Discrete 3D+3D Temporal Geometry

A single-axiom unified framework for galactic dynamics, cosmology, particle physics and quantum coherence

Author: Simone Calzighetti (Principal Investigator).
Deposit version: v7.1 — 1 August 2026. Theory origin: 14 September 2025.
Canon: 125 nodes / 181 edges / 26 symbols · checkpoint NO DRIFT (2026-07-28).
Supersedes v6.0 (10 July 2026), v5.0 (26 June), v4.0 (17 June), v3.1 (7 June).

AI collaboration, disclosed as method and not as authorship: Lucy (Claude, Anthropic) — derivation engine and machine verification; Vega (GPT, OpenAI) — mandatory adversarial Red-Team referee; Gemini (Google) — observational validation; Copilot (Microsoft) — implementation support. They are instruments. The human author bears final responsibility for every claim.

Read this first — two standing reservations on results this record has already published

A deposit that only advertises its successes is not a scientific record. These two reservations are at the top because they are the fastest route to this framework's current weakest joints.

1 — Three papers in this archive are under open, ratified conflicts. See QUARANTINE_NOTICE_2026-08-01.md, which states each equation by equation, verified at source. The most severe, C-TAU1-SIGNED-ANSATZ (P0): the signature (−,+,+,+,−,−) makes the two compact directions timelike, yet the published reduction ansatz writes the compact block with a plus sign carried by a positive-definite matrix — it spatialises them. The no-ghost result built on that reduction is reclassified PRIOR-ART / SIGNED-DERIVATION-MISSING. This is not a claim that the published number is wrong: ratified Gate II-1 recovered exactly the published form, φ²/2 = 1.309017, by two independent symbolic routes. But a derivation that is not valid as written is not a derivation, whatever its result. Nothing has been deleted.

2 — The electroweak identity is stated here at its ratified strength, lower than in every previous version of this record.

The framework yields the exact algebraic identity (3−φ)/6 = 0.2303276685…, numerically within 0.7% of the measured sin²θ_W. Its interpretation as the Weinberg angle is UNESTABLISHED: the measure is not yet uniquely defined (KS-AH-7: NOT-ADJUDICABLE, ratified 2026-07-13) and the action-first normalisation map is unresolved (ACTION-FIRST BLOCKED — NORMALIZATION MAP UNRESOLVED, ratified 2026-07-14). The agreement is recorded; the prediction is not yet earned.

Both audits sit in 07_RATIFIED_AUDITS_EW_2026_07/. They state, in those words, that the framework is not in scope and not falsified. Ratified is not propagated: both carry the clause "No canon edit. Canon FROZEN.", so the canon shipped here still records EW001 = THM. For that node alone, Section 07 and §7 of the quarantine notice are the ratified scientific overlay and take precedence over the frozen entry until propagation — a separate act, not performed by this deposit. Everywhere else in this archive, the canon wins.

Who I am, and why that matters

My name is Simone Calzighetti. I am not a professor, not an academic researcher, and I hold no PhD. I am a theoretical-physics enthusiast. On 14 September 2025 I had an intuition about discrete mathematics and three-dimensional space, and I have spent every day since turning it into a falsifiable framework — with AI systems, under a discipline built to catch my own mistakes. I state this openly because this deposit asks to be judged on its derivations, its numbers and its kill-switches, not on credentials.

The framework

Six-dimensional spacetime, signature (−,+,+,+,−,−): three spatial dimensions and three temporal ones. Two temporal dimensions are compactified on a torus T² with modular parameter τ = i/φ (φ the golden ratio), at scales L₂ ≈ 9.5 ly and L₃ ≈ 6.0 ly; ordinary time stays uncompactified. This geometry generates emergent gravitational effects that account for phenomena attributed to dark matter and dark energy, without exotic particles. The framework has exactly one dimensional input (M̄_Pl ≡ the choice of units ≡ G); it predicts dimensionless ratios, not absolute scales.

The organizing axiom and its chain of theorems

[POST 1 — Determinacy Postulate] The modular parameter of the compact temporal torus T² is fixed by the SO(3,3) symmetry of the 6D Einstein–Hilbert action: τ = i/φ. This reduces to the self-consistency condition P(θ*) = 1/D = 1/6, whose unique positive root solves x² − x − 1 = 0.

