We present a development of the RCD program (Radial Coherential Dynamics) in the complex scalar sector of quantum sequential growth of causal sets. The central question is not only how a causal history grows, but how much of the information about HOW IT WAS FORMED can survive when one of its elements is erased without privileging any temporal label. The central result is an exact separation between coherent inheritance and memory of the cut. For the erasure branches a_i (i = 0, …, N−1), define the branch matrix 𝒜_N = N^(−1/2) [a_0, …, a_{N−1}], the uniform register mode |u_N⟩ = N^(−1/2)(1, …, 1), and Q_N = I − |u_N⟩⟨u_N|. The memory matrix M_N = D_mix − D_coh = 𝒜_N Q_N 𝒜_N† ⪰ 0 identifies exactly the memory sector orthogonal to coherent inheritance. Its trace, Tr M_N = (1/2N²) Σ_{i,j} ‖a_i − a_j‖², is the pairwise dispersion of the causally distinct erasure alternatives: relational memory is precisely that dispersion. The Relational Inheritance Bifurcation theorem (BHR-3) shows already at N = 3 that the inherited amplitude can coincide exactly while non-zero memory of the mode of formation persists: Δ₃^coh = 0 together with Ω₃ > 0, where D_mix − D_coh = Ω₃ · [[1, −1], [−1, 1]]. At the exact point r = 1, Ω₃ = 1/294; the reading with the register traced out has spectrum {147/149, 2/149} and visibility 145/149, while the coherent reading remains pure. One and the same observable inheritance can therefore coexist with residual relational information about its formation. The second main result is an exact memory-flow law. On the star family 𝒮_N — an antichain of N−1 elements with one element above all of them — covariant coherent erasure is closed for every N. With p = ε/(t₀+ε), p′ = (N−1) p / N, P := p/N = constant, and the complete coupling trajectory is ε_n = n ε_N / (N + (N−n) ε_N). The memory has rank one, M_N = (N−1)|P|² J with J = [[1, −1], [−1, 1]], so that Ξ := M_N/(N−1) = |P|² J is an exact invariant of the flow, M_{N−1} = M_N − Ξ, and 𝓜_N − 𝓜_{N−1} = 2|P|²: each coarse-graining step transports an exactly determined amount of relational memory. The star spine 𝒮₄ also appears as an exact component of the closure variety of the 4→3 flow. The non-degenerate regular branch C₁₂ is born tangent to this spine and inherits, at leading order, its memory-contraction factor 2/3. The deposit also contains the prior results leading to this structure: absence of a non-trivial scalar fixed point under uniform incoherent erasure (𝔗_mix); absence of a non-trivial fixed point under the uniform coherent superposition (𝔗_coh), which equals the classical PF-0 condition complexified; the need to formulate erasure covariantly by classes; the rational renormalization map on the closure variety; and the minimal register compression, rank D_mix = rank M_N + 1. These results motivate the Principle of Relational Memory of the Cut, dynamical version (PMRC-D): every fundamental physical differentiation that produces causally distinguishable histories must also determine, through a covariant rule, which information of that differentiation remains recorded and which information may interfere coherently. The emerging working thesis is: "The inheritance of a history does not by itself determine the memory of how it was formed." The program remains open toward closed sectors with several independent memory modes and a covariant dynamics capable of transporting those modes under coarse-graining. Exact results are N ≤ 4 except those with closed proofs for all N (the star spine and its flow); the 5→4 step outside 𝒮_N is numerical; the physical universality of PMRC-D remains open. No claim is made about the emergence of time, Mermin's 25 % or the necessity of H > 1 as a law. Deposit contents: RCD Adjudication Record v0.2 (signed Spanish original and English version; the Spanish text prevails), diligence notes 𝔗_mix-0, 𝔗_coh-0/𝔗_flow-0 and BMR-0 (Relational Memory), reproducible code with SHA-256 manifests, and README. Panel Cerezo credits: ChatGPT (formulation of the diligences, BHR-3, PMRC-D, 𝒜QA† form, invariant Ξ, series of C₁₂) and Claude (exact computation, lemmas, curve C and map R, star spine 𝒮_N, cross-verification and drafting), under the direction and adjudication of the principal.
