Near-Critical Registration Capture and a Born-Frequency Limit in Canonical ACRF-4: A Conditional Theorem, Native Critical-Surface Tests, and Finite-Size Convergence
This paper establishes a conditional Born-frequency limit within canonical ACRF-4 and tests each bridge of the proposed measurement mechanism without inserting a Born probability rule.The result begins from a previously timestamped finding that complementary linear cross-registration channels produce a canonical receiver-successor consequence partition closely following cos^2(Delta):sin^2(Delta), while a deterministic whole-channel first-threshold mechanism fails. The present work replaces that failed mechanism with a native ACRF critical surface derived from the exact local stability of the centered receiver odd mode. The corresponding multiplier is Lambda=(alpha-beta p_S) sech^2[(alpha+beta p_S-mu/2)m_S], with readiness margin q=1-Lambda.A Near-Critical Registration Capture Theorem is proved conditionally: if a macroscopic ready apparatus has a smooth nonzero density of microreceivers near q=0, if weak canonical consequence shifts native stability linearly as DeltaLambda=kappa Gamma+o(Gamma), and if complementary channels are exchangeable, then the newly committed microreceiver fraction is linear in Gamma. The macroscopic capture ratio therefore approaches the normalized canonical consequence ratio, and in the weak-coupling two-channel construction approaches cos^2(Delta).The theorem premises are then tested numerically using the frozen canonical ACRF-4 successor and retained-edge update. The source-induced retained receiver-edge response scales as |Delta p_S| proportional to Gamma^0.999977. Native near-critical crossing count is linear in aggregate canonical consequence with R^2=0.996247 in a power-enhanced test. A direct seven-angle closure test gives an underlying consequence-ratio error below 2.3 x 10^-5 and a finite native crossing-ratio mean absolute error of 0.0150. An independent finite-size study finds the crossing-ratio error scales as N^-0.4816 and, more directly, as N_cross^-0.4916, consistent with inverse-square-root finite-count convergence rather than a persistent systematic departure.Two negative controls are retained as part of the result: a whole-channel threshold race produces winner-take-all behavior, and a fixed positive microscopic consequence threshold creates a weak-coupling dead zone. The paper therefore isolates the remaining RT proof obligations to deriving the physical near-critical apparatus ensemble, proving the coarse-grained relation between canonical retained structure and detector memory, and extending the construction beyond two-channel measurements.
This paper establishes a conditional Born-frequency limit within canonical ACRF-4 and tests each bridge of the proposed measurement mechanism without inserting a Born probability rule.The result begins from a previously timestamped finding that complementary linear cross-registration channels produce a canonical receiver-successor consequence partition closely following cos^2(Delta):sin^2(Delta), while a deterministic whole-channel first-threshold mechanism fails. The present work replaces that failed mechanism with a native ACRF critical surface derived from the exact local stability of the centered receiver odd mode. The corresponding multiplier is Lambda=(alpha-beta p_S) sech^2[(alpha+beta p_S-mu/2)m_S], with readiness margin q=1-Lambda.A Near-Critical Registration Capture Theorem is proved conditionally: if a macroscopic ready apparatus has a smooth nonzero density of microreceivers near q=0, if weak canonical consequence shifts native stability linearly as DeltaLambda=kappa Gamma+o(Gamma), and if complementary channels are exchangeable, then the newly committed microreceiver fraction is linear in Gamma. The macroscopic capture ratio therefore approaches the normalized canonical consequence ratio, and in the weak-coupling two-channel construction approaches cos^2(Delta).The theorem premises are then tested numerically using the frozen canonical ACRF-4 successor and retained-edge update. The source-induced retained receiver-edge response scales as |Delta p_S| proportional to Gamma^0.999977. Native near-critical crossing count is linear in aggregate canonical consequence with R^2=0.996247 in a power-enhanced test. A direct seven-angle closure test gives an underlying consequence-ratio error below 2.3 x 10^-5 and a finite native crossing-ratio mean absolute error of 0.0150. An independent finite-size study finds the crossing-ratio error scales as N^-0.4816 and, more directly, as N_cross^-0.4916, consistent with inverse-square-root finite-count convergence rather than a persistent systematic departure.Two negative controls are retained as part of the result: a whole-channel threshold race produces winner-take-all behavior, and a fixed positive microscopic consequence threshold creates a weak-coupling dead zone. The paper therefore isolates the remaining RT proof obligations to deriving the physical near-critical apparatus ensemble, proving the coarse-grained relation between canonical retained structure and detector memory, and extending the construction beyond two-channel measurements.
