Consideration — The yield envelope is the join
Logarithmic source times \(Y\) · ultraviolet integral \(1/(2p+1)\) · \(p\ge 2\) on the open fast face · \(Y\to 1\) reinstalls the infrared end
CPT Framework — FT-YIELD-ENVELOPE / FT-NEST-STRAIN / FT-AXIS-YIELD-SYM / FT-AXIS-PHASE1-YIELD-SPLIT Date: 2026-10-02 Status: CONSIDERATION ·PLACED_FT_YIELD_ENVELOPE_IS_THE_JOIN
Stimulus: Principal Investigator — activate the nest density only below the yield surface of §0bn986, \(\rho_{\mathrm{nest}}=\sqrt{GM a_{\mathrm{sym}}}/(4\pi G r^2)\,Y(a_{\mathrm{sym}}/g)\), and test whether \(Y\sim y^p\) for \(p\ge 2\) passes the perihelion gate, whether the outskirts satisfy \(v^4=GM a_{\mathrm{sym}}\), and whether the Solar System is then strictly Newtonian.
Parents: a_geom_ft_nest_strain_is_one_member_consideration.md (§0bn1090) · a_geom_ft_axis_yield_symmetric_consideration.md (§0bn986) · a_geom_ft_axis_phase1_yield_split_consideration.md (§0bn1023) · a_geom_ft_parameter_corridor_map.md (§0bn1080) · a_geom_ft_axis_mu_narrow_results.md · a_geom_ft_nested_layer_reinstalls_ir_consideration.md (§0bn1088) · a_geom_ft_f_target_withdrawn_consideration.md (§0bn1087) · a_geom_ft_axis_carrier_nest_consideration.md (§0bn984)
Algebra: Poisson's equation with \(Y\) on the logarithmic source. The frozen perihelion estimator applied to the local deficit. No new elliptic mill. No parent rewritten.
Tags: [E honesty] · [E algebra] · [Programme] · [O residual]
§0bn1090, §0bn986, §0bn1023, §0bn1080, §0bn1088, §0bn1087, and §0bn984 stay as written. \(a_{\mathrm{sym}}\) is not selected. \(p\) is not crowned. Public chips stay OPEN. Phase 2 is not run. No promote.
0. One-line
The envelope \(Y\) is the join, written onto the logarithmic source. For \(Y\sim y^p\) the ultraviolet integral is \(\delta g/g_N=1/((2p+1)x^{p+1/2})\). For \(p\ge 2\) that tail lies on the open fast face of the perihelion gate. The step is the corner §0bn1023 already installed. \(Y\to 1\) returns \(v^4=GM a_{\mathrm{sym}}\).
1. The threshold is already placed
§0bn986 writes one acceleration from the unit-pair excess,
\[ a_{\mathrm{sym}}=(2-\sqrt{2})\,\frac{c^2}{R}=1.605\times 10^{-10}\,\mathrm{m\,s^{-2}}. \]
The banked yield surface stays \(g=c^2/R\). No mass enters the threshold. The shape of the passage between the Newtonian end and the deep end remains the join \(F\). The symmetric point fixes the location of one threshold. It does not choose the probe. The rotation-curve mill stays unrun.
The proposal multiplies the density of §0bn1090 by an envelope \(Y(a_{\mathrm{sym}}/g)\). That density remains the Poisson source of \(\Phi=A\ln r\) with \(A=\sqrt{GM a_{\mathrm{sym}}}\). The factor in front of \(Y\) is \(A/(4\pi G r^2)\). A numerator \(GM a_{\mathrm{sym}}\) does not have the dimension of \(A\), and it is not the source that produces \(v^4=GM a_{\mathrm{sym}}\). Putting \(a_{\mathrm{sym}}\) inside \(A\) and inside the argument of \(Y\) is the unit choice §0bn986 withheld. This note records that choice. The banks do not adopt it.
