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The Bare Binary Biconditional: Its Quantum-Calculus Correspondence and Implications for the Equal-Weight Envariance Gap

Christopher M Struck

Published Jul 30, 2026• 0 citations
Publisher

Abstract

This paper presents a cross-disciplinary synthesis connecting bare binary symmetry, half-probability, Bernoulli product measure, Lebesgue measure, integration, and existing symmetric binary quantum measurement. Within the Bernoulli family, it establishes an exact correspondence between exchange invariance of a two-outcome law, the value p = 1/2, Lebesgue measure under dyadic pushforward, and almost-sure convergence to half-frequency. It then addresses the equal-weight envariance gap by stating an explicit physical-bareness premise, supported by but not derived from the cited foundational results, and shows that the value this premise selects is the same one the correspondence independently distinguishes. The contribution is the connection of these independently developed mathematical and physical results into a single bounded structure. It also presents the binary-Lebesgue construction as a clearer, exact replacement for the familiar but misleading Zeno picture of calculus and limits.

DOI
10.5281/zenodo.21705118

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Definite Outcomes at Infinity: The Bare Binary Biconditional, Repeated QND Measurement, and the Quantum Convergence Conjecture

Christopher M Struck

The Bare Binary Biconditional selected 1/2 as the unique exchange-invariant law on a bare binary alternative and connected independent repetitions of that law to half-frequency and Lebesgue measure. This paper develops the separate dynamic question that originally motivated the biconditional and the present research series: in what precise sense can an accumulating observation record approach a definite binary outcome? A self-contained classical result is given first. Let Θ ∈ {0,1} have the symmetric prior, let observations be conditionally independent with strictly positive and distinguishable likelihoods, and let 𝑞𝑛 = 𝑃(Θ = 1 ∣ 𝑌1,…,𝑌𝑛). Then (𝑞𝑛) is a bounded martingale beginning at 1/2, and identifiability, not martingale convergence alone, gives 𝑞𝑛 → 1{Θ=1} almost surely. Its two endpoints therefore have equal probability. A binary-probe specialization places the observation record itself in {0,1}ℕ; a Gaussian model is retained as a continuous-observation comparison. Repeated independent runs, rather than the dependent observations within one run, produce the fair binary endpoint record connected to [1]. Bauer and Bernard’s repeated quantum non-demolition measurement theorem supplies a bounded quantum counterpart: pointer weights are martingales; under their nondegeneracy condition they converge to pointer-state endpoints; the endpoint probabilities are the initial pointer weights; and relative entropy governs the asymptotic decay of losing weights. Their theorem does not choose 1/2. This paper imports that starting value only under the explicit bare symmetric premise of [1]. Reproducible classical simulations illustrate finite-depth posterior behavior and information-rate scaling without simulating quantum dynamics or treating finite posteriors as completed limits. The remaining claim that observation gives physical significance to this convergence structure is stated separately as the author’s quantum convergence conjecture. Note clarifying this paper’s conclusion: “Particle-like” refers here to a definite observation outcome. In the repeated-QND setting analyzed, the quantum wave-function measurement model mathematically predicts convergence to a definite, particle-like observation outcome. The phrase “not evidence that the object has ceased to be a wave” means only that a particle-like observation is consistent with how the quantum wave function predicts the system should behave upon measurement. This clarification does not deny wave-function collapse or address wave or particle behavior outside the observation process discussed.

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Quantum probabilities from quantum entanglement: experimentally unpacking the Born rule

Jérémie Harris, Frédéric Bouchard, Enrico Santamato +3 more

The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying the quantum foundational bedrock. A major family of derivations is based on envariance, a recently discovered symmetry of entangled quantum states. Here, we identify and experimentally test three premises central to these envariance-based derivations, thus demonstrating, in the microworld, the symmetries from which the Born rule is derived. Further, we demonstrate envariance in a purely local quantum system, showing its independence from relativistic causality.

