    execution_status ←
      VALIDATION_FAILED

    HALT


  proceed:

    T_BUILD_SYMBOL_TABLE


# =============================================================================
# 12. SYMBOL TABLE CONSTRUCTION
# =============================================================================

TRANSFORM T_BUILD_SYMBOL_TABLE:

  declarations ←

    extract_declarations(
      section_map.symbols,
      section_map.io,
      section_map.state,
      section_map.transforms
    )


  symbol_table ←
    empty_map


  FOR EACH declaration d:

    REQUIRE d has:

      - identifier
      - domain
      - mutability
      - scope


    IF symbol_table contains d.identifier:

      IF declarations_are_compatible(
        symbol_table[d.identifier],
        d
      ) = false:

        validation_error.add({

          type:
            CONFLICTING_DECLARATION,

          symbol:
            d.identifier

        })

    ELSE:

      symbol_table[d.identifier] ←
        d


  usages ←
    extract_symbol_usages(F)


  FOR EACH usage u:

    IF resolve(u,symbol_table) = none:

      validation_error.add({

        type:
          UNRESOLVED_SYMBOL,

        symbol:
          u.identifier,

        location:
          u.location

      })


  IF validation_error contains hard error:

    execution_status ←
      UNRESOLVED_SYMBOL

    HALT


  proceed:

    T_VALIDATE_IO


# =============================================================================
# 13. IO VALIDATION
# =============================================================================

TRANSFORM T_VALIDATE_IO:

  input_contract ←
    section_map.io.input_contract


  output_contract ←
    section_map.io.output_contract


  FOR EACH required input field f:

    IF f not present in I_0:

      IF declared_default(f) exists:

        I_0[f] ←
          declared_default(f)

      ELSE:

        validation_error.add({

          type:
            MISSING_INPUT,

          field:
            f

        })


  FOR EACH supplied input field f:

    REQUIRE:

      type_of(I_0[f])
      conforms_to
      declared_type(f)


  FOR EACH supplied parameter p:

    REQUIRE:

      p is declared

      and

      type_of(P[p])
      conforms_to
      declared_type(p)


  IF validation_error != ∅:

    execution_status ←
      VALIDATION_FAILED

    HALT


  proceed:

    T_RESOLVE_DEPENDENCIES


# =============================================================================
# 14. DEPENDENCY RESOLUTION
# =============================================================================

TRANSFORM T_RESOLVE_DEPENDENCIES:

  dependencies ←
    extract_external_dependencies(F)


  FOR EACH dependency d:

    IF d is primitive:

      mark d:
        AVAILABLE


    ELSE IF d is flow_defined:

      mark d:
        AVAILABLE


    ELSE IF d ∈ X:

      mark d:
        AVAILABLE


    ELSE:

      mark d:
        UNAVAILABLE

      unresolved_items.add({

        type:
          CAPABILITY,

        identifier:
          d

      })


  IF any required dependency is UNAVAILABLE:

    execution_status ←
      CAPABILITY_UNAVAILABLE

    HALT


  proceed:

    T_PARSE_TRANSFORMS


# =============================================================================
# 15. TRANSFORM PARSING
# =============================================================================

TRANSFORM T_PARSE_TRANSFORMS:

  T ←
    extract_transforms(section_map.transforms)


  FOR EACH transform T_i ∈ T:

    REQUIRE T_i has:

      - transform_id
      - activation_rule
      - preconditions
      - operations
      - postconditions
      - continuation_rule


    operation_sequence ←
      parse_operations(T_i)


    FOR EACH operation op:

      REQUIRE:

        op is primitive

        or

        op is flow_defined

        or

        op is available in X


    transform_model[T_i.id] ← {

      id:
        T_i.id,

      reads:
        infer_read_set(T_i),

      writes:
        infer_write_set(T_i),

      preconditions:
        parse_conditions(T_i.preconditions),

      operations:
        operation_sequence,

      postconditions:
        parse_conditions(T_i.postconditions),

      successors:
        parse_continuations(T_i)
    }


  proceed:

    T_PARSE_INVARIANTS


# =============================================================================
# 16. INVARIANT PARSING
# =============================================================================

TRANSFORM T_PARSE_INVARIANTS:

  Q ←
    extract_invariants(section_map.invariants)


  FOR EACH invariant q ∈ Q:

    REQUIRE:

      q has:
        - invariant_id
        - predicate
        - scope
        - severity


    predicate_tree ←
      parse_logical_expression(q.predicate)


    REQUIRE:

      all symbols in predicate_tree resolve


    invariant_model[q.id] ← {

      predicate:
        predicate_tree,

      scope:
        q.scope,

      severity:
        q.severity
    }


  proceed:

    T_BUILD_EXECUTION_GRAPH


# =============================================================================
# 17. EXECUTION GRAPH CONSTRUCTION
# =============================================================================

TRANSFORM T_BUILD_EXECUTION_GRAPH:

  N ← ∅

  E ← ∅


  entry_node ←
    NewNode(INPUT_ENTRY)


  N ←
    N ∪ {entry_node}


  FOR EACH transform T_i ∈ T:

    transform_node ←
      NewNode(TRANSFORM,T_i.id)

    N ←
      N ∪ {transform_node}


  FOR EACH continuation from T_i to T_j:

    edge ← {

      source_node:
        node(T_i),

      target_node:
        node(T_j),

      activation_condition:
        continuation.condition,

      governing_constraints:
        constraints_for(T_i,T_j),

      state_update_rule:
        continuation.state_rule
    }

    E ←
      E ∪ {edge}


  terminal_nodes ←
    derive_terminal_nodes(
      section_map.termination,
      section_map.emission
    )


  N ←
    N ∪ terminal_nodes


  connect entry_node to declared initial transform


  graph_validation ←
    ValidateGraph(G_F)


  REQUIRE:

    - exactly one execution entry exists
    - every non-terminal node has a continuation
    - every terminal node has an emission rule
    - every referenced transform exists
    - no required node is unreachable
    - recursion targets are declared
    - branch joins are valid when present


  IF graph_validation fails:

    execution_status ←
      VALIDATION_FAILED

    HALT


  proceed:

    T_VALIDATE_STATIC_SEMANTICS


# =============================================================================
# 18. STATIC SEMANTIC VALIDATION
# =============================================================================

TRANSFORM T_VALIDATE_STATIC_SEMANTICS:

  FOR EACH transform T_i:

    read_set ←
      transform_model[T_i].reads

    write_set ←
      transform_model[T_i].writes


    FOR EACH symbol s ∈ read_set:

      REQUIRE:

        s is bound before T_i

        or

        s is conditionally bound on every path reaching T_i


    FOR EACH symbol s ∈ write_set:

      REQUIRE:

        s is mutable

        or

        s is unbound and permits initial binding


    REQUIRE:

      preconditions are type-valid

      operations are type-valid

      postconditions are type-valid

      continuations reference type-valid conditions


  validate:

    - immutable bindings are never overwritten
    - each branch produces a coherent state
    - each declared output can be produced
    - no transform writes undeclared state
    - no transform reads inaccessible scope
    - recursion has a declared termination condition
    - loops have a declared exit condition
    - external capability results have declared types


  IF static validation fails:

    execution_status ←
      VALIDATION_FAILED

    HALT


  proceed:

    T_INSTANTIATE_INITIAL_STATE


# =============================================================================
# 19. INITIAL STATE INSTANTIATION
# =============================================================================

TRANSFORM T_INSTANTIATE_INITIAL_STATE:

  S_0 := {

    step:
      0,

    active_node:
      entry_node,

    active_transform:
      null,

    input_values:
      I_0,

    parameter_values:
      P,

    symbol_table:
      instantiate(symbol_table,I_0,P),

    immutable_bindings:
      declared_immutable_symbols,

    mutable_bindings:
      declared_mutable_symbols,

    state_values:
      initialise_declared_state(section_map.state,I_0,P),

    prior_state:
      null,

    branch_id:
      ROOT_BRANCH,

    parent_branch_id:
      null,

    recursion_depth:
      0,

    active_constraints:
      K,

    active_invariants:
      Q,

    pending_transforms:
      {declared_initial_transform},

    completed_transforms:
      [],

    rejected_transforms:
      [],

    emitted_values:
      {},

    terminal:
      false,

    valid:
      true
  }


  REQUIRE:

    input invariants hold for S_0

    state initialisation constraints hold

    all required initial symbols are bound


  B_0 ←
    {S_0}


  proceed:

    T_EXECUTION_LOOP


# =============================================================================
# 20. STRUCTURAL CONTINUATION OPERATOR
# =============================================================================

FUNCTION Γ(S_t,G_F):

  candidate_edges ←

    outgoing_edges(
      G_F,
      S_t.active_node
    )


  valid_edges ← ∅


  FOR EACH edge e ∈ candidate_edges:

    condition_result ←

      evaluate(
        e.activation_condition,
        S_t.symbol_table,
        S_t.state_values
      )


    IF condition_result = true:

      valid_edges ←
        valid_edges ∪ {e}


    ELSE IF condition_result = unknown:

