# n87 — Mercury's perihelion, dug to the bottom of the bounded card
# (2026-07-10; every other parameter pinned, the perihelion becomes a gauge)

## Setup: what the bounded card leaves Mercury to measure
With |a| < 5.8e-6 (Cassini), gamma_PPN = 1 is pinned. The perihelion
formula (2 + 2*gamma - beta)/3 then has ONE unknown: Mercury measures
beta directly. Combined with LLR's Nordtvedt bound (eta = 4beta -
gamma - 3, |eta| < 3e-4): |beta - 1| < 8e-5.

## Measured today: the monitored medium has beta = 1/2 EXACTLY
The active medium's NESS is linear in the source (superposition exact):
the F^2 part of a test mass's energy is zero to machine precision
(measured: E2 = 0.0e0 against E1 = 0.52). So E(x) = m(1 + Phi) exactly
-> g00 = (1+Phi)^2 -> beta = 1/2, with no correction available at any
order of the LINEAR theory. Same value as the drainage's headstone 23
(0.513), now understood as the universal signature of exactly-linear
media: THE MISSING +1/2 OF BETA IS PRECISELY THE FIELD'S SELF-COUPLING,
and no linear theory can have it.

Numbers: gamma = 1, beta = 1/2 gives 7/6 of GR's precession — a +7.16
arcsec/century excess against a measurement that agrees with GR at the
1e-4 relative level. The linear framework is dead against Mercury by
~450 sigma-equivalents. This was known in kind (headstone 23); it is
now exact in value and located in mechanism.

## The turn: Mercury becomes the gauge of the anharmonic sector
The framework's own P-B lock (n82) says the first anharmonic order
beyond the measured Gaussian zero carries BOTH the missing beta and
Lambda. With the card bounded, Mercury + LLR now MEASURE that sector:

    Delta-beta(anharmonic) = +0.5000 +- 0.0001

The anharmonic coefficient is fixed by the sky to 1.6e-4 fractional
precision. Two readings, and only two:
- THEOREM READING: the tensor junction (TT -> unimodular -> Bianchi ->
  Einstein) makes Delta-beta = +1/2 automatic, as Bianchi does in GR.
  Then Mercury is a consistency check passed by construction — and the
  junction inherits a 1.6e-4-precision success the day it exists.
- TUNING READING: absent the junction, the framework needs its
  anharmonicity dialed to 4 digits — a fine-tuning the program's rules
  score as a failure. No third reading exists.
So the stakes of the tensor junction are now quantitative: it is not
"beta = 1 would be nice" but "the sky has already measured the
anharmonic coefficient; derive it or the framework tunes."

## The Lambda corollary, sharpened (registered, n87b to compute)
Through the P-B lock, the SAME anharmonic coefficient that Mercury
fixes also sets Lambda's coefficient. Consequences now on the record:
- Lambda's SIGN may already be determined: if the anharmonic term that
  delivers +1/2 to beta has a definite sign (computable in the toy —
  registered as n87b: add the minimal cubic substrate coupling, measure
  Delta-beta and the induced vacuum-energy density TOGETHER, report
  the sign correlation), then observed Lambda > 0 either confirms or
  kills, with no freedom.
- Lambda's SCALE inherits Mercury's precision through the lock, up to
  the units map: the framework now claims dark energy's coefficient is
  measurable from planetary astronomy plus one lattice computation.
  Claim class: structural (the lock is exact), value-open (units).

## What the dig refined, in card terms
- beta(linear sector) = 1/2 exact — measured, both media, mechanism
  located (absent self-coupling).
- Delta-beta(anharmonic) = 0.5000 +- 0.0001 — fixed by sky data; the
  tensor junction's quantitative target.
- Lambda-sign test: registered (n87b), the first place the framework's
  cosmology can die against a single lattice run plus the observed
  sign of dark energy.