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Symmersion and the Born Rule A Structural Reconstruction from Conjugate Binary Recurrence to Quadratic Event Frequency

Eric Needham

Symmersion and the Born Rule develops a conditional structural reconstruction of quantum probability from conserved binary distinction, exchange completion, reciprocal conjugation, quadratic norm, orthogonal outcome channels, and stochastic event closure. The Born rule assigns the probability of an outcome represented by an orthogonal projector Π_j as: P_j = ⟨ψ|Π_j|ψ⟩ For a state expanded in an orthonormal basis, |ψ⟩ = Σ_j c_j |j⟩ the corresponding outcome weights are: P_j = |c_j|² Complex amplitude, conjugation, quadratic norm, orthogonal decomposition, normalization, and stochastic realization are individually standard. The question addressed here is why they may belong to one structural architecture, and why the reciprocal pair underlying complex conjugation should exist in the first place. The paper introduces symmersion as the minimal exchange-completion of a realized asymmetric law. If J is an involutive exchange satisfying J² = I, then the exchange action on a law A is: A -> JAJ and its symmersion is: Symm_J(A) = {A, JAJ} with the singleton case understood when JAJ = A. For a positive binary projective law with ratio coordinate r, the exchange-conjugate map satisfies: f_JAJ(r) = 1 / f_A(1/r) Whenever the original map has a positive fixed point r*, the exchange-conjugate map has the reciprocal fixed point: r*^-1 The distinguishability tension tau(r) = r / (1 + r)² is invariant under inversion: tau(1/r) = tau(r) Symmersion therefore produces reciprocal conjugacy base-generally. It does not by itself select the golden ratio. Applied to the conserved binary recurrence L_(n+1) = L_n + S_nS_(n+1) = L_n with matrix A_+ = [[1, 1], [1, 0]] the projective ratio satisfies: r_(n+1) = 1 + 1/r_n Its unique positive fixed point is the golden ratio: phi = (1 + sqrt(5)) / 2 The exchange-conjugate law is: A_- = J A_+ J = [[0, 1], [1, 1]] and its positive fixed point is: phi^-1 The completed law orbit therefore contains reciprocal equilibria: {phi, phi^-1} The golden ratio enters only through the upstream selection of the conserved binary recurrence. The later reciprocal and complex constructions do not independently privilege phi. Symmersion initially produces an orbit of laws, while the amplitude construction requires a simultaneous reciprocal pair within one state. The Equivariant Pairing Proposition supplies this step. Under the exchange embedding x -> (x, Jx) the paired evolution A_+ ⊕ A_- preserves the relation: x_n^- = Jx_n^+ Consequently, the projective coordinates remain exactly reciprocal at every step: r_n^- = 1 / r_n^+ This is not merely an equilibrium relation. The reciprocal pair is maintained throughout the paired evolution. One interpretive commitment remains explicit: the complete physical state is represented by the exchange orbit, realized as the equivariantly paired state, rather than by one arbitrarily selected representative. The reciprocal fixed-point pair is then extended continuously. If F satisfies F(0) = 1F(s + t) = F(s)F(t)F(1) = phi and F is continuous and positive, then: F(s) = phi^s with: F(-s) = phi^(-s) = 1/F(s) The continuous reciprocal branches are therefore: r_+(s) = A phi^sr_-(s) = A phi^(-s) Their product is invariant: r_+(s) r_-(s) = A² and their ratio carries signed asymmetry: r_+(s) / r_-(s) = phi^(2s) After phase calibration, the reciprocal pair can be mapped to a complex amplitude. Under this correspondence: branch product -> squared modulusbranch ratio -> phasebranch exchange -> complex conjugation The state is linear in its amplitude carrier but quadratic in its invariant completion: Psi* Psi = |Psi|² For two orthogonal amplitude magnitudes a and b, Euclidean composition gives: combined magnitude = sqrt(a² + b²) If a non-negative observable weight W satisfies: W(sqrt(a² + b²)) = W(a) + W(b) and W(1) = 1 then the unique admissible weight is: W(a) = a² The exponent two is therefore fixed by Euclidean orthogonal composition together with non-negative additivity and normalization. Repeatable resolution requires an idempotent channel map: P² = P If the resolved component and its complement also satisfy the Pythagorean identity ||ψ||² = ||Pψ||² + ||(I - P)ψ||² for every ψ, then P must be an orthogonal projector: P = P† For a complete family of mutually orthogonal projectors satisfying Π_j Π_k = delta_jk Π_j and Σ_j Π_j = I the norm decomposes as: ||ψ||² = Σ_j ||Π_j ψ||² For a normalized state: Σ_j ||Π_j ψ||² = 1 and the quadratic