Research-Stack/0-Core-Formalism/otom/docs/audit/InvertedFAMMStructure.md

Inverted FAMM Structure

Purpose

This note defines an inverted FAMM structure as the dual of ordinary FAMM route-memory.

Ordinary FAMM records how validated routes become basins, hazardous failures become scars, and unresolved adapter stress becomes torsion.

Inverted FAMM reverses the question:

Instead of asking: which routes became lawful basins?
Ask: which basins, scars, or torsion fields predict missing routes, hidden constraints, or adversarial shortcuts?

Core Statement

FAMM = forward route memory from traversal outcomes
Inverted FAMM = inverse inference from memory geometry back to missing laws, hidden bottlenecks, and absent receipts

The inverted structure is not anti-evidence. It is an error-field and negative-space analyzer.

Forward FAMM

Forward FAMM update:

route traversal -> receipt -> PASS/HOLD/SCAR/QUARANTINE/MISSING -> basin/scar/torsion update

Forward state:

FAMM_r = {
  basin_strength,
  scar_strength,
  unresolved_torsion,
  evidence_mass,
  provenance_integrity,
  loss_budget,
  repair_capacity,
  finite_witness_count,
  validator_status,
  last_outcome
}

Inverted FAMM

Inverted FAMM performs inverse reconstruction:

basin/scar/torsion field -> infer missing receipt / hidden constraint / absent bridge / unstable law boundary

Inverted state:

IFAMM_q = {
  missing_law_pressure,
  hidden_constraint_score,
  inverse_route_likelihood,
  scar_shadow_strength,
  basin_boundary_gradient,
  torsion_source_estimate,
  adversarial_shortcut_risk,
  counterexample_pressure,
  receipt_deficit,
  suggested_probe
}

Interpretive Meaning

Basin inversion

A strong basin with a sharp boundary implies an undiscovered constraint:

high basin_strength + sharp basin_boundary_gradient -> hidden law boundary candidate

Use:

find the invariant that explains why lawful routes cluster here and fail nearby

Scar inversion

A repeated scar pattern implies a possible negative theorem, forbidden transition, or missing adapter:

repeated scar geometry -> obstruction candidate

Use:

extract common failure invariant

Torsion inversion

Unresolved torsion that accumulates without becoming a scar implies an incomplete bridge:

persistent torsion + nonfatal loss -> missing receipt / missing metric / missing transfer operator

Use:

ask what receipt would convert HOLD into PASS or SCAR

Quarantine inversion

Repeated quarantine can indicate adversarial compression, invalid evidence promotion, or route spoofing:

quarantine cluster -> trust-boundary defect

Use:

separate genuine missing evidence from unsafe inference route

Inverted Update Law

Given a region of route-memory state R, compute:

MissingLawPressure(R) =
  w_t * unresolved_torsion
  + w_s * scar_shadow_strength
  + w_b * basin_boundary_gradient
  + w_m * missing_receipt_count
  - w_v * validator_coverage

A high value means the system should search for a missing law, invariant, proof obligation, or empirical measurement.

Negative-Space Probe

The inverted system emits probes instead of conclusions:

probe = {
  suspected_missing_receipt,
  route_region,
  failure_cluster,
  candidate_invariant,
  minimum_test,
  expected_disambiguation,
  risk_if_ignored
}

Probe examples:

Need DifferenceSetReceipt for this spectral candidate.
Need finite-window active-cell count before density promotion.
Need LandauerCostReceipt because route claims information deletion without dissipation.
Need dimensional calibration because raw temperature was mixed with dimensionless loss.
Need nonseparable encoding because spectral voids do not prove Sidon sums.

Relation to Current Pipeline

Forward pipeline:

finite signal
-> FFT/Hodge/Dirac filter
-> remainder resonance candidate set
-> void/topological defect receipt
-> active-cell set
-> nonseparable encoding
-> Sidon audit
-> crossing receipt
-> FAMM update

Inverted pipeline:

FAMM memory field
-> locate scar/torsion/basin gradients
-> infer missing receipt or hidden invariant
-> generate finite probe
-> run measurement/proof/search
-> update FAMM with result

Main Use Cases

counterexample discovery
missing receipt detection
hidden invariant extraction
adversarial route detection
negative-space search
proof-obligation prioritization
model drift detection
research question refinement

Inverted FAMM Gates

IFAMM_PASS: predicted missing receipt was found and validated
IFAMM_HOLD: missing route is plausible but underconstrained
IFAMM_SCAR: predicted route collapses into a repeatable obstruction
IFAMM_QUARANTINE: inverse route exploits evidence gap or unsafe promotion path
IFAMM_MISSING: not enough finite evidence to form a probe

Boundary

Inverted FAMM cannot promote a claim by itself.

Correct role:

Inverted FAMM finds where the project should look next.
It identifies missing laws, missing receipts, hidden constraints, and likely counterexamples.

Incorrect role:

Inverted FAMM proves the theorem.
Inverted FAMM converts scars into evidence.
Inverted FAMM treats absence of failure as proof.

Audit Classification

Receipt: InvertedFAMMStructure
Status: ARCHITECTURE_DRAFT
Gate: U_scope
Reason: coherent as a negative-space route-memory analyzer, but it requires explicit finite scoring functions, probe-generation rules, validator wiring, and safeguards against promoting absence-of-evidence into evidence.

Required Receipts

FAMMStateReceipt
BasinBoundaryGradientReceipt
ScarClusterReceipt
TorsionSourceReceipt
MissingReceiptDetector
CounterexampleProbeReceipt
AdversarialShortcutReceipt
FiniteProbeReceipt
ValidatorReceipt
FAMMUpdateReceipt