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Consideration — The geometric-mean layer reinstalls the infrared end

Logarithmic exterior is the deep-MOND splice · \([K]\) does not enforce it · \(M\to 0\) collapses the sphere · \(R\to\infty\) removes the layer

CPT Framework — FT-NESTED-LAYER / FT-F-TARGET-WITHDRAWN / FT-AXIS-K-DICT / TF-ADM-CL Date: 2026-10-01 Status: CONSIDERATION · PLACED_FT_NESTED_LAYER_REINSTALLS_IR Stimulus: Principal Investigator — write the nested boundary-value problem left unwritten in §0bn1087, on the geometric-mean sphere \(r_{\mathrm{trans}}=\sqrt{r_s R}\), with a linear interior and a logarithmic exterior, and read the junction from \([K]=(2-\sqrt{2})/R\). Parents: a_geom_ft_f_target_withdrawn_consideration.md (§0bn1087) · a_geom_ft_axis_projection_curvature_dictionary.md (§0bn985, PLACED_FT_AXIS_K_DICT) · a_geom_ft_axis_yield_symmetric_consideration.md (§0bn986) · a_geom_compact_core_draft.md Appendix D · a_geom_tf_adm_clifford_distortion_consideration.md (PASS_TF_ADM_CLIFFORD) · a_geom_ft_axis_dtn_consideration.md (§0bn981) Algebra: the splice of \(\Phi=-GM/r\) to \(A\ln(r/r_t)\). No new script. No \(F\) varied.

Tags: [E honesty] · [E algebra] · [Programme] · [O residual]

§0bn1087, §0bn985, §0bn986, Appendix D, and the two-cap transmission stay as written. Both \(a_{\mathrm{sym}}\) and \(a_\star\) stay. Public chips stay OPEN. Phase 2 is not run. No promote.

0. One-line

The geometric-mean sphere with a logarithmic exterior reinstalls the infrared end. Once \(\Phi\) is made continuous, the radial slopes meet only by selecting \(a_{\mathrm{sym}}\) or \(a_\star\). \([K]=(2-\sqrt{2})/R\) writes \(a_{\mathrm{sym}}\) and does not enforce the splice. \(M\to 0\) collapses the sphere onto the origin. \(R\to\infty\) removes the layer. The nested junction kept off \(F\) stays unwritten.

1. The three scales stay three books

The work order names three radii. The banks already keep them apart.

| Book | Radius on the page | What it is | |------|--------------------|------------| | Local core | \(a=3.862\times 10^{-13}\,\mathrm{m}\) | Dictionary seat. Appendix D's \(I_p\simeq S^3(a_p)\). \(R_{\mathrm{match}}/a_e\) is not derived. | | Container | \(R=3.28\times 10^{26}\,\mathrm{m}\) | Dictionary container. \(a_\star=c^2/R=2.740\times 10^{-10}\,\mathrm{m\,s^{-2}}\). \(a_{\mathrm{sym}}=(2-\sqrt{2})a_\star=1.605\times 10^{-10}\,\mathrm{m\,s^{-2}}\). | | Collective mass | \(r_s=GM/c^2\) | A length built from a mass. No mass enters \(a_{\mathrm{sym}}\) (§0bn986). |

PASS_TF_ADM_CLIFFORD puts the cone page on the Clifford torus of a local core. The moduli \((\eta,p)\) are local. Appendix D's explicit non-claim leaves local host \(=\) cosmic \(S^3\) unasserted, and it leaves the exterior matching theorem unasserted. §0bn985 records the same refusal: one \(\Delta\ell\), read on \(a\) and on \(R\), is two rows, and the seat is not fused with the container leaf.

The sphere \(r_{\mathrm{trans}}=\sqrt{r_s R}\) is a fourth length. It is not \(a\), not \(R\), and not \(R_{\mathrm{match}}\).

2. The sphere is the radius where Newtonian \(g\) equals a unit

Write \(r_s=GM/c^2\) and \(r_\star=\sqrt{r_s R}\). The Newtonian acceleration of a point mass is \(g_N=GM/r^2\), so

\[ g_N(r_\star)=\frac{c^2}{R}=a_\star. \]

The radius where the same field equals \(a_{\mathrm{sym}}\) is larger by the pure factor \(1/\sqrt{2-\sqrt{2}}\):

\[ r_{\mathrm{sym}}=\sqrt{\frac{GM}{a_{\mathrm{sym}}}}=\frac{r_\star}{\sqrt{2-\sqrt{2}}},\qquad \frac{1}{\sqrt{2-\sqrt{2}}}=1.3065629648763766. \]

The parenthesis that offers \(a_{\mathrm{sym}}\) at \(r_\star\) does not hold. \(g_N(r_\star)=a_\star\).

