nf.fsp.bssnf

Binary Sign-Switching Normal Form (BSS-NF)

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

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1. Executive Summary & Metadata

Binary Sign-Switching Normal Form (BSS-NF) enforces structural normalization and explicit defect routing in the representation regime.

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

Foundational Treatise Specification

• Domain: Real symmetric matrices H in R^{N x N} in a fixed discrete basis.
• Transformation Group: Diagonal sign changes G_gauge = {diag((-1)^{alpha_1}, ..., (-1)^{alpha_N})}.
• Target Property: Stoquasticity (H_ij <= 0 for all i != j).
• Anomaly Excluded: Quantum Monte Carlo on artificially non-stoquastic presentation when diagonal gauge eliminates negative signs.
• Necessary And Sufficient Criterion: Cycle Holonomy Theorem: sum_{e in C} s_e == 0 mod 2 for every cycle C in the graph.
• Reconstruction Theorem: Transformed matrix H' = D_alpha H D_alpha is stoquastic; partition function sign weights are positive.
• Positive Witness: Spanning forest rooted potential alpha in {0, 1}^N satisfying edge equations.
• Negative Witness Obstruction: Explicit cycle C with odd sign-bit holonomy (sum_{e in C} s_e == 1 mod 2).

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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 (BSS-NF Candidate)	Anomaly risk: a Monte Carlo sign ; problem that a diagonal sign gauge ; removes
2. Normalization / Gate	BSS-NF Verification Gate	off-diagonal non-positive... (certified_implies_authorized: false)
3. After / Certified Output	Certified BSS-NF Output	Invariant: ['edge_sign_bits', ; 'gauge_potential', 'cycle_space']

Contract-Declared Worked Cases

- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '4.1'

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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.bssnf
Pillar I: Excludes Anomaly	Name and eliminate a specific class of data or state defects.	a Monte Carlo sign problem that a diagonal sign gauge removes
Pillar II: Preserves Invariant	State the structural invariant and protected skeleton preserved across transitions.	Invariant: off-diagonal non-positive entries after a diagonal sign conjugation, with the spectrum unchanged by that conjugation Protected Skeleton: ['edge_sign_bits', 'gauge_potential', 'cycle_space']
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)

• an odd cycle of positive-sign edges is a witness that no such gauge exists. The form does not remove that obstruction
• the matrix is real and symmetric in a fixed finite basis. Continuous basis changes are outside the form
• the normal-form packaging is local to the source treatise. The holonomy criterion itself is the standard GF(2) switching condition
• does not imply authorization

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5. Operational Verification & Terminal Routing

5.1 Required Declarations

• matrix_dimension
• nonzero_off_diagonal_entries
• edge_sign_bits

5.2 Required Evidence

None declared.

5.3 Validation Criteria

Criterion ID	Name	Condition
criterion	Check	H is real and symmetric
criterion	Check	each nonzero off-diagonal entry contributes its edge bit
criterion	Check	a proposed gauge satisfies alpha_i XOR alpha_j equals the edge bit on every edge
criterion	Check	the gauged matrix has non-positive off-diagonal entries
criterion	Check	if no gauge exists, an odd cycle is exhibited

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
AcceptStoquastic	Repair
RejectOddCycle	Reject
RejectNonSymmetric	Refuse

5.5 Declared Failure Modes

• symptom: a cycle has odd edge-bit sum, diagnosis: no diagonal sign gauge makes H stoquastic, terminal: RejectOddCycle
• symptom: an off-diagonal pair disagrees, diagnosis: H is not symmetric, terminal: RejectNonSymmetric

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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.bssnf (BSS-NF) against its declared schema, boundary constraints, and witness obligations in this model.
   • Supports deterministic terminal routing (AcceptStoquastic, RejectOddCycle, RejectNonSymmetric) 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 at_rest, this contract enforces that a Monte Carlo sign problem that a diagonal sign gauge removes is either excluded under off-diagonal non-positive entries after a diagonal sign conjugation, with the spectrum unchanged by that conjugation 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.bssnf.

