18 July 2026. A foundations companion. It asks what the Teotl field is (not just what it does), and — with equal weight — where the framework fails or bottoms out. It contains the sharpest honest negative in the program.
Read this first. Most results here reproduce standard physics and are, by construction, empirically degenerate with it — their value is conceptual: a single circle-valued field is shown to unify time, charge, the galactic acceleration scale, and dark energy, and to recast the measurement problem as a problem of time. But two results are limits, not victories: the “classical” version of the field cannot reproduce generic quantum entanglement (§4), and the dark-energy fit needs a super-Planckian decay constant (§5). Both are stated plainly. Nothing here claims to beat quantum mechanics or to solve the cosmological-constant problem. Pre-registrations and gates are in the
*_prereg.mdfiles; every verdict was fixed before computing.
pw_emergent_time.py)TFT says “time = phase cycling.” We make this precise via the Page–Wootters mechanism. A globally timeless constraint state (Ĥ_C + Ĥ_S)|Ψ⟩ = 0 — built from the Hamiltonians alone, no external time — with the S¹ phase as the clock, reproduces Schrödinger evolution of the rest when conditioned on the phase-clock (fidelity 1.000000). The emergent time is cyclic and the clock spectrum is a comb — the same compact-time structure that appears elsewhere in the program.
The payoff is ontological: the internal phase circle (whose winding is charge) and the time circle (whose cycling is time) are one structure. Time is the phase read relationally. Honest floor: this reproduces standard quantum mechanics exactly, so “the phase is time” is an identification, not a forced result — an ordinary external-time reading fits the same data.
scale_darkenergy.py)If that same S¹ is the cosmological dark-energy field (an axion-like pNGB), a single circle at the Hubble scale ties together four things:
A0_STATUS.md); the proportionality is, and is Milgrom’s (1983).
scale, the 2π from the circle;Because one field sets both, a₀ and the dark-energy w(a) are locked — a joint, DESI-testable signature no standard framework offers (ΛCDM has neither; MOND has a₀ but no dark-energy dynamics). A confirmed phantom crossing (w < −1) would falsify it. There is even a natural reading of a₀ itself: it equals the field’s own de Sitter quantum fluctuation δφ ~ H₀/2π (§5), i.e. the acceleration below which the field’s quantum fluctuation dominates the dynamics.
Honest floors: the absolute scale (ρ_DE / H₀) is an input — this does not solve the cosmological-constant problem; and the quantum-time = dark-energy-circle identity is a hypothesis (empirically degenerate), tested by consistency and consequence, not proven. The falsifiable content (no-phantom, wₐ) is the physics of the dark-energy companion; here it is attached to the one circle that is also time.
meas3_selection.py, meas4_classical_arrow.py)Recasting measurement as a problem of time: TFT does not replace decoherence, it completes it. Only definite (decohered) branches can close as single-valued histories — a coherent “cat” of macroscopically-distinct branches cannot close (its closure amplitude → 0 with size). The loop’s seam phase then selects one outcome per run (no branching), with Born frequencies. Einselection (the pointer basis) is reproduced, and energy positivity (E>0) supplies a microscopic clock direction (matter vs antimatter = opposite winding).
Honest floors: definiteness and the pointer basis follow, but which outcome is realized (the seam phase) is a boundary condition, not derived — collapse is reframed as deterministic loop-closure, not conjured away. And the thermodynamic arrow is a low-entropy boundary condition (the past hypothesis), correctly not derived from the time-symmetric dynamics — deriving it would violate T-symmetry.
A support withdrawn (added 5 August 2026). It is natural to assume this account can lean on the field’s dissipative behaviour — that an amplitude which decays is already halfway to a classical world. Calculated: it cannot, and the result is a theorem rather than an estimate. Under any deterministic decay of branch amplitudes,
rho_00 -> rho_00 e^(-b0 t), rho_11 -> rho_11 e^(-b1 t), rho_01 -> rho_01 e^(-(b0+b1) t/2)
the off-diagonal exponent is forced to the arithmetic mean of the two diagonal ones — exactly what renormalisation divides out. The normalised coherence C = |ρ₀₁|/√(ρ₀₀ρ₁₁) is therefore invariant for every choice of decay rates and every time (computed deviation 0.0e+00, against a Lindblad comparison falling 1.000 → 0.018 over the same interval). No choice of rates evades it, because the amplitude equation forces that mean.
A decaying amplitude produces global loss, not loss of coherence between branches — after renormalisation it decoheres exactly nothing. Genuine decoherence requires a stochastic element or real entanglement with an environment; a deterministic sink supplies neither.
Consequence for this section: the loop-closure account above stands on loop-closure alone. It cannot draw support from amplitude dissipation, and any reading of the form “the framework is natively dissipative, so it natively explains the classical world” is false. Whatever produces the classical limit here, it is not a decaying amplitude. The floors stated above are unchanged; one assumed prop behind them is removed.
tens_completeness.py)Can a single economical (polynomial-resource) circle-valued field realize the full 2ⁿ-dimensional Hilbert space of n subsystems? No. A single field profile has polynomially many parameters; a general n-qubit state needs 2ⁿ amplitudes. The field is therefore entanglement-bounded (matrix-product-state-like): random-state fidelity collapses with n, and the entanglement it can carry is capped.
