One page. What is claimed, what is open, what was withdrawn. Everything here links to
DERIVED_SUMMARY.md for the derivation and to a runnable script for the number.
Last audited 4 September 2026.
Physical reality is a single complex field ψ = ρe^{iθ}. Its phase turns on a circle, and time is the cycling of that phase — so mass is a rate, not a substance. The circle is not assumed: it is the field’s vacuum manifold, so compactness is earned. From that one premise a large part of quantum mechanics follows as consequence rather than postulate, gravity appears as the geometry the field’s energy induces, and the dark sectors become aspects of one field. The programme’s discipline is that every claim is gated, and negative results are recorded as prominently as positive ones.
These rest only on the low-energy field living on a circle. Audited 3 Sep 2026: none is regime-dependent; the teotl-quanta reframing changes none of them.
| claim | where |
|---|---|
| Charge quantisation — integer, from single-valuedness of the phase | §2, §7d |
| Number–phase uncertainty ΔN·Δθ ≥ ½ — a theorem, not a postulate | §7d |
| CHSH closure → 2√2, and the Born rule |c|² from envariance | §7d |
| Spin-statistics — spin-½ from ℤ₂ self-linking of vortex lines (a within-QM result: the Finkelstein–Rubinstein argument needs a wavefunction, so it cannot be used as a route TO quantisation — scope narrowed 5 Sep 2026, claim itself unaffected) | §7c |
| Gravity’s shape and sign — 1/r, universal attraction, matter and antimatter alike | §4 |
| Solar-system reproduction — Kepler + Mercury’s 42.9″, one frozen constant (⚠ Mercury needs the spatial metric, which costs one inserted posit — see the note below) | §5 |
| a₀ ∝ cH₀ scaling — because the field is the dark energy, so Λ cancels | §6 |
The metric sector, completed 4 September 2026 — and priced. One inserted condition, f·h = 1 (the fundamental cell occupying a fixed coordinate extent per axis), fixes the spatial metric and gives γ = 1 (Cassini 0.91 σ, against 4.35×10⁴ σ before). Lorentz covariance — which the framework already had — then supplies the vector sector for free: frame dragging 39.2 mas/yr against GP-B’s 37.2 ± 7.2 (0.28 σ). All four classic tests now clear: dilation, deflection 1.7512″, Mercury 42.98″/cy, Shapiro 1.0000, dragging 39.2.
It is NOT in the DERIVED table above, and deliberately. f·h = 1 does not follow from the framework’s primitives, and its reading is chosen: the 4-volume reading gives γ = 1/3 and is excluded. One inserted posit plus one chosen reading — cheaper than importing the field equations, not free.
And these four tests do not discriminate. The resulting 1PN metric is GR’s, which is the stated aim (reinterpret GR so it can meet the quantum sector, not refute it) — but it means clearing them is a consistency floor, not evidence for the framework. The first real difference is at 2PN: 1.6×10⁻¹⁶ at Mercury. Real, unmeasurable.
⚠ REVISED 5 Sep 2026 — f·h = 1 is a CHECK, not the mechanism. Two limitations, found the day after the above was written. (1) It cannot radiate. GR has 2 propagating degrees of freedom — the tensor polarisations LIGO observes. A metric built from one function carries 1: a scalar breathing mode. The construction reproduces the static 1PN metric and cannot reproduce gravitational waves at all. (2) The dynamics were already located elsewhere and make the posit redundant. EGC0 identified gravity’s coefficient as the Sakharov induced-gravity coefficient ((1/6 − ξ) = 1/6 for minimal coupling, which is how the phase couples); induction generates an Einstein–Hilbert term, so γ = 1 and β = 1 follow from varying an action — with both tensor polarisations. Posits: 1 + 1 reading → 0, and the price is not new (G was already open). ⇒ Read the note above as an independent consistency check on the induced route, not as the mechanism. Numbers stand; role demoted.
Consolidated 1–3 Sep 2026. A dozen previously-declared “floors” reduce to three numbers.
| number | what it is | status |
|---|---|---|
| Λ ≈ 8.7×10⁻¹²² | the cosmological constant | shared with all of physics — not TFT’s alone |
| σ | the soliton-interior shape | the mass hierarchy and the metric coefficient both reduce to it — but see the note below: σ alone is not a physical number |
| Nξ | the kinetic normalisation | physical but UV-anchored; inert at low energy |
The cutoff scale is not among them — E₀, ℓ₀, τ₀ follow from measured {ħ, c, G}. What is undetermined is σ’s value at that scale, which is the ordinary effective-field-theory situation.