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

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

schema_version: smplx.nf.contract.v0
schema_ref: ./normal_form_contract_schema.yaml
contract_kind: normal_form_contract
identity:
  id: nf.fsp.bssnf
  name: Binary Sign-Switching Normal Form
  short_name: BSS-NF
  version: v0
  theory_status: proposed
  contract_status: draft
  verification_status: absent
  family: representation
  status: draft
classification:
  hierarchy_level: normal_form
  subject_kind:
  - real_symmetric_matrix
  - diagonal_sign_gauge
  data_posture: at_rest
  mathematical_posture: formal_algebraic
purpose:
  excludes_anomaly: a Monte Carlo sign problem that a diagonal sign gauge removes
  preserves_invariant: off-diagonal non-positive entries after a diagonal sign conjugation,
    with the spectrum unchanged by that conjugation
  protected_skeleton:
  - edge_sign_bits
  - gauge_potential
  - cycle_space
  erased_structure: []
  non_claims:
  - an odd cycle of positive-sign edges is a witness that no such gauge exists. The
    form does not remove that obstruction
  - the matrix is real and symmetric in a fixed finite basis. Continuous basis changes
    are outside the form
  - the normal-form packaging is local to the source treatise. The holonomy criterion
    itself is the standard GF(2) switching condition
  - does not imply authorization
definition:
  source: coda_all_universe/mssn_sn_tower/definitions.md
  section: '4.1'
  formal_statement: 'Let H be a real symmetric N by N matrix. A diagonal sign gauge
    is a diagonal matrix D with diagonal entries +1 or -1. The gauged matrix is D
    H D. It is stoquastic when every off-diagonal entry is less than or equal to zero.
    For each off-diagonal nonzero entry, the edge bit is 1 when the entry is positive
    and 0 when the entry is negative. A gauge with bits alpha exists if and only if
    alpha_i XOR alpha_j equals the edge bit on every edge, if and only if every cycle
    has edge-bit sum even. A positive witness is a 0-1 potential on a spanning forest
    that meets every edge equation. A negative witness is one cycle whose edge-bit
    sum is odd.

    '
  quadrilateral:
    qualifies_as_true_normal_form: false
    qualification_gap: reconstruction_is_a_diagonal_similarity_not_a_relational_join
    qualifies_as_extended_form: true
    anomaly_exclusion: running a sign-problem method on a presentation that a diagonal
      sign gauge makes stoquastic
    lossless_join:
      status: similarity
      statement: D H D has the same spectrum as H, and the gauge is inverted by applying
        it twice
    dependency_preservation:
      status: edge_equations_retained
      statement: every nonzero off-diagonal edge keeps its sign equation under the
        gauge search
    syntactic_decision:
      status: matrix_decidable
      statement: 'The decision reads the sign pattern of H. It builds a spanning forest,
        propagates the edge equations, and checks the remaining edges. A Monte Carlo
        sample is not the decider.

        '
declarations_required:
- matrix_dimension
- nonzero_off_diagonal_entries
- edge_sign_bits
evidence_required: []
validation_criteria:
- H is real and symmetric
- each nonzero off-diagonal entry contributes its edge bit
- a proposed gauge satisfies alpha_i XOR alpha_j equals the edge bit on every edge
- the gauged matrix has non-positive off-diagonal entries
- if no gauge exists, an odd cycle is exhibited
terminal_routes:
- AcceptStoquastic
- RejectOddCycle
- RejectNonSymmetric
terminal_route_map:
  AcceptStoquastic: Repair
  RejectOddCycle: Reject
  RejectNonSymmetric: 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: a cycle has odd edge-bit sum
  diagnosis: no diagonal sign gauge makes H stoquastic
  terminal: RejectOddCycle
- symptom: an off-diagonal pair disagrees
  diagnosis: H is not symmetric
  terminal: RejectNonSymmetric
examples:
- ref: coda_all_universe/mssn_sn_tower/definitions.md
  section: '4.1'
references:
- coda_all_universe/mssn_sn_tower/definitions.md
obstruction_substrate:
  policy_ref: ../substrate/default_finite_obstruction_policy.yaml
  finite_carrier:
    description: real symmetric matrix H over finite basis V
    max_rank: null
  cochain_complex:
    C0: vertex_gauge_potentials_alpha_in_Z2
    C1: edge_sign_bits_s_in_Z2
    C2: cycle_holonomies_in_Z2
    d0: 'maps vertex signs to edge differentials: (d0 alpha)_ij = alpha_i XOR alpha_j'
    d1: maps edge bits to cycle sums modulo 2
    quotient: H1(G, Z2) = ker(d1) / im(d0)
  obstruction_class:
    notation: '[holonomy(C)]'
    exact: 'all_cycles_even: [s] == 0'
    residual_closed: 'frustrated_odd_cycle: [s] != 0'
    open: unbounded_graph_or_non_hermitian
proof_obligations:
- id: bssnf.holonomy_soundness
  status: theorem_shaped
  ref: ../demos/demo_2/sign_switching.py
  section: 4. Binary graph switching