13 July 2026 (CHSH closure); 14 July 2026 (Born rule). A quantum-foundations
companion, extending the CHSH result in teotl chsh.py.
Framing superseded, 17 August 2026 — the results are unaffected. This note (and its title) attribute the results to compact time, “the ℝ³×S¹ structure, time as a closed loop.” An audit of what each result actually uses found that S¹ is the TARGET space of the phase field, not a dimension of spacetime — internal in the same sense the U(1) of electromagnetism is internal. Every result here — the CHSH closure, the Born rule, charge quantisation, number–phase uncertainty — follows from single-valuedness of the PHASE on its circle, and none of them requires time to be compact. The correct reading throughout is therefore “compact phase,” not “compact time”; see the corrected substrate row in
DERIVED_SUMMARY.md. No number, derivation or conclusion in this note changes — what changes is the attribution. A full retitling is deferred; this note records the correction in the meantime.
The qubit emulator (teotl_qc.py) reproduces single qubits faithfully — a lone
qubit is just a phase and an amplitude, which a classical field carries. The
hard part is entanglement: the Bell/CHSH correlations that are provably too
strong for any local classical description. teotl chsh.py reported the
honest result — TFT’s local field saturates CHSH at exactly S = 2.0000, the
classical bound, while the exact quantum reference reaches 2√2 ≈ 2.828.
This note asks whether compact time (the ℝ³×S¹ structure, time as a closed
loop) changes that — and finds that, worked through honestly, it derives the
quantum value 2√2 with no free parameters, reframing quantum non-locality as
compact-time coherence. It also states, prominently, what that does and does not
buy. Pre-registration: CHSH0_prereg_compact.md.
Read this first. The result below reproduces quantum mechanics; it does not beat it and, by a Bell test, cannot be distinguished from it. Its value is conceptual — a deterministic, single-valued-phase account of why the quantum correlation has the value and the ceiling it does. The Born rule (single- outcome |ψ|²), previously flagged here as open, is now derived from the same closure (§6,
born1..5_*.py) — assumption-conditional, and still degenerate with QM. The distinguishing-experiment question is now itself answered (§7): a search of the natural channels finds the theory empirically degenerate with QM, with one open falsifiable edge (tensor-completeness). It is not a claim to have surpassed quantum mechanics.
chsh_compact_time.py)First, the honest trap. Bell’s S ≤ 2 rests on three assumptions: locality, single outcomes, and measurement independence (the hidden variable is uncorrelated with the settings). The quantum value 2√2 is no-signaling — the measurement marginals stay flat — so exceeding 2 does not require faster-than-light signaling; it requires giving up measurement independence. Compact time offers a non-conspiratorial way to do that: on a closed time loop the “past” state and the “future” setting lie on the same manifold, so the periodicity condition ties them (the time-symmetric / two-state-vector picture), not superdeterministic fine-tuning.
But a posited time-loop reweighting of the hidden variable is unconstrained — we checked, and a naive one reaches S = 2.90, above the quantum bound (a no-signaling super-quantum, PR-box-like correlation). That over-reach is the tell: an arbitrary measurement-dependence can give any value up to 4, so “compact time lets S exceed 2” is, alone, vacuous. The physics has to come from the actual field, not a chosen weight.
chsh_closure.py)The fix is that TFT’s S¹ variable is not an arbitrary weight — it is a single-valued complex phase. That one physical fact does all the work:
The three cases line up exactly: dephase the S¹ (lose the phase) → 2, classical; treat it as an arbitrary weight → 2.90, unphysical; keep it a single-valued coherent phase → 2√2, quantum.
