Manuscript scope ledger

These 73 overlapping scopes retain the accepted manuscript correspondence and explicit input boundaries. This public summary does not reproduce private review histories or count full theorems. Fresh release receipts check the exported proof source; they do not by themselves re-review prose fidelity.

Counts: 47 verified / 23 conditional / 3 excluded.

Proposition 1.1 — verified

Reference: p. 5; paper3/P:review-prop-2/mode-sign-core.

V2 scope: The semigroup and nonlinear PDE stability conclusions in the proposition are excluded; this row covers only modewise ordered-field algebra.

Theorem 1.1 — outside-agreed-scope

Reference: p. 6; paper3/T:local-bifurcation/part-1-pde.

V2 scope: Requires center-manifold construction, finite-regularity remainder control, implicit bifurcation, and PDE spectral stability.

Theorem 1.1 — verified-conditional

Reference: p. 6; paper3/T:local-bifurcation/part-1-scalar-core.

V2 scope: The finite scalar side, derivative-sign, and amplitude conclusions are proved from the explicit reduced equation and separate remainder interfaces. Center-manifold reduction, branch existence, and PDE stability lifting remain outside this row.

Supplied inputs:

• identification of η=χ₀−χ* and s=u_n₀ with the reduced scalar equation
• α>0 and β≠0
• fixed C≥0 and 0<r≤1

• uniform bound

  R(η,s)

  ≤C(

  s

  η²+

  s

  ³

  η

  +s⁴) on

  η

  ,

  s

  ≤r, or its stronger Eq. 2.43 version

• HasDerivAt (R η) (Rs η s) s throughout that box

• uniform derivative bound

  Rs(η,s)

  ≤C(η²+s²

  η

  +

  s

  ³), or the stronger matching version

Theorem 1.2(1) — verified-conditional

Reference: p. 8; paper3/T:local-bifurcation/part-2.

Case (iii) is covered for the corrected statement (negativity for beta >= 0, beta != beta_0); see finding v1-misprint-thm12-iii-beta-range.

Supplied inputs:

• identification of the reduced cubic coefficient with E:beta-n0
• quadratic graph coefficients E:a_0,i and E:a_k,i>0 and cubic assembly E:Gamma-n0-explicit
• fixed-mass coefficient identification E:beta-n0-minimal with a=b=a₀=0

Theorem 1.2(2) — verified-conditional

Reference: p. 8; paper3/T:local-bifurcation/part-3.

V2 scope: The all-parameter physical rational formula, degree/cancellation bounds, admissible mode corridors, roots, limits, and finite sign consequences are proved. Excluded tie bifurcations and PDE branch/stability conclusions are not classified.

Supplied inputs:

• identification of the non-minimal reduced cubic coefficient with E:beta-n0
• its quadratic graph and cubic assembly formulas E:a_0,i, E:a_k,i>0, and E:Gamma-n0-explicit

Remark 1.1(1) — verified-conditional

Reference: p. 7; paper3/R:local-bifurcation-minimal.

V2 scope: scalar zero exclusion, sign, local branches and stronger value/derivative remainder estimates from the explicit reduced equation, C4 jets and exact trivial-branch linearization. The actual fixed-mass PDE reduction and PDE spectral stability remain inputs.

arXiv v2: Step 4’ of the proof of Theorem 1.1 (minimal case) no longer states that the same Taylor argument gives the derivative remainder estimate, as the October 20 source did; the remark is unchanged. The derivative bound remains an explicit conditional input here.

Supplied inputs:

• identification of the minimal-model scalar equation and its coefficients, including the fixed-mass reduction
• α>0 and β≠0
• fixed C≥0 and 0<r≤1

• uniform weak mixed value bound on

  η

  ,

  s

  ≤r

• differentiability in s and the uniform weak derivative bound for the full mapped scalar interface
• For the stronger v2 value and derivative bounds: C4 reduced-field JetHyp and exact TrivialBranchLinearization in the physical Fourier amplitude and original time coordinate.

Theorem 1.3 — outside-agreed-scope

Reference: p. 10; paper3/T:global-extension.

V2 scope: Requires unilateral global bifurcation, compactness, and PDE a priori theory.

Theorem 1.4 — verified

Reference: p. 11; paper3/T:spiky-solu-thm.

the whole frozen Theorem 1.4 is proved for given actual positive nonconstant classical stationary families: gradient blowup for every gamma>0 in both models, with prescribed mass only in the minimal model and actual PDE-derived mass in the logistic model; for logistic gamma<1+alpha the same subsequence has constant C1 signal limit, every conjugate-Lq source and signal derivative test, and 0<=c<=uStar^gamma. Actual maxima and all analytic compactness/extraction inputs are constructed, not assumed. Weak Lq/W2q is stated by EVERY conjugate-Lq integral test of actual classical source/derivative representatives on interval Lebesgue measure; no named Mathlib WeakSpace/bundled Sobolev equivalence is claimed.

Remark 1.3(2) — verified

Reference: p. 11; paper3/R:no-delta-concentration.

The exact quantitative local-mass bounds, uniform decay and atomless-limit/no-Dirac wrappers together prove the whole exclusion remark for all given stationary spike families. The remark does not assert such a family exists; the manuscript explicitly leaves nonminimal-family existence open. Promotion of this parent exclusion to verified is honest within that universal exclusion scope.

