Position coupling, real remainders and equation markers

Position Ward-column and formal-jet recovery.
Keep the free degree-four coordinates and
lift the frame as P Z
s = J4Ps. Its inverse on the horizontal image gives exactly

The summands are complementary projections, giving inverse A1/x,1/y. It fixes unit and expectation,
commutes with rotations and conjugation, and scales scalar and stress by x, y respectively, without
feeding centered components into the unit. For P ′
s = PsAxs,ys exact multilinearity gives

The three R factors are one stress and two scalar labels.
With real nonzero b = B0, choose
eventually
xs =
q
b/ Re Bs,
ys = b/(x2
s Re Rs).
(TR84)

TR78 makes these positive and tending to one. Any total frame choice outside the eventual interval
suffices. They impose exactly
Re B′
s = b,
Re R′
s = b.
(TR85)

Conversely any positive TR82 adjustment satisfying these identities has x2
s →1 and hence xs →1,
followed by ys →1. It preserves all limiting frame, unit, symmetry and centering laws. Finite-
basis expansion of Axs,ysvj preserves every compact separated jet comparison: coefficients converge,
finite sums commute with derivatives, and bounded free derivatives on compact sets control the
coefficient errors alongside the original comparison errors. No further extraction is made. Define
the measured position charge by

It uses the separated derivative test, not the physical stress coefficient as its definition. TR80 with
independently calibrated Wilson charge gW and xs, ys →1 gives Re E′
s/g2
W →c. Thus λX/g2
W →1,
so it is eventually positive, and

The positive root in TR86 fixes the branch and calibrates gX to Wilson charge before using Ward
equations. In the adjusted frame retain the exact TR76 coefficients; d/g2
X →d2 still holds. Taking
real parts of TR77, with TR85 and qs = Re Q′
s →q0, gives

The first uses d/g2
X →d2 and gives κ →1; substitution of TR88 proves the second. Bounded
convergent quotients therefore give

Dividing this error by g3
X gives the signed cubic little-o statement. No full series is required.

TR2 at each fixed separated shape, with its fixed-distance logarithm, gives normalized trace limit
cG|x1 −x2|−8. Divide by the square of TR91’s nonzero limiting coefficient to get the free scalar
limit for ws. The uncentered unit remainder gives ms = O(s4/|ds|) →0, since s4 log(1/s) →0.
Unit insertion gives
⟨vs(sx1)vs(sx2)⟩χ = ⟨ws(sx1)ws(sx2)⟩χ + m2
s,
(TR92)

It is convex: the prefactor is nonnegative and TR3 integrates convex finite exponentials.
For
differentiable convex Us converging pointwise to V differentiable at one, fixed secants from a < 1
and to b > 1 bracket U′
s(1). First pass to limits in these slopes, then let a ↑1, b ↓1, giving
derivative convergence. Here V (t) = cGt−8, hence

Smooth selected separated moments identify this with the real first-position scalar derivative Bsel
s .
Physical conservation in TR76, tested against the selected scalar, gives exactly κsEsel
s + dsBsel
s
= 0;
unit derivatives vanish. Thus κs →1, ds/g2
W →d2 and TR94 imply

Pass first to the scalar pointwise limits, then let a ↑1 and b ↓1. Both limiting secants approach
−8cG. All quantities in this inequality are identical after horizontal readout, so TR94 is recovered
without differentiating a pointwise limit. Smooth separated moments identify this derivative with
Re Bsel
s .
Finally the selected conservation reader has the exact scalar form κsEsel
s
+ dsBsel
s
= 0. With
real tensor coefficients its real part gives

Write Ak(c) = AgeK(c, k), with calibrated bβk/2 ≤βk. For fixed c,

because Ak(c) ≤bβ1/2
k
and its product with β−1
k
is at most 2 bβ−1/2
k
. In one transported degree-
five chart and the same coupling, let Pk be the principal logarithmic orbit projection and Ek the
uncorrected stress-divergence orbit projection. Set

For the second limit the principal term is (Ak(1/5)/βk) Lease(Pk) →0 by TS1; the other term is its
paid divergence error. Before transport this is β−1
k
times the orbit of Xk −βkZk, an exact served
equation insertion by the finite local divergence identity. Logarithmic insertion comparison gives
common coefficient/function limits; core projection commutes with orbit and coordinate change is
linear. Thus TS2 is exactly this normalized transformed limit. The retained equation estimate
gives
bβp
kisi(vki) →0
for every integer p ≥0.
(TS4)

Estimate the first component before the finite real orbit; symmetry supplies the others. The other
required estimate concerns the measured vector si(vki), where ki is the common retained scaling
sequence. Taking p = 1 and Aki(1/5) ≤bβki yields the other weighted leak. Together with TS3,
TR50’s repair gives bounded functionals μi, si →Vbase and Aki(1/5)∥μi ◦si −Lease ∥op →0. The
growing precision is paid before solving the stress correction.

Near-equation vector and weighted sensor recovery.
The source retains both degree-five
orbit projections in the same chart and with the same extracted coupling. For their horizontal
copies P Z
k = JPk, EZ
k = JEk, the exact vector reader is

The calibration bound gives explicitly Ak(c)/βk ≤2 bβ−1/2
k
→0. Hence ∥β−1
k Pk∥≤CAk(1/6)/βk →
0, recovering the vector limit in TS3. The scalar payment is separately

TS5 is the actual fixed-placement limit at the prescribed near point and origin, distinct and non-
wrapping eventually. Its spectator is the retained corrected lift on the same refinement. Under the
scaling frame, the first insertion for bβp
kisi(vki) is exactly

at physical scale aibi, by chart linearity, exact coupling cancellation and degree-five scaling. Thus
each tested response is bounded by bβp
ki tiltCap(ji)5 Tau(ji, −100) →0: clock and tilt have rate zero
and Tau has strictly negative rate. The real invariant-vector detector from these degree-four tests
and the same frame comparison recovers vector convergence, proving TS4 on the exact combination.
This is (TS4); the estimate is proved on the exact degree-five near-kernel combination rather than
inferred from its unweighted limit. At summit k = k(i), x = bβk, fixed surgery order m, write the
finite marker
Mm(U) =
X
wm,α
ρ(x) Lα(U).
(TS6)

Addresses lie in the prescribed box with finite orbit indices.
Census gives P |wm,α|
≤
cardB(x) wheat(m), and subpower box growth with the surgery parameter makes this di-
vided by ρ(x) at most x2 eventually, hence at most Tau100000
i
on the calendar.
Each lens has
common radius-2 coast(x) support fitting the chamber, and fine pullback root norm at most
Tau−1900000
i
. Its beacon lies in the enlarged box within allowed distance, so every address has that
estimate. Seminorm linearity and finite pullback commutation give marker root bound Tau−1800000
i
.
Pointwise |Mm| ≤x2 also holds. Thresholds fix m first, then are uniform over finer cutoffand
volume.

with the source’s transported coordinate convention.
The inverse coupling and renormaliza-
tion coupling cancel before estimating.
Consequently each spectator response is bounded by
bβp
ki tiltCap(ji)5 Tau(ji, −100), whose first factors have rate zero and whose last factor has strictly
negative rate.
The same finite invariant-vector detector and frame comparison recover norm
convergence in TS4.
This uses the actual weighted insertion, not only its previously known
unweighted limit.
For fixed surgery order m keep the full finite address/orbit list and every coefficient wm,α/ρ(x).
Pullback commutes with the marker sum exactly: