nf.fsp.micnf

Meron Involutive Cancellation Normal Form (MIC-NF)

Contract ID: nf.fsp.micnf | Family: representation | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

Meron Involutive Cancellation Normal Form (MIC-NF) enforces structural normalization and explicit defect routing in the representation regime.

Property	Value
Canonical ID	nf.fsp.micnf
Short Name	MIC-NF
Full Name	Meron Involutive Cancellation Normal Form
Family	representation (Data in Representation, Canonical Wire Grammars, Orbit Representatives & Gauge Fixing)
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	in_transformation
Index Status	draft
Theory Status	proposed
Verification Status	absent
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

Foundational Treatise Specification

• Domain: Finite signed configuration path sums Z = sum_{x in S} w(x) sgn(x).
• Transformation Group: Involutive sign-reversing pairing Xi: S_0 -> S_0.
• Target Property: Exact cancellation of negative path sectors leaving non-negative survivor sector.
• Anomaly Excluded: Identity map fallacy, same-sign pairing, and unbudgeted global annihilation (Z == 0).
• Necessary And Sufficient Criterion: Exists cancellation subset S_0 and bijection Xi: S_0 -> S_0 such that Xi(Xi(x)) = x, Xi(x) != x, sgn(Xi(x)) = -sgn(x), w(Xi(x)) = w(x), and sgn(y) = +1 for all y in S \ S_0.
• Reconstruction Theorem: Z = 0 + sum_{y in S_surv} w(y) >= 0. Strict positivity requires non-empty survivor sector.
• Positive Witness: Explicit involution table Xi and non-empty survivor set S_surv.
• Negative Witness Obstruction: Fixed points (Xi(x) = x), same-sign pairs, or weight mismatch (w(Xi(x)) != w(x)).

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2. Visual Architecture & Normalization Diagram

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3. Concrete Worked Example & Mechanism

Visual Transition Summary

Stage	Carrier / Operator	Concrete State / Invariant
1. Before / Candidate Input	Input Carrier (MIC-NF Candidate)	Anomaly risk: an identity map ; presented as a sign reversal, a ; same-sign pairing presented as...
2. Normalization / Gate	MIC-NF Verification Gate	the cancellation subset... (certified_implies_authorized: false)
3. After / Certified Output	Certified MIC-NF Output	Invariant: ['cancellation_subset', ; 'involution', 'survivor_sector']

Contract-Declared Worked Cases

- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '4.2'

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4. The Four-Pillar Quadrilateral & Non-Claims

Every contract in simplex-normal-forms is evaluated against the Four-Pillar Quadrilateral defined in contracts/_golden_nfs/standard.yaml:

Pillar	Requirement	Contract Specification for nf.fsp.micnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	an identity map presented as a sign reversal, a same-sign pairing presented as cancellation, and a pairing of the whole configuration space that erases the partition function
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: the cancellation subset carries an involution with no fixed point that reverses sign and preserves weight, and the survivor sector is positive Protected Skeleton: ['cancellation_subset', 'involution', 'survivor_sector']
Pillar III: Lossless Reconstruction	State the non-loss reconstruction operator (join, direct sum, pullback, or replay).	Lossless reconstruction under declared projection, direct sum, or pullback operators.
Pillar IV: Syntactic Decidability	State how conformance is decided from declared constraints without unbudgeted scans.	Decidable from declared schema and witness metadata within bounded evaluation budgets.

Explicit Non-Claims (certified_implies_authorized: false)

• does not claim that every signed sum admits such an involution
• a survivor sector that is empty makes the partitioned sum zero. That total cancellation is the anomaly named in the source, so this form refuses it
• expectation values of observables need their own numerator check
• does not imply authorization

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• configuration_set
• cancellation_subset
• involution
• weights
• signs

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	S0 and the survivor sector partition S
criterion	Check	Xi composed with itself is the identity on S0
criterion	Check	Xi has no fixed point on S0
criterion	Check	Xi reverses sign and preserves weight
criterion	Check	every survivor has sign +1
criterion	Check	the survivor sector is non-empty

5.4 Terminal Route Map

Every evaluation terminates in a declared local route mapped to a canonical terminal in standard.yaml:

Local Terminal Route	Canonical Terminal
Accept	Preserve
RejectFixedPoint	Reject
RejectWeightMismatch	Reject
RejectSameSign	Reject
RejectEmptySurvivor	Refuse

5.5 Declared Failure Modes

• symptom: Xi(x) = x for some x in the cancellation subset, diagnosis: a fixed point cannot reverse its own sign, terminal: RejectFixedPoint
• symptom: Xi preserves sign, diagnosis: the pair adds the weights, terminal: RejectSameSign
• symptom: the weights of a pair differ, diagnosis: the pair leaves a nonzero remainder, terminal: RejectWeightMismatch
• symptom: the cancellation subset is the whole configuration set, diagnosis: the pairing annihilates the partitioned sum, terminal: RejectEmptySurvivor

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6. Epistemic Claim Hygiene Quadrant

In accordance with the repository's epistemic claim discipline:

1. What it supports:
   • Supports syntactic and operational checking of nf.fsp.micnf (MIC-NF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (Accept, RejectFixedPoint, RejectWeightMismatch, RejectSameSign) when required declarations or invariants fail.
2. What it does not support:
   • Does not confer operational authority (AuthorityEffect: None, certified_implies_authorized: false).
   • Does not claim semantic truth of undeclared external processes or unverified upstream inputs beyond the declaredprotected skeleton.
3. The narrowest defensible claim:
   • Within the declared substrate in_transformation, this contract enforces that an identity map presented as a sign reversal, a same-sign pairing presented as cancellation, and a pairing of the whole configuration space that erases the partition function is either excluded under the cancellation subset carries an involution with no fixed point that reverses sign and preserves weight, and the survivor sector is positive or routed to an explicit refusal/recovery terminal.
4. The next test needed:
   • Property-based falsification and mutation testing across boundary edge cases and composed multi-step pipelines for nf.fsp.micnf.