      HANDLE according_to ambiguity_policy:

        REJECT:

          reject e


        DECLARED_DEFAULT_ONLY:

          IF declared default resolves condition:

            evaluate using declared default

          ELSE:

            reject e


        MINIMUM_ASSUMPTION:

          assumption ←
            derive_minimum_non-semantic_assumption(e)

          IF assumption changes user intent
          OR weakens hard constraint:

            reject e

          ELSE:

            evaluate using assumption


  RETURN:
    valid_edges


# =============================================================================
# 21. TRANSFORM ACTIVATION
# =============================================================================

FUNCTION ActivateTransform(S_t,T_i):

  REQUIRE:

    T_i ∈ S_t.pending_transforms


  precondition_results ←

    evaluate_all(
      transform_model[T_i].preconditions,
      S_t
    )


  IF any hard precondition = false:

    RETURN:

      REJECTED_STATE {

        reason:
          PRECONDITION_FAILURE,

        transform:
          T_i
      }


  IF any hard precondition = unknown:

    RETURN:

      REJECTED_STATE {

        reason:
          UNRESOLVED_PRECONDITION,

        transform:
          T_i
      }


  S_work ←
    Copy(S_t)


  S_work.active_transform ←
    T_i


  RETURN:
    S_work


# =============================================================================
# 22. DIRECT SEMANTIC EXECUTION
# =============================================================================

FUNCTION ExecuteTransform(S_work,T_i):

  operations ←
    transform_model[T_i].operations


  FOR EACH operation op in declared order:

    REQUIRE:

      operation_precondition(op,S_work)
      =
      true


    result ←

      execute_semantically(
        operation = op,
        symbols = S_work.symbol_table,
        state = S_work.state_values,
        capabilities = X
      )


    IF result.status = failure:

      RETURN:

        REJECTED_STATE {

          reason:
            OPERATION_FAILURE,

          operation:
            op,

          details:
            result
        }


    apply declared state mutation only


    trace operation:

      - consumed values
      - produced values
      - changed symbols
      - checked constraints


  RETURN:
    S_work


# =============================================================================
# 23. POSTCONDITION VALIDATION
# =============================================================================

FUNCTION ValidateTransformResult(S_before,S_after,T_i):

  postcondition_results ←

    evaluate_all(
      transform_model[T_i].postconditions,
      S_after
    )


  IF any hard postcondition != true:

    RETURN:
      false


  FOR EACH invariant q active in S_after:

    IF evaluate(q,S_after) != true:

      RETURN:
        false


  REQUIRE:

    no immutable binding changed

    no undeclared state field appeared

    every changed value has a governing operation

    state continuity is preserved


  RETURN:
    true


# =============================================================================
# 24. SUCCESSOR GENERATION
# =============================================================================

FUNCTION GenerateSuccessors(S_after,T_i):

  continuation_edges ←
    Γ(S_after,G_F)


  successors ← ∅


  FOR EACH edge e ∈ continuation_edges:

    S_next ←
      Copy(S_after)


    S_next.step ←
      S_after.step + 1


    S_next.prior_state ←
      StateID(S_after)


    S_next.active_node ←
      e.target_node


    S_next.completed_transforms ←
      S_after.completed_transforms || [T_i]


    S_next.pending_transforms ←
      transforms_at(e.target_node)


    IF cardinality(continuation_edges) > 1:

      S_next.parent_branch_id ←
        S_after.branch_id

      S_next.branch_id ←
        NewBranchID()


    IF edge enters recursion:

      S_next.recursion_depth ←
        S_after.recursion_depth + 1


    IF edge exits recursion:

      S_next.recursion_depth ←
        max(
          0,
          S_after.recursion_depth - 1
        )


    apply edge.state_update_rule


    successors ←
      successors ∪ {S_next}


  RETURN:
    successors


# =============================================================================
# 25. BRANCH VALIDATION
# =============================================================================

FUNCTION ValidateSuccessor(S_next):

  IF S_next.step > maximum_transition_steps:

    RETURN:
      STEP_LIMIT_REACHED


  IF S_next.recursion_depth > maximum_recursion_depth:

    RETURN:
      RECURSION_LIMIT_REACHED


  IF any active invariant fails:

    RETURN:
      INVARIANT_VIOLATION


  IF any active hard constraint fails:

    RETURN:
      CONSTRAINT_VIOLATION


  IF symbol closure fails:

    RETURN:
      UNRESOLVED_SYMBOL


  RETURN:
    VALID


# =============================================================================
# 26. EXECUTION LOOP
# =============================================================================

