HDBLAST

26 October 2026: conditional registered screen magnetic curvature. On the exterior angular sphere at (x,r)=(1/4,5/4), before shell contact, the projection of the second parameter derivative of the screen magnetic tensor onto the selected TT pair is Π_pair H″(0)=h(T+ − T−). The calculation combines the linear second-response TT term with the complete scalar quadratic term, retaining the observer, radial-normal, connection and epsilon corrections. The complete first variation of this screen tensor vanishes, and exact angular identities close the quadratic term on T+ − T−.

The coefficient h is enclosed in [-0.024929793753, -0.004128493946] and is strictly negative under the retained assumptions. Its scalar quadratic contribution is enclosed in [-0.000000000000000000690462, 0.000000000000000001996501]. These are outward rational bounds at the registered point. The projected angular field vanishes on the two registered great circles S=0 and S=1, where S=n0²+n1². The future observer is the unit normal to t=constant; the radial normal is the positive unit normal to y=constant within that spatial slice, with screen axes and orientation fixed by the registration.

This is a conditional curvature diagnostic for the registered formal response. The parent background, master, gauge, coefficient-existence and subsidiary assumptions remain in force. No gauge-invariant observer or clock anchoring has been constructed. The result establishes no physical radiation, force or detection, no finite-amplitude nonlinear family and no cosmological origin. The full second-order Einstein–matter boundary problem remains unfinished. No new Zenodo version or journal submission accompanies this checkpoint.

Derivation and scope · Source, data and replay

26 October 2026: exterior tensor derivatives enclosed at one point. Under the retained model assumptions, the tensor coefficient of the actual total formal response and all its first and second scaled coordinate derivatives are enclosed at (x,r)=(1/4,5/4), before shell contact. The two TT coefficients remain equal.

Their common value u lies in [−0.000207982516, −0.000010433243], and its scaled time derivative u_x lies in [−0.000280539069, −0.000046459086]. Both are strictly negative. The radial first derivative and all three second-derivative intervals contain zero, so their signs remain inconclusive.

This is a local result about metric coefficients in the registered gauge. It does not establish a full-domain C2 approximation, an Einstein residual bound or a physical curvature or gravitational-wave observable. Nonlinear physical realization and cosmological origin remain open. No new Zenodo version or journal submission accompanies this checkpoint.

Derivation and scope · Source, data and replay

26 October 2026: a full angular response approximation passes its accuracy target. One continuous C1 first-feedback approximation on the retained local domain D+ keeps all 23 angular slots. Its complete conditional X-error bound for values and scaled first x/r derivatives is below 1.1787044291e28, meeting the 1e29 target in the positive angular H norm.

At the exterior point (x,r)=(1/4,5/4), a separate targeted calculation of the actual total response encloses each of the two equal TT coefficients in [−0.001311761536, −0.000001509027]. This is a metric coefficient in the registered gauge, not an observed gravitational wave or a gauge-invariant tensor observable.

These numerical approximation and integral-defect certificates retain the conditional parent coefficient-existence and subsidiary assumptions. They establish no pointwise C2 Einstein residual, nonlinear or finite-amplitude physical family, or cosmological origin. No new Zenodo version or journal submission accompanies this checkpoint.

Derivation and scope · Source, data and replay

Conditional shell and anchored-volume sign bounds — 25 October 2026. Conditional bounds place N0(x,0) and J0(x,0) below zero at the retained scaled times x=9/8 and x=5/4. Under the retained parent projection assumptions and a smooth kinematical completion, the anchored proper-time volume second parameter derivative is also negative at x=9/8. Finite-amplitude volume change, a full angular Einstein solution and cosmological conclusions remain unestablished. No new Zenodo version or journal submission accompanies this checkpoint.

Source capsule · Observable bound and assumptions

25 October 2026: a coupled five-mean response approximation passes its accuracy target. One continuously defined C1 approximation now includes the first coupled bulk feedback and shell boundary lift. Its complete conditional error bound is below 1.6090323e28 in the retained scaled X norm, meeting the 1e29 target. The 64 and 128 meshes gave insufficient bounds; the declared 256 limit passed. Both execution modes agree, and an independent exact integrator reproduces the stored feedback samples.