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7. Canonical YAML Contract (contracts/nf.fsp.micnf.yaml)

Click to expand full canonical YAML contract (contracts/nf.fsp.micnf.yaml)

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.fsp.micnf
  name: Meron Involutive Cancellation Normal Form
  short_name: MIC-NF
  version: v0
  theory_status: proposed
  contract_status: draft
  verification_status: absent
  family: representation
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - signed_configuration_sum
  - involution
  data_posture: in_transformation
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: an identity map presented as a sign reversal, a same-sign pairing
    presented as cancellation, and a pairing of the whole configuration space that
    erases the partition function
  preserves_invariant: the cancellation subset carries an involution with no fixed
    point that reverses sign and preserves weight, and the survivor sector is positive
  protected_skeleton:
  - cancellation_subset
  - involution
  - survivor_sector
  erased_structure:
  - cancelling_pairs
  non_claims:
  - does not claim that every signed sum admits such an involution
  - a survivor sector that is empty makes the partitioned sum zero. That total cancellation
    is the anomaly named in the source, so this form refuses it
  - expectation values of observables need their own numerator check
  - does not imply authorization
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  section: '4.2'
  formal_statement: 'The partitioned sum is Z = sum_x w(x) sgn(x) over a finite set
    S, with w(x) positive and sgn(x) equal to +1 or -1. A cancellation subset S0 and
    its complement, the survivor sector, partition S. The map Xi on S0 satisfies Xi(Xi(x))
    = x, Xi(x) different from x, sgn(Xi(x)) = -sgn(x), and w(Xi(x)) = w(x). Every
    survivor has sign +1. The sum over S0 is then zero, and Z equals the sum of the
    survivor weights. The survivor sector is required to be non-empty, so that sum
    is positive.

    '
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_the_survivor_sum_not_a_relational_join
    qualifies_as_extended_form: true
    anomaly_exclusion: identity pairings, same-sign pairings, and cancellation of
      every configuration
    lossless_join:
      status: survivor_sum
      statement: Z equals the sum of the positive survivor weights
    dependency_preservation:
      status: pair_axioms_retained
      statement: involution, absence of fixed points, sign reversal, and weight equality
        are all required
    syntactic_decision:
      status: finite_set_decidable
      statement: the five conditions are checked on the declared finite set, the declared
        subset, and the declared map
declarations_required:
- configuration_set
- cancellation_subset
- involution
- weights
- signs
evidence_required: []
validation_criteria:
- S0 and the survivor sector partition S
- Xi composed with itself is the identity on S0
- Xi has no fixed point on S0
- Xi reverses sign and preserves weight
- every survivor has sign +1
- the survivor sector is non-empty
terminal_routes:
- Accept
- RejectFixedPoint
- RejectWeightMismatch
- RejectSameSign
- RejectEmptySurvivor
terminal_route_map:
  Accept: Preserve
  RejectFixedPoint: Reject
  RejectWeightMismatch: Reject
  RejectSameSign: Reject
  RejectEmptySurvivor: Refuse
certification:
  certificate_required: false
  certificate_kind: null
  obstruction_class_required: false
  loss_ledger_required: false
  verifier_required: false
  loss_ledger_trigger_id: null
authorization:
  authorized_by: null
  certified_implies_authorized: false
failure_modes:
- symptom: Xi(x) = x for some x in the cancellation subset
  diagnosis: a fixed point cannot reverse its own sign
  terminal: RejectFixedPoint
- symptom: Xi preserves sign
  diagnosis: the pair adds the weights
  terminal: RejectSameSign
- symptom: the weights of a pair differ
  diagnosis: the pair leaves a nonzero remainder
  terminal: RejectWeightMismatch
- symptom: the cancellation subset is the whole configuration set
  diagnosis: the pairing annihilates the partitioned sum
  terminal: RejectEmptySurvivor
examples:
- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '4.2'
references:
- coda_all_universe/mssn_sn_tower/definitions.md
obstruction_substrate:
  policy_ref: ../substrate/default_finite_obstruction_policy.yaml
  finite_carrier:
    description: finite set of signed paths S with weights w(x) > 0 and signs in {-1,
      +1}
    max_rank: null
  cochain_complex:
    C0: path_weights_and_signs
    C1: paired_cancellation_edges
    C2: closure_obstructions
    d0: maps path pairs (x, Xi(x)) to weight differences and sign sum
    d1: checks completeness and fixed-point freedom
    quotient: Q_meron = S0 / <x ~ Xi(x)>
  obstruction_class:
    notation: '[meron_defect]'
    exact: 'involution_valid_and_weights_equal: [defect] == 0'
    residual_closed: 'sign_mismatch_or_weight_delta: [defect] != 0'
    open: unpaired_negative_paths_remain
proof_obligations:
- id: micnf.involution_cancellation_soundness
  status: theorem_shaped
  ref: ../demos/demo_2/meron_witness.py
  section: Meron pairing gate