The 0 and 1 doctrine
The primitive is literally {one, zero} = {1, 0}: zero is the finite-transition floor — the fastest a state can change to another state (the minimum interval), from which the speed of light c emerges (light is not a separate primitive; c is the late expression of the finite-transition limit, §9) — and one is everything else: the superforce, a single unified interaction. Every dimensionless quantity is forced by their involution, tick, and loop closure: 2 (involution period), 3 (minimal non-retracing loop), 6 = 2×3, 12, and the 8+3+1 census all fall out (sections 0–29, kernel-checked). Classification: the coupling itself is input by construction. The observed 137 and 1836 are running (low-energy) values, not inputs — the fundamental input is the superforce "1", and the renormalization-group flow from 1 down to low energy is outside the current formal scope.
The method, not the numbers, is the result
The kernel cleanly separates the Standard Model's structural content from its numerical content. Machine-verified: the fermion census (12), the gauge dimensions (8·3·1), charge quantization, the confinement triples, Lorentz kinematics, the inverse-square law, the gauge group {±1}, the curvature spectrum, and closure⇔charge. The dynamical identification of dimensionless constants is a separate part of the program: the physical identification of the α⁻¹ = 137 and mₚ/mₑ = 1836 arithmetic identities requires a trace through seed evolution. Their arithmetic verification alone does not establish that identification. The existing audit grades are retained; no new fitting claim is inferred here. The result is a proof-of-concept that particle-physics structure is finitely axiomatizable: countable, decidable, and kernel-verifiable on a six-element lattice.
Closure, charge, and gap are one phenomenon
Sections 7, 20c and 24 compose into the theory's deepest statement: a phase closes with a distinct partner if and only if the gauge group moves it, if and only if it carries charge. The gauge-invariant phases {0,3} are exactly the ones that never close — the massless sector. This is a discrete shadow of the center-symmetry mechanism of confinement: in QCD the ℤ₃ center of SU(3) governs confinement, and the photon stays massless because U(1) is unbroken. Here the CRT split ℤ₆ ≅ ℤ₂ × ℤ₃ carries the same two factors — ℤ₃ matches the color triplet, ℤ₂ the matter/antimatter role. The phase action distinguishes invariant and moving phases. Gap operators give that distinction spectral meaning; a propagation law determines its long-range physical expression.
Why Lorentz invariance is so robust
The Euclidean budget shows the light cone and time dilation are an exact Pythagorean identity, scale-covariant at every refinement. There is no O(γ) correction to the dispersion relation because the constraint is refinement-invariant — the same identity at every scale. That is why Lorentz invariance survives to extreme precision: it is not an emergent approximation but a symmetry of the counting. Formal status: the Euclidean (L2) versus taxicab (L1) choice is an input, not a derivation.
The arrow of time and the repeating seed state
The primitive update is a period-2 involution — time-reversible by construction. Direction cannot come from the map; it requires a monotone quantity outside the current formal scope. Classification is precise: the arrow of time is a consequence, not a primitive.
Scope: what the kernel excludes
The current formal record separates arithmetic identities, exchange-sign definitions, charge assignments and dynamics from their physical identification through seed evolution. A definition in a formal file may encode a derivation made elsewhere; its provenance must be traced before classifying it as an independent input. The new amplitude and shared-time constructions below extend the program through explicit mathematical bridges. A finite formal kernel can encode dynamics; the question is which dynamics the seed determines.
The grid theorem and the free-angle empirical status
On the six-fold cosine grid, both this construction and ordinary quantum correlations cap CHSH at 5/2; the unrestricted Tsirelson bound is 2√2. The grid ceiling alone does not distinguish the construction from quantum mechanics. The unrestricted-basis empirical fact, stated once and cleanly: free-angle Bell experiments measure ~2.73. The ℤ₆ structure is therefore classified as a substrate symmetry, not a measurement-basis symmetry — a precise scope statement on what the lattice governs. On the restricted grid itself, the bound is now hardware-confirmed: 96/96 saturating configurations censused on ibm_marrakesh, |S| ∈ [2.33, 2.55], controls ≤ 1.99.
Causal order: priority proposal · updated 2026-10-21
External evidence. Richter et al. (PRX Quantum 7, 010354 (2026)) report 1.8328 ± 0.0045 against a definite-order bound of 1.75 (18σ), with experimental loopholes. Qu et al. (2025 preprint) report a 24σ violation using a photonic switch and a spacelike-separated observer. These implement device-independent certification protocols subject to their causal assumptions and experimental limitations; they are not loophole-free demonstrations. The isolated quantum switch does not violate standard causal inequalities: the extended protocol uses relativistic causal constraints and free interventions (van der Lugt et al., 2023).
Valibouse et al. (2026) demonstrate time-delocalized local readout with post-selected optical gates and a quantum eraser preserving path coherence, with causal witness CW ≈ −0.305(1). This witness is a separate certification from the 18σ and 24σ inequality tests. See also the review Indefinite Quantum Causality (2026). These results motivate a sector-resolved test; they do not experimentally confirm this lattice mapping.
