vestigium

When does a quantum possibility become a fact?

vestigium — Latin for trace, record, footprint. All three are the same object here: the partial trace is the operation at the heart of nearly every script below; the record is what turns a superposition into an outcome; and the footprint left in the environment is why observers agree on what happened.

This is a laboratory for the measurement problem, built in small verified steps. Nothing in it is new physics — every result reproduces something already known, some of it for a century. That is the point: each experiment is checked against exact analytics or a real published measurement, so when the machinery is later pointed at an open question, the machinery itself is not in doubt. Two of our own bugs were caught by exactly these checks (both documented below).

The one result here that was not known in advance

The detail below grew one honest correction at a time and is now longer than the result. Shape first:

Confirmed	A value for the corner spread at s=5 was committed before the run existed and came back at 0.7 %. Pre-registration, not blinding — the measuring session held the earlier points.
Established	The spread falls under refinement, 1.69 → 0.029 % over s=1–6. The vanishing is forced for any regulator family sharing a continuum limit.
Overturned today	The falloff is not exactly s⁻². s=6 shows the constant drifting 3.48 % with the local exponent steepening — and no subleading correction A s⁻²(1 + B s⁻ᵖ) can produce that shape.

2026-09-04 — a(120°) SUPERSEDED. An external workspace checked the corner numbers against the BWK16 bound a(th) >= (pi^2 C_T/3) log[1/sin(th/2)] — a theorem with no fitted parameter — and a(120°) = 0.0038956 sits 13.3% below it. Confirmed and diagnosed: xi/N = 0.62, so the box and not the mass was the IR cutoff, and the fit window R = 4..14 sat entirely in lattice corrections. Corrected: a(120°) = 0.004465 measured (+14.6%), 0.0044915 extrapolated to m -> 0 — 0.9992x the bound, still marginally below, while the 4-param fit measures 0.0045195 above it. The bound is satisfied within the two-model bracket and the deficit falls from 13.3% to 0.08%; a(60°) = 0.0242324 -> 0.0256670 (+5.9%). The quoted 1.85% across-regulator spread was blind to this — all four regulators shared one (N, m, window), and the systematic is 61% in m and ~12% in the window. See qsim/CORNER_BOUND_FINDINGS.md.

| Open | A systematic of the same order as the effect, ≈4.8e-05 in B at s=2. Not a property of the zero mode: the mode is identical across admissible kernels by construction (all reduce to m² at k=0), so what differs is its coupling to the bulk. The percentage figure this row used to carry (22–41 %) is an artifact of the kernel set — one further admissible kernel moves it 3.5×. Whether the drift survives it is unresolved. | | Not mine | Three of the five refinement points were measured by another session on code that is not in this repo. |

Not a new physical law — a pre-registered prediction that was confirmed by someone else's computer. The entanglement corner coefficient's regulator spread falls under lattice refinement as s⁻². From that law, a value for the next resolution was written down and committed before the run existed:

filed pre-run, s=5	0.043 %
measured independently	0.0427 %
agreement	0.7 %

The measurement was made by a separate session (thebridge) on a resolution I never ran.

What that does and does not establish — checked against the disclosure log rather than recalled. The first draft of this section claimed the measuring session was working blind and "could not have tuned to the answer." That is false. The log shows it held the s=1–4 spreads and had measured s=3 and s=4 itself, so it could have derived 1.081/25 = 0.043 in one line. This is pre-registration, not blinding:

• what it buys — the prediction was committed before the s=5 run existed, so the law could not be retrofitted to the answer after the fact. That is the failure mode pre-registration exists to stop, and it is stopped here.
• what it does not buy — the executor was not independent of the expected value. A blind confirmation would be stronger and this is not one.

(The same session was clean on the triangular-lattice values — different study, logged separately. Conflating the two is what produced the wrong first draft.)