The round figures \(r_s\approx 3\,\mathrm{km}\) and \(r_s\approx 0.01\,\mathrm{pc}\) are \(2GM/c^2\). On that sphere \(g_N=c^2/(2R)\). The check uses the IAU nominal \(GM_\odot=1.3271244\times 10^{20}\,\mathrm{m^3\,s^{-2}}\), \(c=299792458\,\mathrm{m\,s^{-1}}\), the dictionary \(R\), the astronomical unit \(149597870700\,\mathrm{m}\), and the parsec \(648000/\pi\) of that unit. With those inputs, \(c^2/R=2.7401072522463955\times 10^{-10}\,\mathrm{m\,s^{-2}}\), which meets the printed \(2.740\times 10^{-10}\) at the printed digits.

| Mass used | \(GM/c^2\) | \(\sqrt{(GM/c^2)R}\) | \(\sqrt{(2GM/c^2)R}\) | |-----------|------------|----------------------|------------------------| | \(1\,M_\odot\) | \(1476.6250380501249\,\mathrm{m}\) | \(4652.074101850824\,\mathrm{AU}\) | \(6579.026288002072\,\mathrm{AU}\) | | \(10^{11}\,M_\odot\), as named | \(0.004785415841587841\,\mathrm{pc}\) | \(7.132166787632891\,\mathrm{kpc}\) | \(10.086407000177385\,\mathrm{kpc}\) |

The second row is the mass named in the work order. It is not a banked galactic mass. \(g_N=a_\star\) on the third column and \(g_N=a_\star/2\) on the fourth. Neither column is the Appendix D match shell.

3. The transmission of the two potentials

The interior potential of the work order is the Newtonian monopole, \(\Phi_{\mathrm{int}}=-GM/r\), so \(\partial_r\Phi_{\mathrm{int}}=GM/r^2\). Outside the origin it is harmonic. The exterior potential of the work order is

\[ \Phi_{\mathrm{ext}}=A\ln(r/r_t)+C,\qquad A=\sqrt{GM\,a_{\mathrm{sym}}},\qquad \partial_r\Phi_{\mathrm{ext}}=A/r. \]

In three dimensions \(\nabla^2\ln r=1/r^2\), so

\[ \nabla^2\Phi_{\mathrm{ext}}=\frac{A}{r^2}\neq 0. \]

The logarithm is not an exterior solution of \(\nabla^2\Phi=4\pi G\rho\). There is no Dirichlet-to-Neumann map of a linear vacuum exterior that returns it. The booked DtN sits on a different sphere: the container boundary and the cap equator, with transmission \(2n/R\) (§0bn981, §0bn995). That transmission is not a condition at \(r_t\).

The additive constant can make \(\Phi\) continuous at one chosen radius. It cannot make the slopes continuous at the same time, unless \(A\) and \(r_t\) are tied to one unit.

At the geometric-mean sphere \(r_t=r_\star\), with the coefficient kept at \(\sqrt{GM\,a_{\mathrm{sym}}}\),

\[ \partial_r\Phi_{\mathrm{ext}}=a_\star\sqrt{2-\sqrt{2}},\qquad \partial_r\Phi_{\mathrm{int}}=a_\star. \]

\[ \sqrt{2-\sqrt{2}}=0.7653668647301795. \]

The Neumann jump, exterior minus interior, is

\[ [\partial_r\Phi]_{r_\star}=-(1-\sqrt{2-\sqrt{2}})\,\frac{c^2}{R}, \]

\[ 1-\sqrt{2-\sqrt{2}}=0.23463313526982055. \]

The value that matches \(\Phi\) at the same sphere is \(C=-\sqrt{GM\,a_\star}\), which uses \(a_\star\), while \(A\) was written with \(a_{\mathrm{sym}}\).

The slopes and the values become continuous together in exactly one of these two splices:

| Choice inserted by hand | Sphere | Coefficient \(A\) | What is selected | |-------------------------|--------|-------------------|------------------| | Container threshold | \(r_\star=\sqrt{GM/a_\star}\) | \(\sqrt{GM\,a_\star}\) | \(a_\star\) | | Symmetric threshold | \(r_{\mathrm{sym}}=\sqrt{GM/a_{\mathrm{sym}}}\) | \(\sqrt{GM\,a_{\mathrm{sym}}}\) | \(a_{\mathrm{sym}}\) |

On either consistent splice the second radial derivative still jumps. With \(g_N(r_t)=a_0\) and \(A=a_0 r_t\),

\[ \partial_{rr}\Phi_{\mathrm{int}}=-\frac{2a_0}{r_t},\qquad \partial_{rr}\Phi_{\mathrm{ext}}=-\frac{a_0}{r_t}. \]

For a circular orbit the consistent logarithm is \(g^2=g_N a_0\), hence \(v^4=GM a_0\). With \(x=g/a_0\) that is \(\mu=x\). It is the infrared end of the class §0bn1087 withdrew as the working target. Writing \(v^4=GM a_{\mathrm{sym}}\) selects \(a_{\mathrm{sym}}\) inside that end.

The linear interior \(\nabla^2\Phi=4\pi G\rho\) has no nonlinear quadrupole. A uniform external acceleration is pure gauge for that equation. The equation then contains no anomalous quadrupole to carry inward from \(r_t\). A harmonic quadrupole that is regular at the origin has constant second derivatives, so its value at Saturn is the value on the sphere. Cassini stays the catalogue quadrupole \(Q_2=(3\pm 3)\times 10^{-27}\,\mathrm{s^{-2}}\). This note does not set \(Q_2=0\).