Quantum coherence is the phase closing single-valuedly on the compact time circle. The “spooky” Bell correlation is, in this account, the deterministic geometry of a closed-time phase seen from ℝ³ — the hidden phase is perfectly definite, it just cancels from the observable correlation, so ℝ³ looks uncertain. The quantum value (cos) and the quantum ceiling (Tsirelson) both follow from single-valuedness, with nothing tuned. This makes precise the thesis that what is usually called quantum uncertainty is ordinary S¹ behavior.
| statement | label |
|---|---|
| local field saturates CHSH at S = 2.0000 | computed (the honest baseline) |
| S > 2 is no-signaling → needs measurement-dependence, not signaling | derived |
| an arbitrary time-loop weight is unconstrained (reaches 2.90) | computed (a warning) |
| single-valued S¹ phase ⇒ hidden variable cancels ⇒ E = cos(a−b), no tuning | derived |
| coherent phase ⇒ Tsirelson ⇒ CHSH capped at 2√2 automatically | derived (2.828 numerically) |
| “quantum coherence = the phase closing on the S¹ time circle” | interpretation (coherent) |
| a measurement that distinguishes compact-time TFT from standard QM | searched (§7) — none feasible; empirically degenerate; one edge (tensor-completeness) open |
| single-outcome probabilities (the Born rule = |c_k|²) | derived (§6, assumption-conditional) |
born1..5_*.py)The closure gives the two-point correlation coherently; that single measurement
outcomes follow |ψ|² is a separate step, taken here. Pre-registration:
BORN0_prereg.md. Five pre-committed stages, each with a runtime gate — verdict
PASS (structural, assumption-conditional), all gates met at machine precision,
no tuning.
born1). For an
entangled two-channel state, the system swap 0↔1 is undone by a purely
environmental unitary iff |c₀| = |c₁| (counter-op unitarity defect ~1e-16
equal-moduli; 0.6–8 for the unequal control). A pure-environment operation
cannot change a system-local probability, so the swap leaves them unchanged ⇒
P(0)=P(1) — with no |c|² inserted (Zurek envariance, realized on the field).born3, load-bearing).
Fine-grain a rational weight m/n into n branches; the branches come out all
equal-modulus (1/√n), so envariance makes them equiprobable ⇒ P = m/n = |c_k|²,
independent of n. The exponent 2 is coherent-superposition normalization —
equal branches must carry amplitude 1/√n, so branch-count = 1/amplitude² — not
a charge postulate. An independent route via the field’s own conserved U(1)
charge on the Haar S¹ measure agrees (born2), with the Born additivity of
exclusive outcomes tracing to winding-orthogonality (the interference cross-term
vanishes).born4). A spin-½ winding measured at relative
angle θ has phase-geometry overlap cos(θ/2) (the SU(2) half-angle), so the
weight rule gives P(+|θ) = cos²(θ/2). It is uniquely pinned by consistency
with the closure’s E = cos θ: no other exponent reproduces the coherent
correlation (q = 1, 3, 4 break it).born5). A single object, P = |⟨·|Ψ⟩|², yields the
marginals, no-signaling, E(a,b)=cos(a−b), Tsirelson S=2√2, and Malus (its
product-state limit) — mutually consistent, no second mechanism glued on.What this buys, and its floor. The Born rule reduces to *the S¹ swap symmetry
dis1, dis2)Both results above reproduce QM, so the standing open question was: is there ANY
observable where compact-time TFT and standard QM differ? We searched the
channels where compact time can differ, with the gates fixed in advance
(DIS0_prereg.md). The honest answer: there is one in principle, but the theory
is empirically degenerate with QM for any feasible experiment.
dis1). Discretizing /
compactifying the time circle to any size N leaves E(a,b)=cos(a−b) and S=2√2
unchanged (machine precision, N-independent), because the hidden time-phase
cancels (correlations see only setting differences). CHSH can never
distinguish them — upgrading the numerical 2√2 to a structural statement.dis1). Single-
valuedness on a period-T loop forces an energy comb Eₙ=2πn/T and revival that is
exact and strictly periodic at T. But the effect scales as 1/T. The only T
consistent with observed continuously-tunable spectra is cosmological
(T~1/H₀ → comb spacing ~10⁻³³ eV, revival ~ age of the universe = unobservable);
a microscopic T is excluded — it would quantize energy in mc²≈511 keV units,
forbidding eV atomic lines. Time-winding sectors are degenerate (phase 2πn).dis2). The coherent-phase
closure, extended to three parties, reaches Mermin M = 4 = QM: the full GHZ
paradox and contextuality irreducible to 2-body (all pairwise correlations
vanish, yet M=4). So even the “does TFT fall short of QM?” test comes back
degenerate.Sharpest in-principle falsifier: forbidden transition frequencies between comb teeth → observed continuous spectra already bound T ≳ 1/H₀ (satisfied). The QC arc’s old caveat (“reproduces QM, no distinguishing test”) is now a result, not a hand-wave — the reinterpretation is not experimentally separable from QM by any feasible measurement. Its value is conceptual/foundational.