Proposition 1.2 — verified

Reference: p. 12; paper3/T:semidiscrete-threshold.

V2 scope: The discrete eigenvalue/threshold functions, changing-mode minima, physical identification, and convergence are proved together with the concrete scheme-to-modal bridge. No semidiscrete trajectory existence or nonlinear stability theorem is asserted.

Equation 1.24 (read with -Delta_h) — verified

Reference: p. 12; paper3/T:semidiscrete-threshold/discrete-stencil-spectrum.

Verified for the corrected statement: Equation 1.24 read with -Delta_h (eigenvalues 4 h^-2 sin^2(n pi/(2N)), n=0..N, of the reflected Neumann stencil).

Equation 6.1 — verified

Reference: p. 36; paper3/T:semidiscrete-threshold/discrete-linearization.

V2 scope: Directional derivatives of both literal schemes, the conservative all-node identity, fixed-mass composition, elliptic elimination, modal growth, and threshold minimum are proved. No ODE trajectory existence, nonlinear stability, or PDE semigroup conclusion is asserted.

Lemma 2.1, Equations 2.1-2.4 — verified

Reference: p. 13; paper3/L:trig-integrals.

V2 scope: Current scope is finite identities only; the former ell-one infinite-series lemma is a different statement.

Equation 2.16 — outside-agreed-scope

Reference: p. 15; paper3/E:center-manifold-eq.

V2 scope: The center graph is an analytic input to conditional finite-jet theorems.

Equation 2.8 — verified

Reference: p. 14; paper3/E:B0-Bu-Bv-eq.

V2 scope: The retained declarations are the derivative formulas of the mobility from which B0, Bu and Bv follow at (u, 1+v); the formulas are identical with the same positive-base domain. Their later infinite-mode use does not change the primitive derivative types.

Equation 2.9 — verified

Reference: p. 14; paper3/E:Buu-Buv-Bvv-eq.

V2 scope: The factorial-normalized Buu, Buv and Bvv formulas and positive-base hypotheses are unchanged.

Equation 2.10 — verified

Reference: p. 14; paper3/E:L1-L2-eq.

V2 scope: A-prime, A-second and A-third retain the same factorial convention (Taylor coefficients of the logistic source at u*).

Equation 2.11 — verified

Reference: p. 14; paper3/E:C-const.

V2 scope: The three signal-source Taylor coefficients are formula-identical and retain the same factorial normalization.

Equation 2.20 — verified

Reference: p. 16; paper3/E:Ck-restated.

actual nonlinear Neumann signal response: all three physical Ck coefficients are identified by a derived actual cubic response jet with quartic bound, preserving zero/positive mode normalization and manuscript factors2 and4. The first actual normalized modal derivative is derived, including the Mathlib-only actual_modal_derivative interface, without a projected balance, response differentiability, response Taylor remainder or abstract resolvent-response assumption. Positive base/equilibrium/mu/L and ordinary given actual solution-family regularity/PDE/Neumann data are explicit. Equation2.20(p16) is covered at this coefficient/actual-family scope; center-manifold or response-family existence is not claimed.

Equation 2.32 — verified-conditional

Reference: p. 19; paper3/E:Gamma2n0-explicit.

V2 scope: The displayed second-harmonic coefficient is identical. Current infinite convolution formulas reduce at quadratic center order to the same constant/second-mode support; no infinite-series theorem is claimed by this row.

Supplied inputs:

• audited primitive Taylor coefficients and quadratic elliptic signal jet
• positive equilibrium with signal equilibrium relation and nu,gamma>0
• L>0 and n0>0 with x=lambda_n0

Equation 2.33 — verified-conditional

Reference: p. 19; paper3/E:a_0,i.

All original quadratic coefficient extraction and logistic formula scope are retained; center-graph existence and actual invariance remain conditional. Separately, the minimal fixed-mass constant coefficient is now derived: an actual mean-zero weak-H2 perturbation family with the prescribed cubic phase-norm Peano expansion and eventual zero population mass has jet.a0=0. Positive length and mode remain explicit; no mass or critical-kernel hypothesis on H3 is needed. For a physical positive fixed-mean population, the mean-zero family is obtained by the existing homogeneous subtraction dictionary; the theorem does not claim that a positive population has zero mass. This is a conservation-based minimal consumer, not a proof of the logistic coefficient value, actual graph construction, or dynamical invariance. No primary promotion is inferred.

Supplied inputs:

• degree-two constant-mode invariance residual obtained from the center-graph equations
• zero mean of the quadratic graph profile for the separate fixed-mass consumer
• positive domain length and positive mode index when applying that consumer

Further accepted input reductions:

{
  "date": "2026-10-22",
  "date_basis": "UTC",
  "source_gate": "d0a898282bdb2e08b283eda564439fe36e1cb08e",
  "boundary": "Scoped actual complement, elliptic-family and conservation ingredients are supplied by the new exact types. Population family/jet construction, critical sensitivity/mode, actual operator/branch identification, global bifurcation and PDE stability remain. No parent status change."
}

Supporting ingredients and their precise inputs:

Equation 2.36b — verified-conditional

Reference: p. 20; paper3/E:a_k,i>0.

V2 scope: The bA’‘/2 term is mandatory. Any legacy wrapper omitting it is rejected.