channel weights are: P_j = ||Π_j ψ||² = ⟨ψ|Π_j|ψ⟩ At this stage the Born weights are established as normalized channel weights. To obtain terminal event frequencies, the paper enters a finite-dimensional, norm-preserving Wiener closure class with a single closure rate Gamma_cl > 0. Within this class, Ito norm preservation fixes the drift-diffusion relation, each spectral weight is a bounded martingale, and absorption into one terminal event sector yields: Prob(event j) = |⟨j|ψ_0⟩|² The numerical value of Gamma_cl affects the temporal profile of closure but not the terminal Born frequencies. The stochastic mechanism is mathematically continuous with the established collapse-model and quantum-state-diffusion literature. The proposed contribution is not the discovery of the martingale route itself. It is the structural argument motivating the reciprocal geometry, the transition into a norm-preserving closure class, and the exclusion of an independent rate arising solely from branch orientation. The full conditional chain is: unity-> co-constituted distinction-> conserved asymmetric recurrence-> exchange-conjugate law orbit-> equivariantly paired state-> exact reciprocal coordinates-> continuous reciprocal flow-> complex conjugate amplitude-> quadratic norm-> orthogonal channel weights-> Born event frequency The claim boundary is explicit. The matrix identities are elementary and their novelty is claimed architecturally rather than computationally. Symmersion produces reciprocal structure but does not independently select phi. The physical representation of the complete state by the exchange orbit remains an interpretive premise. The Euclidean composition law, the selected stochastic closure class, and the single source-level closure rate are stated assumptions or inherited structural premises. The paper does not claim an unconditional derivation of all quantum mechanics, a new physical sector, or an experimental confirmation of symmersion. The companion validation ledger, validate_symmersion.py, implements 63 checks keyed to the paper’s numbered equations and results. SymPy verifies the implemented algebraic identities exactly, including exchange conjugacy, equivariant pairing, reciprocal flow, amplitude reconstruction, quadratic composition, orthogonality checks, and symbolic Ito drift-diffusion relations. NumPy tests the numerical consistency of selected dynamical and stochastic claims through direct iteration, random-law checks, and Monte Carlo simulation. The stochastic simulations reproduce two-state and three-state Born frequencies within the stated Monte Carlo tolerances, support the martingale behaviour of spectral weights, and are statistically consistent with closure-rate independence of terminal frequencies. The script does not mechanically prove the functional uniqueness arguments, interpretive premises, imported literature results, or physical validity. 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We study a conditional uniqueness problem for event weights in finite quantum contexts and a protected physical model in which its orbital premise can be made explicit. A finite contextual game consists of a normalized state and a finite PVM, with event size x(E) = ∥PEψ∥². We define stabilized orbital equivalence of binary terminal states by isometries on nonconsequence factors while fixing a binary consequence register. For orthogonal binary terminals the stabilized orbit is completelyclassified by the two branch norms. It follows that any positive, normalized, finitely additive event valuation that is invariant on these orbits and is defined on a ternarily rich class must equal x(E). The five assumptions are logically independent. We then distinguish geometric orbital equivalence, physical inaccessibility, and the separate normative premise of equal value. Uniform physical sealing is characterized by factorization through the conditional expectation onto an accessible algebra, with dimension-free diamond-norm stability bounds. Finally, a four-qubit collective-noise sector supplies a noiseless logical qutrit realizing ternary richness and an ideal binary readout at the selected grain. Under explicit post-readout sealing and interface-closure assumptions, same-size acts have the same admissible future consequence statistics. In a restricted instantaneous- protection model, total 2-local Hamiltonians cannot perform an informative protected readout, whereas an exact 3-local construction exists. The physical results do not derive event additivity or orbital value indifference; those remain explicit representation and normative premises.