The one surviving falsifiable edge (honest, open). The GHZ computation used the full 2ⁿ-dim tensor Hilbert space the closure claims to be; CHSH established only the 2-body sector. A field-theoretic proof that n windings realize the full 2ⁿ tensor space is undone — and if the single-field S¹ construction secretly saturates at 2-body it would give Mermin M~0 and be falsified by real GHZ experiments. So TFT must be full-tensor to survive, and there it is degenerate with QM. That tensor-completeness question is the genuine remaining frontier.
uncertainty_s1.py)The two results above are the correlation and probability pillars of quantum mechanics, from the S¹. The third pillar — uncertainty — follows from the same single-valued phase, as a theorem rather than a postulate.
On S¹ the single-valued phase is e^{iθ}; its conjugate is the winding number
N = −i∂_θ, whose spectrum is the integers — this is charge quantization
(verify_charge_quantization.py). The commutators are exact: [N, cosθ] = i sinθ,
[N, sinθ] = −i cosθ (verified to 1e-13). The Robertson bound then gives the
Carruthers–Nieto number–phase uncertainty
| ΔN · Δ(sinθ) ≥ ½ | ⟨cosθ⟩ | , |
which we verify holds for every state, is saturated by the von Mises (circular minimum-uncertainty) family, and reduces to ΔN·Δθ = ½ (Heisenberg) in the phase-localized limit (numerically 0.500). The tradeoff is physical and forced: a definite charge/winding (ΔN=0) has a uniform, undefined phase; sharpening the phase spreads the charge.
So the same single-valued S¹ phase underwrites all three pillars — charge quantization and the coherent correlations/Born rule and the uncertainty principle. One structure, three pillars — with uncertainty derived, not assumed (and, unlike §1–7, this one is a clean theorem with no degeneracy caveat: it is the quantum uncertainty relation for the field’s own conjugate variables).
This is a reinterpretation with a derived correlation structure, not a surpassing of quantum mechanics. Two things it is not:
What it does buy: the compact-time conjecture, previously ill-posed (any S reachable), is now well-posed and constrained — the field’s own single-valuedness fixes the answer to the quantum value and the quantum ceiling, deterministically and with no free parameter. That is a real step, honestly bounded.
pip install numpy
mkdir -p outputs
python3 chsh_compact_time.py # local S=2; the naive time-loop overshoots (a warning)
python3 chsh_closure.py # single-valued phase -> cos(a-b), Tsirelson-capped 2sqrt2
python3 born1_envariance.py # equal amplitudes -> equal weights, by exact symmetry
python3 born2_measure.py # U(1)-charge measure -> |c|^2, additivity from orthogonality
python3 born3_finegrain.py # |c|^2 for all amplitudes from envariance alone (exponent 2)
python3 born4_malus.py # continuous Born law P(+|theta)=cos^2(theta/2)
python3 born5_closure_knit.py # one rule -> marginals + E=cos(a-b) + Tsirelson + Malus
python3 dis1_distinguish.py # search for a distinguishing observable: none feasible (1/T)
python3 dis2_ghz.py # GHZ/Mermin: coherent closure reaches M=4=QM (degenerate)
python3 uncertainty_s1.py # the uncertainty principle DERIVED: dN*dtheta>=1/2 from the S^1
Each prints its pre-registered gate and verdict; JSON lands in outputs/.