Why do conformal field theories that share almost nothing else agree on the normalised corner function a(θ)/C_T — and by exactly the amount they disagree?
cuspis — Latin, a point or spear-tip. In d = 3 the entangling corner turns out to be a cusp
on the replica twist line, Γ⁽ⁿ⁾ = (1 − n)aₙ, and recognising that is what unlocked the second of
the two results below.
The collapse across free scalars, free fermions, holographic models and Wilson–Fisher fixed points is an observation with no explanation. There is a proven lower bound that all of them nearly saturate, and no account of why. That gap is the task.
Two results, and an instrument.
① The known entropic constraints do not force the collapse. Analytic, with a falsifiable
numerical companion, and independently verified by a separate project. Reframed, the collapse is
the statement that κ/C_T ∈ [3.672, 4.179] plus a separate ≈1% shape residual — and every value
of κ/C_T in (0, ∞) is admissible under the general constraints, with a(θ)/C_T unbounded above at
every angle. This answers a question left open in the literature, in the negative.
② The residual's sign is not universal — it is set by one number at each end. At the smooth end
by the trial function's own fourth-order coefficient; at the sharp end by a₀ − ã₀, which changes
sign inside the physical band because κ/σ straddles 3π. For the n = 1 free scalar the two ends
disagree, so a sign change was predicted before the run — and the instrument found it, at
27 ± 3°, with a₀ ≈ −0.3 C_T (sign robust, magnitude model-dependent). Why the scalar has such a
constant and holography does not is explained by the operator content of the twist-pair fusion, which
predicts it vanishes exactly for the Dirac fermion — a prediction then confirmed by a run whose verdict
rule was frozen beforehand (a₀ = 0.00 ± 0.02 C_T).
③ The instrument. An arbitrary-precision solver for the Casini–Huerta–Leitao ODE system, validated to ≤ 5·10⁻⁵ against published values from 20° to 170°. It was built because no four-digit free-field a(θ) below 45° appears in the literature the sweep could find, and its 20°–45° values are, on that basis, the first independent determination there.
Read RESULT.md first — §10 is the handover.
- The sub-45° values have no external referee. That is why the instrument exists, and it means those numbers are simultaneously the result and the only check on the result. The pre-registered predictions that passed were sign tests, which do not validate a magnitude.
- The upper edge of the band is explained by nothing known. The rigorous window after ② is
κ/C_T ∈ [2.39, ∞)against an observed[3.67, 4.18]. - The Dirac branch is parked. Its known-answer control failed twice; the diagnosis is exact and
both failed runs are kept. See
report.md.
This is machine-generated research, produced by large language models working in a repository
with a written operating contract, a verification protocol frozen before any result, and a
requirement that every claim carry a grade (verified / partially verified / unverified) and
every citation record what was actually read — abstract, table, or full text.
The failures are in here too, deliberately: two failed control runs, a self-reported violation of the repo's own one-variable rule and its exact cost, and a prior-art filing that excluded the paper which answered one of the targets. A failed run you keep is a control for the fix; a failed run you discard is a rumour.
RESULT.md — the general entropic constraints do not localise κ/C_T. Reframed, the collapse is the statement that κ/C_T lies in a 13% band, plus a separate ≈1% shape residual. The band is not a consequence of the known constraints: every value of κ/C_T in (0, ∞) is admissible, and a(θ)/C_T is unbounded above at every angle. This answers, in the negative, a question left open in the literature, and it means the collapse needs an explanation the constraints cannot give. Phase 2 (2026-09-05): the residual's sign is set by one number at each end of the interval and is not universal; the free scalar's sharp-end constant was measured and its predicted node found at 27°; and the missing bridge for a bound on κ exists in the 2026 cusp-bootstrap literature — a rectangle crossing giving κ ≥ (2π/3)a(π/2), hence κ/C_T ≥ 2.39 — and is provably one-sided, so the upper edge of the band is still explained by nothing. Start at RESULT.md; §10 is the handover.
| file | what it is |
|---|---|
| PHASE2.md | what Phase 1 left open — the two targets, quoted from RESULT.md §10 rather than reframed. Read RESULT.md first. |
| RESULT.md | the result — standalone: statement, scope, proof, mechanism, prior work, handover |
| TASK.md | the problem, what is believed known (and must be verified), the frozen three-tier verification protocol, and the five reportable outcomes |
| CLAUDE.md | the operating contract — loads every session, deliberately short |
| SISTERS.md | the four sibling repos: what each holds, which file to open for what, and the read-only rule |
| TODO.md | living list of open questions and unverified claims |
| report.md | one entry per experiment (EXP-001 … EXP-011), required fields enforced |
| references.md | 52 verified references, each with what was actually read (abstract / table / full text) |
| scripts/README.md | the checks, the parked instrument, and how to re-fetch the source texts |
S = B · (perimeter/δ) − a(θ) · ln(perimeter/δ) + O(1)
a(θ) is universal — independent of regulator, lattice, and microscopic detail. Normalise it
by C_T and theories with different field content, different spectra, and different couplings
land on nearly the same curve. Nearly, not exactly — and the residuals are the whole
quantitative content of the problem.
The obstruction is believed to be conceptual rather than computational — the numbers exist for several theories; the account of why they agree does not. That is a description of the problem's current state, not a restriction: if a large computation turns out to be the way in, take it and say why the reasoning route was insufficient.
The point of this setup is that a wrong answer is detectable, which is not true of most open problems.
TIER 1 both known limits reproduced ANALYTICALLY exact, symbolic, unfoolable
TIER 2 structural constraints a(θ) must satisfy exact, once established with sources
TIER 3 matches ../quantum's a(60°), a(90°), a(120°) numerical, three points, weak
Tier 3 alone can be passed by a wrong formula. Tier 1 is the gate.
Tier 3's referee did not survive the run. ../quantum's a(120°) sits 13% below a rigorous
lower bound that follows from strong subadditivity, Lorentz invariance and C_T alone — so the
lattice number, not the bound, is the suspect. Details and extraction parameters are in
TODO.md under Handed off; the sibling is read-only and was not modified. Use
[CHL09] Table 1 and [HHCWM16] Tables 1–2 as the referee instead.
Five projects with deliberately independent roots, kept ignorant of each other on purpose — so that when two of them agree, the agreement is evidence rather than echo. All read-only from here, and every number imported from one is cited by repo and file.
All five are public. The ../ paths in SISTERS.md are local checkout names; the links are the
same repositories.
| local path | name | what it is |
|---|---|---|
../quantum |
vestigium | verified QM lattice numerics — the original referee for a(60°), a(90°), a(120°) |
../conjecture_machine |
ansatz-machine | exact symbolic GR over finite fields; the model for a validated instrument |
../SpaceTime |
tabula-geometrica | what is discoverable in principle — and silent_nulls.md, 50+ ways a bug reads as a result |
../BlackHole |
DeepStrain | real LIGO strain, honest error bars; source of the "plateau" failure mode |
../TheBridge |
trivium | the cross-validation layer and the lessons ledger |
SISTERS.md says which file to open for what. Nothing there needs to be loaded by default —
go and get what you need when you need it.
- Prior art first. Two recent efforts in this family were correct computations of results that already existed. Finding this one is already solved is a result, not a failure.
- Grade every claim
verified/partially verified/unverified, and never let an unverified one travel without its label. - Do not fit a formula to three points and call it verified. That is inference, not verification.
- The siblings are read-only. Cite them by repo and file in the same sentence as the number.