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This paper presents a cross-disciplinary synthesis connecting bare binary symmetry, half-probability, Bernoulli product measure, Lebesgue measure, integration, and existing symmetric binary quantum measurement. Within the Bernoulli family, it establishes an exact correspondence between exchange invariance of a two-outcome law, the value p = 1/2, Lebesgue measure under dyadic pushforward, and almost-sure convergence to half-frequency. It then addresses the equal-weight envariance gap by stating an explicit physical-bareness premise, supported by but not derived from the cited foundational results, and shows that the value this premise selects is the same one the correspondence independently distinguishes. The contribution is the connection of these independently developed mathematical and physical results into a single bounded structure. It also presents the binary-Lebesgue construction as a clearer, exact replacement for the familiar but misleading Zeno picture of calculus and limits.
The Bare Binary Biconditional selected 1/2 as the unique exchange-invariant law on a bare binary alternative and connected independent repetitions of that law to half-frequency and Lebesgue measure. This paper develops the separate dynamic question that originally motivated the biconditional and the present research series: in what precise sense can an accumulating observation record approach a definite binary outcome? A self-contained classical result is given first. Let Θ ∈ {0,1} have the symmetric prior, let observations be conditionally independent with strictly positive and distinguishable likelihoods, and let 𝑞𝑛 = 𝑃(Θ = 1 ∣ 𝑌1,…,𝑌𝑛). Then (𝑞𝑛) is a bounded martingale beginning at 1/2, and identifiability, not martingale convergence alone, gives 𝑞𝑛 → 1{Θ=1} almost surely. Its two endpoints therefore have equal probability. A binary-probe specialization places the observation record itself in {0,1}ℕ; a Gaussian model is retained as a continuous-observation comparison. Repeated independent runs, rather than the dependent observations within one run, produce the fair binary endpoint record connected to [1]. Bauer and Bernard’s repeated quantum non-demolition measurement theorem supplies a bounded quantum counterpart: pointer weights are martingales; under their nondegeneracy condition they converge to pointer-state endpoints; the endpoint probabilities are the initial pointer weights; and relative entropy governs the asymptotic decay of losing weights. Their theorem does not choose 1/2. This paper imports that starting value only under the explicit bare symmetric premise of [1]. Reproducible classical simulations illustrate finite-depth posterior behavior and information-rate scaling without simulating quantum dynamics or treating finite posteriors as completed limits. The remaining claim that observation gives physical significance to this convergence structure is stated separately as the author’s quantum convergence conjecture. Note clarifying this paper’s conclusion: “Particle-like” refers here to a definite observation outcome. In the repeated-QND setting analyzed, the quantum wave-function measurement model mathematically predicts convergence to a definite, particle-like observation outcome. The phrase “not evidence that the object has ceased to be a wave” means only that a particle-like observation is consistent with how the quantum wave function predicts the system should behave upon measurement. This clarification does not deny wave-function collapse or address wave or particle behavior outside the observation process discussed.
The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying the quantum foundational bedrock. A major family of derivations is based on envariance, a recently discovered symmetry of entangled quantum states. Here, we identify and experimentally test three premises central to these envariance-based derivations, thus demonstrating, in the microworld, the symmetries from which the Born rule is derived. Further, we demonstrate envariance in a purely local quantum system, showing its independence from relativistic causality.