This advances the numerical construction of five mean coefficients. The full angular second-order spacetime, intrinsic observable sign, nonlinear family and physical origin remain unestablished. These are mathematical bounds under the stated model assumptions, not physical measurements. No new Zenodo version or journal submission accompanies this checkpoint.

Derivation and scope · Frozen data, source and replay · Source archive (ZIP)

23 October 2026: the initial-data precision gate passes. An exact residual comparison sharpens the same retained regular-shell branch to a beta interval of radius 9.505e-18. A separately registered recentered rational approximation has a certified initial/free-wave error below 8.519e27, meeting the 1e28 target across all 96 cells. The old candidate and its failed bound are preserved. Some illustrative component allocations are exceeded; the exact combined inequality passes.

This clears an input-accuracy bottleneck under the stated model and theorem assumptions. The evolved numerical total response and intrinsic observable sign remain unevaluated. It establishes no nonlinear physical family or higher-dimensional Big Bang origin. No new Zenodo version or journal submission accompanies this update.

Working note (PDF) · Derivations, source and checks · Source archive (ZIP).

23 October 2026: tighter response bounds and an intrinsic shell-volume prescription. A zero-region estimate tightens the conditional local response contraction bound to q = 0.000426125. A complete initial-data pilot defines one differentiable rational candidate but its certified error bound fails the chosen accuracy gate. This identifies the next precision task; it proves no impossibility. A separate derivation defines intrinsic shell volume at anchored proper time, including clock, slicing and quadratic metric terms. Its value remains unevaluated.

The numerical total response and invariant observable sign remain unevaluated. These conditional results establish no nonlinear physical family or higher-dimensional Big Bang origin. Earlier checkpoints are preserved; no new Zenodo version or journal submission accompanies this update.

Working note (PDF) · Derivations, source and checks · Source archive (ZIP).

October 2026: a positive auxiliary mean after causal contact with the shell. At scaled (x,r) = (3/2,1/4), after the past causal cone reaches the timelike shell, interval integration of the actual quadratic source and an exact closed system of five rotational means prove that the probability-normalized S³ mean of the auxiliary covariant yy response is positive. The bounds include all feedback in that mean system, including the Robin shell condition. All quadratic mean contributions of the first-order harmonics are retained before exact projection.

This conditional result concerns an auxiliary coefficient in the registered frame and gauge. The full evaluated numerical waveform remains unfinished. The retained total response U already has a conditional analytic construction and physical constraint completion; this auxiliary mean does not determine its physical interpretation. No physical force, exact nonlinear spacetime family or evidence of a higher-dimensional Big Bang origin is inferred. No new Zenodo version or journal submission accompanies this checkpoint.

Technical working note (PDF) · Editable source and reproducible calculations · Source archive (ZIP) · Verification evidence.

October 2026: certified causal bounds for an auxiliary response. Interval evaluation certifies that the angular mean of the covariant yy component of the auxiliary forced metric response is strictly positive throughout a continuous neighborhood of scaled (x,r) = (1/4,5/4), before signals from the shell can reach it. The calculation includes 20 clipped causal source cells, the certified uncertainty in the actual first-order fields and all 23 bulk amplitudes with their couplings.

This conditional, model-specific result concerns a coordinate coefficient in the stated frame and gauge. The full numerical second-order waveform, including shell interaction, remains unfinished. The auxiliary response is not established as a separate solution of the complete physical constraint system; no physical force, exact nonlinear spacetime family or evidence of a higher-dimensional Big Bang origin is inferred. No new Zenodo version or journal submission accompanies this checkpoint.

Technical working note (PDF) · Derivations, editable source and reproducible calculations · Verification evidence.

16 October 2026: complete quadratic geometric sources and an intrinsic shell diagnostic. The actual full Ricci/Einstein–scalar source adapter and exact 23-amplitude projection retain both transverse-traceless orientations. Certified interval evaluation covers 81 causal cells, 64 evolved radial-profile cells and 6 shell time cells. The bulk forcing norm satisfies F_H ≤ 17387513156573, improving the retained upper bound by more than 9.8 times.