Supplied inputs:

• vanishing degree-two center-graph invariance polynomial and projected stable-mode balances
• audited quadratic forcing and physical coefficient identification
• positive physical parameters and equilibrium relations
• unique strict critical mode, supplying nonzero stable growth rates, particularly sigma_(2n0)

Equation 2.42 — verified-conditional

Reference: p. 20; paper3/E:E-F-G-H.

All original finite E, F, G, Q, R and S algebra and prior H evidence are retained. The critical cubic H signal response now follows for an actual Neumann elliptic signal family from a genuine weak-H2 population cubic norm Peano expansion with prescribed constant/second-harmonic quadratic term and critical-orthogonal H3. Weighted integrability, projection algebra and negligible modal residual are derived; no projected cubic signal balance or signal response jet is assumed in this consumer. Positive length/mode, positive uStar and mu, SignalEquilibrium, exact representatives and eventual actual elliptic solutions remain explicit. The given a0/a2 and population jet are not derived from center-graph invariance. The conclusion is a critical Fourier-coordinate asymptotic at fixed physical parameters and two-sided amplitude nhds0, not a full signal-space jet, mixed parameter jet, or graph/branch/stability construction. Parent scope remains conditional and earlier histories are retained.

Supplied inputs:

• projected cubic elliptic signal balance
• weighted interval integrability of h3φₙ, (φh2)φₙ, and φ³φₙ
• zero critical projection of h3
• identification of φ and h2 with the critical cosine and quadratic graph profile giving Π(φh2)=a0+a2/2 and Π(φ³)=3/4
• physical signal-equilibrium relation and positive parameters for the scaled-to-physical identification

Further accepted input reductions:

{
  "date": "2026-10-22",
  "date_basis": "UTC",
  "source_gate": "d0a898282bdb2e08b283eda564439fe36e1cb08e",
  "boundary": "Scoped actual complement, elliptic-family and conservation ingredients are supplied by the new exact types. Population family/jet construction, critical sensitivity/mode, actual operator/branch identification, global bifurcation and PDE stability remain. No parent status change."
}

Supporting ingredients and their precise inputs:

Equation 2.41 — verified-conditional

Reference: p. 20; paper3/E:Gamma-n0-explicit.

V2 scope: Conditional on weighted interval integrability and the analytic center-graph condition that the critical projection of h3 is zero; all remaining cubic extraction/projection algebra is proved.

Supplied inputs:

• audited primitive and quadratic-jet coefficients and elliptic signal aggregates
• projected cubic elliptic balance with nonzero denominator
• weighted interval integrability of h3, phi*h2 and phi^3 against the critical mode
• zero critical projection of h3
• identification phi=cos(n0pix/L) and h2=a0+a2cos(2n0pix/L)
• positive physical equilibrium and L>0,n0>0

Equation 2.44 — verified

Reference: p. 21; paper3/E:alpha_n0.

V2 scope: The linear crossing coefficient and positivity hypotheses are unchanged.

Equation 2.45 — verified-conditional

Reference: p. 21; paper3/E:beta-n0.

V2 scope: The corrected finite cubic assembly is derived from the accepted quadratic and cubic projection components. It proves the coefficient identity, not center-manifold or PDE branch existence.

Supplied inputs:

• accepted quadratic center jet and cubic critical projection, including their analytic application inputs
• positive physical equilibrium satisfying logistic and signal equilibrium relations
• nonzero critical eigenvalue and valid stable-mode denominators when identifying the jet with the center graph

Equation 2.43 — verified-conditional

Reference: p. 20; paper3/E:center-ODE-final.

Supplied inputs:

• identification of the reduced field with Eq. 2.43
• α>0 and β≠0
• fixed C≥0 and 0<r≤1

• uniform bound

  R(η,s)

  ≤C(s²η²+

  s

  ³

  η

  +s⁴) on the local box

• HasDerivAt (R η) (Rs η s) s there

• separate uniform bound

  Rs(η,s)

  ≤C(

  s

  η²+s²

  η

  +

  s

  ³)

Equation 2.49 — verified-conditional

Reference: p. 22; paper3/E:beta-n0-minimal.

V2 scope: The fixed-mass coefficient remains -chi-star Gamma3 with a0=0 and no logistic channels. The constant signal correction is still retained.

Supplied inputs:

• fixed-mass phase-space identification of the population constant correction with zero
• a=b=0 and positive signal equilibrium
• audited fixed-mass quadratic jet and cubic projection, including their analytic application inputs and nonzero modal denominators

Equation 3.1 — verified-conditional

Reference: p. 23; paper3/E:beta-n0-pq.

V2 scope: The p(beta), q(beta), beta-independent Pi and coefficient decomposition are unchanged after movement to Section 3.

Supplied inputs:

• audited identification of the assembled cubic coefficient with the reduced-equation coefficient
• quadratic graph coefficient formulas and the model-appropriate constant mode
• positive equilibrium relations and strict critical-mode guards supplied in the surrounding manuscript

Equation 3.3 — verified

Reference: p. 23; paper3/E:A-B-C-simple.

V2 scope: The exact derivative formulas A=-q’‘(0)/2, B=bA’‘p’(0)-q’(0), and C at beta=0 are unchanged.

Equation 3.4 — verified

Reference: p. 23; paper3/E:A-closed-form.

V2 scope: The closed form and strict negative sign under a strict critical mode are unchanged for both model classes.

Equation 1.15 and Section 3.2 — verified

Reference: p. 8; paper3/E:beta-n0-eq1/general-rational.