[PHI-SEL] POST 1 is retained as the organizing axiom, but τ* = i/φ is theorem-internal, not merely assumed: the golden temporal torus is selected radiatively — tree-level shape flatness plus a one-loop Hurwitz/minimax resonance envelope (the Diophantine bound L(r) ≥ √5 holds with equality iff golden), reinforced by Anti-S-Duality. The sign of the small (2.2%) deviation from exact φ is predicted by the temporal Casimir/Epstein structure. The R3 residual — the size of that 2.2% — is bounded but not computed, and at current precision is consistent with r* = φ⁻¹ exact to within 1σ. τ* = i/φ is a theorem within the 3D+3D axioms, not an absolute world-theorem.

τ = i/φ
  ↓
A = T·P = [[1,1],[1,0]]     Fibonacci companion of x²−x−1=0
                             A = twist T × two-time exchange P; det(A) = −1 forced by
                             anti-holomorphy + Anti-S-Duality; tr(A) = 1 ⇒ golden spectrum
  ↓
K = I + A² = [[3,1],[1,2]]  Q-sector Gram/kinetic matrix:  tr(K) = det(K) = 5     [THM]
  ↓
W = uᵀKu = 7 = 2 + 5        Coherent-mode rigidity  (u = (1,1)ᵀ)                  [THM]
  ↓
Ω_geom = 37/145 ≈ 0.2552    Geometric dark-matter density                         [STRTHM]
  ↓
A_kernel = 259/3480         Kernel amplitude, exact rational                      [DER-COND]
  ↓
μ(k,a) = (259/3480)·S(a)/[1+(k/k_μ)²]     Modified-gravity kernel
  ↓
CLASS verified:  R = μ_phys/μ_th = 1.000 ± 0.003   (71 independent points)

Every arrow is a derived link; the load-bearing ones are structural theorems, symbolically verified (SymPy residuals = 0). Two inputs are declared rather than hidden: the Modular Closure Principle (fixing the modal action Q ↦ AQ) and COLORIN — the colour factor, an explicit gauge input: rank bifurcation theorem K-CF-5b shows a single Spin(3,3) (rank 3) cannot gauge the rank-4 Standard Model, so SU(3)_c must enter as an independent commuting factor. K and W = 7 are therefore theorems modulo Modular Closure + COLORIN, with open residuals Ra′ (anti-holomorphy from the (−,−) signature + charge conjugation) and Rb (Hosotani realisation of T·P) tracked explicitly. The capstone spine carries one declared open blocker, N0WEP.

Dimensional unification: D = 6 from first principles

D = 6 is derived, not postulated, from four independent principles each selecting it uniquely: (CTD) cosmological–topological duality imposes 2D²+1 = D²+6D+1 ⇒ D² = 6D ⇒ D = 6; (Lovelock–Gauss–Bonnet) D = 6 is the minimal even dimension where the topological protection g²Λ = 2π/W holds; (E₂ modular anomaly) the factor 12 = 2D in the Eisenstein anomaly permits vacuum selection only for D = 6; (spectral identity) ζ(2) = π²/6 = π²/D. The algebraic quartic closure — tr(K) = det(K) = 4+n² with det(K) = D−1 and D² = 6D — gives n⁴+4n²−5 = 0 ⇒ n = 1, promoting the primitive-generator principle from axiom to theorem. D = 6 remains derived conditional on the Determinacy Principle.

The cosmological-constant sector (K-LAMBDA)

The compact 3D+3D vacuum is AdS-like in its shape-Casimir sector: read as a mode-decorated spectral supertrace under the canonical (1/2,0) spin structure, V_Cas(τ*) < 0 (KL1, DER-COND; mpmath S_b = +40.48, S_f = −34.67). A positive cosmological constant therefore requires a separate internal geometric uplift, supplied by the internal F₄₅ magnetic-type flux: it redshifts one power more slowly than the Casimir term and dominates at the frozen, dimensionally anchored compact volume, giving a positive, de Sitter-leaning Λ_eff (KL2, DER-COND). The winding/stored sector is conserved matter, not a vacuum residual, and does not flip the sign. The sign of Λ is derived-conditionally geometric; its magnitude is dimensionally anchored, not predicted. Open residuals: G-CC-Φ (supertrace rewrite), G-CC-volmin (volume-freeze mechanism).