The intrinsic material-shell quadratic scalar has exactly two simple mean zeros in each scaled-time bracket [5/4,11/8] and [13/8,7/4] on the specified background time sections: four distinct zeros are certified, with others elsewhere not excluded. A separate derivative-bound corollary preserves opposite evolved second scalar-mean signs on two complete radius cells for scaled times [1/1000,1/200]. The full numerical second waveform and continuous response residual remain unfinished. These conditional formal-coefficient results establish no exact nonlinear family, physical force or higher-dimensional Big Bang origin.

Technical working note (PDF) · Derivations, editable source and reproducible calculations · Verification evidence.

15 October 2026 (Pacific): certified numerical inputs for the selected second-order response. Continuous error bounds now control the actual first master and gauge fields through scaled derivative order three. An exact 23-amplitude homogeneous bulk/boundary operator and a sharp selected-space evaluation constant, sqrt(321/19), support the next numerical calculation.

Whole-cell evaluation of the same second initial profiles tightens the global formal-response norm bound from 1.02 × 10³⁶ to 1.03 × 10³³, about 990 times smaller. These are conservative coefficient bounds, not measured field amplitudes. The full quadratic metric-source adapter and a continuously certified numerical second-response waveform remain unfinished. Conditional mathematics does not establish an exact nonlinear family, invariant force or a higher-dimensional Big Bang origin. No new Zenodo version or journal submission accompanies this checkpoint.

Technical working note (PDF) · Derivations, editable source and reproducible calculations · Verification evidence.

16 October 2026: constructive second-order continuation for one aligned quadrupole: A convergent causal integral series defines the full formal second metric–scalar coefficient for the selected registered state on the controlled shell-contact domain. Its operator has contraction factor at most 0.002109 on a 287-dimensional angular tensor-section space, retaining every generated tensor sector.

The construction joins the complete bulk and shell forcing in one common first relative-harmonic gauge, compatible second initial data, physical shell laws and subsidiary gauge constraints. Qualitative smoothness is proved separately from numerical bounds for values and first scaled derivatives.

The coarse comparison-norm cap, ‖U₂‖_X < 1.02 × 10³⁶, is dominated by the inherited second initial scalar data and carries no physical signal interpretation. No candidate numerical waveform has been evaluated. This conditional coefficient construction does not establish an exact nonlinear family, a finite-amplitude remainder, an invariant force or a higher-dimensional Big Bang origin. Internal AI-assisted review does not replace external specialist review. No new Zenodo version or journal submission accompanies this checkpoint.

Research source and exact reports · Second-response construction · Bulk forcing proof · Physical shell forcing · Result status and literature context · Source ZIP · Replay and publication evidence.

16 October 2026: certified first relative-harmonic gauge wave: For the retained regular background and quadrupole, a convergent construction now gives the first coordinate gauge vector with continuous derivatives through order three. An explicit approximation has uniform error below 5.47 × 10⁻⁹ in its radial coordinate amplitude on the controlled causal triangle.

This conditional first-order result preserves the shell’s normal Dirichlet and tangential Neumann conditions. Transforming the quadratic bulk and shell sources and solving the second-order Einstein–scalar evolution remain unfinished. No invariant force or higher-dimensional Big Bang origin is established. Internal AI-assisted review does not replace external specialist review. No new Zenodo version or journal submission accompanies this checkpoint.

Research source and exact reports · Construction and error proof · Derivative bounds · Source ZIP · Replay and publication evidence.

15 October 2026: second initial norms and initial harmonic gauge: Exact rational calculations give coarse certified upper bounds for the values and first spatial covariant derivatives of all four physical second initial fields: the spatial metric, extrinsic curvature, scalar and unit-normal scalar velocity. The initial constraint-source cancellations remain exact. Explicit local initial relative-harmonic lapse and shift velocities include the quadratic terms.

These are conditional initial-data bounds in the stated frames and model units. They do not demonstrate a response, establish a physical force, solve the full second-order spacetime evolution or construct an exact finite-amplitude family. Common boundary/gauge completion and validated coupled evolution remain open. Earlier frozen checkpoints are preserved; internal AI-assisted checks do not replace external peer review.