V2 scope: The general physical numerator/denominator construction, positive denominator, degree-five bound, and balanced degree-four cancellation are proved; the normalized example is only a downstream specialization.

Equation 3.6 — verified

Reference: p. 26; paper3/E:n0-branch.

V2 scope: The unique minimizing mode cells and exactly two adjacent modes at tie lengths are unchanged.

Theorem 1.2(2) interval display — verified

Reference: p. 8; paper3/E:lambda-k.

V2 scope: The exact lambda corridor is unchanged but no longer has its own label; it appears in Theorem 1.2(2) and again in the proof near line 2564. Large-length convergence to sqrt(a alpha mu) is unchanged along admissible lengths.

Equation 3.7 — verified

Reference: p. 26; paper3/E:new-beta-eq1/normalized-rational-chain.

V2 scope: This is a downstream specialization and regression test, never a replacement for the all-parameter M4 theorem.

Equation 3.8 — verified-conditional

Reference: p. 27; paper3/E:new-beta-eq2.

V2 scope: The normalized large-domain quadratic, positive root beta_ld and sign split are unchanged.

Supplied inputs:

• the all-ones physical coefficient identification
• mode data with lengths tending to infinity and eventually strict critical modes
• β≥0 for the stated nonnegative-parameter sign classification
• the local bifurcation theorem converting a nonzero cubic-coefficient sign into bifurcation direction

Table 3.2 — verified-conditional

Reference: p. 27; paper3/Tb:polynomial-roots.

V2 scope: Exact rational isolating intervals, exhaustive admissible-root counts, rounding cells, physical signs, and finite mode comparisons are proved. Decimal displays are presentation corollaries.

Supplied inputs:

• identification of the normalized rational expression with the reduced cubic coefficient on λ>1/2
• interpretation through untied critical-mode intervals
• the local bifurcation theorem converting coefficient signs into super-critical or sub-critical bifurcation

Lemma 4.1 — verified

Reference: p. 28; paper3/L:spiky-solu-lm1.

to the COMPLETE frozen Lemma4.1(p28), L:spiky-solu-lm1, not merely its Holder proof step. Part(1), for every alpha,gamma,m>0, follows from StationaryMassIdentity.ClassicalProfile.interval_bounds: actual stationary population PDE and Neumann data derive mass_balance, then strict min/equilibrium/max and strict first/(1+alpha)-moments, with NO m>=1 restriction on this part. Part(2) for 0<gamma<=1+alpha and1<=m<=alpha+2 follows from ClassicalHolderNorm.logistic_uniform_norm_bound on literal physical classical solutions. It proves actual C2+theta norms bounded by C, with theta,C existential BEFORE all sensitivities and solutions; a,b,alpha,mu>0,nu>=0,beta>=0,abs(chi)<=X. The frozen nu>0 and0<=chi<=X,X>0 are contained. Ordinary positive nonconstant population/classical solution hypotheses remain; no bound, target regularity, coefficient-Holder estimate, extraction or Schauder conclusion is assumed. Supplied continuous second derivatives and closed-interval derivative identities are ordinary C2 classical data; their Holder regularity is DERIVED. The full parent can be promoted after required gates/registry binding, preserving the exact two-part frozen statement rather than conflating historical extra subrows.

Lemma 4.2 — verified

Reference: p. 29; paper3/L:minimal-spike-bounds.

to the COMPLETE frozen Lemma4.2(p29), L:minimal-spike-bounds. Standing1<=m<=2,0<gamma<=1,a=b=0,positive prescribed-mass solutions. Part(1) is exactly IntervalMoments.minimal_interval_bounds for continuous positive nonconstant u with actual model mass int u=LuStar: strict min/mean/max and strict quadratic moment. There is no missing PDE conclusion in that mass hypothesis; it is literally the model assumption. Part(2) follows from ClassicalHolderNorm.minimal_uniform_norm_bound: theta,C are chosen BEFORE every actual physical solution; all actual C2+theta norms bounded, including m=2. The theorem even covers constant positive families and gamma=0, stronger than the frozen nonconstant/gamma>0 statement. alpha=0 in the minimal physical profile is harmless because a=b=0 removes the reaction term; no alpha dependence enters C. The positive constant vStar need not be constructed: neither assertion uses it except the ordinary specified positive mass uStar. No independent flux, gradient, moment estimate or target regularity assumption remains. The full parent can be promoted after required gates/registry binding, while historical extra support subrows retain their own scope.

Lemma 4.1 and Lemma 4.2, bounded-sensitivity proof context — verified

Reference: p. 28; paper3/L:bounded-sensitivity-compactness.

actual bounded-sensitivity stationary-family compactness in the genuine all-weak-H2 Neumann phase norm. Both physical logistic and fixed-mass minimal families yield strict parameter/phase norm-convergent subsequences to original classical PDE profiles. The all-positive logistic extension derives the constant case from actual population mass balance, removing the earlier all-nonconstant family restriction.