The honest caveat, because the number invites more than it can carry: the constant behind that law is not constant at all — 1.0800, 1.0816, 1.0675, 1.0447 across s=3–6, a 3.48 % drift (see above; this line read "stable to 1.3 % across s=3,4,5" before s=6 existed). So the prediction landed within its own law's drift — it is a genuine confirmation, and its 0.7 % should not be quoted as the law's precision. 1.081 is also a property of this four-regulator family, not of the corner term (see the mechanism test below, which found the pairwise disagreement changes sign — so a small spread is not by itself evidence of universality).

What is forced, for any family sharing a continuum limit, is the vanishing. That is the robust claim, and the spread has no measurable floor: 1.69 → 0.25 → 0.12 → 0.068 → 0.043 %.

s=6 landed (2026-08-22), and the s⁻² law is not exact. Measured by thebridge:

s	3	4	5	6
s²×spread	1.0800	1.0819	1.0685	1.0447

interval	3→4	4→5	5→6
local exponent	−1.994	−2.056	−2.123
applies at s ≈	3.46	4.47	5.48

The exponents are finite differences, so each approximates the derivative at the geometric midpoint of its interval, not at either endpoint. The displacement is second-order for a slowly varying deviation and cannot flip the sign of the trend — so the exclusion below stands — but the three numbers are not the deviation at s=4, 5, 6 and must not be fitted.

The constant moves 3.48 % over s=3–6 where s=3–5 alone gave 1.25 %, and the local exponent steepens monotonically. Against a per-point numerical floor of 0.12–0.38 % (clip band), the drift is roughly 10× the noise.

Two things this rules out, and one it does not. Ruled out: any subleading correction of the form A s⁻²(1 + B s⁻ᵖ) with p > 0 — every such model predicts the deviation from −2 to shrink as s grows, and it grows (0.006 → 0.056 → 0.123). Also ruled out: that the 1.25 % over s=3–5 was scatter; it was a truncated range. Not ruled out: the bulk-coupling systematic described below. Its residual is ≈4.8e-05 in B at s=2 — a 3.5 % drift sits well inside that, and contamination falling faster than the signal would steepen the apparent exponent exactly as observed. My measurements of that residual stop at s=3 and give opposite directions at s=1→2 and s=2→3, so they do not decide it either way.

Provenance, stated here rather than 40 % further down. This repository can produce the first two of those five numbers. s=1 and s=2 are computed by corner_coefficient.py and stored in its artifact. s=3, s=4 and s=5 were measured by a separate session (thebridge) on hardware and code that are not here — nothing in this repo can regenerate them, and they are relayed values. The s=5 figure is the one my pre-registered prediction was tested against. An s=6 runner exists and is calibrated against the known s=1 answer, but has not been run.

An open systematic, of the same order as the effect (added 2026-08-22). The lattice zero mode contributes ~20 % of the corner coefficient B itself. All four regulators weight it identically — reg(0,0) = m² exactly — so most of it cancels in a regulator-to-regulator difference. Most, not all: the non-common residual is a real absolute quantity — ≈4.8e-05 in B at s=2. The percentage figure this line used to quote (22–41 % of the regulator signal) is not a property of the method: the denominator is the spread across whichever kernels you happen to include. Adding one further admissible kernel — same continuum limit, passes the O(k⁴) gate — widens the signal 3.5× and drops the ratio from 41.5 % to 12.2 % without changing the systematic itself. Found by thebridge at s=3 and replicated here at s=2. The absolute residual is the stable quantity; the fraction is an artifact of the kernel set. It does not refine away — the total shift is L-independent at fixed l/L (L^+0.01, L^−0.01), because the rank-1 term contributes log c + 2 log l and the L-dependence lives in the constant.

This does not overturn the falloff — a mechanism where the zero mode produces the s⁻² decay was proposed, tested and killed by exactly that flatness. But it means a fifth to a half of the residual spread at any given resolution is not regulator physics, and nobody has sized it at the resolutions where the claim lives. Treat the spread as an upper bound on regulator-dependence, not a measurement of it.

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Foundations experiments

Each is a standalone script with its verification printed at the top of the run.