Research source and exact reports · Second initial norms · Initial harmonic completion · Spacetime initial one-jet.

15 October 2026: explicit compact second initial constraint data: For the registered aligned packet, explicit local formulas now cancel the Hamiltonian and all four momentum constraint sources through second order. The correction remains in the original packet annulus and vanishes near the initial shell. A certified nonzero Hamiltonian source at the packet maximum rules out zero second initial data.

The construction allows nonzero second scalar and unit-normal scalar-velocity data. Physical Cauchy fields and spacetime coordinate velocities are distinguished, with the lapse conversion included. This is a conditional second initial jet, not a finite-amplitude exact initial family or a second-order spacetime evolution. Initial norms, the common gauge and boundary completion, and validated U2 evolution remain unfinished; no physical force or higher-dimensional Big Bang origin is established. Checks and independent internal AI-assisted review do not replace external peer review.

Explicit formulas, proof and reproducible checks.

15 October 2026: interior derivatives and bulk forcing bounds: For the registered aligned quadrupole, exact rational estimates now bound every mixed first-master derivative through order three on the controlled causal triangle. A new actual-potential second-derivative bound and the reflected shell terms close this derivative calculation. Reconstruction then gives coarse continuous envelopes for all fifteen longitudinal Einstein quadratic-source components and the complete scalar source over the triangle times S3.

These are conditional bounds in the stated first longitudinal representative and adapted background frames, in model units. They do not compute the second-order response or a physical force. The earlier conditional coefficient construction retains its stated scope; quantitative U2, compatible second initial and common-gauge boundary data, and a validated coupled evolution remain unfinished. Internal AI-assisted review does not establish external peer review, novelty or a higher-dimensional Big Bang origin.

Derivations, exact reports and reproducible source.

15 October 2026: continuous coordinate shell sources: The certified quadrupole now gives enclosures over 4353 continuous time cells for the actual metric trace at the shell, its proper-time derivative, and the complete one-copy coordinate scalar and isotropic Israel coefficients Sp and SK, with scalar degrees 0, 2 and 4. The metric correction has L2 norm less than 9e-13 of the one-copy material scalar source, yet changes the coordinate source signs at a scalar zero. Each of Sp and SK consequently has at least two distinct zeros inside the certified central bracket.

These results are conditional on the retained analytic inputs and the stated first longitudinal representative and derivative Taylor convention. The reduced Robin forcing norm is evaluated only for degree 4; the degree-2 Robin row is algebraic only. The norms use the background proper shell measure, and coefficient signs depend on the representative. The earlier conditional construction of the complete second coefficient retains its stated scope. No effective nonlinear amplitude or remainder, observational scale or cosmological origin is established. Independent internal checks do not replace external peer review. Full U2 remains quantitatively unresolved.

Derivation and bounds · Source and replay guide.

See the separate publication evidence for replay, CI and live-readback status. Publication evidence.

15 October 2026: quadratic material tension forcing: For the previously certified wider quadrupole (delta = 1/1000, ell = 10^-6 in model coordinates), the explicit material scalar-junction tension contribution is Q_phi = -4 psi_1^2 in the outward-normal SUM convention. It has scalar degrees 0, 2 and 4. Exact continuous enclosures give -2.3 × 10^-19 < ell^2 mean(Q_phi) < -6.7 × 10^-20 throughout 2 ≤ x ≤ 3 and 5.778 × 10^21 < ||Q_phi||_L2(dmu_0)^2 < 8.644 × 10^21 over 0 ≤ x ≤ 3, using the background proper-time measure and model units. The angular mean stays negative throughout the tail; the field vanishes on Y = 0.

This isolates the explicit material tension contribution; the complete quadratic source for bulk-coordinate unknowns and full U2 are not numerically evaluated. The earlier conditional construction of the complete second coefficient is preserved under its stated data and evolution assumptions. No effective finite amplitude, nonlinear remainder, observational scale or cosmological origin is established. Independent internal exact checks are distinct from external peer review and formal PDE verification. Earlier dated entries retain their scope. No new Zenodo version or APS submission was made.