Lemma 5.1 — verified

Reference: p. 34; paper3/L:minimal-spike-limit.

actual positive classical stationary solution families and the manuscript uniform Lr bound yield one strict subsequence with constant source c>=0, uniform C1 signal limit, and EVERY conjugate-Lq test for the source and signal derivatives of orders zero, one and two. Population first integral, mass/source bounds and denominator bound are derived from actual PDE/Neumann data; no weak or extraction premises remain. Weak Lq/W2q is represented explicitly by all conjugate integral tests of classical derivative representatives, not a claimed Mathlib WeakSpace/bundled Sobolev equivalence. Weak Lq/W2q is stated by EVERY conjugate-Lq integral test of actual classical source/derivative representatives on interval Lebesgue measure; no named Mathlib WeakSpace/bundled Sobolev equivalence is claimed.

Equation 6.2 — verified

Reference: p. 36; paper3/E:main-DIS-conservative.

V2 scope: The literal finite face-flux divergence telescopes under trapezoidal weights, and its directional derivative is connected to the reflected fixed-mass modal system. Numerical trajectory existence and differentiability are not asserted.

1.2 — verified

Reference: p. 8; paper3/new-minimal-rational-degree-three.

Minimal-model degree-three numerator/denominator clause; existing non-minimal bindings do not certify it.

Lemma 2.1, Equations 2.1-2.4 — verified

Reference: p. 13; paper3/infinite-fourier-convolution.

ell1 summability, exchange of sums/integrals, and convolution multiplicities; finite identities are ingredients only.

2.43 — verified-conditional

Reference: p. 20; paper3/current-remainder-consumer.

Supplied inputs:

• identification of the application remainder and scalar coefficients
• α>0 and β≠0
• fixed C≥0 and 0<r≤1
• either stated uniform value-bound interface on the local box
• actual differentiability in s
• the corresponding independently supplied uniform derivative bound

Theorem 1.3(1) — verified-conditional

Reference: p. 10; paper3/relative-component-topology.

All original parent scope and earlier evidence are retained. The actual cosine Neumann jet and its positive-mode coefficient=1 are now constructed in the genuine weak-H2 phase. The literal Fourier kernel is a closed complement of its span; every phase point splits continuously and uniquely into its physical amplitude and orthogonal remainder. The canonical remainder supplies the profile expansion without a separate remainder-existence or orthogonality assumption. The minimal fixed-mean population has the unique affine decomposition with mean-zero orthogonal remainder. This is the actual weak-H2 complement, not fractional-domain X_theta, physical operator range, stable spectral identity, center-manifold or branch construction. A curve still needs actual amplitude=t; that fact is not implied for arbitrary curves. Earlier curve topology retains half-curve continuity, stationary membership and open-neighborhood exhaustion. Critical sensitivity/mode identity, curve/chart construction, manuscript weak/operator/classical identification, global alternatives or noncompactness of the same component, and PDE stability remain explicit missing or supplied inputs. No full parent promotion or extension of the ambient m-range is inferred.

Supplied inputs:

• the actual admissible subtype V and its topology
• the solution closure
• base membership from the local branch
• relative openness of the removed punctured arc from local uniqueness
• connectedness and solution-closure containment of the retained arc with its base
• disjointness from the removed arc
• a selector on the retained arc
• not proved here: the unilateral alternatives and their bifurcation foundations

Further accepted input reductions:

{
  "date": "2026-10-22",
  "date_basis": "UTC",
  "source_gate": "d0a898282bdb2e08b283eda564439fe36e1cb08e",
  "boundary": "Scoped actual complement, elliptic-family and conservation ingredients are supplied by the new exact types. Population family/jet construction, critical sensitivity/mode, actual operator/branch identification, global bifurcation and PDE stability remain. No parent status change."
}

Supporting ingredients and their precise inputs:

Theorem 1.3(3) — verified

Reference: p. 10; paper3/tail-mass-dirac-consumer.

to the scoped finite-density-measure implication, now with its actual-PDE tail producer. PhysicalEndpointConcentration constructs the genuine finite measure q.u(x)dx on the closed interval from ordinary continuous nonnegative classical solutions and proves full finite-measure weak convergence to L*uStar times the endpoint Dirac. No finite measure, common-mass identification, sine bound, pointwise decay, tail bound, neighborhood containment or density/test integral identification is assumed. Minimal population and signal equations, actual Neumann data, positive population, nonnegative signal, fixed prescribed mass, chi->infinity and ordinary strict decreasing/increasing orientation are explicit. Parameter range m>=1,beta>=0,mu,nu>0,0<gamma<=1 contains the entire frozen1<=m<=2 range. Scoped consumer/tail production can be certified after gates; deriving that orientation from branch membership is not this row and remains separate.

Lemma 4.1(1), Equation 4.1 — verified

Reference: p. 28; paper3/L:spiky-solu-lm1/part-1.

ClassicalProfile.interval_bounds proves all four strict extremum/moment conclusions for every given positive nonconstant classical stationary profile. The literal flux, equation and both u/v Neumann data derive mass balance; no independent mass identity remains. Verified within this entire classical-solution statement, without asserting existence.

Supporting ingredients and their precise inputs:

Lemma 4.1(2), Equation 4.6 — verified

Reference: p. 28; paper3/L:spiky-solu-lm1/part-2-moment-step.

The new direct ClassicalProfile.power_moment supplies the exact gamma-moment bound for0<gamma<=1+alpha from the actual stationary equation through interval_bounds; the higher moment is no longer independently assumed. Its current kernel type was printed and read.

Equation 4.14 (proof of Lemma 4.2(2)) — verified

Reference: p. 30; paper3/L:minimal-spike-bounds/part-1.