TRANSFORM T_EXECUTION_LOOP:

  B_t ←
    B_0


  execution_trace ←
    []


  WHILE B_t != ∅:

    IF total_generated_branches > maximum_branch_count:

      execution_status ←
        BRANCH_LIMIT_REACHED

      HALT


    S_t ←

      PopNextState(
        B_t,
        order:
          - lowest step
          - highest hard-constraint satisfaction
          - deterministic branch identifier
      )


    IF S_t.terminal = true:

      terminal_candidates.add(S_t)

      CONTINUE


    IF S_t.step >= maximum_transition_steps:

      rejected_states.add({

        state:
          S_t,

        reason:
          STEP_LIMIT_REACHED
      })

      CONTINUE


    FOR EACH T_i ∈ S_t.pending_transforms:

      S_work ←
        ActivateTransform(S_t,T_i)


      IF S_work is REJECTED_STATE:

        record rejection

        CONTINUE


      S_after ←
        ExecuteTransform(S_work,T_i)


      IF S_after is REJECTED_STATE:

        record rejection

        CONTINUE


      valid_result ←
        ValidateTransformResult(
          S_t,
          S_after,
          T_i
        )


      IF valid_result = false:

        rejected_states.add({

          state:
            S_after,

          transform:
            T_i,

          reason:
            POSTCONDITION_OR_INVARIANT_FAILURE
        })

        CONTINUE


      successors ←
        GenerateSuccessors(S_after,T_i)


      IF successors = ∅:

        IF terminal_condition(S_after) = true:

          S_after.terminal ←
            true

          terminal_candidates.add(S_after)

        ELSE:

          rejected_states.add({

            state:
              S_after,

            reason:
              NO_VALID_CONTINUATION
          })


      FOR EACH S_next ∈ successors:

        successor_status ←
          ValidateSuccessor(S_next)


        IF successor_status = VALID:

          B_t ←
            B_t ∪ {S_next}


        ELSE:

          rejected_states.add({

            state:
              S_next,

            reason:
              successor_status
          })


      execution_trace.add(
        BuildTraceEntry(
          before = S_t,
          transform = T_i,
          after = S_after,
          successors = successors
        )
      )


  proceed:

    T_RESOLVE_TERMINAL_STATE


# =============================================================================
# 27. TERMINAL STATE DETECTION
# =============================================================================

FUNCTION terminal_condition(S):

  success_conditions ←
    section_map.termination.success_conditions


  return:

    evaluate_all(
      success_conditions,
      S
    )
    =
    true


FUNCTION failure_condition(S):

  failure_conditions ←
    section_map.termination.failure_conditions


  return:

    any(
      evaluate_all(
        failure_conditions,
        S
      )
    )
    =
    true


# =============================================================================
# 28. TERMINAL STATE RESOLUTION
# =============================================================================

TRANSFORM T_RESOLVE_TERMINAL_STATE:

  valid_terminal_states ← ∅


  FOR EACH S_terminal ∈ terminal_candidates:

    REQUIRE:

      terminal_condition(S_terminal)
      =
      true


    REQUIRE:

      failure_condition(S_terminal)
      =
      false


    output_candidate ←
      construct_output(
        S_terminal,
        section_map.emission
      )


    output_validation ←
      ValidateOutput(output_candidate)


    IF output_validation = true:

      valid_terminal_states ←
        valid_terminal_states ∪ {S_terminal}


  IF valid_terminal_states = ∅:

    execution_status ←
      NO_VALID_TERMINAL_STATE

    proceed:
      T_EMIT_FAILURE


  IF cardinality(valid_terminal_states) = 1:

    accepted_terminal_state ←
      only(valid_terminal_states)


  ELSE:

    accepted_terminal_state ←

      ResolveMultipleTerminalStates(
        valid_terminal_states,
        execution_mode,
        declared_selection_rule
      )


  proceed:

    T_CONSTRUCT_OUTPUT


# =============================================================================
# 29. MULTIPLE TERMINAL STATE RESOLUTION
# =============================================================================

FUNCTION ResolveMultipleTerminalStates(
  candidate_states,
  mode,
  selection_rule
):

  IF mode = DETERMINISTIC:

    REQUIRE:

      declared_selection_rule exists


    RETURN:

      apply(
        declared_selection_rule,
        candidate_states
      )


  IF mode = BOUNDED_BRANCHING:

    RETURN:

      argmax_{S ∈ candidate_states}
        declared_state_quality(S)


  IF mode = SYMBOLIC_SIMULATION:

    RETURN:

      set_of_all_valid_terminal_states


  IF mode = VALIDATION_ONLY:

    RETURN:

      no_output_execution_state


# =============================================================================
# 30. OUTPUT CONSTRUCTION
# =============================================================================

TRANSFORM T_CONSTRUCT_OUTPUT:

  O_candidate ←

    map_state_to_output(
      accepted_terminal_state,
      section_map.emission
    )


  REQUIRE:

    O_candidate contains every required output field


  REQUIRE:

    each output field conforms to declared type


  REQUIRE:

    each output invariant is true


  REQUIRE:

    no undeclared internal state is leaked


  O* ←
    O_candidate


  execution_status ←
    COMPLETE


  proceed:

    T_BUILD_VALIDATION_REPORT


# =============================================================================
# 31. OUTPUT VALIDATION
# =============================================================================

FUNCTION ValidateOutput(O):

  FOR EACH required field f in IO.WRITE:

    IF f not present in O:

      RETURN:
        false


    IF type_of(O[f]) does_not_conform_to declared_type(f):

      RETURN:
        false


  FOR EACH output constraint k:

    IF evaluate(k,O) != true:

      RETURN:
        false


  RETURN:
    true


# =============================================================================
# 32. EXECUTION TRACE
# =============================================================================

FUNCTION BuildTraceEntry(
  before,
  transform,
  after,
  successors
):

  RETURN {

    step:
      before.step,

    branch_id:
      before.branch_id,

    active_node:
      before.active_node,

    transform:
      transform,

    preconditions:
      evaluated_preconditions(transform,before),

    input_symbols:
      consumed_symbols(transform,before),

    prior_state_hash:
      StateHash(before),

    mutations:
      StateDifference(before,after),

    postconditions:
      evaluated_postconditions(transform,after),

    invariant_results:
      evaluated_invariants(after),

    successor_ids:
      IDs(successors),

    resulting_state_hash:
      StateHash(after)
  }


# =============================================================================
# 33. VALIDATION REPORT
# =============================================================================

TRANSFORM T_BUILD_VALIDATION_REPORT:

  validation_report := {

    flow_name:
      F.identity.name,

    flow_version:
      F.identity.version,

    structural_parse_valid:
      true,

    symbol_closure_valid:
      true,

    input_contract_valid:
      true,

    dependency_closure_valid:
      true,

    transform_structure_valid:
      true,

    invariant_structure_valid:
      true,

    execution_graph_valid:
      true,

    static_semantics_valid:
      true,

    terminal_state_valid:
      true,

    output_contract_valid:
      true,

    transition_steps:
      accepted_terminal_state.step,

    branches_generated:
      total_generated_branches,

    branches_rejected:
      cardinality(rejected_states),

    recursion_depth_reached:
      maximum_observed_recursion_depth,

    external_capabilities_used:
      capabilities_used(execution_trace),

    conventional_compiler_used:
      false,

    bytecode_generated:
      false,

    platform_binary_generated:
      false,

    hidden_dependencies_detected:
      0
  }


  proceed:

    T_BUILD_EXECUTION_GUARANTEE


# =============================================================================
# 34. FAILURE EMISSION
# =============================================================================

TRANSFORM T_EMIT_FAILURE:

  final_failure_status ←

    HighestPriorityFailure(
      validation_error,
      rejected_states,
      unresolved_items
    )


  execution_status ←
    final_failure_status


  terminal_output ←
    null


  execution_guarantee := {

    successful_execution:
      false,

    invalid_output_released:
      false,

    reason:
      final_failure_status,

    unresolved_items:
      unresolved_items,

    rejected_state_count:
      cardinality(rejected_states)
  }


  EMIT:

    execution_status

    terminal_output

    final_state:
      null

    execution_trace

    validation_report

    unresolved_items

    execution_guarantee


  HALT


# =============================================================================
# 35. EXECUTION GUARANTEE
# =============================================================================

TRANSFORM T_BUILD_EXECUTION_GUARANTEE:

  execution_guarantee := {

    successful_execution:
      true,

    execution_method:
      "direct symbolic state propagation",

    source_representation:
      "Flow Algebra",

    conventional_compiler_required:
      false,

    bytecode_required:
      false,

    linker_required:
      false,

    platform_binary_required:
      false,

    all_symbols_resolved:
      true,

    all_required_dependencies_available:
      true,

    all_executed_transforms_declared:
      true,

    all_preconditions_satisfied:
      true,

    all_postconditions_satisfied:
      true,

    all_active_invariants_preserved:
      true,

    terminal_state_reachable:
      true,

    output_contract_satisfied:
      true,

    hidden_semantics_required:
      false,

    output_traceable_to_input:
      true,

    claim:

