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000093 | 4. Countermodel R: deterministic relational readout traces
000094 | 
000095 | Primitive description: two preparation events r0,r1 and a repeatable read
000096 | event q. Allowed finite traces after rb are exactly q:b repeated n times,
000097 | for every n>=0. Composition concatenates compatible traces. There is no
000098 | primitive random choice or probabilistic mixture. This is a complete
000099 | deterministic prediction for every experiment in its declared vocabulary,
000100 | not an unweighted envelope with unspecified branching probabilities.
000101 | 
000102 | Its two residual classes can be represented by B={0,1}; this is a minimal
000103 | realization of the trace rule. A read relates b to continued b and report b.
000104 | The selective branches M_a are partial identities on {a}. They satisfy
000105 | M_a M_a=M_a and M_0 M_1=M_1 M_0=empty. Identity and these branches exhaust
000106 | the continued-system maps generated here. In particular they commute.
000107 | 
000108 | Preparations occur at the beginning. Erasing/discarding the continuing
000109 | system, re-preparing it mid-protocol, arbitrary functions, flips, external
000110 | coins and parallel subsystem operations are NOT part of this minimal model.
000111 | The common benchmark did not require them. They can be added as explicit
000112 | extensions, but that changes the countermodel. R realizes repeatable
000113 | distinctions with no nontrivial probability/convex structure or noncommuting
000114 | continued-system action. Delta-probability notation can describe it; this
000115 | does not make a full probabilistic mixture space necessary.
000116 | 
000117 | Discriminator: a controlled rb followed by the opposite report falsifies R
000118 | as specified. Reproducible nontrivial statistical mixtures would require an
000119 | extension of its preparation menu. Nominal variability alone cannot exclude
000120 | every larger deterministic model with hidden preparation contexts.
000121 | 
000122 | 5. Countermodel I: quantitative classical information with joint access
000123 | 
000124 | Primitive quantitative preparation data are distributions p_ab on four
000125 | joint report assignments (a,b), a,b in {-1,+1}. All convex mixtures are
000126 | allowed. The full joint readout is available. The minimal operational state
000127 | is exactly p because joint outcomes distinguish all its entries. Normalized
000128 | states form a simplex with four vertices, affine dimension3, unique pure
000129 | mixture coefficients. Real probabilities and randomization are explicit
000130 | model assumptions, not derived from observation.
000131 | 
000132 | X_a(p)_cd=indicator(c=a) p_cd; Z_b similarly filters d=b. Total X and total Z
000133 | are identity channels; selective X/Z branches commute. They repeat their own
000134 | results, preserve the other value, and permit joint readout/parity Y=ab.
000135 | Allow stochastic transformations and resets as an explicit extension beyond
000136 | readout. Erasure and observation then coexist, so observation does not require
000137 | global information conservation under every allowed transformation.
000138 | 
000139 | Define even preparation=(delta_+++delta_--)/2 and odd preparation=
000140 | (delta_+-+delta_-+)/2. Their individual X and Z marginal distributions agree,
000141 | but parity is always+1 in the first and-1 in the second. The serial protocol
000142 | X,Z,X preserves the first X report with probability1.
000143 | 
000144 | Discriminator: a genuine loss of the earlier X record caused by intermediate
000145 | Z falsifies these specified nondisturbing I instruments. It does not falsify
000146 | all classical dynamics; an ordinary classical disturbing instrument is an
000147 | alternative explanation.
000148 | 
000149 | 6. Countermodel D: operation-generated square with repeatable disturbance
000150 | 
000151 | Specify report probabilities and continued responses rather than importing
000152 | a particle or quantum algebra. The complete normalized single-system
000153 | response profile is (x,z) in [-1,1]^2, with
000154 |  P(a|X)=(1+a x)/2, P(b|Z)=(1+b z)/2.
000155 | All square-corner preparations, convex mixtures, square symmetries and the
000156 | following instruments are admitted, closed under finite serial/adaptive
000157 | composition and probabilistic coarse-graining. These choices define a
000158 | counterworld; they are not claimed logically necessary.
000159 | 
000160 | Use unnormalized v=(t,x,z), t>=0, |x|<=t, |z|<=t. The mass functional is t.
000161 |  X_a(v)=((t+a x)/2,(a t+x)/2,0),
000162 |  Z_b(v)=((t+b z)/2,0,(b t+z)/2).
000163 | Each branch is positive: s=(t+a x)/2>=0 and the output is (s,a s,0), a cone
000164 | element. The branch masses sum to t. Subsequent same-test outcomes repeat
000165 | with certainty, including X_a squared=X_a and X_-a X_a=0 here. Those stronger
000166 | idempotences are choices in this example, not the general lemma of section3.
000167 | 
000168 | Total channels are X(v)=(t,x,0) and Z(v)=(t,0,z). Each destroys the other
000169 | coordinate. The total channels commute (both orders give (t,0,0)); selected
000170 | branches generally do NOT. Do not report all measurement orderings as
000171 | noncommuting. For example X_+ Z_+(1,0,0)=(1/4,1/4,0), whereas
000172 | Z_+ X_+(1,0,0)=(1/4,0,1/4), distinguishable by a subsequent readout.
000173 | 
000174 | After X gives a, the normalized profile is (a,0); after Z gives b it is
000175 | (0,b); the last X is uniform. Thus X,Z,X preserves the first X result with
000176 | probability1/2, including on every corner preparation. This distinguishes D
000177 | from the specified I model through an actual common serial protocol.
000178 | 
000179 | D has four extreme profiles but affine dimension2. Its diagonal mixtures
000180 | are equal: (++ + --)/2=(+- + -+)/2=(0,0). Every allowed continuation acts
000181 | linearly on unnormalized profiles, so those mixtures give identical answers
000182 | for ALL allowed protocols, not just a sampled list of words. Their images
000183 | in I have different parity. An affine parity effect on D is impossible:
000184 | every affine g satisfies g(++)+g(--)=g(+-)+g(-+), contrary to the desired
000185 | parity indicator values1,1,0,0. Similarly no sharp joint X/Z readout exists:
000186 | its ++ branch would take corner values1,0,0,0, violating the same identity.
000187 | 
000188 | The simplex I and square D cannot be isomorphic preserving mixtures and
000189 | all operational answers: their affine dimensions and decomposition fibers
000190 | differ. R has only two predictive classes and no mixture operation, so is
000191 | inequivalent to either enriched theory. Embedding R as a restricted fragment
000192 | of I does not make the complete declared theories equivalent. The models
000193 | are deliberately different operational predictions, not three incompatible
000194 | claims about the hidden ontology of Nature.
000195 | 
000196 | D uses the familiar single-system square response geometry of generalized
000197 | probabilistic frameworks, not a derived quantum system. Only the generated
000198 | operations above are admitted: positivity on this cone does not establish
000199 | validity of every positive map on arbitrary multipartite composites. No
000200 | bipartite, spacetime, Bell, locality or thermodynamic completion is claimed.
000201 | 
000202 | Indeed D has a classical hidden implementation for these generators: keep
000203 | two hidden bits; an X read reports the first and independently reseeds the
000204 | second uniformly, with the roles exchanged for Z. Readout marginals and
000205 | their future answers reproduce D. Different corner-mixture procedures can
000206 | then have different inaccessible hidden parity although D identifies their
000207 | operational states. No preparation-noncontextuality condition was imposed.
000208 | This is why disturbance or a non-simplex operational quotient alone does
000209 | not exclude all classical implementations or establish fundamental quantum
000210 | structure. I and D remain distinguishable operational theories because I
000211 | admits nondisturbing joint/parity access and D does not.
000212 |