ClassicalMinimalEstimates.flux_identity derives the literal flux identity from the actual a=b=0 stationary population equation and Neumann data. No first-integral assumption remains; classical derivatives and the original equation are standing solution data.

Lemma 4.2(1) — verified

Reference: p. 29; paper3/L:minimal-spike-bounds/part-2.

Actual prescribed-mass positive nonconstant classical solution supplies both strict extrema and strict second moment; prescribed mass is the ordinary model hypothesis.

Proof of Lemma 4.2(2) (applies Equation 4.6, p. 28, TeX line 2687) — verified

Reference: p. 29; paper3/L:minimal-spike-bounds/jensen-step.

ClassicalMinimalEstimates.power_moment proves the exact prescribed-mass concave moment estimate for all0<=r<=1. Prescribed mass is an explicit model hypothesis of the manuscript, not an unresolved analytic inference. In particular r=m-1 covers both endpoints m=1 and m=2.

Supporting ingredients and their precise inputs:

Proof of Lemma 4.2(2), m=1 case of the variation-of-constants display — verified

Reference: p. 29; paper3/L:minimal-spike-bounds/part-5-exp-representation.

Literal minimal PDE derives flux at m=1; integrating-factor continuity follows from actual signal regularity and1+v>0. The exact exponential representation is concluded.

Remark 1.3(2) — verified

Reference: p. 11; paper3/R:no-delta-concentration/local-mass.

ClassicalProfile.window_mass_bounds states both the exact Holder-type displayed inequality and its strict equilibrium envelope; mass balance is derived. window_mass_decay_of_classical_profiles proves uniform decay in the family index and center. This covers the quantitative local-mass row end to end for given profiles.

Supporting ingredients and their precise inputs:

Theorem 1.3(3) — verified

Reference: p. 10; paper3/T:global-extension/part-3-concentration.

to the concentration clause under its explicitly separate ordinary monotonicity input. For actual positive strictly decreasing/increasing fixed-mass minimal classical stationary families, m>=1 and0<gamma<=1, the code DERIVES the sine-test obstruction, sensitivity-uniform actual signal envelope, every interior u_n(delta)->0, tail/head decay, every continuous-test integral limit, actual finite-density-measure Dirac convergence, and endpoint population value->infinity. Endpoint values are the genuine maxima by monotonicity, so the frozen max->infinity conclusion follows. The right endpoint is proved directly using the positive-gradient ODE and a right interior interval, then the exact right-endpoint measure consumer; no reflected solution or reflection-PDE premise is assumed. Strict orientation is NOT inferred from membership in C1+/- by these four modules; full Theorem1.3(3), branch membership/orientation, noncompactness and alternatives(ii)/(iii) remain outside this scoped row. Thus do not promote the whole global-extension parent solely from this review.

1.1 — verified-conditional

Reference: p. 6; paper3/T:local-bifurcation/branch-curves.

Supplied inputs:

• identification of the scalar reduced field and its coefficients
• α>0 and β≠0
• a common box with C≥0 and 0<r≤1
• the stated value remainder bound
• HasDerivAt identifying Rs
• the stated derivative remainder bound
• continuity in η for every fixed small s. Center-manifold construction and lifting to PDE curves remain excluded

Supporting ingredients and their precise inputs:

Lemma 2.1 and Equation 2.17 — verified

Reference: p. 13; paper3/L:trig-integrals/phase-space-bridge.

Genuine weak-H2 functions and their Neumann trace data produce the absolutely continuous NeumannW21 representatives; all actual normalized cosine coefficients are unchanged by a.e. equality. The four exact function/integral identities are proved for the original representatives with original coefficients. Absolute summability and reconstruction are derived. Pointwise reconstruction of the original function on the closed interval is proved when it is continuous, exactly as in the old classical-function scope; for arbitrary raw H2 representatives reconstruction correctly targets the constructed continuous representative. The prior external one-dimensional Sobolev/trace identification is discharged, without asserting any Banach-space topology or phase-E map.

Proof of Lemma 4.2(2), Equations 4.7-4.8 — verified

Reference: p. 29; paper3/L:minimal-spike-bounds/part-3-signal.

Actual second PDE and both Neumann conditions supply the exact signal sup-plus-L1 component bound and displayed constant; no separately assumed signal estimate.

Supporting ingredients and their precise inputs:

Proof of Lemma 4.2(2): the uniform bounds on u and u_x from the flux identity (4.14), before the elliptic step to (4.13) (alternative estimate for 1<=m<2) — verified

Reference: p. 30; paper3/L:minimal-spike-bounds/part-4-sup.

Literal minimal PDE derives flux; actual signal PDE derives gradient; M is the actual gamma moment. The complete original sharp power envelope and derivative estimate hold for0<m<2.

Proof of Lemma 4.2(2), m=1 case — verified

Reference: p. 29; paper3/L:minimal-spike-bounds/part-5-sup-bound.

Actual PDEs and prescribed mass derive flux, signal-gradient and integrating-factor regularity; the exact historical exponential supremum envelope follows for m=1.

Lemma 4.1(2), Equation 4.11 (alternative route for 1<=m<(alpha+2)/2) — verified

Reference: p. 29; paper3/L:spiky-solu-lm1/part-2-C1-chain.

actual given classical stationary-system C1 consumers. c1_bound discharges mass/moment inputs over the old energy range0<m<(alpha+2)/2; c1_gronwall_bound independently covers the current full range1<=m<=alpha+2, including both endpoints. Supremum and Gronwall conclusions are derived from actual equations and moments, not assumptions. The higher Holder/Schauder C^(2+theta) conclusion remains excluded.