Material tension theorem, normalization and assumptions · Reproducible calculations and independent internal checks.

Publication checks remain pending until documented in the separate final additive evidence receipt. Final additive publication evidence.

Conditional inertial shell waveform: For a newly specified wider aligned quadrupole (delta = 1/1000, ell = 10^-6 in model coordinates, T = 3ell and Z = 8ell), exact interval bounds resolve the shell response at first variation. The normalized trace H(x) = ell D(ell x) satisfies 3.2 × 10^-8 < H < 5.3 × 10^-8 throughout 2 ≤ x ≤ 3. The negative physical gain G makes the corresponding scaled physical first variation ell S negative there. At least one zero lies in each of (1.49998, 1.49999) and (1.96, 1.97); uniqueness is not asserted. The leading proper-time response coefficient satisfies 6.06 × 10^11 < C < 7.44 × 10^11 in R_epsilon = epsilon^2 C + o(epsilon^2), in the stated model normalization.

The actual-background, exact-data, gauge, regularity and whole nonlinear correction-support hypotheses remain explicit. This first-variation result does not close the full second-order nonlinear waveform or provide an effective epsilon or nonlinear remainder bound. No observational scale or higher-dimensional Big Bang origin is established. Internal AI-assisted review and finite checks are distinct from external peer review and formal PDE verification. Earlier entries retain their dated scope. No new Zenodo version or APS submission was made.

Waveform theorem, normalization and assumptions · Reproducible calculations and independent reviews.

Publication checks remain pending until documented in the separate final additive evidence receipt. Final additive publication evidence.

Conditional first-variation shell waveform: For the same specified narrow aligned quadrupole (delta = 1/1000, ell = 10^-10 in model coordinates), a continuous 4098-cell enclosure resolves the physical scalar’s linear first variation. The normalized trace H differs from its explicit comparator by less than 1.0000004 × 10^-7 on 0 ≤ x ≤ 3; the negative physical gain reverses its sign. The leading proper-time response coefficient now satisfies 6.06 × 10^15 < C < 7.44 × 10^15 in R_epsilon = epsilon^2 C + o(epsilon^2).

The retained background, exact-data, gauge, regularity and nonlinear support hypotheses remain explicit. The post-pulse estimate |H| < 4.752 × 10^-14 on 2 ≤ x ≤ 3 is a small-tail bound. No finite-amplitude nonlinear waveform, observed time scale or higher-dimensional Big Bang origin is established. Next: a wider state with a validated background and an effective nonlinear remainder estimate. Internal AI-assisted review and finite checks are distinct from external peer review. Earlier entries retain their original scope. No new Zenodo version or APS submission was made.

Waveform theorem, normalization and assumptions · Reproducible calculations and independent reviews · Continuous-cell enclosure data.

Conditional quantitative shell-response bound: For the retained regular branch at delta = 1/1000 and one newly specified narrow aligned quadrupole (ell = 10^-10 in model coordinates), a free-wave comparison and exact-rational bounds give 10^15 < C < 4 × 10^17 for the leading proper-time response coefficient in R_epsilon = epsilon^2 C + o(epsilon^2). The estimate controls the physical derivative at the shell, with its positive boundary inertia.

This is a coarse conditional analytic enclosure in the stated model normalization. It gives no numerical waveform, effective nonlinear amplitude range, observational scale or evidence for a higher-dimensional Big Bang origin. The background, exact-data, gauge, regularity and full nonlinear support hypotheses remain explicit. Internal AI-assisted reviews and finite arithmetic checks are distinct from external peer review. Earlier entries retain their checkpoint scope. No new Zenodo version or APS submission was made.

Quantitative theorem, normalization and assumptions · Reproducible arithmetic and independent reviews.

Conditional nonzero physical shell response: For one explicitly specified aligned quadrupole with nonzero displacement and velocity, the full coupled Einstein–scalar shell equations now imply a nonzero physical first scalar response. In the retained conditional exact nonlinear family, a squared scalar-response measure over a finite intrinsic proper-time window has a strictly positive leading quadratic coefficient.

