nf.physics.pachner_lp_scaling

4D Pachner Moves and Joint LP Scaling Normal Form (L1PSNF)

Contract ID: nf.physics.pachner_lp_scaling | Family: physics | Index Status: draft | AuthorityEffect: None

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1. Executive Summary & Metadata

4D Pachner Moves and Joint LP Scaling Normal Form (L1PSNF) enforces structural normalization and explicit defect routing in the physics regime.

Property	Value
Canonical ID	nf.physics.pachner_lp_scaling
Short Name	L1PSNF
Full Name	4D Pachner Moves and Joint LP Scaling Normal Form
Family	physics (Discrete Differential Geometry, Regge Calculus, Gauge Theory, Lattice Fermions & Quantum Gravity)
Qualification	Extended Form Contract (Normal Form Contract)
Data / Math Posture	in_motion
Index Status	draft
Theory Status	experimental
Verification Status	absent
Contract Version	v0
Governance Fence	AuthorityEffect: None (certified_implies_authorized: false)

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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 (L1PSNF Candidate)	Anomaly risk: topology corruption ; under unchecked Pachner moves, ; Euler characteristic drift,...
2. Normalization / Gate	L1PSNF Verification Gate	verified bistellar... (certified_implies_authorized: false)
3. After / Certified Output	Certified L1PSNF Output	Invariant: ; ['simplicial_complex_cells', ; 'pachner_replacement_witnesses'...

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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.physics.pachner_lp_scaling
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	topology corruption under unchecked Pachner moves, Euler characteristic drift, uncertified LP duality gaps, and breach of empirical benchmark scaling targets
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: verified bistellar replacement legality preserving PL 4-manifold topology, Euler characteristic conservation, exact rational LP duality gap closure, and empirical solver time scaling within the declared target exponent alpha <= 3.5 on tested instances Protected Skeleton: ['simplicial_complex_cells', 'pachner_replacement_witnesses', 'boundary_matching_isomorphisms', 'euler_characteristic', 'primal_objective_phi', 'dual_objective_psi', 'duality_gap_zero', 'empirical_scaling_exponent']
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)

• finite empirical log-log regression across a tested instance range does not establish an asymptotic complexity theorem; exceeding threshold alpha > 3.5 denotes an empirical scaling-target breach, not super-polynomial divergence (e.g., polynomial N^4 exceeds 3.5)
• bistellar move validity requires local replacement witnesses and link conditions; Euler characteristic and cell intersection checks alone do not certify topology preservation
• does not support polynomial time complexity for general non-sparse linear programs
• does not claim recovery of smooth 4D general relativity without non-perturbative scaling

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• simplices_4d: list of 5-tuples representing 4-simplices
• pachner_move_witnesses: sequence of local replacement records with declared removed subcomplex, inserted subcomplex, and boundary isomorphism
• euler_characteristic: alternating sum of cell counts chi = N_0 - N_1 + N_2 - N_3 + N_4
• joint_lp_results: primal and dual objectives with duality gap g = |Phi - Psi| == 0 over Q
• benchmark_run_specification: input family, coefficient bit lengths, solver version, tolerance, and measurement range [N_min, N_max]
• empirical_scaling_exponent: estimated power-law slope alpha from regression of log(solve_time) vs log(N_4)

5.2 Required Evidence

• pachner_replacement_witnesses: list
• boundary_subcomplex_matches: list
• euler_characteristic_series: list
• primal_dual_gap_series: list
• benchmark_measurement_range: {'min_n4': 'integer', 'max_n4': 'integer', 'step_count': 'integer'}
• solver_environment_metadata: {'solver_name': 'string', 'solver_version': 'string', 'coefficient_bit_lengths': 'integer'}
• empirical_regression_exponent: float
• empirical_r_squared: float

5.3 Validation Criteria

Criterion ID	Name	Condition
``	****
``	****
``	****
``	****
``	****

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
PachnerLPScalingPreserved	Preserve
TopologyBreach	Reject
DualityGapViolation	Refuse
ScalingTargetBreach	Escalate
IllegalPachnerReplacement	Refuse
Unsupported	Unsupported

5.5 Declared Failure Modes

All defects route explicitly through the Terminal Route Map above.

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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.physics.pachner_lp_scaling (L1PSNF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (PachnerLPScalingPreserved, TopologyBreach, DualityGapViolation, ScalingTargetBreach) 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_motion, this contract enforces that topology corruption under unchecked Pachner moves, Euler characteristic drift, uncertified LP duality gaps, and breach of empirical benchmark scaling targets is either excluded under verified bistellar replacement legality preserving PL 4-manifold topology, Euler characteristic conservation, exact rational LP duality gap closure, and empirical solver time scaling within the declared target exponent alpha <= 3.5 on tested instances 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.physics.pachner_lp_scaling.