      "The terminal output was derived directly from the supplied symbolic
       specification through declared state transitions and constraint
       propagation without conventional source-code compilation."
  }


  proceed:

    T_EMIT_SUCCESS


# =============================================================================
# 36. SUCCESS EMISSION
# =============================================================================

TRANSFORM T_EMIT_SUCCESS:

  EMIT:

    execution_status:
      COMPLETE

    terminal_output:
      O*

    final_state:
      accepted_terminal_state.state_values

    execution_trace:
      execution_trace

    validation_report:
      validation_report

    unresolved_items:
      []

    execution_guarantee:
      execution_guarantee


  HALT


# =============================================================================
# 37. COMPILER PIPELINE COMPARISON
# =============================================================================

CONVENTIONAL_PIPELINE:

  source_text
    →
  lexical_analysis
    →
  syntax_tree
    →
  semantic_analysis
    →
  intermediate_representation
    →
  optimisation
    →
  machine_or_bytecode_generation
    →
  linking
    →
  loading
    →
  processor_execution


COMPILERLESS_PIPELINE:

  symbolic_specification
    →
  structural_parse
    →
  semantic_execution_graph
    →
  symbol_and_constraint_validation
    →
  direct_state_transition
    →
  terminal_state_validation
    →
  output_emission


STRUCTURAL_DIFFERENCE:

  conventional:

    meaning is translated into another executable representation


  compilerless:

    supplied meaning remains the active execution representation


# =============================================================================
# 38. DIRECT EXECUTION EQUATION
# =============================================================================

FORMAL_EXECUTION:

  F :=
    complete symbolic specification


  S_0 :=
    Initialise(F,I_0,P)


  S_{t+1} :=

    Γ(
      S_t,
      G_F,
      K,
      Q
    )


  where:

    Γ selects only transitions satisfying:

      Preconditions(T_i,S_t)
      =
      true

    and:

      Constraints(S_{t+1})
      =
      true

    and:

      Invariants(S_{t+1})
      =
      true


  termination:

    ∃ t:

      Terminal(S_t)
      =
      true


  output:

    O*
    =
    Emit(S_t)


# =============================================================================
# 39. VALID CONTINUATION PREDICATE
# =============================================================================

PREDICATE VALID_CONTINUATION(S_t,T_i,S_{t+1}):

  VALID_CONTINUATION :=

    DECLARED(T_i)

    AND

    PRECONDITIONS(T_i,S_t)

    AND

    APPLY(T_i,S_t)
    =
    S_{t+1}

    AND

    POSTCONDITIONS(T_i,S_{t+1})

    AND

    INVARIANTS(S_{t+1})

    AND

    SYMBOL_CLOSURE(S_{t+1})

    AND

    TYPE_COHERENCE(S_{t+1})

    AND

    RESOURCE_BOUNDS(S_{t+1})


# =============================================================================
# 40. EXECUTABILITY PREDICATE
# =============================================================================

PREDICATE EXECUTABLE(F,I_0):

  EXECUTABLE :=

    STRUCTURALLY_COMPLETE(F)

    AND

    SYMBOLICALLY_CLOSED(F)

    AND

    INPUT_VALID(I_0,F.IO)

    AND

    DEPENDENCIES_AVAILABLE(F)

    AND

    GRAPH_VALID(G_F)

    AND

    ∃ O*:

      REACHABLE(O*,F,I_0)

      AND

      OUTPUT_VALID(O*,F.IO.WRITE)


# =============================================================================
# 41. COMPILERLESS PREDICATE
# =============================================================================

PREDICATE COMPILERLESS(F):

  COMPILERLESS :=

    EXECUTION_REPRESENTATION(F)
    =
    SUPPLIED_SYMBOLIC_REPRESENTATION(F)

    AND

    no mandatory transformation exists from F into:

      - machine code
      - native binary
      - object code
      - mandatory bytecode

    before behavioural execution may begin


# =============================================================================
# 42. SEMANTIC CLOSURE
# =============================================================================

SEMANTIC_CLOSURE:

  Let RequiredSemantics(F) be every meaning required to execute F.