4. The dictionary jump does not enforce the splice

§0bn985:

\[ [K]_{\Delta}=\frac{\Delta\ell}{\rho},\qquad [g]_{\Delta}=c^2[K]_{\Delta}=\frac{c^2\Delta\ell}{\rho}. \]

On the container, \(\Delta\ell=2-\sqrt{2}\) and \(\rho=R\), so \([K]_{\Delta}=(2-\sqrt{2})/R\) and \([g]_{\Delta}=a_{\mathrm{sym}}\). The printed curvature is \(1.786\times 10^{-27}\,\mathrm{m^{-1}}\). The same excess on the seat \(a\) is the other row, \(1.363\times 10^{29}\,\mathrm{m\,s^{-2}}\). The Israel prefactor and the lapse factor stay unpaid. The dictionary does not choose \(F\), and it does not deposit a pair on the empty region.

The Neumann mismatch of §3, divided by \(c^2\), is \((1-\sqrt{2-\sqrt{2}})/R\). That pure number is \(0.23463313526982055\). The dictionary pure number is \(2-\sqrt{2}=0.5857864376269049\). The L1 hemisphere is \(1\), and the principal jump is \(2\). Four different numbers. The curvature jump writes the threshold \(a_{\mathrm{sym}}\) on the container radius \(R\). It does not write \([\partial_r\Phi]\) at a mass-dependent sphere, and it does not choose which of the two splices in §3 is the boundary condition.

5. The two limits

Keep the work order's simultaneous choice: \(r_t=\sqrt{(GM/c^2)R}\) and \(A=\sqrt{GM\,a_{\mathrm{sym}}}\). The Neumann jump at that moving sphere is \(-(1-\sqrt{2-\sqrt{2}})\,c^2/R\). It does not depend on \(M\).

\(M\to 0\) at fixed \(R\). Then \(r_t\to 0\) and \(A\to 0\). The sphere collapses onto the origin. The hosted interior loses its volume. The slope jump sampled on the collapsing sphere stays the container number above. A junction of an extended body does not remain. The dictionary threshold never took a mass.

\(R\to\infty\) at fixed \(M\). Then \(a_\star\to 0\), \(a_{\mathrm{sym}}\to 0\), \(r_t\to\infty\), and the Neumann jump tends to \(0\). Every fixed ball is eventually interior to the receding sphere, and the logarithmic coefficient tends to \(0\). The condition on that ball is the Newtonian monopole.

The same two limits remove either consistent splice of §3. A finite layer that survived \(M\to 0\) or \(R\to\infty\) is not what these potentials do.

6. What stays unwritten

A nested junction whose unknown is not \(F\), and not the logarithm of §3, is still unwritten. The surface it would use has to be a surface already on the page. The surfaces on the page are the Compton core, the unpaid Appendix D shell, the cap equator, and the Clifford leaf. \(\sqrt{r_s R}\) is not one of them.

This note does not supply a replacement transmission. It does not select \(a_{\mathrm{sym}}\) or \(a_\star\). It does not fuse \(I_p\) with \(S^3(R)\). It does not vary an \(F\). It does not rewrite §0bn1087 or §0bn985.

7. Integrity

| Check | Result | |-------|--------| | Two potentials and both limits written out | YES | | Logarithmic exterior identified as the infrared end \(\mu=x\) | YES | | \([K]=(2-\sqrt{2})/R\) enforced the splice | NO | | \(a_{\mathrm{sym}}\) or \(a_\star\) selected | NO | | \(Q_2\) set to \(0\) | NO | | Local core fused with cosmic \(S^3\) | NO | | \(\sqrt{r_s R}\) identified with \(R_{\mathrm{match}}\) | NO | | A replacement transmission written | NO | | An \(F\) varied, or \(\mu=F'(X)\) returned | NO | | §0bn1087 or §0bn985 rewritten | NO | | Phase 2 run | NO | | Promote | NO |

8. First task

Load §0bn1088. The geometric-mean sphere with a logarithmic exterior reinstalls the infrared end. \([K]=(2-\sqrt{2})/R\) writes \(a_{\mathrm{sym}}\) and does not enforce the splice. At \(\sqrt{(GM/c^2)R}\) the slope jump is \(\bigl(1-\sqrt{2-\sqrt{2}}\bigr)\,c^2/R\). \(M\to 0\) collapses the sphere onto the origin. \(R\to\infty\) removes the layer. Do not select \(a_{\mathrm{sym}}\) or \(a_\star\). Do not set \(Q_2=0\). Do not fuse a local core with the cosmic container. Do not ask this junction to return \(\mu=F'(X)\). Do not rewrite §0bn1087 or §0bn985. Phase 2 is not run. Public chips stay OPEN.

§FT-NESTED-LAYER: the geometric-mean logarithm reinstalls the infrared end; \([K]=(2-\sqrt{2})/R\) does not enforce the splice; \(M\to 0\) collapses the sphere; \(R\to\infty\) removes the layer; promote NO.
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