We present a development of the RCD program (Radial Coherential Dynamics) in the complex scalar sector of quantum sequential growth of causal sets. The central question is not only how a causal history grows, but how much of the information about HOW IT WAS FORMED can survive when one of its elements is erased without privileging any temporal label. The central result is an exact separation between coherent inheritance and memory of the cut. For the erasure branches a_i (i = 0, …, N−1), define the branch matrix 𝒜_N = N^(−1/2) [a_0, …, a_{N−1}], the uniform register mode |u_N⟩ = N^(−1/2)(1, …, 1), and Q_N = I − |u_N⟩⟨u_N|. The memory matrix M_N = D_mix − D_coh = 𝒜_N Q_N 𝒜_N† ⪰ 0 identifies exactly the memory sector orthogonal to coherent inheritance. Its trace, Tr M_N = (1/2N²) Σ_{i,j} ‖a_i − a_j‖², is the pairwise dispersion of the causally distinct erasure alternatives: relational memory is precisely that dispersion. The Relational Inheritance Bifurcation theorem (BHR-3) shows already at N = 3 that the inherited amplitude can coincide exactly while non-zero memory of the mode of formation persists: Δ₃^coh = 0 together with Ω₃ > 0, where D_mix − D_coh = Ω₃ · [[1, −1], [−1, 1]]. At the exact point r = 1, Ω₃ = 1/294; the reading with the register traced out has spectrum {147/149, 2/149} and visibility 145/149, while the coherent reading remains pure. One and the same observable inheritance can therefore coexist with residual relational information about its formation. The second main result is an exact memory-flow law. On the star family 𝒮_N — an antichain of N−1 elements with one element above all of them — covariant coherent erasure is closed for every N. With p = ε/(t₀+ε), p′ = (N−1) p / N, P := p/N = constant, and the complete coupling trajectory is ε_n = n ε_N / (N + (N−n) ε_N). The memory has rank one, M_N = (N−1)|P|² J with J = [[1, −1], [−1, 1]], so that Ξ := M_N/(N−1) = |P|² J is an exact invariant of the flow, M_{N−1} = M_N − Ξ, and 𝓜_N − 𝓜_{N−1} = 2|P|²: each coarse-graining step transports an exactly determined amount of relational memory. The star spine 𝒮₄ also appears as an exact component of the closure variety of the 4→3 flow. The non-degenerate regular branch C₁₂ is born tangent to this spine and inherits, at leading order, its memory-contraction factor 2/3. The deposit also contains the prior results leading to this structure: absence of a non-trivial scalar fixed point under uniform incoherent erasure (𝔗_mix); absence of a non-trivial fixed point under the uniform coherent superposition (𝔗_coh), which equals the classical PF-0 condition complexified; the need to formulate erasure covariantly by classes; the rational renormalization map on the closure variety; and the minimal register compression, rank D_mix = rank M_N + 1. These results motivate the Principle of Relational Memory of the Cut, dynamical version (PMRC-D): every fundamental physical differentiation that produces causally distinguishable histories must also determine, through a covariant rule, which information of that differentiation remains recorded and which information may interfere coherently. The emerging working thesis is: "The inheritance of a history does not by itself determine the memory of how it was formed." The program remains open toward closed sectors with several independent memory modes and a covariant dynamics capable of transporting those modes under coarse-graining. Exact results are N ≤ 4 except those with closed proofs for all N (the star spine and its flow); the 5→4 step outside 𝒮_N is numerical; the physical universality of PMRC-D remains open. No claim is made about the emergence of time, Mermin's 25 % or the necessity of H > 1 as a law. Deposit contents: RCD Adjudication Record v0.2 (signed Spanish original and English version; the Spanish text prevails), diligence notes 𝔗_mix-0, 𝔗_coh-0/𝔗_flow-0 and BMR-0 (Relational Memory), reproducible code with SHA-256 manifests, and README. Panel Cerezo credits: ChatGPT (formulation of the diligences, BHR-3, PMRC-D, 𝒜QA† form, invariant Ξ, series of C₁₂) and Claude (exact computation, lemmas, curve C and map R, star spine 𝒮_N, cross-verification and drafting), under the direction and adjudication of the principal.