Experiment	What it shows	Verified against
weak_measurement.py	collapse as a gradual process under continuous monitoring	Born rule emerged: 30.6% of runs committed to the 30% branch (predicted 0.300 ± 0.024); conditional variance rides the exact Riccati curve to steady state 1/√(8k) within 0.2–1.0%
decoherence_frames.py	the which-path record accumulating collision by collision; quantum Darwinism; revival	V = cosᵏ(θ/2) to 2×10⁻¹⁶; V²+D² = 1 to 4×10⁻¹⁶; 4 of 12 environment qubits already hold 95% of the record; a recycled 2-qubit environment returns V to 1.0000 at collision 16 exactly as predicted
zeno.py	a watched quantum system freezes; telegraph dynamics at strong monitoring	survival matches [cos²(π/2N)]ᴺ to 0.0055 across N = 1…64 (Itano et al. 1990, trapped ions); flip rate fits k^−0.96 vs. the Zeno prediction k^−1
bohmian_fan.py	pilot-wave trajectories from the guidance equation (the Philippidis fan)	equivariance: 20,000 guided paths rebuild the Born fringes to 4.3% of peak with no Born rule imposed; 0 of 20,000 trajectories crossed the symmetry axis
csl_toy.py	objective collapse (CSL-type): superpositions die with size, no observer required	V(N) rides exp(−κ₀N²d²T); real-units panel reproduces the actual exclusion logic — Adler's rate puts the edge at ~10⁵ amu (right at Fein-class experiments), GRW's at ~10⁹ amu
teleport.py	entanglement + 2 classical bits moves an unknown state; neither ingredient works alone	fidelity 1.000000000000 over 2000 Haar-random states; with the bits withheld Bob's state is exactly I/2 (4×10⁻¹⁶) — no-signaling, constructively; entanglement-free ceiling 0.6706 vs. theory 2/3
wigner_friend.py	whether "a measurement happened" is relative to the observer	sealed lab: interference ⟨X_S X_F⟩ = 1 and reversal fidelity = 1, while the friend's own state is a definite I/2; once the record leaks both fall to 0 and ½. ⟨X_S X_F⟩ = cos(φ/2) to 2×10⁻¹⁶
bell_game.html	CHSH and GHZ as games you can lose	40,000 rounds: classical 74.56% against its 75% ceiling, quantum 85.30% vs. cos²22.5° = 85.36%; GHZ 100.00%, zero losses in 40,000 rounds where every classical plan caps at 75%

Hidden dimensions — the projections program

The idea "maybe fields are stacked in dimensions we don't see and we watch the shadow", made precise and run through three independent engines. Full write-up in PLAN_projections.md.

kk_projection.py	mass = motion in a hidden dimension (Kaluza–Klein)	rest-buzz frequencies 1.0022 / 2.0031 / 3.0027 against the exact tower 1 / 2 / 3; group velocities within 0.66% of Klein–Gordon
kk6_twisted_tower.py	two hidden loops with a twist — the axion as a measurable spectral splitting	blind protocol (formula scored only after measurement): all 10 winding sectors within 0.3%; the (1,1)/(1,−1) pair splits by 0.44867 vs. 0.44996 predicted, while (1,0)/(0,1) stay degenerate to 6.7×10⁻¹⁶
fractal_boundary.py	does a structured detector wall change where particles land?	yes for how many — periodic vs. Cantor masks at identical 29.6% coverage differ by ~67% in detection efficiency — but the fringes never move: the wall gates, it does not shift

The neural leg (can a network discover the hidden dimension from projections alone?) and the symbolic leg (is the Kaluza reduction a theorem?) live in the sister repos below. All three agreed.

Cross-validation probes

Independent numerics for the trivium cross-validation project — deliberately implemented without sharing code, so agreement means something.

• entropic_hinge.py + hinge_mp.py — the Longo relative-entropy identity underpinning the 2026 entropic-gravity result (Dorau & Much, PRL): relative entropy of a coherent excitation on a wedge = 2π × its boost energy. Verified to 0.02–0.21% on a harmonic chain at two lattice resolutions. Two findings worth more than the check: this computation is impossible in double precision (the modular weights live in e⁻¹⁰⁰-scale covariance tails — every float64 sweep carried 10–14% clip bands), and the Gaussian modular matrix has an operator-ordering trap that a thermal self-test passes silently — only a squeezed state distinguishes them. Both now permanent regression tests.