⚠ σ ALONE IS NOT A PHYSICAL NUMBER (5 Sep 2026). Two studies here quote σ thresholds that look contradictory — σ ≥ 3/16 and σ > 1/2 — and are not. For a general V = ½m²ρ² − bρ⁴ + σρ⁶ the Q-ball window opens at ω_min² = m² − b²/(2σ), i.e. σ > b²/(2m²): b = 1 gives 1/2, b = √(3/8) gives 3/16. Same physical condition, two normalisations of the quartic. The invariant is the dimensionless combination b²/(σm²), not σ. Quoting a numerical value for σ without stating the normalisation is meaningless — and this file, and the “one dimensionless number” framing elsewhere, should name b²/(σm²).
⚠ σ’s status is contingent on the particle model (noted 4 Sep 2026). The sextic term whose coefficient σ is exists for exactly one reason: Derrick’s theorem, which forces a static 3-D lump to carry a stabiliser. That requirement comes from modelling the particle as a Q-ball / amplitude soliton. Under a particle model of persistent excitations — a mode rather than a lump — Derrick never applies, no stabiliser is required, and σ is not a parameter of the theory at all. Study IMPORT0 (4 Sep 2026) found σ is therefore an artefact of a modelling choice, not a floor the framework itself imposes.
This is NOT yet a withdrawal. The corpus still records the amplitude-soliton reading, and which particle model TFT actually asserts is unsettled. Recorded here so the reader knows σ’s status depends on that open question, and that everything reducing to σ — the mass-hierarchy angle, the metric coefficient, the fission window — inherits the same contingency.
⚠ CORRECTED the same day (PART0, 4 Sep 2026): σ does NOT dissolve. The note above named only one of the sextic’s two jobs. Derrick is one. Bounding the potential below is the other, and it applies to time-dependent excitations too. With the sextic removed, V = ½ρ² − ρ⁴ is unbounded below and the field runs away; requiring V ≥ 0 with a unique vacuum at ρ = 0 needs the discriminant 1 − 2σ < 0, i.e. σ > 1/2 — verified numerically (at σ = 0.25, min V = −1.104 and a test excitation collapsed into the well rather than decaying). So: σ’s DERRICK role dissolves under a persistent-excitation particle model; its BOUNDEDNESS role does not, and that role is independent of which particle model is chosen. σ stays on this list.
(What the same study did establish for the particle model: persistent, localised, uncharged 3-D excitations do exist in this potential and survive to t = 2400 retaining 34% of core energy, against 0.2% for a pure-phase control — so “an excitation that persists” is a real object here. It is long-lived, not eternal: the late-time decay rate is −2.0×10⁻⁵ per unit time and still falling.)
What sets that lifetime, and what makes anything here exactly stable (LIFE0, 4 Sep 2026). The excitation oscillates at ω ≈ 0.66 m — below the mass gap — so radiation at the fundamental is kinematically shut. The only open channel is the second harmonic (2ω ≈ 1.32 > m), suppressed by P(2ω)/P(ω) ≈ 2×10⁻⁷. That suppression is the lifetime, and the single ratio ω/m fixes both facts: how far below the gap it sits decides both that the fundamental is closed and how weak the escaping harmonic is.
A charged configuration is a different case: it is absolutely stable. Coleman’s criterion — a charge-Q lump cannot decay into Q free quanta of mass m if E/Q < m — is satisfied with margin: E/Q = 0.681 and 0.695 against m = 1. Decay is forbidden, not slow. The conserved quantity doing the work is this framework’s own U(1) Noether charge, which is electric charge (CHRG1). So the stable particle here is the charged one, for the same reason the electron is stable in nature: it is the lightest charged thing and charge conservation has nowhere to send it.
The constraint that follows, and it is a real one. The uncharged excitation has Q = 0, so the criterion is not merely unmet but undefined — there is no conserved quantity to obstruct decay, which is therefore rate-limited and never forbidden. Nothing in this framework’s inventory supplies exact stability except the Noether charge. A purely uncharged persistent excitation cannot be a stable particle here.
Caveat: the charged runs still shed ~4% of core energy, consistent with relaxation off the initial profile plus absorber losses rather than an open decay channel — but that run does not separate the two, and E/Q was evaluated at a single time. The stability claim rests on the criterion, not on a converged profile.
