01-core-formalization.md

Core Formalization — The Logarithmic Value Law

Thread A from 00-thesis-seed.md §9.1: state the axioms tightly and derive what functional form they force. This is the "Shannon Section 1" of value — one clean definition and one provable result.

Motivating premise (why this matters now)

In the near future, all AI agents are driven by values. If that is true, a theory of value is not philosophy — it is the control theory for agent populations. To route, align, and govern many separated AI individuals we need value to be a measured quantity with laws, not a verdict. Everything below is built to that end.

1. The object being measured

Shannon measures a probability distribution p = (p₁,…,pₙ) over possible messages. We measure the structurally-parallel object on the value side:

An agent A holds a finite, scarce resource budget E (free energy, or its proxy: compute, time). It commits that budget across n goal-channels as an allocation e = (e₁,…,eₙ), eᵢ ≥ 0, Σeᵢ = E.

Same simplex geometry as Shannon's p; different semantics. eᵢ is resource committed to channel i, not probability of message i.

Each channel carries a goal-weight kᵢ ≥ 0: how much advancing channel i serves agent A's goal, per unit of effective progress. The vector k = (k₁,…,kₙ) is the agent-frame — change the agent, change k. The budget E is frame-independent (the cross-frame invariant of 00 §3); k is what varies between agents.

2. Two confirmations of the logarithmic form — kin, not independent

The form V(e) = Σᵢ kᵢ ln eᵢ is reached by two structurally different arguments, one static and one dynamic. Honesty note (2026-06-07, after external review): an earlier version called these "independent, sharing no premises." That overstates. Axiom A3 (scale-invariant proportional increments) is the static shadow of multiplicative compounding — "equal multiplicative increments yield equal value increments" already encodes the logarithm, as §3's proof makes visible (a parallel additive axiom would force a linear v by the identical Cauchy argument). So the two routes share their load-bearing premise in static vs dynamic form, and their agreement is a consistency check on the functional form — evidence the choice is coherent, not an independent over-determination.

Confirmation A (static): axioms uniquely determine the functional form

The question here is purely static: what functional form for V is consistent with the basic properties a value measure must satisfy? The answer is uniquely logarithmic, proved via Cauchy's theorem in §3.

We want V(e) = Σᵢ vᵢ(eᵢ) (separability across independent goal-channels — Axiom S). For each vᵢ:

• (A1) Continuity. vᵢ is continuous on e > 0.
• (A2) Monotonicity. vᵢ is non-decreasing: committing more resource toward a goal-relevant channel never reduces value.
• (S) Separability. Value over independent goal-channels adds: V(e) = Σᵢ vᵢ(eᵢ).
• (A3) Scale-invariant proportional increment. The value added by a proportional increase in resource is the same regardless of the current level: $$v_i(\lambda e) - v_i(e) = \varphi_i(\lambda)\quad\text{for all } e>0,\ \lambda>0,$$ with φᵢ independent of e.

A3 is the load-bearing one for this route: it encodes the absence of a privileged resource-unit — value has no absolute zero of scale, just as information has no privileged base of logarithm. It is Cauchy's functional equation in disguise (§3 makes this explicit). This is the rigorous static determination: given these axioms, the only consistent form is logarithmic.

Confirmation B (dynamic): compounding forces the same form — rigorous in 02

The question here is dynamic: when value compounds, what is the correct measure of long-run gain? The answer, via the ergodicity argument of Peters (2019) and Kelly (1956), is the expected log of the multiplier. 02 §1–3 develops this rigorously via the repeated game: the long-run value growth rate is

$$G(b) = \mathbb{E}_{s \sim q}!\Big[\log \sum_i b_i, o_i(s)\Big],$$

and the optimal allocation in the horse-race case recovers V* = Σᵢ kᵢ ln eᵢ — the same logarithmic form, now derived from dynamics with no scale assumption. The two confirmations address different questions: Confirmation A fixes the static functional form; Confirmation B shows the dynamic optimum lands on the same shape.

3. The theorem (Confirmation A — rigorous static proof)

Claim. Any vᵢ satisfying (A1)–(A3) has the form vᵢ(eᵢ) = kᵢ ln eᵢ + cᵢ with kᵢ ≥ 0. The same form is reached by Confirmation B's dynamic route; per §2's honesty note, their agreement is a consistency check, not independent over-determination.

