Editing Openai/69596681-3f7c-8006-b9fd-01d9d8191763 (section)

=== Assistant: Proceeding. ===
Proceeding. Below is METHODS BLOCK 14 — PRECISION SCALING, expanded to maximum algorithmic depth using machine-learning–style language (asymptotic limits, monotone refinement, adversarial resolution escalation, and hard-collapse triggering). This block formalizes what happens as measurement precision increases. It is the bridge between robustness testing and final falsification.

METHODS BLOCK 14 — PRECISION SCALING

OBJECTIVE
Determine whether the theory remains viable under arbitrarily increasing measurement precision and numerical resolution. This block enforces the principle that a valid theory must not rely on finite-resolution artifacts. Precision escalation is adversarial, monotone, and irreversible.

SCOPE AND AUTHORITY
• Executed after Monte Carlo robustness (Block 13)
• May only introduce PrecisionFailure flag
• Cannot rescue prior collapse
• Does not alter data provenance or structure

ML analogy: robustness under shrinking noise radius / increasing bit-depth.

PRECISION PARAMETER
Define precision level p ∈ ℕ, p ≥ p₀.

Interpretation of p (pre-declared and frozen):
• Instrument resolution (e.g., σ → σ / 2^p)
• Numerical precision (e.g., float32 → float64 → arbitrary precision)
• Sampling density
• Temporal resolution

The exact semantics of p must be declared in Model Map or Data Types and hashed.

PRECISION OPERATOR
Define precision-scaling operator:

Ψ_p : (D, Σ) → (D_p, Σ_p)

Constraints:
• D_p represents the same underlying physical data at higher resolution
• Σ_p monotonically decreases or refines uncertainty representation
• No new data points fabricated
• No reweighting or smoothing
• Provenance preserved

Formally:
Σ_{p+1} ⪯ Σ_p in Loewner order (uncertainty shrinks or refines).

MONOTONICITY REQUIREMENT
For any collapse indicator C ∈ {StructuralViolation, FeasibleFail, EvidenceFailure}:

If C(D_p)=1 for some p, then ∀ q>p: C(D_q)=1.

No “collapse at intermediate precision then recovery”.

PRECISION ESCALATION LOOP
Define precision sequence {p₀, p₁, …, p_max}.

For each p_k:
# Apply Ψ_{p_k}
# Re-evaluate residuals, structural violation, feasibility, evidence (no structure changes)
# Record collapse indicators

Loop terminates when:
• Collapse occurs
• p_max reached
• Convergence detected (see below)

CONVERGENCE CRITERION
Define convergence if collapse indicators stabilize:

∃ p'' such that ∀ p ≥ p'':
C(D_p) constant.

If constant = 0 ⇒ survives precision escalation up to tested limit.
If constant = 1 ⇒ collapse confirmed.

No averaging across p.

INFINITE-PRECISION LIMIT
Define asymptotic verdict:

lim_{p→∞} Verdict(D_p)

Operational rule:
If ∃ finite p such that Verdict(D_p)=1 ⇒ asymptotic Verdict=1.

If ∀ tested p up to p_max Verdict=0 and no robustness failures detected ⇒ asymptotic Verdict remains undecided but provisionally STAND.

PRECISION FAILURE FLAG
Define PrecisionFailure = 1 iff either:
• RobustnessFailure=1 AND precision escalation induces collapse
• StructuralViolation emerges under precision refinement
• EvidenceFailure emerges monotonically under decreasing Σ

Else PrecisionFailure = 0.

ML analogy: adversarial limit where noise → 0.

NO NUMERICAL ESCAPES
Forbidden:
• Adding regularization at higher p
• Introducing noise floors to prevent collapse
• Switching kernels or tolerances
• Changing τ_struct

The theory must survive its own exact limit.

FAILURE MODES
• Non-monotone Ψ_p ⇒ INVALID
• Precision-dependent thresholds ⇒ INVALID
• Precision-dependent priors ⇒ INVALID
• Adaptive stopping based on favorable outcomes ⇒ INVALID

ARTIFACT EMISSION
Emit immutable PrecisionScalingRecord:

⟨
p_sequence,
CollapseAt_p (if any),
PrecisionFailure,
ConvergenceFlag
⟩

Hashed and audit-logged.

PIPELINE ROLE
PrecisionFailure feeds directly into:
• Global Collapse (Block 12)
• Failure Attribution (Block 15)
• Certification logic (Block 16)

ML INTERPRETATION
This block enforces resolution-invariant correctness:

• A valid classifier must not flip under shrinking perturbations
• A valid probabilistic model must not rely on finite-noise artifacts
• A valid theory must survive its own exacting limit

This is equivalent to requiring uniform stability in the zero-noise limit.

END METHODS BLOCK 14.

If you confirm, the next block will be METHODS BLOCK 15 — FAILURE ATTRIBUTION GRAPH (ARTIFACT #5), expanded into DAG semantics, causal labeling, and queryable failure diagnostics.

© Robert R. Frost
2026-01-03