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 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.
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 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 Cinematic Leaf I–II — Unified Abstract This two-note collection develops an exact mathematical framework for three distinct questions: when a restricted readout of a discrete system supports autonomous dynamics, which hidden distinctions must be retained for complete prediction, and when the resulting discrete history can be realized by a continuous flow. The framework keeps the internal carrier, observable readout, sampling grain, and physical interpretation separate; the integers 24 and 48 remain fixed rather than serving as parameters in a continuum limit. **Part I — Factor Closure, Minimal Predictors, and Flow Embeddability in Exact 24/48 Dynamics** establishes the factor-closure criterion for compact Hausdorff systems and constructs the canonical predictor as the quotient by equality of complete observable futures. In a cyclic torus skew-translation laboratory, it derives exact block-closure conditions, minimal future-relevant register size, torus-versus-lift recurrence laws, and 24/48 Fourier and correlation-revival identities. Every closed torus block is an exact sample of a translation flow, although its continuous generator is not uniquely determined. A parabolic counterexample proves that an autonomous canonical predictor need not embed in a continuous flow on the same state space, even when a larger unrolled cover supports one. **Part II — Minimal Amplitude Predictors, Fine-Compatible Flows, and 24/48 Half-Step Lifts** applies these gates to a frozen unitary weighted cycle T=DX24 with a phase-sensitive rank-eight primitive readout. It obtains an exact accumulated-word criterion for block closure and computes canonical amplitude predictors by positive observability Gramians. The simple spectrum of the weighted cycle ensures that every Hermitian logarithm of the fine transaction preserves the one-click predictor's invisible subspace. A coarsely sampled predictor admits a flow of that same fine transaction precisely when it retains the full one-click predictive subspace. An exact example yields complex predictor dimensions 20, 16, and 8 at sampling intervals of 1, 12, and 24 clicks. A separate population-record analysis proves an obstruction to continuous all-state population autonomy along unitary paths ending in a nontrivial permutation. The 48-register analysis distinguishes central-cover memory, adjacent-pair fine-step parity, and square-root spectral branches. Opposite square roots at fixed squared eigenvalue have the same central deck parity, while a prescribed half-step fixes only the parity of each integer logarithm branch. A positive sharp-record-preserving encoder cannot reproduce a coherence-sensitive half-step through monomial fine evolution for all inputs. An explicit coherent doubled realization exists, but changes the ordinary paired population camera. Together, the notes distinguish predictive closure, continuous interpolation, and fidelity to the same underlying transaction. Proofs are accompanied by reproducible exact certificates and numerical flow checks. The native Bank dynamics, physical preparation and readout, generator selection, click duration, locality, Lorentzian structure, and spacetime attribution remain open; no emergence of physical spacetime or experimental confirmation is claimed. **Keywords:** factor closure; canonical predictor; observable-future equivalence; observability; unitary dynamics; flow embeddability; stroboscopic sampling; logarithm branches; quantum records; 24/48 clock; Bank–Leaf framework.
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
We develop a mathematical framework in which physical description is organized through predictively sufficient closure of richer realization spaces. A closure map identifies realizations that are equivalent relative to specified observables, while dynamic sufficiency requires the induced reduced state to support autonomous evolution. This yields a precise formulation of the principle that law can close before realization does: a law-level state may be fully predictive within a declared domain even when multiple underlying realizations remain possible. We distinguish quotient closure, realization fibers, idempotent closure maps, refinement, partial closure, closure memory, obstruction, and saturation of multiple closure constraints. These structures are then tested against explicit physical examples involving compact phase winding, topological defects, gauge compensation, exchange statistics, spinorial closure, quantum measurement channels, and effective propagation geometry. The analysis exposes several limitations of a motivating one-scalar coherence construction. In particular, a smooth exact phase connection has vanishing local curvature; a relative phase does not generate orthogonality or Pauli exclusion; winding alone does not supply spin- or the observed electric-charge spectrum; and a single scalar gradient cannot define a generic nondegenerate four-dimensional metric. Rather than abandoning the coherence hypothesis, these obstructions motivate a minimally enriched effective architecture involving amplitude/phase, gauge connection, spinor, and geometric frame sectors. We distinguish this effective reconstruction from the stronger hypothesis that all such structures descend from a common parent coherence object. The resulting framework does not claim a completed derivation of the Standard Model, general relativity, or quantum measurement. Instead, it identifies the mathematical and physical gates such derivations must satisfy and isolates noncommuting saturation, closure obstruction, representation reconstruction, stable particle sectors, and universal emergent geometry as principal theorem and physics programs. Keywords: closure mathematics; predictive sufficiency; quotient dynamics; gauge theory; topological defects; particle identity; quantum foundations; emergent geometry; noncommuting closure; unification.
This preprint develops the theory of spectral closure systems: an operator-algebraic framework for local access to a global compatibility object through typed, lossy closures, organized as a calculus of admissible sections of access maps. Truth is extendability: a local datum is globally valid exactly when it extends along the inverse system, with finite-stage obstruction certificates by compactness. Recovery of local charts is canonical exactly on the sufficiency sector (Petz recovery, conditioning); off it, every totalization is impotent or pays an exact relative-entropy price. Time is the Radon–Nikodym cocycle of the failure of a nonsingular dynamics to preserve belief, realized as the modular flow of its crossed-product envelope; a nontrivial clock forces a noncommutative envelope, with Murray–von Neumann–Connes type read off the ratio set and the flow of weights as envelope invariant. In the quantum perimeter, point-valued global truth is empty (Kochen–Specker), state-valued compatibility survives (Gleason, Christensen–Yeadon), Bell violations measure the classical gluing defect (Fine), and stable classical recordability is characterized as the commutative, modular-invariant, asymptotically sufficient chart sector; on the reflection-positive sector the record process reconstructs a positive-generator dynamics. Globality is relative but closure is terminal: stage sets collapse inward under directed exhaustion, the envelope tower closes outward at the second step by Takesaki duality, unforced finite-entropy residuals decay at a rate uniform in the modular cocycle (a complete logarithmic Sobolev inequality by transference from the circle), and the forcing cost balances exactly in dilation against system–environment correlation. Further results include an independence theorem for the underlying skeleton, a two-tier classification theorem with provable Foreman–Rudolph–Weiss wildness inside its fibers, an envelope universality theorem (the temporal envelopes are exactly the injective von Neumann algebras carrying a Cartan subalgebra), and a sector-split Orlicz envelope for relative entropy beyond bounded densities. Every external input is a named theorem-import with matched hypotheses; the interpretive vocabulary is proved to be a conservative definitional extension of the formal core.