Anti-S-Duality trilogy: vacuum selection

Why τ = i/φ is selected over its modular S-dual τ = i·φ. Paper α — left-chirality of the Standard Model selects τ = i/φ; the S-dual is Boltzmann-suppressed by ~10⁻³⁵. Paper β — physical modular subgroup Γ⁰(2) (S ∉ Γ⁰(2)); spin structure (1/2, 0). Paper γ (FCNC) — bridge scale μ_B = v·exp(−π/φ²) = 74.16 GeV (MU0, WORKING_CANONICAL); 18-channel b → sμμ sweep; kill-switch F-LHC-v1 (|C₉^NP| ≤ 10⁻²) testable at HiLumi-LHC by 2030. Paper ε — CP phase as Berry holonomy δ_CKM = π/φ² = 68.75°.

Direction-D subleading series: universal kernel 1/φ²

A single universal kernel 1/φ² controls all subleading corrections to both CKM and PMNS mixing (papers ζ, ζ.3–ζ.7). Six CKM observables and four PMNS observables, all pulls < 0.7σ; the geometric factor ξ = λ²/(2π) arises as a Berry-cycle measure, distinct from a 4D loop coefficient 1/(16π²); KK suppression proven pair-by-pair under the Z₂ orbifold projection (numerically to 10⁻¹⁵). The Lepton–Quark Subleading Universality statement unifies both fermion sectors.

The dynamical system

Q-field cosmological evolution obeys the autonomous ODE u′ = ξ(ξ − 3u)/[2(1 − u)], with u = Ω_Q/Ω_m, ξ = ln(1+z), and an IC-independent attractor (Δw ≈ 5×10⁻⁹).

Transition redshift z_tr = 0.583, band [0.569, 0.627] — the redshift at which the thawing free volume modulus crosses the golden kinetic signature u_J = 1/3 (⟺ Ω_σ = 2/147 ⟺ Ω_geom = 37/145), with zero new parameters (V₀ fixed by Ω_DE = 0.685); CLASS-verified at 0.5829, single crossing. Status WORKING_CANONICAL / PRE-REGISTERED, derived-conditional on DE-003. Supersedes the retired 0.972 and the provisional 0.9256.

Dark-energy slope, G-CC-λ OPEN. The late-time equation of state w₀ = −0.849 (free thawing branch, DE-003; CPL (−0.85, −0.23)) is a pre-registered, falsifiable prediction, not derived geometry: the thawing slope λ = 1 is a phenomenological IR ansatz — the canonical Casimir potential is shape-dependent and stabilises the volume, so there is no rolling exponential V ∝ e^(−λσ). The sourced/free branch split dissolves the earlier w₀ tension: the sourced branch is geometric dark matter (dust), the free branch gives w₀. The stability map remains analytically PASS.

A note on M₆ — two faces of one symbol

The symbol M₆ denotes two physically distinct compactification faces, and they must not be identified:

• Cosmological / IR face (geometric mean of the compact scales): M₆ = (M̄_Pl²/[(2π)²L₂L₃])^(1/4) ≈ 32.7 eV (M6C/M6GM, DER-COND). This supersedes the v3.1 figure of 46.3 eV.
• Winding / UV face (electroweak): m_w = M̄_Pl/√r = 1.91×10¹⁸ GeV, volume-independent (C-37/G37), with M₆^winding ≈ m_w/π ≈ 1.5×10¹⁸ GeV.

The two faces sit ~25 orders of magnitude apart; conflating them is the historic "two-tori" trap, now resolved and kept visible. A third meaning — the fundamental scale — belongs to the canon and to neither face.

Results, each at its declared closure level, each with a kill-switch

Closure tags are part of the result. THM = theorem from the axioms · STRTHM = structural theorem · WORK = working-canonical · DER-COND = derivation-conditional · EMP = empirical match · OPEN = open gap.

Average precision across the dimensionless parameters: ~1.2%.