2. Poisson's equation
Keep \(\nabla^2\Phi=4\pi G\rho\) and \(g=\partial_r\Phi\). The activated source is
\[ \nabla^2\Phi=4\pi G\rho_{\mathrm{baryon}}+\frac{A}{r^2}\,Y\!\left(\frac{a_{\mathrm{sym}}}{g}\right),\qquad A=\sqrt{GM a_{\mathrm{sym}}}. \]
For a point mass the exterior identity is
\[ \partial_r\!\left(r^2 g\right)=A\,Y\!\left(\frac{a_{\mathrm{sym}}}{g(r)}\right). \]
\(Y\) depends on the total field, so the first integral is not \(g=g_N+(A/r)Y\). The member scored in §0bn1090 is the special case \(Y=1\) at every radius. Where the envelope is still small, \(r^2\delta g\) is the integral of \(A Y\) from the origin, and the point mass supplies the constant \(GM\).
Two asymptotes do different jobs. \(Y\to 0\) for \(g\gg a_{\mathrm{sym}}\) is the ultraviolet tail. \(Y\to 1\) for \(g\le a_{\mathrm{sym}}\) is the isothermal density. A pure power \(Y=y^p\) has the first asymptote and misses the second: it grows with radius and never saturates.
3. The ultraviolet integral
Iterate on \(g_N=GM/r^2\) where \(g_N\gg a_{\mathrm{sym}}\), with \(Y\sim y^p\) and \(y=a_{\mathrm{sym}}/g_N\). Then \(Y\sim(a_{\mathrm{sym}} r^2/GM)^p\), and
\[ r^2\delta g=\int_0^r A\left(\frac{a_{\mathrm{sym}}}{GM}\right)^p s^{2p}\,\mathrm{d}s=\frac{A}{2p+1}\left(\frac{a_{\mathrm{sym}}}{GM}\right)^p r^{2p+1}. \]
Dividing by \(g_N\) gives
\[ \frac{\delta g}{g_N}=\frac{1}{(2p+1)\,x^{p+1/2}},\qquad x=\frac{g_N}{a_{\mathrm{sym}}}. \]
The power \(x^{-(p+1/2)}\) is the power in the work order. The coefficient \(1/(2p+1)\) is that integral. A larger total \(g\) makes \(Y\) smaller, so the iteration bounds the first correction from above. The integral converges at the origin for \(p>-1/2\).
4. Perihelion sees Saturn, and it does not select \(p\)
PASS_FT_AXIS_MU_NARROW_STILL_MULTIPLE fires a row when the worst of Mercury, Earth, and Saturn exceeds \(0.003''/\mathrm{cy}\). The estimator is \(\Delta\varpi=2\pi(\delta g/g_N)\) per orbit, with the planets and the conversion of ft_axis_mu_narrow.py. The comparison scale of that gate remains \(1.2\times 10^{-10}\,\mathrm{m\,s^{-2}}\). \(a_{\mathrm{sym}}\) is not its unit. Replacing the comparison scale by \(c^2/R\) still fires the slow face in that note and does not remove a fast row.
The table below is that estimator applied to the local deficit of §3, with the printed \(a_{\mathrm{sym}}=1.605\times 10^{-10}\,\mathrm{m\,s^{-2}}\) inside \(x\). The elliptic mill was not rerun. The frozen-mill solar mass parameter is \(GM=6.67430\times 10^{-11}\times 1.9884\times 10^{30}\,\mathrm{m^3\,s^{-2}}\), and the semi-major axes are \(0.387098\), \(1\), and \(9.537\,\mathrm{AU}\). An order-unity change in the radial weight of a different power does not close a gap of many orders. It can move a result that sits against the gate.
| \(p\) | Mercury \(''/\mathrm{cy}\) | Earth \(''/\mathrm{cy}\) | Saturn \(''/\mathrm{cy}\) | |------:|---------------------------:|-------------------------:|--------------------------:| | \(1\) | \(4.632660044412018\times 10^{-5}\) | \(1.9235303388405797\times 10^{-4}\) | \(5.665219150424835\times 10^{-3}\) | | \(2\) | \(1.1273010716537872\times 10^{-13}\) | \(3.123680983594949\times 10^{-12}\) | \(8.367734902776909\times 10^{-9}\) | | \(4\) | \(1.0301106804674452\times 10^{-30}\) | \(1.2712381979432081\times 10^{-27}\) | \(2.8171845875506646\times 10^{-20}\) |
Saturn is the worst of the three, because its \(x\) is the smallest. On this \(GM\), Mercury has \(g_N=0.0395746241616179\,\mathrm{m\,s^{-2}}\) and \(x=2.4657086705057886\times 10^{8}\). Saturn has \(x=4.0621883099174994\times 10^{5}\). The work order's \(4\times 10^{-2}\) and \(2.5\times 10^{8}\) are that Mercury order.
