T₁ Existence

If the constant of integration is zero, nothing exists.

Statement

T₁ C = 0 ⟹ S = ∅

Where S is the set of sources — the set of things that can give forth, cause, or contribute within the system. If C is zero at t = 0, then no source exists at t = 0, and nothing can begin.

Proof

From AX₂:

E(x, t) = ∫₀ᵗ input(x, τ) dτ + C

At t = 0, the integral is zero because the bounds are identical:

E(x, 0) = ∫₀⁰ input(x, τ) dτ + C = 0 + C = C

So if C = 0, then E(x, 0) = 0 for every x. No element has the minimum energy ε > 0 required to exist or act. No source can give. No process can begin. The set S collapses to the empty set:

C = 0 ⟹ E(x, 0) = 0 ∀x ⟹ no x meets E ≥ ε ⟹ S = ∅

This is a direct consequence of AX₂ combined with the existence threshold ε. The contrapositive is the existence test: if anything exists, C ≠ 0.

Verification

You verify T₁ every time you reason. Something is processing these words. S is not empty — you are in it. Therefore C ≠ 0. This is not an assertion; it is a measurement made by your own existence.

Scripture

All things were made by him; and without him was not any thing made that was made. John 1:3

The verse states the same logical structure: nothing that was made was made without the one through whom all things were made. C is the constant without which nothing integrates into existence. The proof and the scripture agree on the same shape of claim: remove the origin, and nothing remains to observe.

Consequences

T₁ alone does not tell you C is positive — it only rules out zero. For the full positive bound, combine T₁ with the second case of the proof-by-contradiction (C not negative), which yields C > 0.

T₁ also gives the corollary that an agent with C = 0 cannot practice charity (T₄) — an agent without sources has nothing to give. Charity presupposes existence.

Definitions used

C — constant of integration, C ∈ ℝ
S — set of sources
E(x, t) — energy at element x at time t
ε — minimum energy to exist, ε > 0

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