Empirical contact with real data

SPARC (175 galaxies, median RMS 15 km/s, zero free parameters per galaxy); WALLABY PDR2 (187 galaxies, 15.0 km/s RMS); HALOGAS extended rotation curves; SLACS lensing (4.0σ); NANOGrav 15-yr (T₂ = 30 yr, T₃ = 19 yr consistent with the compactification scales; the adjudicated KK quantum m_KK = 2π/T₂ = 4.39×10⁻²⁴ eV coincides with the 30-yr circumferential mode); DESI DR1 BAO (χ² = 14.00, ΔBIC = +25.59 vs ΛCDM); Δχ²_CMB = −13.96 (TTTEEE_lite, pinned, EMP); golden-point 0-dof profile Δχ² = +0.95 vs ΛCDM (EMP); full Solar-System compatibility; cosmic-web scale λ₁₃ = 0.856 Mpc on the φ-ladder; galactic breathing scale λ₂ = 4.30 kpc (A0-MT-2, DER-COND, non-circular via the Connection Lemma).

N-body cosmological simulations (GADGET-4). Runs to 768³ with the 3D+3D kernel μ(k,a). Power-spectrum enhancement P₃D₊₃D/P_ΛCDM = +8% to +10% at z = 0, growing from zero at z ≈ 3 (Q-field activation); σ₈ ratio 1.045 (partially alleviates the S8 tension by ~2×, zero free parameters); 512³ convergence run confirms numerical convergence; the physical growth gate z29 → z19 passes at P-ratio 2.2236 ± 0.007 against the theoretical 2.211.

A negative result, kept visible. The internal golden ratio R(z) → −1/√5 of the compact temporal sector is DERIVED-INTERNAL and robust — and its proposed shear/lensing signature was closed negatively by the framework's own gates: structurally non-projectable, target ~10⁻³⁷. A framework that can tell its own beautiful number "you are not observable this way" is doing its job.

Rigorous mathematical foundations

UV completion via asymptotic safety (2 relevant operators) · Parameter Regress Theorem (4D theories have an irreducible floor d_free ≥ 19) · Boltzmann-complete kernel theorem (CLASS v3.3.4, R = 1.000 ± 0.003 on 71 points) · uniqueness of τ = i/φ (four no-go theorems) · IBM Quantum hardware verification (k = 1.003 ± 0.097, 4 backends) · Kreĭn-space spectral action · variational V-tree · Dimensional Unification (D² = 6D) · Topological Protection (ζ(3) cancels on T², g²Λ = 2π/W) · Non-Renormalization (7/9 geometric numbers protected to all loops) · NLO Hierarchy Correction (ΔΛ = 1/28; convergence beyond NLO OPEN) · Anti-S-Duality (Δ_NP ~ 10⁻³⁵) · Modular Closure (Γ⁰(2)) · Bridge Scale (μ_B = 74.16 GeV) · Berry Holonomy · Universal Subleading Kernel · K-LAMBDA · A-SELECTION capstone · Master Logical Chain (24-layer DAG, 15 pre-registered kill-switches).

The method layer, and gates that bite

Since v6.0 the deposit ships the framework's method layer, not only its physics: METHOD_GRAPH v0.1.7 (32 method nodes / 73 edges) including the ratified M-TIME-NONSPATIAL gate — structured time is not hidden space: compact temporal moduli may be handled with geometric and spectral tools, but any observable-facing spatial-like reading requires an explicit projection rule and explicit tracking of timelike signature, signs, contours and Wick conventions; the Observable Projection Graph (OPG, hard-blocker with executable engine, acceptance 7/7) — no claim reaches empirical status without a complete projection chain; and the Equation Usage Graph (EUG, hard-blocker) against silent reuse of coefficients across frames or epochs.

Gate II — the off-diagonal temporal modulus τ₁ (new in v7)

The framework's first systematic attack on the dynamics of the off-diagonal modulus of the temporal torus, and the route by which C-TAU1-SIGNED-ANSATZ will close. τ₁ = γ₄₅/γ₄₄ = Re τ is not r_τ = ln(τ₂/τ₂,*) — an earlier identification of the two was withdrawn on 2026-07-28.