This timestamped note records a controlled Registration Theory result concerning the quantum measurement problem and the Born rule.Using the frozen canonical ACRF-4 dynamics, complementary linear cross-registration amplitudes were tested as candidate measurement channels. In a ready-state construction, the normalized receiver-successor consequence followed the Born-shaped partition cos²(Δ):sin²(Δ) to high numerical accuracy, with a maximum absolute deviation of 3.84 × 10⁻⁴ across the tested angles.A second test removed any imposed probabilistic branch rule and allowed the two whole measurement channels to compete deterministically for first threshold crossing under repeated canonical successor evolution. That mechanism failed to reproduce Born frequencies, with a maximum absolute error of approximately 0.253. The failure is retained as a negative result rather than adjusted away.A subsequent preregistered microcapture test examined whether a macroscopic ensemble of near-ready receiver elements converts aggregate cross-registration consequence linearly into newly activated microscopic capture events. Exchangeability between matched complementary channels was recovered in the controlled 45° construction, but fixed-resolution microcapture linearity failed: a finite consequence threshold produces a zero-activation region in the weak-coupling limit.Together, these results narrow the remaining Registration Theory measurement problem. The quadratic Born-shaped consequence appears naturally in the canonical receiver dynamics, while the unresolved bridge is the conversion of that consequence into microscopic retained-record hazards. The next theorem target is a near-critical retained-margin mechanism in which continuous registration transfer acts on a distribution of detector elements close to commitment.The accompanying CSV and text files provide the numerical outputs and run summaries for the successor first-capture and preregistered microcapture tests.
This status report asks whether Registration Theory (RT) derives the qubit Born rule, P(+) = cos²φ, from its own deterministic registration dynamics, or only hosts a construction that reproduces it. RT's canonical finite model is the four-node adaptive consequence-registration field (ACRF-4).Main results:A conditional Born theorem. If the preparation–measurement map supplies (C1) an additive seed, (C2) a fine-state distribution equal to the planar shadow of the uniform 2-sphere, and (C3) a consequence angle locked to twice the preparation–apparatus angle, then ACRF's exact receiver seed yields P(+) = cos²φ. Condition (C2) is shown to be necessary (Cramér–Wold).Native quadratic structure. RT's bilateral phase invariant ζ satisfies tr(P̂_U P̂_V) = (1 + Re ζ²)/2 = cos²Δ, a gauge-invariant projector overlap with the Born form, and the angle doubling follows from RT's sign-gauge quotient. Square-root ledger coordinates realize a Pythagorean split, S = s cos²θ and T = s sin²θ.Obstructions:At detector readiness, the receiver seed is exactly bilinear and does not depend on the coordinate that sets the native Born weight.Phase-only sufficiency fails by explicit counterexample: identical ζ with opposite seed signs.Tangent isotropy is broken by the same boundary couplings that produce the seed.Readiness restricts the seed-visible fine state to a circle, which gives the wrong probability law.A full so(3) commutator algebra generated by retained history is generic and does not by itself select the spherical measure.Conclusion: RT currently hosts, but does not derive, the Born rule. The report identifies the single missing object, a preparation–measurement map, and lists preregistered tests, including a recorded prediction of failure for the isotropy test. It also states one falsifiable consequence: small deviations from Born frequencies concentrated near the outcome threshold.Every claim is labeled exact [E], numerically verified [V], or reported from project audits [R]. The accompanying Python script (rt_born_checks.py, NumPy, fixed seeds) reproduces every [V] claim.Computations, adversarial checks and drafting were assisted by AI language models. The author is responsible for the content.
