Dependency and Scope

What this paper does

AP12 derives the uncertainty principle from the axioms.

Three results.

1: ℏ is forced as the action-scale of the minimum record. The five-step forcing argument closes KS-Q.8 (left LIVE — HARD by AP09 7.3), conditional on KS-L.4 (a new and strictly smaller exposure).

2: The position-momentum commutation relation [x̂, p̂] = iℏ is derived from Stone’s theorem applied to the local translation group on the AP09 manifold. The Heisenberg uncertainty bound ΔxΔp ≥ ℏ/2 follows from Cauchy-Schwarz.

3: The energy-time uncertainty relation ΔE·Δt_A ≥ ℏ/2 is derived in its Mandelstam-Tamm form, using Axiom R as the time direction and Stone’s theorem applied to time evolution.

The uncertainty principle is a theorem of the record algebra read on the derived manifold. No physics imported.

Dependencies

AP20 4 (axioms; fenced KS-P.1 / KS-P.2). Load-bearing for the indivisibility of ε (Axiom B) and the single time direction (Axiom R).

AP10 4 (the Lorentzian signature; formal metric-level derivation fenced KS-D.4). Load-bearing for the manifold structure.

AP09 3.2 (Hilbert space from axioms). Load-bearing for the operator algebra.

AP09 7.3 (Stone’s theorem applied to time evolution; Schrödinger equation). Load-bearing for the Hamiltonian as generator and for the form U(t) = e^{−iHt/ℏ}.

AP10 (N = 3 spatial dimensions, AS = manifold). Load-bearing for the translation group structure.

AP20 5 (AS = manifold, EH and QRA proven). Load-bearing for the smooth manifold structure on which translations act.

AP06 4 (Landauer bound). Referenced for the thermodynamic cost of records (k_BT ln 2 per bit), companion to the quantum cost ℏ.

Axiom mapping

Axiom S → not directly load-bearing in 1–3. Enters the Cauchy-Schwarz factor of ½ indirectly via the two-sector structure.

Axiom B → ℏ. The minimum viable splinter ε is indivisible; the action of one minimum record is ℏ. Forced by Stone + the indivisibility of ε + the absence of dimensionless parameters in {S, B, R, C}.

Axiom R → time evolution. Time is the direction of record accumulation. The Hamiltonian is the generator of time evolution. The Mandelstam-Tamm energy-time bound follows.

Axiom C → finite causal bound. Operates indirectly through the Lorentzian manifold structure on which translations act.

Kill switch summary

KS-L.1 (CLOSED — was KS-Q.8): ℏ is derived as the action-scale of the minimum record. The five-step forcing argument shows: Stone’s theorem gives a unique scale factor, the axioms produce a unique action scale, no dimensionless parameter intervenes, and the indivisibility of ε pins the correspondence.

KS-L.2 (NEW): The commutation relation [x̂, p̂] = iℏ depends on the manifold admitting a well-defined local translation group. If the manifold has structure that modifies the translation generator at the Planck scale, the relation might receive corrections. LIVE.

KS-L.3 (NEW): The energy-time uncertainty relation depends on the Mandelstam-Tamm derivation, which requires the Heisenberg equation of motion. If the axioms produce corrections to unitary evolution, the energy-time relation inherits those corrections. LIVE.

KS-L.4 (NEW): Step 3 of the 1.3 forcing argument claims no dimensionless parameter is available from the axioms to modify the relation α = (minimum action). If a dimensionless constant derivable from {S, B, R, C} is shown to enter the relation, the forcing argument weakens. LIVE — strictly smaller exposure than KS-Q.8.

Structural relationships

AP06 (Leakage Constant): ε > 0 is the minimum break. AP12 1.3 uses the indivisibility of ε to force ℏ.

AP09 (Quantum mechanics): provides the Hilbert space, the Born rule, the Schrödinger equation. AP12 closes the ℏ identification AP09 left open (KS-Q.8).

AP10 (Dimension): N = 3 is load-bearing for the translation group structure in 2.

AP15 (Connection): the photon’s wavelength is bounded below by the uncertainty bound. ℏ is the scale factor in the path sum.

AP20 (Proof): EH and QRA proven. The AS IS the manifold; translations are well-defined.