Cite non-vacuous kernel results in place of readWriteSourced_not_qm (#1188)

papers/GR.md

@@ -257,7 +257,7 @@ $$\boxed{\begin{aligned}&\text{implementation locality} + \text{phase-free richn
 
 holds on every nonempty finite carrier. It refines the OI⁺ characterization, and it is not a claim that bare observation incompleteness entails these principles: the normalization half of operational validity is a consequence of implementation generation, an intervention realized by one protocol preserving probability in aggregate, and neither principle is supplied by the others, the witnesses being stated for the Lie-rank condition that phase-free richness supplies (kernel: `redundancy_fails`, `lieRank_not_redundant`). The nonclassical repertoire the equivalence consumes is one continuously driven transition together with the exchanges of distinguishable states; the quarter phase is dispensable at every level. The exchange clause is not reducible to the powers of a single cyclic permutation: on an even carrier one driven transition with one full cycle generates a conjugate of the real antisymmetric matrices, which reaches no population difference (kernel: `not_hControl_evenCycle`), so one full cycle together with one adjacent exchange is the two-element discrete form the kernel formalizes, sufficient for every exchange (kernel: `phaseFree_of_cyclic`), with no claim that a smaller discrete resource is excluded, and no stronger minimality of the one driven transition is claimed. Whether that driven transition can be supplied by a discrete resource is the question of the substratum-source form below. Inverse accessibility is not a hypothesis of the equivalence: Lie-rank richness gives full composite unitary control unconditionally, and on a well-formed theory full control yields inverse accessibility, well-formedness being supplied by implementation locality with embedded observation (kernel: `inverseAccessibility_of_lieRank`). An implementation class closed under the adjoint remains a property of exact quantum mechanics (kernel: `reversibleImplementationLocality_of_qm` in `OIBridge/MicroscopicReversibility.lean`), and, with the other two principles, the class with that closure and the class without it characterize the same theory (kernel: `oiPlusPos_iff_oiPlusElem`).
 