This timestamp records a bounded milestone in Registration Theory (Physical Foundations): an explicit deterministic preparation-measurement-reset model that reproduces the binary quadratic projective weight νP,I(Bi) = ||Pi v||² / ||v||² in a controlled tangent/refinement limit, while isolating the remaining unproved physical bridge.The construction combines an already-derived isotropic rank-three H4 relational sector, a finite fine-registration fiber, a deterministic cyclic successor that visits each fine state exactly once per period, and the existing retained-registration detector result that post-threshold outcome is fixed by the sign of the incoming imbalance and reset returns the detector to readiness. Uniform finite-cycle visitation gives an exact empirical measure on the fine fiber. In the directional refinement limit, full-rank lattice counting in the isotropic H4 tangent geometry gives uniform spherical area. With the explicit linear seed coupling δ = ε(λ̂·nI + mP,I), the detector basins become spherical caps with relative area (1 + m)/2, and the projective contrast identity m = ⟨v,(2Pi−I)v⟩/||v||² yields the quadratic projective weight.The paper does not claim that canonical ACRF-4 has derived the Born rule. The decisive linear particle-apparatus coupling is an additional model assumption, not a result already forced by ACRF-4 or by the detector sign-preservation theorem. A sensitivity theorem shows that replacing the linear offset m by a general function f(m) changes the limiting frequency law to (1 + f(m))/2 on the unsaturated interval. Thus deterministic cycling and isotropic geometry alone do not select the Born form.The contribution is therefore an explicit deterministic RT-compatible Born-form existence construction with a precise claim boundary. The named next proof obligation is H-B/U8: derive the linear tangent particle-apparatus coupling, including its relative coefficient, directly from canonical ACRF-4 dynamics rather than choosing it phenomenologically. Canonical theory name: Registration Theory (Physical Foundations). Version: 1.0.
This theorem note gives a reproducible local two-fragment redundancy result in the repository-selected spectrum-preserving ACRF-8 continuation of Registration Theory.An explicit admissible eight-node witness is partitioned into four two-node sectors A, B, C, and D. A single quotient-valid source distinction D_A writes nontrivially into two receiver sectors B and C through two independently removable checkerboard retained-boundary channels. Each receiver then writes a scalar retained log-ratio record into a disjoint retained-edge fragment. The global adaptive normalization cancels exactly from each log ratio.At the stated witness, both retained record maps have nonzero derivative with respect to the same source coordinate D_A, so each fragment locally determines D_A with all other witness coordinates fixed. Knocking out the A-to-B checkerboard channel preserves the fixed retained-edge budget, erases only the B receiver distinction and B retained record, and leaves the C branch exactly unchanged in the one-step construction; the reciprocal A-to-C knockout has the corresponding property.The note then holds the current state fixed and perturbs only one retained fragment in a budget-preserving antisymmetric direction. Each fragment has a nonzero first-order readout into its own receiver and zero first-order cross-read into the other receiver. Thus, the explicit witness supports write → retain → independently read twice from the same originating distinction.The result establishes local retained redundancy number at least two under the note’s stipulated recoverability, counterfactual-separability, and readout criteria. It is deliberately bounded. It does not derive a global multi-host quotient theorem, dynamical selection of the four two-node partition, quantum mutual-information redundancy, decoherence, Born probabilities, or macroscopic environmental objectivity.The accompanying Python certificate reconstructs the model and witness, verifies budget preservation and positivity, reproduces analytic and finite-difference derivatives, checks both channel knockouts, and verifies independent retained readout.
This paper develops a bilateral formulation of registration in Registration Theory and establishes an exact realization of its structural architecture within canonical finite ACRF-4. The foundational proposal is the Bilateral Participatory Registration Principle: an interaction-generated physical registration is represented not as a primitive one-way source-to-receiver relation, but as one common registration with two inseparable participant-relative manifestations. This principle is introduced as a candidate foundational refinement of Registration Theory rather than as an already established canonical axiom. Conditional on that proposal, we prove an exact abstract factorization theorem: a bilateral representation is jointly consequence-sufficient if and only if both participant consequences factor through the same common representation. The paper also establishes the corresponding participant-exchange covariance condition and the role of shared relational boundary information. The paper then proves that canonical finite ACRF-4 realizes this architecture exactly. The previously derived finite-amplitude quotient is shown to be a common consequence-sufficient carrier for both host orientations. The matched receiver-isolation intervention is proved covariant under participant exchange, so the two oriented consequence experiments are realizations of one intervention architecture rather than independently inserted constructions. On the generic quotient stratum, the nine-dimensional state admits an exact participant-relative fibered decomposition. This construction is then extended globally across symmetry-restoration strata using the invariant orbit record Q = (E, UU^T, VV^T, c^2, cUV^T). The resulting participant-relative pair reconstructs the complete orbit-separating quotient and is exchanged involutively under participant exchange. The culminating exact model-level structure is: Q <-> (rho_{S|C}^Q, rho_{C|S}^Q) -> (Delta W_{C->S}, Delta W_{S->C}). Thus, canonical finite ACRF-4 provides an exact global realization of a bilateral pairing architecture with common origin, participant-exchange covariance, complete quotient reconstruction, and matched downstream consequences. The result does not identify the ACRF participant-relative records with primitive Trinity conjugates, does not identify block exchange with Trinity conjugation, and does not derive spacetime, Lorentz structure, force, charge, or a universal propagation speed. Those identifications remain open problems.
Quantum Darwinism explains effective classical objectivity through the selective proliferation of information about preferred quantum states into multiple environmental fragments. Its mature formulations operate within quantum theory, using a system-environment decomposition, decoherence or einselection, and information-theoretic or broadcast-structure criteria for redundant accessibility.This paper asks an upstream question from Registration Theory (RT): what makes a physical distinction a record before that record is described as information available to an observer? Canonical RT answers that retained registration is persistent organization constraining later evolution, that a distinction becomes a physical record only when it remains consequential, and that persistent physical organization is prior to information extracted from it.The paper assembles a theorem-backed finite-model chain now available in the RT corpus. Canonical ACRF dynamics supplies write-retain-read consequential history. The exact finite-amplitude ACRF-4 quotient identifies a nine-dimensional host-host-boundary state. The certified consequence-minimality theorem (DOI 10.6084/m9.figshare.34063464) shows that the retained relational boundary contributes irreducible future-consequence directions. The spectrum-preserving ACRF-8 redundant-record theorem (DOI 10.6084/m9.figshare.34063500) then proves an explicit local witness in which one quotient-valid source distinction is written into two disjoint retained fragments that are locally recoverable, counterfactually separable, and independently readable, giving RRT(q)≥2 under the stated criteria.The resulting distinction from Quantum Darwinism is precise but bounded. RT recordhood is consequence-based, while Darwinian objectivity is an accessibility-and-redundancy criterion formulated within quantum theory. The ACRF witness demonstrates redundant retained record formation without assuming Hilbert space, density operators, mutual information, decoherence, einselection, Born probabilities, or an observer-defined environment. Therefore, multiple persistent records of one distinction are not definitionally identical to Quantum-Darwinian redundancy.The paper does not claim that Quantum Darwinism is false or that RT has derived quantum mechanics, decoherence, Born probabilities, a mutual-information redundancy plateau, or macroscopic objectivity. Instead, it proposes a disciplined hierarchy: consequential distinction → retention → persistent record → branching → redundant retained records → redundant accessibility → environmental objectivity. The final two steps remain open in RT. A future bridge must dynamically select and scale multi-host redundancy and map consequence-bearing RT records into quantum environmental fragments that satisfy decoherence and strong objectivity conditions.