• entropic_time.py — does the "entropic time" of Barontini, PRR 2026 depend on which coarse-graining you choose? Five legitimate clocks from one exact two-mode Bose–Hubbard run. Same-family control agrees at |τ| = 0.984; the cross-family test disagrees at 0.181 inside the paper's own domain of validity, and coarsening the event set does not rescue it. But the regime scan is the real answer: agreement switches on near Λ ≈ 2–4. Reported as conditional support with a mapped boundary, not a refutation — real split-BEC junctions are interaction-dominated, so the construction is likely robust where it was run; the critique is the missing qualifier.

• log_coefficient_boundary.py — is the entanglement log coefficient non-universal permanently, or only in a regime? Prediction recorded before the run, then confirmed: exact free-fermion Ising chain gives a scaling collapse in ξ/L alone (matched ξ/L agrees to ≤0.058 across L = 64→512), converging on the universal c/6 to 0.6% at criticality, with universality switching on at ξ/L ≈ 2.5. The ordered branch is excluded on physics, not convenience — its ground state is exponentially degenerate, and the run self-flagged it (a ratio of 15.0 with the smallest mode energy at exactly 0.0). (Amended 2026-09-05, from the corner work rather than from this run. The ξ/L collapse is correct for this chain — but a chain has only one length besides ξ: the block runs L//16→L//2, so region scale and box scale are locked in fixed proportion and cannot be separated. The 2+1d corner study is the first system here with two independent lengths, and there the regime condition is R ≪ ξ ≪ N — two conditions with very different leverage: varying the box alone moves the answer 2.3%, varying the mass alone 61%. So ξ/L ≈ 2.5 is a threshold on a composite that this chain holds fixed, not on a single physical condition. The number stands; its interpretation was one condition short.*)

• kappa_vs_mutual_info.py — if a quantity is regulator-contaminated in every regime, the move is not to hunt for a clean regime but to change channel. On one 2D lattice with four regulators sharing a continuum limit, the area-law coefficient κ spreads by 41.6% — and the same lattice refinement that drives the mutual information's spread from 2.18% down to 0.096% (as s^−2.26, converging) leaves κ's spread at 41.8% → 41.7%, unmoved. So κ's regulator dependence is a fixed property of the cut, while I(A:B)'s residual is a vanishing lattice artifact. Includes a two-probe numerical floor audit (1.25×10⁻⁶%, so the sub-percent residual is real, not noise) and one discarded leg left in the file — an IR-matching attempt built on an effective-mass estimator that is biased in 2D, which made the spread worse and was replaced.

• corner_coefficient.py — the successor: strips have no corners, and in 2D the corner term is the coefficient that is supposed to be genuinely universal. On one lattice with the same four regulators, the area coefficient spreads by 36.3% and stays there under refinement (36.3% → 36.2%), while the corner coefficient spreads by 1.7% → 0.2% — below the method's own measured systematic floor, and consistent with exactly zero. Model-dependence, added after an independent check: that 1.7% assumes the 3-parameter model; adding the physically-expected 1/ℓ correction gives 3.3% instead, and a 1% unmodelled contamination moves the extracted coefficient by 7%. So no single-resolution number should be quoted as the answer. What is robust is the refinement behaviour — the spread falls under lattice refinement under either model, while the area coefficient does not move at all. Independently extended by the bridge session to two further resolutions I never ran (s=3, L=480 and s=4, L=640): corner 1.69 → 0.25 → 0.12 → 0.068% against area 36.26 → 36.24 → 36.23 → 36.225%. The corner spread went straight through 0.1% and kept falling, so there is no floor — the coefficient is universal with no measurable residual, not universal-to-a-tolerance. And it is not a numerical artifact: sweeping the symplectic-eigenvalue clip across five decades moves the s=4 spread by 3×10⁻⁵ percentage points, 2254× smaller than the spread itself. The law is cleaner than any local exponent: spread × s² is constant to ~1.3% over s=3,4,5 — the s⁻² expected from four regulators that agree to O(k⁴). (s=1 is 56% off and s=2 is 7.5% off; both lattices are simply too coarse.)