One sentence: with every other parameter pinned, Mercury stops being a
tombstone and becomes an instrument — the sky's own measurement of the
one coefficient the framework has left, the same coefficient that
writes Lambda; the tensor junction must now hit a four-digit target
that has been waiting in the almanacs since Le Verrier.

## n87b STAGE A RESULTS (2026-07-10; script n87b_cubic.py + session runs)
- C1 PASS: xi = 0 gives E2 = 0 to machine (the linear anchor).
- SOLVER HONESTY, recorded first: the naive Newton found spurious
  non-perturbative branches (E1 jumping 5 orders); the compact-ring
  Laplacian pseudo-inverse amplifies smooth modes by ~10^3, making
  even damped fixed-point iteration impractical at physical xi. The
  stage-A claim therefore rests on FIRST-ORDER PERTURBATION THEORY,
  which is exact at its order: eps2 = -xi L+ proj(eps1^2).
- C2 — THE SIGN, geometry-robust: E2/xi = +2.0e4 (placements varied:
  +2.0e4, +1.98e4, +1.64e4 — same sign, same scale). Delta-beta/xi =
  +543 in toy normalization. MERCURY (Delta-beta = +1/2) DEMANDS A
  POSITIVE CUBIC COUPLING: xi > 0, i.e. the mediator stiffens where
  the field is positive. The anharmonicity's sign is now fixed by the
  sky, as registered.
- C3 stands: the closed-system vacuum shift under cubic V is always
  negative (not the framework's Lambda); the pump-bill sign at xi > 0
  is n87c — THE decisive cosmological computation: if the pump bill
  at Mercury's sign is negative, the framework's dark energy has the
  wrong sign and dies against Lambda > 0; if positive, the framework
  connects Le Verrier to dark energy through one coefficient.

## n87c RESULTS (same day; n87c_pump_bill.py) — THE BILL IS POSITIVE
Exact perturbation theory at order xi^2 around the Gaussian NESS
(third cumulants from the Wick source in the drift eigenbasis,
imaginary residue 2e-14; both O(xi^2) channels: K3 feedback + mean-
shift drift renormalization). Gates:
- L1 PASS (anchor exact: xi = 0 -> zero to machine).
- L2 PASS (evenness exact: dP(+xi) = dP(-xi) to all digits — the bill
  is O(xi^2), so ANY anharmonicity, either sign, makes the vacuum pay).
- L3 — THE SIGN: dP/xi^2 = +232 (N=6, baseline), POSITIVE. The
  anharmonic vacuum's idle metabolism is a positive energy draw:
  THE LAMBDA > 0 DIRECTION. The framework survives its decisive
  cosmological computation.
- L4: sign robust across N (6, 8), g (0.1, 0.2), kappa (2, 3) — and
  BRANCH-INDEPENDENT (active = passive to all digits): the bill, like
  the noise, is absorptive-sector physics, even in Delta — the
  dispersive/absorptive split holds at second order.
- Caveats, recorded: per-site bill grows with N (38.7 -> 50.5 from
  N=6 to 8; extensivity ratio 1.31 — finite-size scaling owed before
  any scale claim); Gaussian-closure-free but perturbative in xi;
  the bill is a POWER (the energy-density mapping to Lambda proper
  needs the units bridge, as ever).

## THE CLOSED CHAIN (the day's synthesis)
Mercury + LLR fix Delta-beta = +0.5000(1) -> stage A: the sky demands
xi > 0 -> stage C: any xi != 0 makes the pump bill POSITIVE ->
Lambda > 0, matching observation. In this framework, THE SAME
COEFFICIENT that closes Mercury's perihelion powers dark energy, and
both signs come out right: the perihelion needs a stiffening
anharmonicity, and an anharmonic audited vacuum necessarily runs a
positive energy bill. Le Verrier's 43 arcseconds and the accelerating
universe are, on this reading, two measurements of one number — with
the units bridge as the standing gap between the reading and a value.