The result retains the stated background, exact-data, support, regularity and finite-region hypotheses. It is qualitative: no numerical waveform, response magnitude, effective amplitude threshold or exact nonlinear first-arrival time is supplied. Internal AI-assisted review and finite reproducible checks are distinct; external peer review, global stability and a higher-dimensional origin of the Big Bang are not established. Earlier dated entries retain their original scope. No new Zenodo version or APS submission was made.

Physical response theorem and assumptions · Derivations and reproducible checks.

Exact nonlinear shell evolution at fixed finite regularity: Under the stated background, support and exact-data hypotheses, the selected aligned quadrupole has a small exact Einstein–scalar family on both copies of the controlled positive-cone region for a prescribed finite shell-interaction interval. The construction satisfies the physical shell laws and constraints, with H^14/H^13 evolution and C² parameter dependence in a lower norm.

This is internally reviewed conditional mathematics. The actual first-arrival time and a nonzero gauge-invariant shell response remain to be calculated. No global stability, observational agreement or higher-dimensional origin of the Big Bang is established. The analytic proofs and finite algebra checks are documented separately; no new Zenodo version or APS submission accompanies this checkpoint. Earlier dated entries retain their original scope.

Theorem, proof and assumptions · Reproducible checks and scope.

All-angular linear shell-contact estimate — 14 October 2026 (Pacific): On the registered Einstein–scalar background and a fixed controlled positive-cone collar, the full sixteen-field linear system now has a positive auxiliary Sobolev control estimate independent of angular frequency and of the full tensorial SO(4) cutoff. A bounded change of field variables removes the complete nonsymmetric twelve-row Robin matrix without changing the common wave principal part. Smooth compatible data and forcing give a unique smooth linear solution; the same method controls subsidiary constraint propagation.

Smooth boundary forcing is handled by an explicit lift with its declared regularity cost. No estimate for arbitrary L² Neumann forcing is claimed. The positive norm is analytic control, not physical gravitational energy or a stability result. A separate exact local change of nonlinear boundary variables converts all twelve physical and normal-gauge laws to homogeneous Neumann conditions while preserving the common quasilinear wave principal symbol. This is an algebraic reduction; an exact nonlinear perturbation family through shell contact remains open. The previously established exact initial-data family at fixed high finite Sobolev regularity and the earlier complete second-order coefficient are preserved.

Primary-literature comparisons do not transfer other models' theorems automatically. Internal AI-assisted review is not external peer review; external novelty and a higher-dimensional Big Bang origin remain unestablished. This is a GitHub-only checkpoint, with no new Zenodo version or APS submission. Artifact identifiers use 15 October UTC; this work began 14 October Pacific time. Earlier dated entries retain their original scope.

Estimate, assumptions and reproducible checks · Linear proof · Exact nonlinear boundary reduction.

Complete second-order coefficient through shell contact (14 October 2026): For the specified copy-even commuting quadrupole packet, the unchanged first-order scalar reconstruction cannot satisfy the quadratic degree-four angular equation. The continuation therefore keeps all metric and scalar fields, with their complete quadratic bulk sources and physical shell boundary conditions.

Under the stated background, localization and initial-data hypotheses, a boundary-preserving relative harmonic gauge and a convergent characteristic construction give a smooth complete second Taylor coefficient through first shell contact on a controlled positive-cone collar. It includes scalar degrees zero, two and four and tensor degree two. The construction satisfies all Einstein–scalar equations, induced-field matching and Israel/scalar shell laws through second order, with propagation of the sourced constraints.

This is a conditional coefficient construction. An exact nonlinear family through contact, estimates uniform over all angular modes, nonlinear stability and a higher-dimensional Big Bang origin remain unestablished. Internal AI-assisted proof review is not external peer review, and external novelty is not established. This is a GitHub-only checkpoint, with no new Zenodo version or APS submission. Earlier dated entries, including earlier entries from 14 October, retain their original scope.

Construction, assumptions and reproducible checks · Complete coefficient proof.

Quadratic tensor forcing and a tensor sector through shell contact (14 October 2026): For the specified quadrupole Y = n₁² − n₂² in the fixed longitudinal first-order gauge, the complete gravity-plus-scalar quadratic source has a nonzero transverse traceless angular component wherever either scalar amplitude is nonzero: its coefficient is 16u² + 8v²/3. Thus the two contributions cannot cancel there. A velocity-only packet can have zero source on its initial slice and develop it later.