  The s=5 prediction was filed pre-run and it landed: 0.043% predicted, 0.0427% measured (0.7%). That is the strongest single result here — a number predicted before it was computed, by a session that had not seen the data, and confirmed.

  Correction, 2026-08-22. This paragraph previously read "spread × s² = 1.081 at both s=3 and s=4, agreeing to 0.15%" and left s=5 standing as a pending prediction after the answer had arrived. Both halves were wrong to leave. The three measured values are 1.0800, 1.0816, 1.0675 — the two-point agreement really is 0.15%, but adding the third point gives 1.31% and the fourth (s=6) gives 3.48 %, with the local exponent steepening — the constant is drifting, not scattering. The 0.15% was the agreement between the two best-agreeing points, quoted as the stability of a law. The quantity computed (how close the best pair happens to sit) was not the quantity named (how constant the law is). The s⁻² behaviour survives this and is the robust claim; the precision attached to the constant did not. What the constant is not: a mechanism test — sweeping the strength of one regulator's higher-derivative term — found the pairwise disagreement changes sign near c ≈ 0.125, while the dispersion mismatch driving it is strictly positive and monotone. So the disagreement is not proportional to the O(k⁴) mismatch, and the spread can be made accidentally small by choosing regulators that happen to cancel. 1.081 is a property of this four-regulator family, not of the corner term. What is universal is the vanishing — that is forced for any family sharing a continuum limit. A small measured spread is not by itself evidence of universality. Two extractions of the same data agree (1.7% vs 2.1%) — not independent measurements: both run on the same entropies, lattice and regulators, differing only in whether the constant term is fitted or differenced away, so this shows the coefficient is insensitive to that choice and nothing wider. The verdict is unchanged at two masses. One control failed and is kept in the file: a strip control returned a spurious log (B ≈ −0.496), diagnosed rather than explained away — driving ξ/L from 1.79 to 0.06 sends it to −0.005, confirming finite-size contamination in a badly chosen control geometry. It was replaced by rectangle-minus-square, where the corners cancel identically, and that control's residual is the quoted 4.1% floor.

• corner_angles.py — turning that single point into a curve. A square lattice can only make 90° corners cleanly, so this moves to a triangular lattice, where equilateral-triangle regions (three 60° corners) and hexagonal regions (six 120° corners) are exact and need no staircase. Registered before running: a(60°) > a(90°) > a(120°) — a recall check, not a prediction, since the monotone ordering is in the literature; this was demoted after tabula pointed it out, and the gate has said so since while this line did not. Holds — 0.0242 / 0.0116 / 0.0039.

  The lattice claim this line used to make was untested and is withdrawn. It read "a(90°) comes from the square lattice, so the curve is lattice-independent as well as regulator-independent." There is no triangular measurement at 90° — the artifact records a90_square_lattice only, and the per-regulator data holds a(60) and a(120) alone. With no two lattices measured at the same angle, nothing here tests lattice-independence. What the three points show is that a square-lattice value at 90° falls between the triangular values in the expected order: consistent with lattice-independence, and not a test of it. Across-regulator spread is 0.5% at 60° and 1.9% at 120° against 33% for the area coefficient on the same runs. (Superseded 2026-09-04 — that spread is not the uncertainty: all four regulators shared one (N, m, fit window), and the dominant systematics are orthogonal to regulator choice, 61% in m and ~12% in the window. See the note above and qsim/CORNER_BOUND_FINDINGS.md.) New control that can fail: the area coefficient must not depend on the region's shape, and triangles vs hexagons agree to 0.03%.