Distinct from the constants above: these are not missing numbers but missing physics, and work could still close them.
| question | where it stands |
|---|---|
| Is there an equation of motion for the metric? | A route, not a derivation (revised 5 Sep 2026). The dynamics are the Sakharov-induced Einstein–Hilbert term located by EGC0 — which supplies γ = β = 1 and both tensor polarisations, and makes the f·h = 1 posit redundant. What is still missing is the O(1) coefficient of G (regularisation, species count — EGC0’s floor), and whether one-loop induction is more than a heuristic. (A worry listed here on 4 Sep — whether higher-curvature terms swamp the induced R term — was closed 5 Sep: the suppression parameter R·ℓ₀² is 1.4×10⁻⁹² at the solar surface and 1.8×10⁻⁷⁶ even outside a solar-mass black hole. Not a threat.) The one outstanding structural gap is radiative: every single-function construction here — f·h = 1 and the contraction/Gullstrand–Painlevé form alike — carries 1 propagating degree of freedom where GR has 2 (the tensor polarisations LIGO observes). Only the full induced/ADM route can close it. (A cutoff at Λ = 1/ℓ₀ returns G’s order by construction, since ℓ₀ is defined through G — an identity, not evidence.) The metric is sourced, not yet dynamical from first principles. But the metric’s missing COMPONENTS are no longer missing (4 Sep 2026): one inserted condition, f·h = 1, fixes the spatial sector (γ = 1, Cassini 0.91 σ), and Lorentz covariance then supplies the vector sector free (frame dragging 39.2 mas/yr, GP-B 0.28 σ). All four classic tests now clear. The components are had; the dynamics generating them are not. |
| What carries baryon number? | Nothing identified. Winding was withdrawn as the carrier, and the proton-stability claim went with it. |
| What does the winding number W do? | Unassigned. It carries a local sector label: no monopole moment, confined, no relics. A proposed T-duality role was withdrawn (3 Sep). |
| Does the field account for the dark sector? | Not fully. The phase sector cannot clump on cluster scales; the amplitude sector clumps but is not transparent. Neither supplies the observed lensing offset. |
| What sets the mass hierarchy’s angle? | Structure is exact (√m_k = M(1 + A cos(δ + 2πk/3)) reproduces all three leptons), but the one free angle reduces to σ. Degree-6 dynamics can place it; that is a fit, not a derivation. |
| The particle spectrum | The least-developed sector: no first-principles masses or couplings. Sharpened 9 Sep 2026 — and the sharpening is the useful part. Four independent ladders have now been tried and all four are too flat: excitation counting (ratios fall, 4 → 1.78), binding counting (2.0 → 1.25), the 1-D breather tower (m₂/m₁ ≤ 2 for every coupling), and knot ropelength (widest ratio available 3.94, on verified values). The leptons need μ/e = 206.8 and τ/μ = 16.8. So the problem is not “the right ladder has not been found” — it is that ladders are additive and a hierarchy this steep needs cancellation. That is consistent with the one route that did reproduce the leptons: the generation matrix’s near-singularity (det C ≈ 0, the electron a near-null eigenvector). A fifth ladder is not the thing to try. |
| Is TFT’s phase-regime requirement negotiable? | Unresolved, and it matters — the argument for it is from what other results need, not a theorem. It decides whether the quanta emulator models the theory or is a separate object sharing a Hamiltonian. |
Listed so a reader meets them once, here, rather than discovering them scattered.
| withdrawn claim | why | when |
|---|---|---|
| a₀’s 2π is derived | the field equation gives cω₀; the 2π is a factor of 6.28 the framework cannot supply | Aug 2026 |
| charge = winding | charge is the Noether charge; winding has no monopole moment | Aug 2026 |
| baryon number = winding | no carrier; the proton-stability claim went with it | Aug 2026 |
| metric ansatz g_ij = δ_ij(1+|∇θ|²/E₀²) | wrong shape (1/r⁴); the weak-field form is what works | Aug 2026 |
| T-duality prevents the infinities | a field winds for free; only extended objects stretch | Sep 2026 |
| quanta are two-state (fragmentation bound) | true in the charge regime only; TFT’s regime is harmonic and stable | Sep 2026 |
| question | verdict |
|---|---|
| SU(2)/SU(3) from one circle | no — four independent routes, one wall (added structure required) |
| modified inertia from an action | no — four studies; the mechanism is a horizon heuristic |
| the lepton hierarchy from counting | no — excitation and binding both give ratios that flatten while the data explode |
| compact time replaces quantisation | no — it fixes the spectrum within QM; it does not derive QM |
DERIVED_SUMMARY.md — every derivation, with the script that computes it.PREDICTIONS.md — the falsifiable edges.teotl_rotor_qc.py, teotl_field_qc.py — the quanta emulator (self-test: python3 teotl_rotor_qc.py).
Note: it runs in the charge regime; TFT proper needs the phase regime. It is a working
emulator, not a model of TFT’s regime.teotl_substrate.py — abandoned, marked at the top of the file. Kept for the record.