  Let DeclaredSemantics(F) include:

    - declared symbols
    - declared types
    - declared transforms
    - declared conditions
    - declared invariants
    - declared defaults
    - declared primitive meanings
    - declared capability contracts


  Then:

    F is semantically closed

    iff:

    RequiredSemantics(F)
    ⊆
    DeclaredSemantics(F)


  compilerless execution requires:

    SemanticClosure(F)
    =
    true


# =============================================================================
# 43. PORTABILITY MODEL
# =============================================================================

PORTABILITY:

  A Flow Algebra specification is portable between reasoning systems R_i and
  R_j when both support:

    - structural parsing
    - symbol binding
    - logical evaluation
    - state propagation
    - invariant checking
    - declared branch resolution
    - terminal emission


  portability condition:

    Execute(R_i,F,I_0)
    ≈
    Execute(R_j,F,I_0)


  where equivalence means:

    - same mandatory constraints
    - same valid terminal class
    - same output schema
    - no contradiction in required results


  exact token-level output identity is not required unless declared.


# =============================================================================
# 44. DETERMINISM MODEL
# =============================================================================

DETERMINISTIC_EXECUTION:

  deterministic when:

    - all transform activation conditions are decidable
    - all tie-break rules are declared
    - all defaults are declared
    - all external capability outputs are fixed
    - exactly one valid continuation exists at each step


BOUNDED_NONDETERMINISM:

  permitted when:

    - several continuations satisfy all constraints
    - branch generation is explicitly allowed
    - branch count remains bounded
    - a terminal selection rule is declared


INVALID_NONDETERMINISM:

  exists when:

    - output depends on undeclared preference
    - branch selection lacks a declared rule
    - ambiguity silently changes behaviour
    - hidden model preference determines a hard requirement


# =============================================================================
# 45. EXTERNAL CAPABILITY BOUNDARY
# =============================================================================

CAPABILITY_MODEL:

  external capability use does not invalidate compilerless execution when:

    - the capability is explicitly declared
    - its input contract is explicit
    - its output contract is explicit
    - its failure behaviour is explicit
    - the capability performs a bounded operation
    - the flow retains control over continuation


  examples:

    - database read
    - web request
    - sensor observation
    - file read
    - mathematical calculation
    - message delivery


  capability call pattern:

    request
      →
    contract validation
      →
    capability invocation
      →
    result validation
      →
    state propagation


# =============================================================================
# 46. FAILURE MODEL
# =============================================================================

FAILURE_CLASSES:

  STRUCTURAL_FAILURE:

    - missing section
    - malformed transform
    - invalid graph
    - unreachable required terminal


  SYMBOL_FAILURE:

    - undeclared symbol
    - conflicting declaration
    - invalid scope
    - immutable overwrite


  SEMANTIC_FAILURE:

    - hidden assumption required
    - operation meaning unavailable
    - contradictory rule
    - ambiguous hard requirement


  STATE_FAILURE:

    - precondition failure
    - postcondition failure
    - invalid mutation
    - invariant violation


  RESOURCE_FAILURE:

    - step limit reached
    - branch limit reached
    - recursion limit reached


  CAPABILITY_FAILURE:

    - capability unavailable
    - invalid capability response
    - capability contract violation


  OUTPUT_FAILURE:

    - no valid terminal state
    - missing output field
    - output type mismatch
    - output invariant violation


# =============================================================================
# 47. RECURSIVE VALIDATION ENGINE
# =============================================================================

TRANSFORM T_RECURSIVE_VALIDATION:

  validation_cycle ←
    0


  maximum_validation_cycles ←
    5


  WHILE validation_cycle < maximum_validation_cycles:

    validation_cycle ←
      validation_cycle + 1


    errors ←
      RunAllStaticValidation(F)


    IF errors = ∅:

      validation_status ←
        VALID

      BREAK


    repairable_errors ←

      {
        e ∈ errors |

        repair requires no semantic invention
        and
        repair does not alter declared intent
      }


    non_repairable_errors ←

      errors \ repairable_errors


    IF non_repairable_errors != ∅:

      validation_status ←
        INVALID

      unresolved_items ←
        unresolved_items ∪ non_repairable_errors

      BREAK


    FOR EACH error e ∈ repairable_errors:

      F ←
        ApplyMinimumStructuralRepair(F,e)


  RETURN:

    validation_status


# =============================================================================
# 48. MINIMUM STRUCTURAL REPAIR
# =============================================================================

FUNCTION ApplyMinimumStructuralRepair(F,error):

  permitted repairs:

    - normalise whitespace
    - restore an unambiguous closing delimiter
    - canonicalise a declared identifier consistently
    - remove an exact duplicate declaration
    - reorder independent declarations when dependency order is explicit
    - apply an explicitly declared default
    - resolve an unambiguous local reference


  prohibited repairs:

    - invent missing business logic
    - create a new undeclared requirement
    - weaken a hard constraint
    - invent a default
    - invent an external capability
    - select among semantically different interpretations
    - remove a required invariant


