Displacement

Objects

• S⁰ — ground state. The configuration a system rests in at least cost.
• ξ = ρ(s, S⁰) — displacement. The distance from ground, right now.
• D(ξ) — cost. What sitting at displacement ξ costs each moment.
• Φ = ∫₀ᵀ D(ξ) dt — accumulation. The cost of a whole path.
• δ — net displacement. Where you ended minus where you started.
• G = Φ − k|δ| — the glitch. Cost paid beyond net work done. k is the fair-exchange constant.

Axioms — DC1–DC9

1. Ground exists. D(0) ≤ D(ξ). (DC1*, living: D(0) = θ > 0 — a body pays just to stay alive.)
2. Non-negativity. D(ξ) ≥ 0.
3. Monotonicity. D non-decreasing in ξ. Farther costs more.
4. Path dependence. Φ depends on the path, not the endpoints.
5. Irreversibility. A broken return costs more than the departure: Φ_return > Φ_departure.
6. Loop cost. Any loop that leaves ground pays: ∮ D dt > 0.
7. Fairness. A fair price tracks work Φ_T, not severity ξ. Price on ξ and you mint the glitch, G_econ > 0.
8. Volume. Earn on the volume of real return, Π = m·n·Φ̄, not on the depth of displacement.
9. Threshold. Each system has a critical ξ_c past which a wrong attractor self-reinforces. Period and amplitude are individual.

Results

Wrong attractor. A stable W ≠ S⁰. Suffering is capture in W: a basin that costs Φ and is not where you were going.

Maintenance. Hold a system near ground at the optimal cadence τ* = (3f / aδ²)^⅓. Sooner wastes work; later lets the drift entrench.

Invariance, a priori. At a fold, U = ¼x⁴ − ½ax² − bx, the exponent vector (τ, dU, A, x_h − x_s) = (−½, 3⁄2, 1, ½) is the only domain-free content. Critical slowing τ ∼ μ^−½. The same exponents from a mind to a market to spacetime — the constants carry the domain, the exponents do not. Invariance is real as scaling, not as numbers.

The move

Name the ground. Measure the drift. Pay the honest price of the return.

Two things on this site are this sheet applied to instruments rather than to people. The Frozen Reference is DC1 turned into a rule for measurement: if S⁰ is recomputed from what it measures, ξ stops meaning anything. The Present Holds is the same structure where the ground cannot move at all — the critical line, read as the condition that a state survives its own time evolution.