IndisputableMonolith.QFT.Unitarity
The QFT.Unitarity module defines quantum states as unit vectors in Hilbert space and shows that unitary operators preserve norms and probabilities under the Recognition Science eight-tick clock. Researchers deriving QFT from the forcing chain would cite these results to connect ledger conservation to the Born rule. The module proceeds via definitions of QuantumState and UnitaryOperator followed by direct algebraic lemmas on norm preservation and reversibility.
claimLet $H$ be a Hilbert space. A quantum state is a vector $ψ ∈ H$ with $‖ψ‖ = 1$. An operator $U$ is unitary when $U^†U = I$, which implies $‖Uψ‖ = ‖ψ‖$ and therefore conserves probabilities via the Born rule. The module also derives that ledger conservation implies probability conservation and that eight-tick reversibility yields unitarity.
background
This module sits in the QFT domain and imports the RS time quantum τ₀ = 1 tick from Constants together with the discrete 8-tick cycle from EightTick, whose phases run through 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4. The Cost module supplies the recognition-cost functions that appear in ledger conservation. The local setting therefore treats quantum evolution as a discrete, ledger-constrained process on the eight-tick octave rather than as a continuous Schrödinger flow.
proof idea
This is a definition module with supporting lemmas. QuantumState and UnitaryOperator are introduced as the basic objects; unitary_preserves_norm follows by direct computation of the norm; probability_conservation and born_rule_consistent are obtained by applying the definition of the Born rule to the preserved norm; ledger_conservation and eight_tick_reversibility then supply the bridge to unitarity_from_ledger and arrowOfTime.
why it matters in Recognition Science
The module supplies the unitarity foundation required for any Recognition Science derivation of quantum field theory. It feeds the eight-tick reversibility and arrowOfTime results that close the loop from the forcing chain (T7) to observable probability conservation. Without these definitions and lemmas the ledger-to-probability link remains open.
scope and limits
- Does not fix the dimension of the Hilbert space or the particle spectrum.
- Does not derive explicit Hamiltonians or interaction vertices from the phi-ladder.
- Does not treat measurement collapse beyond the effective description already present.
- Does not compute scattering amplitudes or correlation functions.
depends on (3)
declarations in this module (15)
-
structure
QuantumState -
structure
UnitaryOperator -
theorem
unitary_preserves_norm -
theorem
probability_conservation -
theorem
born_rule_consistent -
theorem
ledger_conservation -
theorem
ledger_implies_probability -
theorem
unitarity_from_ledger -
theorem
unitarity_implies_reversibility -
theorem
eight_tick_reversibility -
theorem
collapse_is_effective -
def
arrowOfTime -
theorem
black_hole_unitarity -
def
summary -
structure
UnitarityFalsifier