-**Substratum-source form.** The three primitive-source principles are jointly supplied by one object: an implementation architecture — a class of admissible implementation operators closed under the operations of a finite operational theory — that is stable under an uncoupled spectator, invariant under relabelling of the carrier, closed under the adjoint, and drives the elementary transitions. A theory generated by such an architecture is exact finite endomorphic operational quantum mechanics on every nonempty finite carrier, and that quantum mechanics is generated by one (kernel: `genTheory_qm_of_quantumArchitecture`, `qm_generated_by_quantumArchitecture` in `OIBridge/SubstratumSource.lean`). The concrete OI substratum — finite states and bijective microscopic dynamics (Axiom 2) — together with one further intervention principle, that a relative phase on a single state is a selectable intervention at every level, supplies the structural part in full. The phase intervention enters as an assumption on the substratum class, not as a consequence of the finite states with the bijective read-write dynamics: every isometry that dynamics and the stated observer access supply carries the all-ones vector to a multiple of itself; on every finite carrier with two or more states, the theory generated by an architecture whose admissible isometries do so fails the requirement of a quarter phase on every state at every level, the failure being exhibited at level one, with no claim that the quarter phase is unavailable at each composite level; and the quarter phase carries that vector off its ray (kernel: `permClass_onesFixing`, `onesFixing_not_phasesAvailable`, `substratumClass_not_onesFixing` in `OIBridge/PhaseSource.lean`). An operator carrying the all-ones vector off its ray is therefore necessary to any access that supplies the phases, and is not shown sufficient; the access assumed is the quarter-phase intervention itself, a member of the class as stated (kernel: `phase_monomial` in `OIBridge/SubstratumInterface.lean`). The interventions of the substratum so stated have monomial observable operators, a permutation of a diagonal (kernel: `bijectiveOperator_monomial`, `phaseOperator_monomial` in `OIBridge/SubstratumInterface.lean`), and that class, with nothing added, is closed under composition, coarse-graining, and ancilla blocks, stable under an uncoupled spectator, invariant under relabelling, and closed under the adjoint — the reversal of a bijective intervention is the inverse bijection and the reversal of a phase is its conjugate — with the exchanges of distinguishable states and the phases among its members (kernel: `substratumClass_structurallyClosed`, `bijectiveOperator_conjTranspose`, `phaseOperator_conjTranspose` in `OIBridge/StructuralClosure.lean`). What it does not supply, under finite bijective read-write dynamics, is the continuously tunable state-mixing operation: a monomial conjugation preserves diagonal states, whereas a continuously driven transition between two distinguishable states passes through operators with two nonzero entries in one row, which no bijection realizes. Finite reversible read-write dynamics, even with genuine hidden-memory and readback behavior, does not itself generate quantum state mixing: a selectable local coupling that is a bijection at every parameter value induces only permutation operators, a strict interpolation toward the exchange is not a bijection, and the memory-swap coupling moves information bidirectionally while remaining monomial (kernel: `readWriteSourced_not_qm`, `offDiagonal_interp_not_monomial`, `readWriteControl_independent` in `OIBridge/ReadWriteControl.lean`; `substratum_residual` in `OIBridge/StructuralClosure.lean`). Adding that one operation as an empirical controllability resource completes the quantum architecture: a structurally closed extension of the substratum class is a quantum architecture exactly when it drives the elementary transitions, it generates exact finite endomorphic operational quantum mechanics on every nonempty finite carrier when it does, and quantum mechanics is generated by such an extension (kernel: `substratum_extension_quantum_iff_drives`, `substratum_plus_control_qm`, `qm_generated_by_substratum_extension`):
+**Substratum-source form.** The three primitive-source principles are jointly supplied by one object: an implementation architecture — a class of admissible implementation operators closed under the operations of a finite operational theory — that is stable under an uncoupled spectator, invariant under relabelling of the carrier, closed under the adjoint, and drives the elementary transitions. A theory generated by such an architecture is exact finite endomorphic operational quantum mechanics on every nonempty finite carrier, and that quantum mechanics is generated by one (kernel: `genTheory_qm_of_quantumArchitecture`, `qm_generated_by_quantumArchitecture` in `OIBridge/SubstratumSource.lean`). The concrete OI substratum — finite states and bijective microscopic dynamics (Axiom 2) — together with one further intervention principle, that a relative phase on a single state is a selectable intervention at every level, supplies the structural part in full. The phase intervention enters as an assumption on the substratum class, not as a consequence of the finite states with the bijective read-write dynamics: every isometry that dynamics and the stated observer access supply carries the all-ones vector to a multiple of itself; on every finite carrier with two or more states, the theory generated by an architecture whose admissible isometries do so fails the requirement of a quarter phase on every state at every level, the failure being exhibited at level one, with no claim that the quarter phase is unavailable at each composite level; and the quarter phase carries that vector off its ray (kernel: `permClass_onesFixing`, `onesFixing_not_phasesAvailable`, `substratumClass_not_onesFixing` in `OIBridge/PhaseSource.lean`). An operator carrying the all-ones vector off its ray is therefore necessary to any access that supplies the phases, and is not shown sufficient; the access assumed is the quarter-phase intervention itself, a member of the class as stated (kernel: `phase_monomial` in `OIBridge/SubstratumInterface.lean`). The interventions of the substratum so stated have monomial observable operators, a permutation of a diagonal (kernel: `bijectiveOperator_monomial`, `phaseOperator_monomial` in `OIBridge/SubstratumInterface.lean`), and that class, with nothing added, is closed under composition, coarse-graining, and ancilla blocks, stable under an uncoupled spectator, invariant under relabelling, and closed under the adjoint — the reversal of a bijective intervention is the inverse bijection and the reversal of a phase is its conjugate — with the exchanges of distinguishable states and the phases among its members (kernel: `substratumClass_structurallyClosed`, `bijectiveOperator_conjTranspose`, `phaseOperator_conjTranspose` in `OIBridge/StructuralClosure.lean`). What it does not supply, under finite bijective read-write dynamics, is the continuously tunable state-mixing operation: a monomial conjugation preserves diagonal states, whereas a continuously driven transition between two distinguishable states passes through operators with two nonzero entries in one row, which no bijection realizes. Finite reversible read-write dynamics, even with genuine hidden-memory and readback behavior, does not itself generate quantum state mixing: a selectable local coupling that is a bijection at every parameter value induces only permutation operators, a strict interpolation toward the exchange is not a bijection, and the memory-swap coupling moves information bidirectionally while remaining monomial (kernel: `readWriteOperator_eq_perm`, `offDiagonal_interp_not_monomial`, `readWriteControl_independent` in `OIBridge/ReadWriteControl.lean`; `substratum_residual` in `OIBridge/StructuralClosure.lean`). Adding that one operation as an empirical controllability resource completes the quantum architecture: a structurally closed extension of the substratum class is a quantum architecture exactly when it drives the elementary transitions, it generates exact finite endomorphic operational quantum mechanics on every nonempty finite carrier when it does, and quantum mechanics is generated by such an extension (kernel: `substratum_extension_quantum_iff_drives`, `substratum_plus_control_qm`, `qm_generated_by_substratum_extension`):
 
 $$\boxed{\begin{aligned}&\text{current OI substratum} + \text{continuous off-diagonal controllability}\\ &\qquad \iff\; \text{exact finite endomorphic operational QM}\end{aligned}}$$
 

papers/GR.tex

@@ -1853,7 +1853,7 @@ \subsubsection{3.3 Gauge-fixing and the dimensional
 operators, a strict interpolation toward the exchange is not a
 bijection, and the memory-swap coupling moves information
 bidirectionally while remaining monomial (kernel:
-\texttt{readWriteSourced\_not\_qm},
+\texttt{readWriteOperator\_eq\_perm},
 \texttt{offDiagonal\_interp\_not\_monomial},
 \texttt{readWriteControl\_independent} in
 \texttt{OIBridge/ReadWriteControl.lean}; \texttt{substratum\_residual}
@@ -5506,4 +5506,4 @@ \subsection{References}\label{references}}
 arXiv:quant-ph/0407049.
 
 \end{document}
-% source-sha256: a475a2a8af62ea262c9981a9e173253105a1234b18259f597f604d5201784992
+% source-sha256: 8b5387b1619b82c12a8c04c724f174d829e16feca6e160f6c575a80bdab73d62