\(p=1\) puts Saturn at \(5.665219150424835\times 10^{-3}''/\mathrm{cy}\), over the gate. \(p\ge 2\) puts all three many orders under \(0.003''/\mathrm{cy}\). That is the open fast face of §0bn1080. Power \(n>1.56\) already stays there, and no survivor is the law. The exponent here is \(p+1/2\) on \(x=g_N/a_{\mathrm{sym}}\), a different parametrization from that \(n\). The gate does not select \(p\).
At \(100\,\mathrm{AU}\), \(x=3.6947377448772268\times 10^{3}\). The planetary region sits above the threshold. For \(p\ge 2\) the tail is small. The strict zero is a separate envelope, scored in §5.
5. The infrared end, and the step
\(v^4=GM a_{\mathrm{sym}}\) is the region \(Y\to 1\). An envelope with both asymptotes, \(Y=y^p/(1+y^p)\) for example, has the ultraviolet integral of §3. Far outside the yield radius the integral of \(Y-1\) converges, and
\[ g=\frac{A}{r}+\frac{GM+AC}{r^2}, \]
with \(C\) an unpaid length of order \(r_y\). The \(A/r\) term dominates, so \(v^4\to GM a_{\mathrm{sym}}\). The \(1/r^2\) coefficient is an \(O(GM)\) shift. The curve is asymptotic. Each \(p\) labels one both-ends member, with \(a_{\mathrm{sym}}\) selected inside \(A\). That infrared end is the end §0bn1087 took off the working target. The members with \(p\ge 2\) that pass §4 remain an infinite set.
The strict sentences — unyielded at every planetary radius, \(\delta g=0\), Mercury's perihelion advance strictly zero — describe the step \(Y=\Theta(a_{\mathrm{sym}}-g)\). For \(r_y=\sqrt{GM/a_{\mathrm{sym}}}\) the identity \(A r_y=GM\) gives
\[ g=\begin{cases} GM/r^2 & r
\(g\) is continuous and equals \(a_{\mathrm{sym}}\) at \(r_y\). The density jumps. The distribution contains no surface delta. Outside the step, \(\mu=g_N/g\) equals \(g/a_{\mathrm{sym}}\) at every radius. That is the corner §0bn1023 installed, \(\mu=1\) above the threshold and \(\mu=x\) below it, and did not adopt. The geometric-mean sphere of §0bn1088 remains the splice whose Neumann data do not meet.
Worked radii at the printed \(1.605\times 10^{-10}\,\mathrm{m\,s^{-2}}\), IAU \(GM_\odot=1.3271244\times 10^{20}\,\mathrm{m^3\,s^{-2}}\), \(1\,\mathrm{AU}=149597870700\,\mathrm{m}\), and \(1\,\mathrm{pc}=(648000/\pi)\,\mathrm{AU}\). The solar yield radius is \(6078.450531242737\,\mathrm{AU}\). The named \(10^{11}\,M_\odot\) yield radius is \(9.318966562022558\,\mathrm{kpc}\). The round figures \(7400\,\mathrm{AU}\) and \(11\,\mathrm{kpc}\) would require \(1.0829224832273194\times 10^{-10}\) and \(1.1519275714336504\times 10^{-10}\,\mathrm{m\,s^{-2}}\). Those accelerations are neither the printed \(a_{\mathrm{sym}}\) nor \(a_\star=2.7401072522463955\times 10^{-10}\,\mathrm{m\,s^{-2}}\). The frozen-mill \(GM\) differs from the IAU value by the factor \(0.9999950358835991\).
The named galactic mass is not a measured baryonic mass, and \(9.318966562022558\,\mathrm{kpc}\) does not place the yield surface at the edge of every optical disk. a_geom_ft_inverse_rotation_results.md already records that one isothermal speed, tuned at \(20\,\mathrm{kpc}\), gives \(255\,\mathrm{km\,s^{-1}}\) against \(220\,\mathrm{km\,s^{-1}}\) at \(8.5\,\mathrm{kpc}\). The point-mass asymptote is not that galaxy. The SPARC bend of about \(2\,\mathrm{dex}\) stays the opposing unpaid test.