For even first-order master data, the corresponding forced second-order tensor sector has homogeneous Neumann shell data and a conditional evolution through first contact on a controlled collar. The full coupled scalar and tensor second-order boundary problem and a nonlinear solution through contact remain open.

Derivation, assumptions and reproducible checks. These statements use the specified background, first-order gauge and parity. They establish no Big Bang origin or external novelty; internal proof review is not external peer review. This is a GitHub-only checkpoint, with no new Zenodo version or APS submission. Earlier dated entries retain their original scope.

Exact localized initial data and short precontact evolution (14 October 2026): Under the stated background assumptions and weighted nonlinear estimates, the reflection-invariant quadrupole sector admits a family of exact nonlinear initial data at a chosen high finite Sobolev regularity. A sufficiently short precontact Cauchy evolution keeps the fields equal to the background in a collar of the shell, so its junction conditions remain unchanged there.

The nonlinear boundary problem through shell contact remains open. The theorem concerns high finite regularity; no spatial smoothness to all orders, Big Bang conclusion or external novelty is established. The proof received internal AI-assisted review, which is not external peer review. Proof, assumptions and limits. This is a GitHub-only checkpoint, with no new Zenodo version or APS submission. Earlier dated entries retain their original scope.

Formal continuation by reflection symmetry (14 October 2026): For the restricted commuting quadrupoles of the localized checkpoint, simultaneous diagonalization makes the first-order data invariant under coordinate reflections. These reflections force all ten local symmetry compatibility pairings to vanish at every finite constraint order. Under the stated localization hypotheses, recursive constraint solving and averaging over the reflection group construct localized formal initial data to all orders. This is a formal series: convergence, an exact nonlinear solution and full evolution through shell contact are not established. Derivation, assumptions and limits.

Localized second-order initial data (14 October 2026): On the time-symmetric regular Einstein–scalar slice, the ten symmetry tests are necessary and sufficient for compactly supported second-order initial-data corrections for smooth packets of fixed reflection parity in an annulus away from the shell and cone vertex, under the stated background and localization assumptions. The note gives an explicit correction for A = B = diag(1,−1,0,0). The correction vanishes near the shell, so the initial second-order junction conditions hold with zero shell displacement and tilt.

An areal-radius-weighted monopole identity fixes a spherical metric moment for first-order packets in the reconstruction gauge. Its negative sign for velocity packets is a statement in that gauge: the moment changes under first-order gauge transformations and is not yet a physical mass or energy observable.

Full second-order evolution through shell contact, quadratic interface sources, convergence of the higher-order formal series and nonlinear existence remain open. This result does not establish a higher-dimensional Big Bang origin. Technical note · Calculations and supporting material · Results and limits. This is a GitHub-only checkpoint; Zenodo 23335485 remains the latest archived spectral edition. No new Zenodo version or APS submission accompanies it.

Shell symmetry obstruction (13 October 2026): On the fixed regular Einstein–scalar two-interior shell, the completed nonlinear Noether identity retains both gravitational boundary terms, moving embeddings and the single common shell action. An independent ADM constraint calculation verifies a nonzero second-order obstruction for an explicit compact rotating quadrupole: it cannot be the tangent to a parameter-C² exact classical family satisfying full shell matching. The example can be chosen among smooth non-gauge linear solutions that reach the shell.

Within degree-two data with one fixed radial profile and reflection parity, all ten necessary Lorentz symmetry tests vanish exactly when the displacement and velocity matrices commute, [A,B]=0. Passing these tests does not establish second-order solvability or nonlinear existence. The result establishes no nonlinear growth, stability, Big Bang origin or APS readiness. Normal and optimized portable replay each check 81 positive algebra checks and 21 wrong-formula controls, counted separately.

Technical note · Proofs and portable replay · Results and limits. This is a GitHub-only research checkpoint. Zenodo 23335485 remains the latest archived spectral edition; no new Zenodo edition or APS submission accompanies this continuation.