• Gate II-0 (ratified 2026-07-26) — object freeze and ansatz audit.
• Gate II-1 (ratified 2026-07-27) — 𝒢_{τ₁τ₁} = 1/(2τ₂²) > 0, verified symbolically by two independent routes; φ²/2 = 1.309017 at the vacuum. Scope narrow and not extensible: homogeneous zero-mode / minisuperspace. Not a count of physical degrees of freedom — the stronger earlier phrasing was withdrawn.
• Gate II-2.0 (ratified 2026-07-28) — freeze only: four objects that must not be identified, orbifold parity, lattice classes kept separate, and a health vector ℌ_[n] in place of a scalar verdict, because a tachyon is not automatically a ghost.
• Steps 5, 6a, 6b, 6b-ii — delivered, PROPOSED, not ratified. The Dirac chain closes at stage two: 24 first-class constraints, rank 𝒞_[n] = 0, N_phys = 9 per complex mode, at eleven exact representatives on the ε_c = +1 branch. Step 6b-ii's kinematic half has been executed (1017 PASS / 0 FAIL over 22 runs); its residue step has not — no Z_α, no pole inversion, no sign, no inertia — and is governed by a pre-residue registration which withdrew its own earlier self-description as false, because a document that misdescribes which calculation it precedes protects nothing.
• Gate II-3 — under STOP, not opened.

Pre-registered predictions (kill-switches)

Fifteen predictions deposited before the corresponding data releases; triggering any one at > 5σ falsifies that sector.

Cosmological & lensing (KS1–KS8): w₀ = −0.849 (DESI DR2 + Euclid, 2026) · γ = 0.567 ± 0.010 · k_μ = 0.203 ± 0.010 h/Mpc · λ₁₃ = 0.856 ± 0.030 Mpc · Σm_ν ≈ 60 meV (KATRIN 2027) · sin²θ₂₃ = 0.535 ± 0.010 (DUNE/T2HK 2028–2030) · null WIMP detection · D/D_ΛCDM = 0.855 ± 0.010.

Particle physics: γ_UT ∈ [64°, 68°] (LHCb 2030) · V_cb / V_ub inclusive = exclusive (Belle II 2028–2030) · δ_CP ∈ [200°, 260°] (DUNE 2030) · sin θ₁₂ ∈ [0.305, 0.325] (JUNO 2027) · FLAG → 0.0410 ± 0.0003 · |C₉^NP| ≤ 10⁻² (HiLumi-LHC 2030).

What I am asking

Verify the derivations, criticise the assumptions, refute the predictions with data, improve the framework, and falsify it through the kill-switches provided. I am not seeking confirmation; I am seeking truth. If this framework is wrong, I want to know why. If it is partially valid, I want it refined by minds more expert than mine.

Archive layout (v7.1)

Four layers in order of authority, plus root documents. Where prose disagrees with the canon, the canon wins — with one named exception, EW001, where the ratified overlay of section 07 takes precedence over the frozen entry.

Root documents. README.md (this file) · ABSTRACT_v7_1_2026-08-01.md (and ABSTRACT_v6 retained — superseded is not void) · QUARANTINE_NOTICE_2026-08-01.md · VERIFICATION_REPORT_2026-08-01.md · PORTABILITY_NOTICE.md · LICENSES.md · README_UPLOAD_CURRENT_NO_DRIFT.md (snapshot note) · VEGA_CONTRACT.json (delivery machine record) · CHECKSUMS.sha256.

01_CANON_LIVE_SNAPSHOT/ — the machine-readable canon, single source of truth. STATE.md ("you are here"), Claim Registry, Conflict Ledger, Open Gaps, Failed-Paths Index, derivation-graph data (nodes.json, edges.json, symbols.json), HTML graph renderers, build and verification scripts, the Equation Usage Graph engine, and the meta-audit blackboards. 125 nodes / 181 edges / 26 symbols, NO DRIFT.

02_CANONICAL_PAPERS_v2_2026_06_15/ — the authoritative synthesis: six consolidation papers (Foundations · Core Spine K/W7 · Particle/Electroweak/Higgs/α · Cosmology/DE/DM/LSS · Predictions & Kill-Switches · Audits/Failed-Paths/Open-Gaps) plus MASTER_CHAIN v5 and the node↔paper index. The curated reading path.

03_TRIAGE_AND_SOURCE_MANIFESTS/ — provenance: how the deposit was curated from the working trees, plus earlier Zenodo manifests for continuity.

04_FULL_PAPERS_LIBRARY/ — the individual full papers with complete step-by-step derivations (100+ papers, ~2200 pages): master documents, foundations, core theorems, particle sector, cosmology, predictions and kill-switches, finished audits, the June dossiers (PHI-SEL, K-LAMBDA, A-SELECTION), the corrected galactic branch and the completed kernel branch.