Technical note BMR-1 of the RCD program (Radial Coherential Dynamics), continuing "Relational Memory under Causal Erasure" (10.5281/zenodo.23630978). The previous record found an exact memory-flow law on the star family, which carries a single memory mode. This note asks for the minimal causal architecture able to carry two independent memories and survive its own coarse-graining. Three exact results. (1) Branch–rank lemma: if the N erasure positions produce r distinct branches, rank M_N ≤ r − 1, with equality under affine independence; two modes require at least three causally inequivalent erasures. (2) The P2_n sector (a chain with one pendant element at height ≤ 2) is closed under erasure for all n ≥ 4, has unbounded height and two exact memory modes: exact transfer map R_n, two flow invariants (y, J), closed-form solution, Hermitian memory M_n with rank 2 iff uv ≠ 0, and linear transport of (|u|², Re ūv, Im ūv, |v|²) with spectrum {a², ac, ac, c²}, a = (n−2)/n, c = (n−3)/n; the relative phase is carried as an independent memory component. (3) Step-closure theorem: the support of every Rideout–Sorkin (RS) amplitude is fixed by the local pair (|↓x|, #max ↓x) and always contains the antichain, so no closed family omitting it, P2_n in particular, admits an RS realization; the star is the only RS-supportable family found. An exact classification of RS closure families 5→4 on restricted supports finds only the star line, plus a certified isolated conjugate pair that closes one step and not the next. The obstruction is against the locality of the RS amplitude support, not against scalar amplitudes as such. Includes the erasure automaton of unlabeled posets up to N = 7 (1, 2, 5, 16, 63, 318, 2045 classes) and all scripts with a SHA-256 manifest. Exact results are stated as such; the enumeration of isolated points on large supports is not certified. English and Spanish versions; the Spanish text prevails. Developed at the Panel Cerezo: ChatGPT (formulation of BMR-1, branch–rank lemma, Hermitian correction), Claude (automaton, P2_n, step-closure theorem, classification, verification and drafting), under the direction and adjudication of the principal.
Technical Note JANO-BR-0 of the RCD program (continues “La doble herencia de la coherencia”, DOI 10.5281/zenodo.23520439; the Adjudication Record “Relational Memory under Causal Erasure”, DOI 10.5281/zenodo.23717352; BMR-1, DOI 10.5281/zenodo.23717139; and REP-0, DOI 10.5281/zenodo.23774093). Question: can the heritage/memory bifurcation of BHR-BMR become a physical readout of one and the same circuit? Results (exact, no hardware): in BHR-3 the mixed relational-memory state ρ_mix(r) has rank 2 whenever Ω₃ > 0, so a single qubit c purifies it, |Ψ(r)⟩ = √λ₊|0⟩|ψ₊⟩ + √λ₋|1⟩|ψ₋⟩, with a three-gate {Ry, CNOT} construction and a closed map r ↦ (θ_c, η); tracing out c returns ρ_mix (R⊥), while measuring c in the HJW basis leaves the system exactly in the coherent-heritage state ρ_coh with probability p_coh = Tr D_coh/Tr D_mix and in the residual-memory state |−⟩ (independent of r) with probability p_M; the three readouts are tied by the exact identity V_mix = p_coh·V_coh − p_M (at r = 1: 145/149 = 147/149 − 2/149). Verified against the original JANO-1 code: the five-qubit JANO-1 circuit with α = α_BR(r) = arcsin √λ₋(r) is literally this purification, and its frozen observables reduce to P_rec = λ₊ and R∥ = R⊥ = √(2λ₊λ₋) = √S_lin(ρ_mix) — a spectral readout of the memory — while the HJW readout is the complementary face that separates heritage from memory. Stated limit: the construction reproduces ρ_mix(r) by design and is not a falsifiable prediction of RCD; hardware stays closed. Scripts with SHA-256 manifest. English and Spanish versions; the Spanish text prevails.