Conditional evolution through shell contact (13 October 2026): On the fixed regular Einstein–scalar background, an explicit characteristic construction gives a smooth family from compact radial data in a sufficiently thin annulus, with a fixed finite set of angular harmonics of degree at least two. The family includes paired solutions that activate the shell after first arrival on a controlled finite slab. Both reflection parities satisfy the complete induced-field, Israel and scalar junctions without harmonic division, and nonzero states are not pure gauge. An explicit pair with the same reflection parity has a nonzero total physical symplectic pairing, conserved after contact with the odd moving-shell cut term included. The pairing is weakly nondegenerate on this smooth thin-annulus data image.

This conditional linearized result concerns a new collar-supported family. It does not continue the earlier near-cone flux pair to contact or establish a nonzero local shell flux, physical energy, unrestricted angular completion, nonlinear stability, cosmological evidence, external originality or APS readiness. Normal and optimized portable replay each check 84 positive exact identities and 14 wrong-formula controls, counted separately.

Technical note · Proofs and portable replay · Results and limits. This is a GitHub-only checkpoint. Zenodo 23335485 remains the latest archived spectral edition; no new Zenodo edition or APS submission accompanies this continuation.

Earlier local coupled cone evolution (13 October 2026): A regular wave equation now reconstructs smooth solutions of all linearized Einstein–scalar equations across the cone. Compact initial data give two non-gauge solutions with nonzero local symplectic flux. Both reflection parities satisfy the shell junctions before the waves reach the shell. The physical Cauchy symplectic form is nondegenerate on the image of smooth compact master data supported in a sufficiently small spatial ball.

Evolution to shell contact of that original near-cone flux pair remains open; the new collar family above has a separate after-contact proof. The full physical state space and nonlinear stability remain open. This antisymmetric symplectic flux is not a positive energy or reheating calculation, and a higher-dimensional Big Bang origin remains unestablished.

Technical note · Proofs and reproducible code · Results and limits. This is a GitHub research continuation. Zenodo 23335485 remains the preceding archived spectral edition; no new APS submission was made.

Current and physical-domain audit (13 October 2026): The Einstein–scalar action now supplies an explicit symplectic current, including unequal spectral values and the doubled-shell balance. Under stated asymptotic hypotheses for actual solutions, its null-cone limit is a scalar symplectic flux. Exact flat test-field examples show that fixed-time radial closure does not imply zero characteristic flux, and that radial and inertial energy completions can differ. The physical state space of the registered gravitational model remains open.

Technical note · Derivations and code · Results and limits. This continuation is published on GitHub; Zenodo 23335485 remains the preceding archived spectral edition. No new APS submission was made. A higher-dimensional Big Bang origin remains unestablished.

Conditional spectral result (13 October 2026): At every fixed 0 < delta ≤ 0.002, the entire scalar spectrum in both reflection parities is separated strictly from zero in the specified self-adjoint, finite-energy regular-cone realization. The proof gives a positive bound as a function of the background; it does not assign a numerical value or a uniform bound as delta approaches zero. An action calculation, including the moving-shell boundary terms, verifies the local kinetic coefficients of generic even and odd scalar modes. Exceptional harmonics are treated separately. The tensor sector has one even massless radial profile, and every nonzero tensor mass obeys m_T ≥ 3H_b/2.

This concerns the regular branch near phi = +1 of the same fixed action. The original hilltop shell remains a distinct unstable solution; no evolution connecting the two is proved. Global physical-state coverage, admissible diffeomorphisms and cone symplectic flux retain the declared assumptions. No nonlinear stability, reheating mechanism or higher-dimensional Big Bang origin is established. Normal and optimized portable Python replay each pass 103 exact checks and 30 wrong-formula controls, counted separately. Proof, assumptions and portable replay · Claim ledger and gap functional.

Verified publication (13 October 2026): Zenodo 23335485, version 2026.10.13-regular-shell-spectra, contains 35 files / 454,964,066 bytes. It preserves all 33 files from 23294979 and adds the new scientific capsule and overview. Complete public readback verified every filename, size and MD5/SHA256 checksum. Publication proof and complete inventory.