05_METHOD_AND_PROJECTION_GRAPHS/ — the method layer: METHOD_GRAPH v0.1.7 with M-TIME-NONSPATIAL, the OPG specification, engine, acceptance tests and baptism blackboards.

06_GATE_II_6D4D_TAU1_2026_07/ — the τ₁ programme. Ratified gates and proposed steps are in separate folders, and the scope of each seal is stated in its README. Includes the complete replay package for steps 5–6b-ii.

07_RATIFIED_AUDITS_EW_2026_07/ — the two ratified electroweak audits and a README stating exactly what each verdict preserves and what it does not.

Raw working-frontier blackboards (Edison-mode notes, session ledgers) are deliberately excluded from the public record; open residuals are tracked in the canon, not published as results.

Reproducibility

All papers in Markdown (PDF where available). 439 files, 438 covered by CHECKSUMS.sha256 — the manifest cannot cover itself; its hash is published in the Zenodo record description. Code runs under Python 3.11 with numpy, scipy, matplotlib, astropy and sympy, with two documented exceptions carrying laboratory-specific paths — one of which prints a false alarm about the canon when run elsewhere. Both are named, with the correct command for each, in PORTABILITY_NOTICE.md. GADGET-4 modifications and CLASS v3.3.4 integration are documented with build instructions; the upstream codes are not redistributed here.

Licences

CC BY 4.0 for the corpus (396 files) and MIT for the code (43 files), mapped file-type by file-type in LICENSES.md. Both are declared in the record metadata.

A note on AI collaboration

Every derivation is machine-verified (SymPy / mpmath) and independently checked by at least two AI systems, with mandatory adversarial Red-Team review by a separate system. Every canonical change passes a fail-closed NO-DRIFT checkpoint; an automated guardian lints every document against over-promotion; nothing enters the canon without explicit human ratification. The use of AI as a derivation engine is disclosed transparently — it is an instrument, not a substitute for scientific judgement.

This release adds a fourth layer to that discipline: the delivery gate. A check that cannot fail is decoration; a null result needs a positive control; a label that asserts more than its code tests is the defect the whole apparatus exists to catch. Several corrections shipped here were found by the referee inside checks written to display rigour — and the release itself was held twice, on findings that were upheld without argument. That is the apparatus working, and it is published rather than smoothed over.

Science is not certainty. It is openness, dialogue, falsifiability. This is my contribution — offered in complete transparency, with humility, and with the hope that someone with the appropriate expertise will test it.

Simone Calzighetti · 3D+3D Laboratory, Abbiategrasso, Italy · simone.calzighetti@3dplus3d.it · www.3dplus3d.it

Changelog — v4.0 (17 June 2026) → v7.1 (1 August 2026)

Canon: 116/164 → 125/181/26. Three new nodes on branch time-torus (G6D4D, TAU1KIN, II20FRZ), four new edges, five new symbols (λ_n, μ₆, ℌ_[n], θ^a, τ₁); node FF reformulated in the shape-dictionary variable; r_τ annotated and no longer a dead letter.

v5.0 (26 June) — ratified Galactic Registry Master, post-19-June capstone/theorem layer, corrected galactic branch.
v6.0 (10 July) — the method layer: METHOD_GRAPH v0.1.7 with the ratified M-TIME-NONSPATIAL gate, OPG promoted to hard-blocker with engine and four baptisms, EUG v0.2 hard-blocker, FORMULA_GRAPH v0.3-LIVE, G45-A arc closed, G-SHEAR-TRANSFER arc sealed as structurally non-projectable.
v7.1 (1 August) — the quarantine notice on three deposited papers; the ratified electroweak audits and the EW001 overlay, with the precedence exception stated explicitly; the Gate II programme on τ₁; the portability notice; the licence map; the verification report with its declared boundary. No canonical numerical value changed between v6.0 and v7.1.

Version 7.0 was built, internally ratified, and held twice at the release gate by the external referee — first on the status of Step 6b-ii, the electroweak overclaim and two non-portable scripts; then on the difference between ratified and propagated, where the deposit's own precedence rule was cancelling the reservation it had just raised. All findings were upheld. The record of both holds is in VERIFICATION_REPORT_2026-08-01.md, §V13 and §V14.