Symmersion and the Born Rule develops a conditional structural reconstruction of quantum probability from conserved binary distinction, exchange completion, reciprocal conjugation, quadratic norm, orthogonal outcome channels, and stochastic event closure. The Born rule assigns the probability of an outcome represented by an orthogonal projector Π_j as: P_j = ⟨ψ|Π_j|ψ⟩ For a state expanded in an orthonormal basis, |ψ⟩ = Σ_j c_j |j⟩ the corresponding outcome weights are: P_j = |c_j|² Complex amplitude, conjugation, quadratic norm, orthogonal decomposition, normalization, and stochastic realization are individually standard. The question addressed here is why they may belong to one structural architecture, and why the reciprocal pair underlying complex conjugation should exist in the first place. The paper introduces symmersion as the minimal exchange-completion of a realized asymmetric law. If J is an involutive exchange satisfying J² = I, then the exchange action on a law A is: A -> JAJ and its symmersion is: Symm_J(A) = {A, JAJ} with the singleton case understood when JAJ = A. For a positive binary projective law with ratio coordinate r, the exchange-conjugate map satisfies: f_JAJ(r) = 1 / f_A(1/r) Whenever the original map has a positive fixed point r*, the exchange-conjugate map has the reciprocal fixed point: r*^-1 The distinguishability tension tau(r) = r / (1 + r)² is invariant under inversion: tau(1/r) = tau(r) Symmersion therefore produces reciprocal conjugacy base-generally. It does not by itself select the golden ratio. Applied to the conserved binary recurrence L_(n+1) = L_n + S_nS_(n+1) = L_n with matrix A_+ = [[1, 1], [1, 0]] the projective ratio satisfies: r_(n+1) = 1 + 1/r_n Its unique positive fixed point is the golden ratio: phi = (1 + sqrt(5)) / 2 The exchange-conjugate law is: A_- = J A_+ J = [[0, 1], [1, 1]] and its positive fixed point is: phi^-1 The completed law orbit therefore contains reciprocal equilibria: {phi, phi^-1} The golden ratio enters only through the upstream selection of the conserved binary recurrence. The later reciprocal and complex constructions do not independently privilege phi. Symmersion initially produces an orbit of laws, while the amplitude construction requires a simultaneous reciprocal pair within one state. The Equivariant Pairing Proposition supplies this step. Under the exchange embedding x -> (x, Jx) the paired evolution A_+ ⊕ A_- preserves the relation: x_n^- = Jx_n^+ Consequently, the projective coordinates remain exactly reciprocal at every step: r_n^- = 1 / r_n^+ This is not merely an equilibrium relation. The reciprocal pair is maintained throughout the paired evolution. One interpretive commitment remains explicit: the complete physical state is represented by the exchange orbit, realized as the equivariantly paired state, rather than by one arbitrarily selected representative. The reciprocal fixed-point pair is then extended continuously. If F satisfies F(0) = 1F(s + t) = F(s)F(t)F(1) = phi and F is continuous and positive, then: F(s) = phi^s with: F(-s) = phi^(-s) = 1/F(s) The continuous reciprocal branches are therefore: r_+(s) = A phi^sr_-(s) = A phi^(-s) Their product is invariant: r_+(s) r_-(s) = A² and their ratio carries signed asymmetry: r_+(s) / r_-(s) = phi^(2s) After phase calibration, the reciprocal pair can be mapped to a complex amplitude. Under this correspondence: branch product -> squared modulusbranch ratio -> phasebranch exchange -> complex conjugation The state is linear in its amplitude carrier but quadratic in its invariant completion: Psi* Psi = |Psi|² For two orthogonal amplitude magnitudes a and b, Euclidean composition gives: combined magnitude = sqrt(a² + b²) If a non-negative observable weight W satisfies: W(sqrt(a² + b²)) = W(a) + W(b) and W(1) = 1 then the unique admissible weight is: W(a) = a² The exponent two is therefore fixed by Euclidean orthogonal composition together with non-negative additivity and normalization. Repeatable resolution requires an idempotent channel map: P² = P If the resolved component and its complement also satisfy the Pythagorean identity ||ψ||² = ||Pψ||² + ||(I - P)ψ||² for every ψ, then P must be an orthogonal projector: P = P† For a complete family of mutually orthogonal projectors satisfying Π_j Π_k = delta_jk Π_j and Σ_j Π_j = I the norm decomposes as: ||ψ||² = Σ_j ||Π_j ψ||² For a normalized state: Σ_j ||Π_j ψ||² = 1 and the quadratic channel weights are: P_j = ||Π_j ψ||² = ⟨ψ|Π_j|ψ⟩ At this stage the Born weights are established as normalized channel weights. To obtain terminal event frequencies, the paper enters a finite-dimensional, norm-preserving Wiener closure class with a single closure rate Gamma_cl > 0. Within this class, Ito norm preservation fixes the drift-diffusion relation, each spectral weight is a bounded martingale, and absorption into one terminal event sector yields: Prob(event j) = |⟨j|ψ_0⟩|² The numerical value of Gamma_cl affects the temporal profile of closure but not the terminal Born frequencies. The stochastic mechanism is mathematically