Supporting ingredients and their precise inputs:

Proof of Lemma 5.1, W^{2,q} step with r=1+alpha — verified

Reference: p. 34; paper3/L:spiky-solu-lm1/part-3-W2q.

retained auxiliary quantitative W2q deduction for actual given classical profiles: actual testing, Holder, Minkowski and population moment premises are all discharged in ClassicalLogisticEstimates.w2q_bound. The old part3 name is historical; frozen v2 Lemma4.1(p28) has no part3. Do not present this auxiliary as a current numbered part3 theorem.

Supporting ingredients and their precise inputs:

Remark 1.3(2) — verified

Reference: p. 11; paper3/R:no-delta-concentration/no-atom-limit.

Every given weak finite-measure limit of an arbitrary family of actual positive nonconstant classical stationary profiles is atomless on the closed interval, including endpoints. The explicit weak convergence premise is the ordinary quantifier of an all-weak-limits claim, not an unproved compactness theorem. The subsequence wrapper also directly rules out any positive multiple of a Dirac measure.

Lemma 5.1 — verified

Reference: p. 34; paper3/L:minimal-spike-limit/deductions.

actual positive classical stationary solution families and the manuscript uniform Lr bound yield one strict subsequence with constant source c>=0, uniform C1 signal limit, and EVERY conjugate-Lq test for the source and signal derivatives of orders zero, one and two. Population first integral, mass/source bounds and denominator bound are derived from actual PDE/Neumann data; no weak or extraction premises remain. Weak Lq/W2q is represented explicitly by all conjugate integral tests of classical derivative representatives, not a claimed Mathlib WeakSpace/bundled Sobolev equivalence. Weak Lq/W2q is stated by EVERY conjugate-Lq integral test of actual classical source/derivative representatives on interval Lebesgue measure; no named Mathlib WeakSpace/bundled Sobolev equivalence is claimed.

Supporting ingredients and their precise inputs:

1.4 — verified

Reference: p. 11; paper3/T:spiky-solu-thm/deductions.

the whole frozen Theorem 1.4 is proved for given actual positive nonconstant classical stationary families: gradient blowup for every gamma>0 in both models, with prescribed mass only in the minimal model and actual PDE-derived mass in the logistic model; for logistic gamma<1+alpha the same subsequence has constant C1 signal limit, every conjugate-Lq source and signal derivative test, and 0<=c<=uStar^gamma. Actual maxima and all analytic compactness/extraction inputs are constructed, not assumed. Weak Lq/W2q is stated by EVERY conjugate-Lq integral test of actual classical source/derivative representatives on interval Lebesgue measure; no named Mathlib WeakSpace/bundled Sobolev equivalence is claimed.

Theorem 1.3(1)-(3) — verified-conditional

Reference: p. 10; paper3/T:global-extension/parts-1-3-logic.

All original parent scope and earlier evidence are retained. The actual cosine Neumann jet and its positive-mode coefficient=1 are now constructed in the genuine weak-H2 phase. The literal Fourier kernel is a closed complement of its span; every phase point splits continuously and uniquely into its physical amplitude and orthogonal remainder. The canonical remainder supplies the profile expansion without a separate remainder-existence or orthogonality assumption. The minimal fixed-mean population has the unique affine decomposition with mean-zero orthogonal remainder. This is the actual weak-H2 complement, not fractional-domain X_theta, physical operator range, stable spectral identity, center-manifold or branch construction. A curve still needs actual amplitude=t; that fact is not implied for arbitrary curves. Earlier actual PDE closure/compactness and selected-component consequences retain all physical ranges and classical-solution hypotheses. Critical sensitivity/mode identity, curve/chart construction, manuscript weak/operator/classical identification, global alternatives or noncompactness of the same component, and PDE stability remain explicit missing or supplied inputs. No full parent promotion or extension of the ambient m-range is inferred.

Supplied inputs:

• Shi–Wang unilateral alternatives for the actual selected branches
• identification of the ambient space, solution set S and admissible set V
• relative closedness and containment in V and closure S
• ambient admissibility closure S contained in V
• precompactness of every bounded-sensitivity slice from the applicable parameter-dependent PDE estimates and compact embedding
• connectedness and base membership of the selected component
• relative clopenness of the correctly oriented monotone locus
• the continuous additive profile representation and constant equilibrium
• orthogonality of W to the first cosine mode
• local simple-eigenvalue expansions when applying the higher-mode perturbation lemma

Further accepted input reductions:

{
  "date": "2026-10-22",
  "date_basis": "UTC",
  "source_gate": "d0a898282bdb2e08b283eda564439fe36e1cb08e",
  "boundary": "Scoped actual complement, elliptic-family and conservation ingredients are supplied by the new exact types. Population family/jet construction, critical sensitivity/mode, actual operator/branch identification, global bifurcation and PDE stability remain. No parent status change."
}

Supporting ingredients and their precise inputs:

2.43 — verified-conditional

Reference: p. 20; paper3/E:center-ODE-final/remainder-from-jet.