Priority claim — proposed correspondence, not a Lean theorem. Robert Fogeborg advances the specific closure/charge-to-causal-order correspondence below as an original proposal. No matching identification was found in the sources reviewed for this update; this is a scoped priority assertion, not proof of absence throughout the literature.
- Proven algebra: the negation action on ℤ₆ moves {1,2,4,5}; each has a unique distinct closure partner. Its fixed phases are {0,3}, which self-close but have no distinct closure partner. “Never-closing” here always means no distinct-partner closure.
- Proposed physical mapping: closing/charged sectors, associated with the color 8 and weak 3 counts, correspond to definite causal order and a gap; the singlet phases {0,3} permit indefinite causal order and are candidates for gapless propagation. Permission for ICO does not require every singlet process to exhibit it.
- Proven update boundary: the primitive update is an involution; a strict arrow is not supplied by that map. An emergent-arrow mechanism remains additional dynamics.
Grade: READING / conditional prediction; under external test. Given an independently specified realization map from lattice sectors to valid experimental process matrices W, the proposal requires every charged-sector realization to have definite order, while allowing causally nonseparable singlet realizations. Definite order means compatibility with A≺B or B≺A; causal separability also permits probabilistic mixtures of those orders. A reliably charged-sector process certified causally nonseparable would falsify the proposed correspondence. Testing the stronger single-order requirement also needs exclusion of mixtures. The realization map, allowed interventions, sector preparation, noise model and witness must be fixed before testing; no such process-matrix bridge is currently derived in Lean.
The existing single-site numerical toy (tools/explore_causal.cjs) splits the six-cycle into a 4-dimensional symmetric subspace and a 2-dimensional antisymmetric subspace. Adding μ(I−N)/2 gaps the latter by μ while preserving the zero mode. μ is an input. Those subspace dimensions are not the 2+4 phase-orbit count, and the toy does not establish causal order, confinement or a continuum gauge field.
Recent external data · October 2026
Hardware follow-up — external capability, not model confirmation. Oxford (15 October) reports a Sr–Ca network with entanglement involving up to four atoms and storage beyond 10 s in calcium. Google’s Willow magic-state cultivation reports a 40-fold error reduction and fidelity 0.9999(1), retaining 8% of attempts; this is a prepared-state result, not a blanket gate-fidelity specification. RIKEN (15 October) reports single-qubit gate fidelities above 99.99% on up to three silicon spin qubits operated simultaneously. These advances motivate feasibility studies for higher-visibility grid-locked CHSH censuses and larger-loop parity tests; achieved precision and platform access still need campaign-specific validation. The existing 50-pass ibm_marrakesh campaign and its grades remain unchanged.
Boundary tests — outside scope. Nayak et al., Science Advances 12, eaeh0566 (2026) report NV-ensemble spin forces driving center-of-mass motion exceeding 100 nm in a 128 mg levitated diamond oscillator. This extends spin-force control into a macroscopic mass regime. It demonstrates a mechanical response to spin forces, not a macroscopic motional superposition, a lattice test or a quantum-gravity result.
Possible kinematic origin of apparent dynamics
Grade unchanged: construction pins w = −1 exactly; under external test / falsifiable. A confirmed dynamical measurement at high significance (the existing 5σ criterion) falsifies that construction. DESI DR2 combinations motivate evolving-dark-energy models; their preference depends on the combined datasets and model. DESI’s April 2026 update expects first full five-year dark-energy results in 2027. This update does not resolve the tension.
Owner-reported numerical exploration; not independently reproduced here. Holding the stated combinatorial growth law fixed (exponential in lattice units, with exact w = −1 in the construction), mild measure drifts or tick modulations yielded only apparent wa ≈ −0.03 to −0.07. Apparent wa ≈ −0.5 to −1 required strong nonlinearity, γ ≳ 0.7–1.0 in the explored continuum-projection family. These are family-dependent numerical ranges, not universal bounds or DESI parameter estimates. The scripts, .npz outputs, map definitions and fitting assumptions are not included in this release.
Expansion under the existing growth law therefore does not automatically produce large apparent dynamics. A strongly evolving map from lattice scale factor to the scale factor inferred with continuous rods can do so in the reported exploration, but that map is an extra assumption, not a consequence of the Lean development. A first-principles derivation and reproducible observational inference remain future work; the w = −1 prediction and falsifiability grade stand.
2026-10-21: ICO literature update + discrete-to-continuum dark-energy exploration. New physical correspondence: READING / conditional prediction. External hardware advances: context only. Boundary test: outside scope. Existing proven, input, numerological and hardware-confirmed grades retained.
Calculated here: what a projection can and cannot change
NUMERICAL / extra input; not Lean-derived or an observational fit. This independent example makes the missing assumption explicit; it does not reproduce the owner's unavailable scripts. Let x = ln aL, y = ln aobs = x + βx²/2 and g = dy/dx. If the lattice vacuum density is simply relabeled as a scalar, ρobs(y) = ρL = constant, the continuity definition w = −1 − (1/3)d ln ρ/dy gives w = −1 for every regular monotone projection. A change of coordinates alone cannot supply evolving dark energy.