continuous with the established collapse-model and quantum-state-diffusion literature. The proposed contribution is not the discovery of the martingale route itself. It is the structural argument motivating the reciprocal geometry, the transition into a norm-preserving closure class, and the exclusion of an independent rate arising solely from branch orientation. The full conditional chain is: unity-> co-constituted distinction-> conserved asymmetric recurrence-> exchange-conjugate law orbit-> equivariantly paired state-> exact reciprocal coordinates-> continuous reciprocal flow-> complex conjugate amplitude-> quadratic norm-> orthogonal channel weights-> Born event frequency The claim boundary is explicit. The matrix identities are elementary and their novelty is claimed architecturally rather than computationally. Symmersion produces reciprocal structure but does not independently select phi. The physical representation of the complete state by the exchange orbit remains an interpretive premise. The Euclidean composition law, the selected stochastic closure class, and the single source-level closure rate are stated assumptions or inherited structural premises. The paper does not claim an unconditional derivation of all quantum mechanics, a new physical sector, or an experimental confirmation of symmersion. The companion validation ledger, validate_symmersion.py, implements 63 checks keyed to the paper’s numbered equations and results. SymPy verifies the implemented algebraic identities exactly, including exchange conjugacy, equivariant pairing, reciprocal flow, amplitude reconstruction, quadratic composition, orthogonality checks, and symbolic Ito drift-diffusion relations. NumPy tests the numerical consistency of selected dynamical and stochastic claims through direct iteration, random-law checks, and Monte Carlo simulation. The stochastic simulations reproduce two-state and three-state Born frequencies within the stated Monte Carlo tolerances, support the martingale behaviour of spectral weights, and are statistically consistent with closure-rate independence of terminal frequencies. The script does not mechanically prove the functional uniqueness arguments, interpretive premises, imported literature results, or physical validity. Those elements are proved in the paper, supported through cited literature, or stated explicitly as assumptions. The result is a completed finite-dimensional structural reconstruction under declared premises, supplying the previously missing exchange-completion origin of the simultaneous reciprocal pair on which the later Born architecture depends. Part of the ENSO Framework corpus. For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com
The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality relation for a pair of arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables. Applications of this new interpretation to measurement theory are discussed briefly.
We study a conditional uniqueness problem for event weights in finite quantum contexts and a protected physical model in which its orbital premise can be made explicit. A finite contextual game consists of a normalized state and a finite PVM, with event size x(E) = ∥PEψ∥². We define stabilized orbital equivalence of binary terminal states by isometries on nonconsequence factors while fixing a binary consequence register. For orthogonal binary terminals the stabilized orbit is completelyclassified by the two branch norms. It follows that any positive, normalized, finitely additive event valuation that is invariant on these orbits and is defined on a ternarily rich class must equal x(E). The five assumptions are logically independent. We then distinguish geometric orbital equivalence, physical inaccessibility, and the separate normative premise of equal value. Uniform physical sealing is characterized by factorization through the conditional expectation onto an accessible algebra, with dimension-free diamond-norm stability bounds. Finally, a four-qubit collective-noise sector supplies a noiseless logical qutrit realizing ternary richness and an ideal binary readout at the selected grain. Under explicit post-readout sealing and interface-closure assumptions, same-size acts have the same admissible future consequence statistics. In a restricted instantaneous- protection model, total 2-local Hamiltonians cannot perform an informative protected readout, whereas an exact 3-local construction exists. The physical results do not derive event additivity or orbital value indifference; those remain explicit representation and normative premises.