6. What stays unpaid
Cassini's catalogue value remains \(Q_2=(3\pm 3)\times 10^{-27}\,\mathrm{s^{-2}}\). At Saturn, \(x=4.0621883099174994\times 10^{5}\). For \(p=2\) the local fractional tail is \(1.9016468379120424\times 10^{-15}\). That addition sits on top of the solar field. It does not compute the quadrupole, and the catalogue value stays where it was. The external-field quadrupole of one AQUAL potential is a different operator. A Newtonian core of this source does not generate that quadrupole.
SILENT still assigns no mass computed from \(H\), from Willmore energy, or from the second fundamental form. \(Y\) is an envelope added to the logarithmic source. Once \(Y\to 1\), the enclosed nest mass still grows without a bound. The square root in \(A\) is still not linear in \(M\). A disk is not this point-mass halo. The Einstein equation with this \(T_{\mu\nu}\) is not solved.
7. Integrity
| Check | Result | |-------|--------| | Poisson equation \(\partial_r(r^2 g)=A Y(a_{\mathrm{sym}}/g)\) written | YES | | Ultraviolet integral \(\delta g/g_N=1/((2p+1)x^{p+1/2})\) for \(Y\sim y^p\) | YES | | \(p\ge 2\) under \(0.003''/\mathrm{cy}\) on the frozen estimator | YES — open fast face; \(p\) not selected | | \(p=1\) over the gate at Saturn on that estimator | YES — \(5.665219150424835\times 10^{-3}''/\mathrm{cy}\) | | Pure power saturates at \(Y=1\) | NO | | \(Y\to 1\) returns the asymptote \(v^4=GM a_{\mathrm{sym}}\) | YES — one member for each \(p\), infrared end reinstalled | | Step \(Y=\Theta(a_{\mathrm{sym}}-g)\) is the corner §0bn1023 installed | YES — \(g\) continuous, no surface delta | | Solar yield radius at the printed \(a_{\mathrm{sym}}\) | \(6078.450531242737\,\mathrm{AU}\) | | Named \(10^{11}\,M_\odot\) yield radius | \(9.318966562022558\,\mathrm{kpc}\) | | \(7400\,\mathrm{AU}\) or \(11\,\mathrm{kpc}\) adopted | NO | | \(Q_2\) set to \(0\) | NO — catalogue \((3\pm 3)\times 10^{-27}\,\mathrm{s^{-2}}\) stands | | \(a_{\mathrm{sym}}\) selected | NO | | \(p\) or this \(\mu\) crowned | NO | | Elliptic perihelion mill rerun | NO | | SPARC catalogue fitted | NO | | SILENT assigned a mass | NO | | Einstein equation solved | NO | | §0bn1090, §0bn986, §0bn1023, or §0bn1080 rewritten | NO | | Phase 2 run | NO | | Promote | NO |
8. First task
Load §0bn1091. The activated density is the logarithmic source times an envelope \(Y(a_{\mathrm{sym}}/g)\). For \(Y\sim y^p\) the ultraviolet integral is \(\delta g/g_N=1/((2p+1)x^{p+1/2})\). For \(p\ge 2\) the frozen estimator puts that tail under \(0.003''/\mathrm{cy}\), on the open fast face. A pure power does not saturate. \(Y\to 1\) reinstalls \(v^4=GM a_{\mathrm{sym}}\). The step is the corner already installed in §0bn1023. Do not set \(Q_2=0\). Do not select \(a_{\mathrm{sym}}\). Do not crown \(p\) or this \(\mu\). Do not rewrite §0bn1090, §0bn986, or §0bn1080. Phase 2 is not run. Public chips stay OPEN.
§FT-YIELD-ENVELOPE: the envelope is the join on the logarithmic source; the ultraviolet integral is \(1/((2p+1)x^{p+1/2})\); \(p\ge 2\) sits on the open fast face; the step is the installed corner; \(Y\to 1\) reinstalls \(v^4=GM a_{\mathrm{sym}}\); promote NO.