The late-time dissipative kernel of the Radial Coherential Dynamics (RCD) program is parabolic and does not generate a Lorentzian causal cone: signature is an input, not an output. This relocates the program's claim to a logically prior, operational question: can a coherence field organize a finite, switchable influence front in a local quantum dynamics, and does the specific spatial order produced by the RCD kernel matter beyond the distribution of coupling intensities? We study a transverse-field Ising chain and measure the infinite-temperature out-of-time-order correlator c(x,t) = ||[Z0(t), Zx]||^2_F / (2·2^N) as the primary observable, with frozen pre-registered gates. By exact operator evolution at N = 10, 12, 13 the front velocity in the clean channel is stable without appreciable size drift (v = 2.000 -> 1.980 -> 1.980 at theta=0.10, consistent with the exact maximal quasiparticle group velocity 2·min(J,h), distinct from and below the rigorous Lieb-Robinson bound), with OFF collapse to ~1e-13. Imposed coherence profiles modulate the front: an incoherent barrier blocks it, a coherence ramp slows it per a local WKB law, a uniform profile recovers the bare velocity. The coherence-to-coupling map J_i = J0·sqrt(C_i·C_{i+1}) is a constitutive ansatz, not a derived or unique law. The decisive test (KS-C1a) was executed on a snapshot of the SEALED 2D Wright-Fisher/Allen-Cahn kernel (de Sitter, H=0.02, seed 20260612, t*=1.0 t_H, central row coarse-grained 120->12). The candidate at t*=0.5 t_H failed the pre-quantum G0 gate; t*=1.0 t_H passed all three amended G0 conditions (CV(J)=0.189, percentile[S(C)]=3.5, xi=2). The result is a PRE-REGISTERED NULL (the RED branch): G1 PASS (real front obeys local WKB, R^2=0.913), G3 PASS (OFF 4.8e-14), but G2 FAIL for all three J-shuffles — cross-prediction differences (dR^2 ~ 0.03) and trajectory differences (Delta ~ 0.03-0.06) fell well below the pre-registered thresholds (0.15 and 0.10) on eight common interior sites. G4 was INDETERMINATE (the coarse-grained frozen snapshot contained no link below J_min=1e-6; declared before execution). A pre-specified diagnostic at row y=32 returned INDETERMINATE by dynamic censorship (extreme weak links left zero common interior sites), documenting an observability limit of the protocol in high-dynamic-range landscapes; implied kappa varied -4.5% to +8.1% across cuts, immaterial to the (differential) null. What survives: the operational front instrument (G1, G3), the perilla as a conditional predictive translator, and the local WKB law. What does NOT survive: the stronger statement that "the coherence-to-coupling map comes from the field." The narrow statement earned by the data is that a static RCD coherence landscape can be inserted into the constitutive ansatz to predict a front locally, but its tested spatial ordering was not operationally distinguishable from shuffled orderings with the same coupling values. Three non-exclusive readings (genuine physical null / coarse-graining loss / cut-dependent dynamic-range limitation) are stated without selection; each requires a new pre-registered follow-up. We claim NO derived metric, NO derived time, NO physical light cone, NO emergent general relativity. The dynamically coupled bridge (KS-C1b) is named as a new hypothesis, not a validated continuation. Execution evidence is sealed by SHA-256 (manifest in the supplementary record). The KS-C1a protocol amendment v0.5 was applied as recorded prior to execution; Appendix B reproduces the ratified pre-registration v0.4 verbatim.