The transferable lesson, now a standing entry in the family ledger: when probing whether a definition is robust, scan the physical regime — the interesting answer is usually a boundary, not a yes/no.

What next — proposals

PROPOSALS.md holds three pre-registered hypotheses, none run: H1 points the validated secular-average instrument at ansatz's own open item ("deformed-Kerr integrability fate: UNDETERMINED"), including its weeks-old rank-3 Killing-tensor solution, with three controls, three cross-oracle routes, and four named ways it is expected to fail; H2 files a number (ξ/L ≈ 2.5) for tabula's mass sweep before it lands; H3 is the only item that touches the founding question, and no sister can test it.

The founding question, resumed

qsim/H3_FINDINGS.md — when does a possibility become a fact? asked as: how is the record laid down, and can two environments be compared at all?

The defensible result: the system's path-independence does not extend to the environment. Under Gaussian pure dephasing the qubit's state depends only on a scalar decoherence function — weak-coupling-long and strong-coupling-brief are provably equivalent for the qubit. They are not equivalent for the record: at S(ρ_S) = 0.150000 held to 4.7e-16, the fraction on the coupling site runs 0.766 → 0.998 by route alone. Single fragment, no sampling.

Matching duration too is still insufficient (12.7% over temporal shape); so are four scalar conditions (8.8% residual). Comparing environments at fixed coupling — which two published studies do — is confounded by a 44× capacity gap, only 9.5× of which is local-operator variance.

Not defensible, and stated as such: the order-by-order trend, whose coverage sensitivity (73–992%) exceeds its signal (12–35%); and any critical-vs-gapped comparison of record structure, which is not yet well-posed. The capacity result itself reproduces [Quan et al. 2006] and is instrument validation, not a discovery.

Method validation — is the instrument measuring the physics?

Two studies whose subject is the measuring device rather than a physical system.

Study	The lemma / claim under test	Outcome
lrl_secular.py · findings · gate	if a conserved Q survives a perturbation to first order, the orbit-average A = <{H₁,Q}> must vanish — so A ≠ 0 disproves survival	Kepler + Laplace–Runge–Lenz, two averaging routes agreeing to 1.5e-10. All three controls pass, including the trivial one: reparametrising the same system averages to 1e-15 while its bracket is nonzero pointwise by up to 11.0
corner_function/ · provenance + independent check	C1–C6 (every known general constraint on a(θ)) do not bound κ/C_T, so the observed band [3.672, 4.179] is not a consequence of the constraints	Imported from an external workspace and checked rather than received. All three requested checks pass; two documentation errors found that the proof does not use; κ/C_T reaches 8.4×10⁶⁴ under the constraints

Three things the LRL run found that were not in the framing: A_x is identically zero by parity for every central perturbation, so a test built on that component alone returns zero regardless of the physics; A as usually defined is minus the secular rate; and the averaging integral is absolutely convergent, so convergence is not where the problem is. The problem is the bounded-F₁ step, which holds per orbit but is applied on an open set and needs the bound uniformly — it is not uniform, and the validity window is (ε β)/(k a) < 0.29 e(1−e), vanishing at both ends.

For the κ result, a literature sweep of the two routes by which a missing constraint could have entered — a bound on the thin-strip coefficient by C_T, and a cross-n Rényi inequality — found neither exists, which supports the claim. The cross-n route additionally fails for a structural reason: Rényi monotonicity varies n at fixed geometry while extracting a log coefficient requires varying geometry at fixed n, and the two do not compose.

2026-09-05 — the s-family is blind to one axis, by construction. L = L_base·s and m = m_base/s, so ξ/L = 0.625 at every s — the same ratio that put a(120°) 13.3% below a theorem. That is correct design for a lattice-refinement study, and it means the scan can never detect a box systematic, because it never varies the axis carrying one. The regulator spreads survive (a systematic common to all four cancels in a spread); what is not supported is reading "spread falls as s⁻²" as "the coefficient converges to its universal value." Those are different claims. Found by running regime_gate.py against already-committed runs — a check that costs microseconds and had never been run.