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7. Canonical YAML Contract (contracts/nf.physics.pachner_lp_scaling.yaml)

Click to expand full canonical YAML contract (contracts/nf.physics.pachner_lp_scaling.yaml)

schema_version: smplx.nf.contract.v0
contract_kind: normal_form_contract
identity:
  id: nf.physics.pachner_lp_scaling
  name: 4D Pachner Moves and Joint LP Scaling Normal Form
  short_name: L1PSNF
  version: v0
  theory_status: experimental
  contract_status: draft
  verification_status: absent
  family: physics
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - 4d_simplicial_complex
  - pachner_moves
  - linear_programming_scaling
  data_posture: in_motion
  mathematical_posture: operational
purpose:
  excludes_anomaly: topology corruption under unchecked Pachner moves, Euler characteristic
    drift, uncertified LP duality gaps, and breach of empirical benchmark scaling targets
  preserves_invariant: verified bistellar replacement legality preserving PL 4-manifold
    topology, Euler characteristic conservation, exact rational LP duality gap closure,
    and empirical solver time scaling within the declared target exponent alpha <= 3.5
    on tested instances
  protected_skeleton:
  - simplicial_complex_cells
  - pachner_replacement_witnesses
  - boundary_matching_isomorphisms
  - euler_characteristic
  - primal_objective_phi
  - dual_objective_psi
  - duality_gap_zero
  - empirical_scaling_exponent
  erased_structure:
  - continuous_diffeomorphism_generators
  - arbitrary_unbounded_solver_relaxations
  non_claims:
  - finite empirical log-log regression across a tested instance range does not establish
    an asymptotic complexity theorem; exceeding threshold alpha > 3.5 denotes an empirical
    scaling-target breach, not super-polynomial divergence (e.g., polynomial N^4 exceeds 3.5)
  - bistellar move validity requires local replacement witnesses and link conditions;
    Euler characteristic and cell intersection checks alone do not certify topology preservation
  - does not support polynomial time complexity for general non-sparse linear programs
  - does not claim recovery of smooth 4D general relativity without non-perturbative scaling
declarations_required:
- simplices_4d: list of 5-tuples representing 4-simplices
- pachner_move_witnesses: sequence of local replacement records with declared removed subcomplex, inserted subcomplex, and boundary isomorphism
- euler_characteristic: alternating sum of cell counts chi = N_0 - N_1 + N_2 - N_3 + N_4
- joint_lp_results: primal and dual objectives with duality gap g = |Phi - Psi| == 0 over Q
- benchmark_run_specification: input family, coefficient bit lengths, solver version, tolerance, and measurement range [N_min, N_max]
- empirical_scaling_exponent: estimated power-law slope alpha from regression of log(solve_time) vs log(N_4)
validation_criteria:
- pachner_move_legality: each bistellar move strictly replaces a standard subcomplex with its dual shell, matching the boundary subcomplex identically
- euler_characteristic_invariance: chi remains strictly invariant under all 1->5, 2->4, 3->3, 4->2, and 5->1 moves
- topological_validity: all generated simplices form an intersection-valid simplicial complex
- strong_duality_certified: duality gap |Phi - Psi| is certified zero in exact rational arithmetic over Q
- empirical_scaling_target: empirical scaling exponent satisfies alpha <= 3.5 over the declared measurement range
terminal_routes:
- PachnerLPScalingPreserved
- TopologyBreach
- DualityGapViolation
- ScalingTargetBreach
- IllegalPachnerReplacement
- Unsupported
terminal_route_map:
  PachnerLPScalingPreserved: Preserve
  TopologyBreach: Reject
  DualityGapViolation: Refuse
  ScalingTargetBreach: Escalate
  IllegalPachnerReplacement: Refuse
  Unsupported: Unsupported
extensions:
  condition_to_local_route_map:
    all_valid: PachnerLPScalingPreserved
    euler_drift: TopologyBreach
    gap_exceeded: DualityGapViolation
    empirical_exponent_exceeded: ScalingTargetBreach
    illegal_replacement: IllegalPachnerReplacement
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  formal_statement: Piecewise-flat 4-manifold topology preservation under verified local
    bistellar replacements with boundary-matching witnesses, exact rational LP duality
    gap certification, and empirical scaling validation against declared benchmark targets.
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_not_a_relational_project_join
    anomaly_exclusion: topology corruption under unchecked Pachner moves, Euler characteristic
      drift, uncertified duality gaps, and scaling-target breaches
    lossless_join:
      status: declared_reconstruction
      statement: verified bistellar move sequence preserving PL topology and exact LP duality closure
    dependency_preservation:
      status: declared_with_the_invariant
      statement: The preserved invariant stays on the declared schema. This form does
        not state a functional-dependency projection of a flat relvar decomposition.
    syntactic_decision:
      status: declaration_decidable
      statement: The check reads the declared constraints and validation criteria.
        One unrecorded runtime image is not the decider.
canonical_terminal_map:
  PachnerLPScalingPreserved: Preserve
  TopologyBreach: Reject
  DualityGapViolation: Refuse
  ScalingTargetBreach: Escalate
  IllegalPachnerReplacement: Refuse
  Unsupported: Unsupported
authorization:
  authorized_by: null
  certified_implies_authorized: false
certification:
  certificate_required: true
  certificate_kind: nf.physics.pachner_lp_scaling_certificate
  obstruction_class_required: false
  loss_ledger_required: false
  loss_ledger_trigger_id: null
  verifier_required: true
evidence_required:
- pachner_replacement_witnesses: list
- boundary_subcomplex_matches: list
- euler_characteristic_series: list
- primal_dual_gap_series: list
- benchmark_measurement_range: {min_n4: integer, max_n4: integer, step_count: integer}
- solver_environment_metadata: {solver_name: string, solver_version: string, coefficient_bit_lengths: integer}
- empirical_regression_exponent: float
- empirical_r_squared: float
schema_ref: ./normal_form_contract_schema.yaml