Theorem 4: Uniqueness of the Cost Functional

J(x) = ½(x + 1/x) under symmetry, finiteness, and normalization

Statement

Uniqueness. The only analytic, dual‑balanced, ledger‑finite cost functional on ℝ>0 is J(x) = ½(x + 1/x).

In plain English

There’s exactly one fair “effort meter.” It treats too much and too little as equally costly, doesn’t blow up faster than your ledger can pay, and prefers balance. That meter is J(x) = ½(x + 1/x). No knobs. No hidden dials.

  • Why inevitable: symmetry (x ↔ 1/x) + smoothness + “no outrageous growth” leave only this form.
  • What it buys: a single convex cost that punishes extremes and underwrites φ in T5.

Formal setting

  • Domain: x > 0; J : (0, ∞) → ℝ.
  • Dual symmetry: J(x) = J(1/x) (ledger dual‑balance).
  • Normalization and positivity: J(1) = 0 and J(x) > 0 for x ≠ 1.
  • Ledger finiteness (growth bound): ∃K > 0 s.t. J(x) ≤ K (x + 1/x) for all x > 0.
  • Regularity: J is analytic on ℂ \ {0} (equivalently, convex on the log‑axis suffices to force analyticity here).

Key lemmas

  1. Symmetric Laurent representation: Dual symmetry and analyticity imply a Laurent expansion J(x) = Σn ≥ 1 cn(xⁿ + x−n).
  2. Finiteness kills higher modes: If any cm with m ≥ 2 is non‑zero, then J(x)/(x + 1/x) ~ cmx^{m−1} as x → ∞, violating the growth bound. Hence cn≥2 = 0.
  3. Normalization/positivity fix c₁: J(1) = 0 enforces no constant term; J(x) > 0 for x ≠ 1 with symmetry implies c₁ = ½.
  4. Convexity barrier: Any piecewise or logarithmic modification breaks analyticity or convexity on the log‑axis, so cannot satisfy all axioms simultaneously.

Proof sketch

Expand any analytic, dual‑symmetric candidate as a symmetric Laurent series. The ledger‑finiteness bound excludes all |n| ≥ 2 terms by asymptotic comparison, leaving only c₁(x + 1/x). Normalization and positivity determine c₁ = ½. Attempts to add piecewise corrections or logarithmic tails either violate convexity on the log‑axis or break analyticity, so the form is unique.

Edge cases

  • Non‑analytic candidates: |log x| or piecewise plasmic forms introduce cusps; they violate the regularity/convexity requirement.
  • Modular costs: Reducing costs modulo m admits nonzero closed loops of net zero, contradicting MP via T1/T3; excluded.
  • Rescaling δ: T1 forbids order‑preserving rescalings of the generator; the ½ factor is fixed by normalization, not by δ drift.

Implications

  • Single convex cost: Establishes a unique convex energy‑like measure governing recognition.
  • Supports T5: Global cost minimization over integer k in xn+1 = 1 + k/xn selects k = 1 and φ.
  • Bridges discrete/continuum: Plays well with T3’s continuity equation and T8’s causal cones.

Related

T1 Ledger Necessity T2 Atomicity T3 Conservation T5 k = 1 and φ

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