Crucially, this explains why the CHSH, Born, and GHZ results elsewhere in the program succeeded — those states are all low-entanglement (GHZ is one ebit, representable at bond dimension 2). But volume-law entanglement is not representable. So the economical “classical” Teotl field is a bounded-entanglement subtheory — and is therefore falsified by quantum-supremacy experiments, where nature realizes exactly the volume-law entanglement the field cannot.
The only way to recover full quantum mechanics is to quantize the field (exponential / Fock degrees of freedom) — at which point it is standard quantum field theory, degenerate with QM and no longer “just a classical circle-valued field.” The framework cannot be both economical-classical and full quantum mechanics. This is the sharpest limit on the program’s quantum ambition, and we state it without softening.
The same limit reached from single-photon optics (added 5 August 2026). The photoelectric effect is usually cited as the proof that light is quantised, and it is tempting to count the framework’s reproduction of it as a success for its account of light. It is not, and the reason is old: a classical field coupled to quantised matter already reproduces the threshold, the KE = hν − φ slope, and the instantaneity (Lamb & Scully 1969). The quantisation that produces the threshold lives in the matter, not the light. This framework quantises matter natively (an integer number/Noether spectrum and a discrete bound tower — noun corrected 18 Aug 2026, UNC1; this read “integer winding”), so it reproduces every photoelectric observable — and that fact carries no weight, because the effect never discriminated.
What discriminates is second-order coherence. Calculated: for any classical field — any theory in which intensity is a non-negative random variable — Cauchy–Schwarz gives g₂(0) ≥ 1. Verified across five distributions (coherent 1.000, thermal 2.008, uniform 1.334, lognormal 4.261, bimodal 1.999; minimum 1.000). The field’s light here is the massless Goldstone, a continuous classical mode, so it falls under the theorem and predicts g₂(0) ≥ 1.
Measured: photon antibunching g₂(0) < 1 (Kimble–Dagenais–Mandel 1977), and beam-splitter anticorrelation α = 0.18 ± 0.06 — 13σ below the classical bound (Grangier–Roger–Aspect 1986). So: the framework reproduces every photoelectric observable and fails the first observable that actually probes the quantisation of light. That is this section’s limit arriving from an independent direction, at one-photon scale, and it was crossed experimentally in 1977. Recovering it requires quantising the field — the same escape, with the same cost, as above.
swmp_tension.py)The dark-energy fit of §2 requires a decay constant f ≳ 1.45 M_Pl — super- Planckian, a genuine swampland concern shared with all thawing quintessence. Clockwork/alignment mechanisms would lower it but need many circles, breaking the one-S¹ picture. Monodromy — winding the single circle Q ~ 15 times — is compatible with one circle and native to TFT (the winding is the theory’s own integer — corrected 18 Aug 2026: this read “winding = charge”; electric charge is the U(1) Noether charge, not the winding, which sources no monopole field. The monodromy argument needs only that the winding is an integer, so it is unaffected), giving an effective super-Planckian f from a sub-Planckian fundamental. So the tension is not eliminated but relocated to a winding-number floor (same class as the framework’s other integer/initial-condition inputs).
Does the tension leak into R³ uncertainty? No — it is decoupled. The field’s
de Sitter fluctuation δφ ~ H₀/2π is f-independent and equals a₀/c; super-
Planckian f only shrinks the angle uncertainty δθ ~ H₀/(2πf) (the phase
classicalizes). So the observable R³ acceleration-uncertainty scale a₀ ∝ cH₀ (coefficient not derived, A0_STATUS.md) is
set by the circle’s size (H₀), protected from the super-Planckian problem, which
stays confined to field space. (The Planck-scale duality still holds abstractly:
super-Planckian f is the field-space mirror of the R³ minimal-length/GUP limit — but
the observable R³ scale is H₀, not M_Pl.)
pip install numpy
mkdir -p outputs
python3 pw_emergent_time.py # time emerges from the S^1 phase (Page-Wootters)
python3 scale_darkenergy.py # one S^1 at H0: a0=cH0/2pi + no-phantom w(a), locked
python3 meas3_selection.py # single-outcome selection by loop-closure
python3 meas4_classical_arrow.py # classical limit (einselection) + arrow of time
python3 tens_completeness.py # the sharp limit: classical field is entanglement-bounded
python3 swmp_tension.py # super-Planckian tension + monodromy + R^3 decoupling
Each prints its pre-registered gate and verdict; JSON lands in outputs/.
Pre-registrations: PW0_prereg.md, SCALE0_prereg.md, MEAS3_prereg.md,
MEAS4_prereg.md, TENS0_prereg.md, SWMP0_prereg.md.