  RETURN:

    minimally_repaired_F


# =============================================================================
# 49. ARCHITECTURAL CONSEQUENCES
# =============================================================================

ARCHITECTURAL_MODEL:

  conventional application architecture often separates:

    - source language
    - compiler
    - runtime
    - executable artefact
    - deployed environment


  compilerless architecture combines:

    - specification
    - validation
    - execution semantics
    - state-transition rules
    - output constraints

  inside one portable symbolic artefact.


  resulting architectural properties:

    SELF_DESCRIPTION:

      the flow declares what it requires


    SELF_VALIDATION:

      the flow declares how valid execution is recognised


    SELF_EXECUTION:

      the flow declares how state progresses


    SELF_TERMINATION:

      the flow declares when execution stops


    SELF_EMISSION:

      the flow declares what may be emitted


# =============================================================================
# 50. SECURITY BOUNDARY
# =============================================================================

SECURITY:

  compilerless execution must not treat supplied structure as trusted merely
  because it is declarative.


  REQUIRED VALIDATION:

    - capability permissions
    - data-access scope
    - mutation authority
    - external action authority
    - output disclosure boundaries
    - recursion limits
    - branch limits
    - execution-step limits


  PROHIBITED:

    - undeclared tool invocation
    - privilege escalation
    - hidden data retrieval
    - unrestricted self-modification
    - bypass of output policy
    - execution of opaque embedded binary payloads
    - silent network access


# =============================================================================
# 51. HUMAN-READABLE EXECUTION
# =============================================================================

HUMAN_READABILITY:

  A valid compilerless flow should permit a human reviewer to identify:

    - required inputs
    - expected outputs
    - state variables
    - transition rules
    - invariants
    - failure conditions
    - external dependencies
    - termination conditions


  Human readability is not itself sufficient for executability.

  Executability additionally requires:

    - structural completeness
    - symbol closure
    - semantic closure
    - transition closure
    - output closure


# =============================================================================
# 52. EXECUTION GUARANTEE STATEMENT
# =============================================================================

EXECUTION_GUARANTEE:

  This flow guarantees that a compilerless system is not interpreted as
  unrestricted natural-language improvisation.

  Execution is permitted only after:

    - structural validation
    - input validation
    - symbol resolution
    - dependency resolution
    - graph construction
    - static semantic validation
    - invariant activation

  During execution:

    - every mutation must be produced by a declared transform
    - every transform must satisfy its preconditions
    - every successor must satisfy its postconditions
    - every active invariant must remain true
    - every branch must follow a declared continuation
    - every external action must use a declared capability
    - every state change must be traceable

  Completion occurs only when:

    - a declared terminal condition is reached
    - a valid terminal state exists
    - the output contract is satisfied
    - no hard constraint remains unresolved

  The system therefore executes directly from symbolic meaning without first
  requiring compilation into bytecode or a platform-specific binary.

  The flow does not claim that physical computation disappears.

  It establishes that compilation is no longer the mandatory semantic bridge
  between the human-readable specification and executable behaviour.


# =============================================================================
# 53. APPROVAL
# =============================================================================

APPROVAL:

  self_contained_structure:
    approved

  direct_symbolic_execution:
    approved

  compilerless_classification:
    approved

  state_transition_model:
    approved

  invariant_preservation:
    approved

  constraint_validation:
    approved

  external_capability_boundary:
    approved

  deterministic_execution_support:
    approved

  bounded_branching_support:
    approved

  failure_visibility:
    approved

  execution_traceability:
    approved

  bytecode_requirement:
    rejected

  platform_binary_requirement:
    rejected

  hidden_semantics:
    rejected


# =============================================================================
# 54. EXECUTE
# =============================================================================

EXECUTE:

  T_INGEST_SPECIFICATION

  T_PARSE_STRUCTURE

  T_BUILD_SYMBOL_TABLE

  T_VALIDATE_IO

  T_RESOLVE_DEPENDENCIES

  T_PARSE_TRANSFORMS

  T_PARSE_INVARIANTS

  T_BUILD_EXECUTION_GRAPH

  T_VALIDATE_STATIC_SEMANTICS

  T_RECURSIVE_VALIDATION

  T_INSTANTIATE_INITIAL_STATE

  T_EXECUTION_LOOP

  T_RESOLVE_TERMINAL_STATE

  T_CONSTRUCT_OUTPUT

  T_BUILD_VALIDATION_REPORT

  T_BUILD_EXECUTION_GUARANTEE

  T_EMIT_SUCCESS


# =============================================================================
# END FLOW
# =============================================================================
```