Proof. Put g(x) := vᵢ(eˣ). A3 with e = eˣ, λ = eʸ gives g(x+y) − g(x) = φᵢ(eʸ) =: ψ(y), independent of x. Setting x = 0: ψ(y) = g(y) − g(0). Substitute: $$g(x+y) = g(x) + g(y) - g(0).$$ Let h(x) := g(x) − g(0); then h(x+y) = h(x) + h(y) — Cauchy's equation. With continuity (A1), h(x) = kᵢx. Hence g(x) = kᵢx + g(0), i.e. vᵢ(eᵢ) = kᵢ ln eᵢ + cᵢ. Monotonicity (A2) forces kᵢ ≥ 0. ∎

The Logarithmic Value Law. $$\boxed{,V(e) = \sum_i k_i \ln e_i + \text{const},}$$

Why this is value, not entropy in disguise

It shares Shannon's logarithm but is a genuinely different quantity:

	Shannon entropy H(p) = −Σ pᵢ ln pᵢ	Logarithmic value V(e) = Σ kᵢ ln eᵢ
variable	probability pᵢ (what we don't know)	resource committed eᵢ (what we spend)
log arises from	additivity over independent events	compounding / ergodicity (A3 is a consequence, not a cause)
directionality	none — symmetric in outcomes	carried by kᵢ (a goal, a sign, a target)
sign of the Σ … ln … term	−	+

The log is the same skeleton reached by two different routes — which is why entropy reappears below as a natural by-product rather than a coincidence.

4. Optimal allocation and the focus penalty

Maximize V(e) = Σ kᵢ ln eᵢ subject to the budget Σ eᵢ = E. Lagrange: kᵢ/eᵢ = μ, so

$$e_i^* = E,\frac{k_i}{K},\qquad K := \sum_j k_j.$$

Optimal resource allocation is proportional to goal-weight — value-proportional betting. This is exactly Kelly's criterion (J. L. Kelly, 1956), and the lineage is not a coincidence: Kelly was Shannon's Bell Labs colleague and derived proportional betting directly from Shannon's channel capacity ("A New Interpretation of Information Rate"). The historical bridge from information to value already exists; the Logarithmic Value Law generalizes it from gambling returns to arbitrary goal-pursuit.

Substituting e* back:

$$V^* = \sum_i k_i \ln!\Big(\frac{E k_i}{K}\Big) = K\ln E + \sum_i k_i \ln\frac{k_i}{K} = \boxed{,K\ln E ;-; K,H(\hat k),}$$

where k̂ = k/K is the normalized goal-weight vector and H(k̂) = −Σ k̂ᵢ ln k̂ᵢ is its Shannon entropy.

Read the result:

• K ln E — realizable value grows with the log of resource, scaled by total goal-stakes K.
• − K·H(k̂) — the focus penalty: the more an agent's goals are spread out (high entropy), the less value it extracts from the same resource. A focused agent (low goal-entropy) realizes strictly more value.

Shannon entropy thus emerges as a dissipation term — not as the value itself. This is the Second Law of Value (00 §4) in microcosm: dispersion wastes value.

5. Consequences for governing AI individuals

1. Give each agent a focused goal. The focus penalty −K·H(k̂) is paid per-agent. Diffuse mandates are not just bad management — they are quantified value dissipation.
2. Allocate resource proportional to goal-weight (eᵢ* ∝ kᵢ), not equally and not winner-take-all. Both uniform and all-in allocations are provably sub-optimal under the log law.
3. Value is comparable within a frame, cardinal up to the constant cᵢ. Across frames (different k), only the shared invariant E is directly comparable — cross-agent value accounting is the open problem (00 §8), and this formalization sharpens why: two agents' Vs live in different k-bases.
4. A fleet has an aggregate V* = Σ_a K_a ln E_a − Σ_a K_a H(k̂_a). Governance = maximize the first sum (route resource to high-stakes agents) while minimizing the second (keep each agent focused).

6. Boundary conditions — where the log law does not hold

Stated plainly, in Shannon's spirit of naming the model's assumptions:

• Threshold / all-or-nothing goals. When value is a step (you need exactly X to clear a bar — a surgery, a quorum, a launch), value is not concave and A3 fails near the threshold. The log law is the smooth-goal regime.
• Satiation. Real value often caps rather than growing unboundedly as ln E → ∞. A bounded variant (e.g. kᵢ(1 − e^{−eᵢ/sᵢ})) may be needed; A3 then holds only below satiation.
• Risk-seeking / convex regions. Gambles, status races, and winner-take-all dynamics produce locally convex value; the concavity that A3 encodes is violated.
• Non-stationary goals. k drifts as the agent learns or the world changes. The static derivation is a snapshot; the dynamic theory (how k evolves) is unwritten.

7. What this bought us (against the tautology objection)

00 §8 warned the theory must do something revealed preference cannot. The log law does:

• it predicts allocation (eᵢ* ∝ kᵢ) from a single axiom,
• it sets a limit (V* = K ln E − K·H(k̂) bounds extractable value), and
• it makes Shannon entropy reappear as a dissipation term — a structural bridge revealed preference has no reason to produce.

8. Next threads

• §9.2 (the coding theorem of value). Push the Kelly connection to a growth-rate limit: under uncertainty, the max long-run growth rate of value is bounded by an information-rate term. This is the closest thing to a Shannon capacity theorem for value, and Kelly already did the gambling case — generalize it.
• Cross-frame transformation. Define the map between two agents' k-bases and find what (beyond E) is invariant. Without it, fleet-level value is only ordinal.
• Dynamics of k. How goal-weights are learned, drift, and are deliberately set (alignment as choosing k).