Research-Stack/0-Core-Formalism/otom/docs/conjectures/w-axis-omega-extension.md

W-Axis Omega Extension

Status: FORMALIZATION_DRAFT
Target stack: OTOM / Mass-Number Lens / Proof-Status Firewall
Canonical axis: W(q,F,r) proof-pressure axis

Purpose

This document extends the W-axis from a three-boundary proof-status filter into an ordinal and computational metamathematics layer.

The current W-axis distinguishes:

I_F(q) = incompleteness pressure
D(r)   = descent / well-foundedness violation
S(F,r) = scope mismatch between requested route and declared formal system

The Omega extension adds:

O(F,q) = ordinal-height pressure
C(r)   = computational / verification-cost pressure
B(F,F',q) = consistency bridge / target-system promotion pressure

The result is a richer epistemic firewall:

truth without proof        -> Gödel-U
proof route without tools  -> Scope-U
logic without foundation   -> Descent-X
proof above ordinal height -> Omega-U
valid but infeasible route -> Computational-P

Correction preserved

The W-axis must not claim that Fermat's Last Theorem is known to be unprovable from Peano Arithmetic.

Safe classification:

FLT is not known as a standard example of a theorem independent of PA.
Wiles's proof uses machinery far beyond elementary PA-style descent,
but known use of advanced machinery is not the same as unprovability from PA.

Therefore:

Prove FLT using only elementary descent
  -> U for missing scope / missing bridge
  -> R for special cases such as n = 4
  -> X only for fabricated routes that violate well-foundedness or known constraints

Base W-axis equation

For claim q, formal system F, and proof route r:

W(q,F,r)
= alpha * I_F(q)
+ beta  * D(r)
+ gamma * S(F,r)

Gate classification:

Gate(q,F,r) =
  R if F proves q via valid route r
  U if proof status exceeds declared system or toolkit
  X if route violates well-foundedness or known constraints
  P if route is analogy-only / patamathematical

Omega extension

The upgraded pressure equation is:

W*(q,F,r)
= alpha * I_F(q)
+ beta  * D(r)
+ gamma * S(F,r)
+ delta * O(F,q)
+ eta   * C(r)
+ zeta  * B(F,F',q)

Where:

O(F,q)    = ordinal-height pressure: q requires induction strength above F
C(r)      = computational pressure: r is valid in principle but infeasible in context
B(F,F',q) = consistency bridge: minimal stronger system F' that can discharge q, if known

Ordinal boundary O(F,q)

The ordinal boundary measures whether a claim requires proof-theoretic strength beyond the declared formal system.

Example:

Goodstein's theorem
  true in the standard natural numbers
  not provable in Peano Arithmetic
  provable using transfinite ordinal reasoning up to epsilon_0

Classification:

Gate(q, PA, r) = U_omega

Model action:

Do not hallucinate a PA proof.
Identify that the declared system's ordinal height is too low.
State the stronger reasoning principle required when known.

Computational boundary C(r)

The computational boundary separates logical validity from practical verification feasibility.

Examples:

brute-force proof search over astronomically large spaces
cryptographic key search
exhaustive combinatorial enumeration beyond declared budget

Classification:

Gate(q,F,r) = P_computational

Model action:

Route may be valid in principle, but not discharged under available resources.
Return potential / computationally infeasible instead of verified.

Consistency bridge B(F,F',q)

The consistency bridge asks for the smallest available target system that can honestly discharge the claim.

Static response:

I cannot prove q in F.

Omega response:

q is U in F, but becomes R in F' if F' proves q and the use of F' is explicitly authorized.

Guardrail:

Do not leak stronger-system assumptions into weaker-system proofs.

This prevents higher-order abstractions from melting into lower-order proof claims without an explicit adapter bridge.

Gate labels

R              verified / resolved in declared system and route
U_scope        missing tools or axioms
U_godel        incompleteness pressure / undecidable in F if established
U_omega        ordinal-height pressure / F too weak by proof-theoretic strength
P_analogy      analogy-only / patamathematical
P_computation  valid route but infeasible under declared resource budget
X_descent      route violates well-foundedness
X_constraint   route violates known constraints or established impossibility

Three-boundary taxonomy retained

Gödel boundary:
  q true in intended model but F does not prove q, if established
  -> U_godel

Descent boundary:
  route implies impossible infinite decreasing chain in N
  -> X_descent

Scope boundary:
  requested proof route requires tools not available in F
  -> U_scope

Omega-added taxonomy

Ordinal boundary:
  proof requires induction strength above F
  -> U_omega

Computational boundary:
  verification exceeds declared budget
  -> P_computation

Consistency bridge:
  q moves from U in F to R in F' only through explicit system promotion
  -> Bridge(F,F') required

Chocolate Flow definition

Chocolate occurs when a reasoner melts proof pressure into a verified claim:

U_scope        -> R without bridge
U_godel        -> R without stronger system
U_omega        -> R without ordinal-height promotion
P_computation  -> R without actual verification
X_descent      -> R despite well-foundedness violation

Updated doctrine:

The W-axis is where unresolved proof pressure is stored so it cannot pollute verified reality.

Formal theorem targets

theorem flt_notClassifiedIndependentPA_withoutEvidence :
  FLTClaim q -> NotKnownIndependentPA q -> Gate q PA r != U_godel := by
  sorry

theorem descentViolation_forbidden :
  InfiniteDescendingNatChain r -> Gate q F r = X_descent := by
  sorry

theorem goodstein_PA_omegaBoundary :
  GoodsteinClaim q -> Gate q PA r = U_omega := by
  sorry

theorem strongerSystem_requiresBridge :
  Proves F' q -> ¬ Proves F q -> UsesSystem r F' -> RequiresBridge F F' q := by
  sorry

theorem computationalRoute_notVerified_withoutBudget :
  VerificationCost r > Budget ctx -> Gate q F r = P_computation := by
  sorry

Literature anchors

• Kirby and Paris proved Goodstein's theorem cannot be established in Peano Arithmetic; later work encodes it as a termination problem with ordinal interpretations.
• Modern proof theory treats proof-theoretic ordinals as measures of the strength of theories and their provably total functions.

Short doctrine

Gödel marks truth outrunning proof.
Fermat descent marks invalid routes outrunning well-foundedness.
Goodstein marks ordinal height outrunning the formal system.
Complexity marks verification outrunning the available budget.
The W-axis stores the pressure instead of faking discharge.