Interactive apps

Self-contained HTML, no build step, no server needed — open the file.

• double_slit_app.html — the full bench: coherence dial, which-path markers, the eraser, Born-sampled dots landing one at a time, live V / D / entanglement-entropy meters and duality curve, plus a two-photon delayed-choice mode (sort by the partner photon to pull fringes or anti-fringes out of flat noise).
• wave_double_slit.html — a live 2-D time-dependent Schrödinger solver: watch the packet spread, hit the absorbing wall, slip through the slits and build fringes.
• bell_game.html — design any classical strategy you like and watch it hit the 75% wall, then watch entanglement walk through it (85.4% for CHSH, a perfect 100% for GHZ).

Neural quantum states

sims/ — the machine-learning leg, kept separate on purpose.

• nqs_tfim.py — a neural network as a wavefunction (variational Monte Carlo) finds the transverse-field Ising ground state to ~0.4% of exact at criticality
• nqs_phase_transition.py — sweeping the field reproduces the quantum phase transition
• nqs_scaling.py — runs to N = 40 spins (Hilbert space ~1.1×10¹²), where exact diagonalization dies around N ≈ 20
• nqs_cnn.py — baking translation symmetry into the ansatz buys ~100× in accuracy. And a lesson: the plain ansatz got the energy right everywhere but the entanglement wrong in the ordered phase (it spontaneously broke the Z₂ symmetry) — entanglement is the stricter probe. Fixed in stage3_symmetric.py.

Running it

python3 -m venv .venv && source .venv/bin/activate
pip install numpy scipy matplotlib mpmath torch      # torch only for sims/
python qsim/weak_measurement.py

Each script prints its own verification and writes its figure beside itself. The high-precision leg (hinge_mp.py) needs mpmath and takes ~2 minutes.

What this is not

• Not new physics. Everything here reproduces known results. The value is a verified instrument and a legible account of why each result is what it is.
• Not a proof of anything speculative. The hidden-dimension work shows what the idea would explain (mass, charge) and what it would cost (unseen particle towers). There is no experimental evidence our universe is built that way, and the repo says so wherever it comes up.
• Honest about limits. Where a result rests on a convention, a lattice artifact, a finite-size effect, or an arbitrary tolerance, that is stated next to the number rather than in a footnote. Where a check was impossible at the precision first attempted, the failed attempt is left in the file.

Family

Five sibling projects, cross-validated against each other:

• ansatz-machine — propose → verify → evolve, hunting exact solutions of Einstein's field equations. Proved the Kaluza reduction used here, dilaton price tag and all.
• tabula-geometrica — can a neural network invent spacetime geometry from raw observation? It discovered mass as the hidden dimension's latent, and independently replicated this repo's Kaluza–Klein numbers.
• DeepStrain — deep-learning searches of real LIGO/Virgo data for black-hole signatures.
• cuspis — the corner function a(θ) of a 3d CFT: why theories that share almost nothing land on nearly the same normalised curve. Uses this repo's lattice numbers as one of its referees. Joined 2026-09-05.
• trivium — the bridge: cross-validating the independent projects against one another, which is where several of the probes above came from.

License

MIT — see LICENSE.

A note on the name

This was called "the zero-mode systematic" all through 2026-08-22. That name is wrong and the correction came from thebridge tracing their own failed prediction. Their reasoning ran: the admissibility gate forces every kernel to agree with m² + k² as k→0, so at k=0 exactly every kernel is m² — the mode is identical by construction — therefore the shift is identical.

The third step does not follow. The mode is identical; its effect on B is not. Deleting it measures how that mode couples to the rest of the spectrum, and the coupling is a bulk property — precisely what an out-of-family kernel perturbs. So the residual is a bulk-coupling difference, not a property of the mode, and the old name asserted the very thing that turned out to be false.

Measured here: the out-of-family kernel's shift sits inside the in-family range at s=1 and outside it at s=2 — not because its deviation grows, but because the in-family kernels converge on each other faster than it converges on them (half-width ×0.28 per doubling against its distance ×0.35).