Coherence First develops a relational reformulation of Radial Coherential Dynamics (RCD) centered on a prior question: what physically selects the factorization of a quantum system into subsystems? Because entanglement is defined only relative to a tensor-product structure or a partition of observable algebras, programs that derive classicality, locality, or geometry from entanglement already presuppose a cut. The paper proposes the Coherential Selection Principle (CSP): the physical factorization is the one that maximizes the persistence of coherent information within its parts while minimizing coherent persistence across their boundaries. In this framework, subsystems are dynamically persistent regions of a relational quantum structure, and the coherential field C(x,t)C(x,t) is reinterpreted as a possible continuous, effective description of a deeper network of selected relations. A numerical proof of concept is presented for an open three-qubit system. Keeping the Hamiltonian fixed while rotating the environmentally monitored algebra shifts the factorization favored by a provisional functional. The effect survives a generic non-commuting control, remains stable under sampling, and follows a continuous five-point rotation of the monitored algebra. The result establishes that open dynamics can transport and recover a recognizable structure of parts that is not fixed by the Hamiltonian alone. The full CSP, the emergence of geometric locality, and the proposed connection to cosmological relaxation remain the next stages of the program. Version 0.2. Preprint; not peer reviewed. This record contains the PDF, LaTeX source, numerical scripts, and reproducibility materials used in the closed- and open-system studies.
We propose to read spatial entanglement and interference between histories as two inheritances of a single coherence prior to the cut: a transverse residue across the partition A|B and a longitudinal residue along the causal order. The hypothesis proper to RCD is the Principle of Double Inheritance of Coherence (PDHC): one rule of differentiation determines at once what stays separated and what stays ordered. To make it an instrument we design JANO-1, a five-qubit circuit for superconducting processors in which a single cut, placed in superposition and then erased, is read twice: as entanglement between two qubits that never interacted and as an interference fringe between the histories “cut” and “no cut”. Exact simulation fixes the three predicted curves with one knob (the amplitude of the cut) and shows that a second knob (the coherence of the cut) makes the pair of residues fill a region, not a curve; the strong hypothesis is therefore stated as a precise question for the amplitude rule of PF2, and the JANO-2 phase returns to design. Nothing has run on hardware; the preregistration is conditional on a second key. Files: English version, Spanish version (the Spanish text is the reference text), and the Python code that reproduces every figure. Developed at the Panel Cerezo with cross-checking among AI tools (ChatGPT, Grok, Claude); the authorship of each contribution is stated in §10.
Action principles, phase accumulation and quantized closure appear across classical mechanics, quantum theory, semiclassical dynamics, superfluidity and geometric phase phenomena. Their coexistence is familiar; their structural relationship is usually taken as part of the established formalism. Here we reconstruct that relationship in the opposite direction. Starting from persistent relational dynamics, we show how a real two-component rotation supplies a phase coordinate without assuming a primitive complex number, how a relational canonical one-form can connect phase increments to action-like increments, and how stationary phase and global phase consistency impose two distinct selection conditions. The local condition yields stationary action, δA_R = 0. The global condition yields integral symplectic periods. Under the explicit convention F_R = −iΩ_R/κ_A for a U(1) connection curvature, global compatibility requires [Ω_R/(2πκ_A)] ∈ H²(M_R,Z), equivalently ∫_Σ Ω_R = 2πnκ_A on closed two-cycles. This separates the origin of discreteness from the origin of the dimensionful conversion scale. Topology fixes integer classes; it does not determine κ_A. We therefore identify a precise remaining problem: what selects a unique, universal and non-zero κ⋆ across coherently composable sectors? The framework makes this question falsifiable by requiring cross-scale closure, causal ancestry, intervention consistency and independence from observer compression. It also distinguishes a universal phase–action conversion scale from scale-dependent effective couplings. The result is a sharpened bridge from relational phase to stationary dynamics and quantized closure, with a mathematically explicit boundary between what is derived, what is inherited from established geometry, and what remains to be selected dynamically. The revision formalizes an Action-Scale Factorization Theorem together with Exact-Action and Curvature-Compatibility no-go results, thereby separating the existence of quantized closed-path sectors from the still-open physical selection of a universal non-zero scale such as ħ.