Additional energy/volume assumption. Instead assume energy E = ρLaL³ is assigned to observed volume aobs³. Then ρeff = ρLexp[3(x−y)], so weff = −1/g = −1/√(1+2β ln aobs). This effective w uses the continuity relation as a definition; no stress tensor, Friedmann dynamics, luminosity-distance prediction or DESI likelihood has been derived. A constant stretch g changes this effective w to another constant; time variation needs an evolving g.
The tangent to the CPL form w(a) = w₀ + wa(1−a) at aobs = 1 has w₀ = −1 and wa = −β. β is this example's free map-curvature parameter, not the γ in the earlier owner-reported family. The exact example is not globally CPL.
Effective w under the additional energy/volume rule. Physical redshift is identified with aobs = 1/(1+z) as another modeling assumption. Last column is an exclusive branch boundary, not an observational bound.
| β | Local wa | w(z=0.5) | w(z=1) | w(z=2) | z must be below |
|---|
| 0.00 | 0.000 | -1.000 | -1.000 | -1.000 | unbounded |
| 0.03 | -0.030 | -1.012 | -1.021 | -1.035 | 17307778.953 |
| 0.07 | -0.070 | -1.030 | -1.052 | -1.087 | 1264.038 |
| 0.50 | -0.500 | -1.297 | -1.805 | outside branch | 1.718 |
| 0.70 | -0.700 | -1.521 | -5.813 | outside branch | 1.043 |
| 1.00 | -1.000 | -2.300 | outside branch | outside branch | 0.649 |
Calculated limitation: monotonicity requires 1+2β ln aobs > 0. For β = 0.7 the branch ends at z ≈ 1.043; for β = 1 it ends at z ≈ 0.649. Thus large negative local wa in this family comes with a singular inverse map at modest redshift. The example cannot serve as a global cosmological explanation without a different, justified projection. Small β gives small local dynamics; strong map evolution is neither automatic nor sufficient for viability.
Reproducibility: download the dependency-free Node calculation and machine-readable results. Run node calculate_projection.cjs; it saves a results JSON beside the downloaded script. Checks include finite differences of the density, map inversion, local slope and invalid-branch rejection. An independent integer enumeration also recovers the cosine-grid CHSH maximum 2.5 and 96 saturators among 1,296 settings. That is a grid identity shared with ordinary quantum cosine correlations, not evidence distinguishing this lattice from quantum mechanics. Hardware results remain the original campaign records.
Derivation targets: status
- Inputs, now forced minima. §30 derives
Role = 2 (any fixed-point-free map needs ≥ 2 elements) and Axis3 = 3 (the minimal non-retracing loop count): the primitive's two numbers are forced minima, classified proven rather than postulated.
- The gapless sector. Section 24 exhibits the never-closing sector; the §30 count match (singlet size = flux exponent = 2) fixes it at the counting level. Proving the actual inverse-square long-range falloff, tying
fluxExponent = 2 to the singlet, is the stated dynamics target — outside current scope.
- Toy dynamics. A discrete transfer matrix on ℤ₆ with closure as weights computes spectra and correlations and tests closure⇔gap numerically — executed across five tools, machine-checked where applicable (see second pass below).
The dynamics frontier — second pass
All three items above are now executed and machine-checked where possible. §30 derives Role = 2 (any fixed-point-free map needs ≥ 2 elements) and Axis3 = 3 (the minimal non-retracing loop count), so the two inputs are forced minima, not postulates. §31 pins the closure structure: on the moving phases {1,2,4,5} closure is a perfect matching (each phase has a unique distinct partner; closing twice returns home), and {0,3} are exactly the self-conjugate phases. §32 derives n² − 1 as the unique quadratic through the closure-forced values 0, 3, 8 — the adjoint census 8+3+0 = 11, with the singlet 1 closing to 12 = |Dir6|. §33 proves the negative boundary: the only bijections of the primitive are id and inverse, so no deterministic map carries an arrow — direction must be extra-dynamical.
The numerics now span five tools. lattice_gauge.cjs (#1) shows 2D ℤ₃ area-law confinement and the 3D ℤ₂ deconfinement transition; polyakov_loop.cjs (#4) reproduces the Polyakov-loop order parameter (⟨P⟩: 0 → 1 across β_c). transfer_matrix.cjs (#2) and spectrum.cjs (#5) make closure⇔gap spectral: the charge gap opens with slope exactly 2 (ΔE_charge = β + 2μ) while the gauge-invariant {0,3} singlet's internal gap stays μ-independent (= β) — the never-closing sector does not confine. running_coupling.cjs (#3) extracts Creutz ratios from ℤ₃ Wilson loops, showing the coupling runs with scale (clean at weak coupling; the 1D toy has zero beta — running is a 2D/3D effect).
Classification, unchanged: μ is an input coupling (structure forced, value outside formal derivation), and none of this derives α⁻¹ = 137 or mₚ/mₑ = 1836 — those remain numerological fits. The §30 count match (singlet size = flux exponent = 2) fixes the gapless sector at the counting level; the actual r⁻² falloff is the stated dynamics target, outside current scope.
