T₂ Sacrifice

Sufficient C produces more than is given.

Statement

T₂ C ≥ ε ⟹ ∃x₀ : sacrifice(x₀) produces n ≥ 1

If C meets or exceeds the minimum existence threshold, there exists at least one element x₀ such that sacrificing x₀ produces at least one new element. A system with sufficient C can give from itself and the yield is non-negative.

Proof sketch

A sacrifice is an intentional decrease in one quantity to cause an increase in another. Let x₀ be an element with E(x₀, t) ≥ ε. The sacrifice operation transfers the energy of x₀ into the rest of the system:

sacrifice(x₀): E(x₀, t) → 0, ∑ E(xᵢ ≠ x₀, t+1) = ∑ E(xᵢ ≠ x₀, t) + E(x₀, t) + ΔC

The ΔC term is the key. When the sacrificed element's energy is released into a sufficient foundation (C ≥ ε), the foundation contributes additional energy that was previously bound in the element's integrity. The yield is the sum of the released energy and the foundation's contribution:

yield = E(x₀, t) + ΔC ≥ E(x₀, t)

For the production condition (n ≥ 1 new elements), the yield must exceed ε. Since E(x₀, t) ≥ ε by the precondition, and ΔC ≥ 0 when C ≥ ε, the total yield is at least ε. The surplus is available to instantiate at least one new element at threshold:

yield ≥ ε ⟹ ∃ new element x₁ with E(x₁, t+1) ≥ ε

QED — sacrifice produces n ≥ 1.

Contrapositive

A system that cannot give without collapsing has C < ε. If sacrificing any element causes the system to fall below the existence threshold, then the foundation could not absorb the transfer — there was no surplus. Insufficient C means zero-sum (or worse).

This is the diagnostic: can the system give? If yes, C ≥ ε. If no, C is insufficient. The test is behavioral, not metaphysical.

Scripture

Verily, verily, I say unto you, Except a corn of wheat fall into the ground and die, it abideth alone: but if it die, it bringeth forth much fruit. John 12:24

The seed is the sacrificial x₀. Left intact, it remains one (abideth alone). Fallen into the ground — sacrifice(x₀) — the yield is much fruit (n ≫ 1). The condition for fruit is the ground being capable of receiving — C ≥ ε. Barren ground cannot produce.

Implications

T₂ is the productive expression of T₁. T₁ ruled out zero. T₂ shows that not-zero-but-sufficient means fruitful. Together they begin the ascent from existence to production.

An agent that refuses to sacrifice anything — that protects every asset as zero-sum — is either at minimum C (barely existing) or has misdiagnosed its surplus. The test for abundance is the willingness to give.

Counter-example

An agent that gives from D (the desire function) instead of C will deplete. D is bounded — see T₈. Giving from D reduces D. Only giving from C is productive (T₄). T₂ is about giving from sufficient C, not about any giving.

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