Supplied inputs:

• C⁴ regularity of the actual reduced remainder on an open neighborhood
• R(η,0)=0 on a small parameter interval
• the five specified point-jet cancellations from the Fourier computation
• identification of the reduced field with αηs−βs³+R and α>0, β≠0
• axis cancellation R_s(η,0)=0 additionally for the Eq. 2.43 bound
• center-manifold existence and transfer of scalar conclusions to the PDE are outside this row

Supporting ingredients and their precise inputs:

Theorem 1.3(2)-(3), ambient-limit proof paragraphs — verified-conditional

Reference: p. 10; paper3/L:ambient-closure/deductions.

All original parent scope and earlier evidence are retained. The actual cosine Neumann jet and its positive-mode coefficient=1 are now constructed in the genuine weak-H2 phase. The literal Fourier kernel is a closed complement of its span; every phase point splits continuously and uniquely into its physical amplitude and orthogonal remainder. The canonical remainder supplies the profile expansion without a separate remainder-existence or orthogonality assumption. The minimal fixed-mean population has the unique affine decomposition with mean-zero orthogonal remainder. This is the actual weak-H2 complement, not fractional-domain X_theta, physical operator range, stable spectral identity, center-manifold or branch construction. A curve still needs actual amplitude=t; that fact is not implied for arbitrary curves. Earlier actual PDE closure/compactness and selected-component consequences retain all physical ranges and classical-solution hypotheses. Critical sensitivity/mode identity, curve/chart construction, manuscript weak/operator/classical identification, global alternatives or noncompactness of the same component, and PDE stability remain explicit missing or supplied inputs. No full parent promotion or extension of the ambient m-range is inferred.

Supplied inputs:

• identification of the manuscript F=0 weak-H2 stationary solution sets and actual branch points with the classical solution sets used by the closure wrappers, at the correct constrained parameters
• actual component relative closedness, inclusion in the admissible set and in closure S, and set-level compactness of bounded-sensitivity slices required by the retained global consumers
• the part (2) noncompactness hypothesis; the part (3) global alternatives, distinguished constant/complement and orthogonality, and strict orientation/local branch foundations; these are not constructed by the analytic closure proof

Further accepted input reductions:

{
  "date": "2026-10-22",
  "date_basis": "UTC",
  "source_gate": "d0a898282bdb2e08b283eda564439fe36e1cb08e",
  "boundary": "Scoped actual complement, elliptic-family and conservation ingredients are supplied by the new exact types. Population family/jet construction, critical sensitivity/mode, actual operator/branch identification, global bifurcation and PDE stability remain. No parent status change."
}

Supporting ingredients and their precise inputs:

Theorem 1.3(3), Equation 4.18, full range 1 <= m <= 2 — verified-conditional

Reference: p. 32; paper3/T:global-extension/monotone-locus.

Actual weak-H2 phase, canonical continuous profile map and the physical signal-continuity/ODE machinery are now constructed internally, so the former generic phase-realization assumptions are removed. The clopen/strict-orientation consumers still require hConst (local constant-limit bifurcation representation) and hStar (local half-branch parametrization), together with component/branch foundations. These are genuine missing local/global bifurcation inputs, not mere constant-state identification. Keep this row conditional.

Supplied inputs:

• ordinary given-component membership and actual classical-PDE profile correspondence; global branch existence/continuation and connected-component foundations remain outside this consumer
• hConst: local constant-limit bifurcation representation; hStar: local half-branch parametrization at the distinguished point; the full weak-H2 population embedding, actual ODE identification, signal C2 topology and pointwise continuity are derived

Supporting ingredients and their precise inputs:

Lemma 2.1 and Equation 2.17 — verified

Reference: p. 13; paper3/L:trig-integrals/phase-space-H2.

Every w in X = {w in H^2(0,L) : w_x(0)=w_x(L)=0}, with H^2 defined by L^2 weak derivatives on (0,L) and the Neumann condition on the continuous representative of w_x, has a representative satisfying NeumannW21; consequently its Neumann cosine coefficients are absolutely summable and its cosine series converges to that representative on [0,L], so Lemma 2.1 applies to elements of X. Includes the 1-D du Bois-Reymond lemma and the W^{1,1} absolutely continuous representative.

2.43 — verified-conditional

Reference: p. 20; paper3/E:center-ODE-final/axis-condition.

Supplied inputs:

• identification of p with a positive physical equilibrium in the logistic or fixed-mass minimal model
• identification x=λ_n₀>0 and of the modal threshold with χ* at the selected critical mode
• existence of the reduced field in the manuscript’s Fourier amplitude and original time coordinate
• C⁴ regularity on an open neighborhood containing a closed box with 0<r≤1
• R(η,0)=0 throughout that box
• the five origin jets R_s=R_ηs=R_ss=R_ηss=R_sss=0, including identification of β_n₀ with the cubic coefficient

• TrivialBranchLinearization throughout

  η

  ≤r

• β_n₀≠0 for the branch conclusions

Equation 2.22 — verified

Reference: p. 17; paper3/E:ode-eq0/projected-remainder.

Supporting ingredients and their precise inputs:

Equations 2.18-2.19 — verified

Reference: p. 16; paper3/E:vk-eq/modal-remainders.

actual nonlinear signal-profile modal residual: exact zero/positive normalization, actual source Taylor estimate, original elliptic projected balance, summability, and full physical-signal equilibrium wrapper are derived. Ordinary given-profile regularity/PDE/small-amplitude inputs remain; no center-manifold or full Fourier-convolution existence is claimed.