Milestone: No exact fixed-point-existence OBJECTIVES.md milestone ID covers this (same status as #167, which this issue is scoped underneath). Informs #167 directly — this is #167's highest-value next target per the linked Exploration's recommendation, not a new direction.
Context. programs/fixed-point-existence/explorations/2026-07-17-M3-hyperboloidal-repair-attempt.md attempted a repair of Gap M3 (the response kernel's Lorentzian-hyperbolic, causal-past-supported structure defeats the existing proof's single-Cauchy-surface spatial-resolvent strategy — full diagnosis in explorations/2026-06-16-FPE4-M3-structural-obstruction.md) using hyperboloidal slicing on asymptotically flat, radiating backgrounds. It closed at Sketch level for the massless sector contingent on an unclosed gap, named G1: the energy-estimate mechanism (§1.4 of the Exploration) needs $\Lambda_{\sup}\equiv\sup_\tau\Lambda(\tau)<\infty$, where $\Lambda(\tau)$ bounds the Grönwall growth rate of the hyperboloidal energy $\mathcal E(\tau)$. This is established (Ma & Huang 2017, arXiv:1711.00498, verified) for the homogeneous wave equation with external source data $J$. It is not shown for the actual case this program needs: $J$ built from $\delta g$ through the coincidence-limit Hadamard point-splitting operator — i.e., the non-local memory kernel M3 originally diagnosed as the whole obstruction, now appearing as the source term feeding this energy estimate.
An independent adversarial stress-test of the construction (also in the linked Exploration) confirmed the mechanism's abstract soundness, found and corrected a real (non-fatal) error in the displayed energy identity — the corrected identity is $\nabla^\mu T_{\mu\nu} = J\nabla_\nu u + \tfrac12\xi u^2\nabla_\nu R$, with the extra term vanishing in the vacuum-near-$\mathcal I^+$ region but not in the bulk, which is exactly where G1 bites — and ranked G1 as the single biggest remaining risk to the whole construction.
Explicit dependency to scope around, not solve here: the Exploration also found (Gap G7) that the Hadamard-renormalized stress tensor's state is not foliation/frame-independent on a genuinely radiating background (BMS-supertranslation vacuum ambiguity). G1's kernel is computed relative to a fixed state. This issue should fix one reasonable state choice (e.g., the standard asymptotic Bondi-frame vacuum, stated explicitly as an assumption) and compute $\Lambda_{\sup}$ relative to that choice — it should not attempt to simultaneously resolve G7's frame-independence question. If closing G1 turns out to depend sensitively on the state choice, that dependency is itself a valuable finding to report, not something to paper over.
Deliverable. Either:
(a) show $\Lambda_{\sup}<\infty$ for the composed estimate — i.e., bound the Grönwall growth contribution from the coincidence-limit-derived source term, using the kernel's known causal-past ($J^-(x)$) support structure and whatever fall-off/decay properties of the retarded Green function are needed (state explicitly what's assumed about propagator fall-off near $\mathcal I^+$, and whether it's established elsewhere in this repo's citation base or needs a fresh citation at exploratory tier); or
(b) a documented obstruction showing $\Lambda(\tau)$ can genuinely diverge (or fails to have a uniform bound) for this source class — e.g. via an explicit construction where causal-past contributions accumulate unboundedly on the hyperboloidal foliation, or a resonance between the retarded propagator's fall-off and the hyperboloidal weight functions. A negative result here is a complete, valuable deliverable per this program's demotion-as-normal-maintenance convention, and would let the massless-sector Sketch claim in the linked Exploration be reassessed (not necessarily withdrawn — see the Exploration's own note that the M3 repair is a family of foliation-specific theorems; a G1 failure specific to hyperboloidal slicing wouldn't necessarily kill the general strategy).
Acceptance criteria. Self-checks per METHODOLOGY (dimensional analysis on any new bound introduced; consistency with the already-corrected §1.4 energy identity in the linked Exploration — do not re-derive it from scratch, cite it; order-of-magnitude sanity on $\Lambda_{\sup}$ against the existing κ-chain scales already discussed under the base paper's M1/M4 gaps, without attempting to resolve those separately-tracked issues). State explicitly which state-choice assumption (per the G7 scoping note above) the bound is computed relative to. Any new citations at exploratory tier unless independently verified per METHODOLOGY's strict tier — this session has twice caught fabricated author attributions on real, correctly-titled hyperboloidal/QFT-in-curved-spacetime papers; verify by direct primary-source fetch, not search-snippet recall.
Declared relations: informs #167 (this is #167's own linked Exploration's top-ranked next target). informs fixed-point-existence's Gap M3 track generally (a documented outcome either way sharpens the "M3 repair is a family of foliation-specific theorems" structural finding). Does not touch Gap G7 (state-selection) — scoped around it per the dependency note above.
Milestone: No exact
fixed-point-existenceOBJECTIVES.md milestone ID covers this (same status as #167, which this issue is scoped underneath).Informs#167 directly — this is #167's highest-value next target per the linked Exploration's recommendation, not a new direction.Context.$\Lambda_{\sup}\equiv\sup_\tau\Lambda(\tau)<\infty$ , where $\Lambda(\tau)$ bounds the Grönwall growth rate of the hyperboloidal energy $\mathcal E(\tau)$ . This is established (Ma & Huang 2017, arXiv:1711.00498, verified) for the homogeneous wave equation with external source data $J$ . It is not shown for the actual case this program needs: $J$ built from $\delta g$ through the coincidence-limit Hadamard point-splitting operator — i.e., the non-local memory kernel M3 originally diagnosed as the whole obstruction, now appearing as the source term feeding this energy estimate.
programs/fixed-point-existence/explorations/2026-07-17-M3-hyperboloidal-repair-attempt.mdattempted a repair of Gap M3 (the response kernel's Lorentzian-hyperbolic, causal-past-supported structure defeats the existing proof's single-Cauchy-surface spatial-resolvent strategy — full diagnosis inexplorations/2026-06-16-FPE4-M3-structural-obstruction.md) using hyperboloidal slicing on asymptotically flat, radiating backgrounds. It closed at Sketch level for the massless sector contingent on an unclosed gap, named G1: the energy-estimate mechanism (§1.4 of the Exploration) needsAn independent adversarial stress-test of the construction (also in the linked Exploration) confirmed the mechanism's abstract soundness, found and corrected a real (non-fatal) error in the displayed energy identity — the corrected identity is$\nabla^\mu T_{\mu\nu} = J\nabla_\nu u + \tfrac12\xi u^2\nabla_\nu R$ , with the extra term vanishing in the vacuum-near-$\mathcal I^+$ region but not in the bulk, which is exactly where G1 bites — and ranked G1 as the single biggest remaining risk to the whole construction.
Explicit dependency to scope around, not solve here: the Exploration also found (Gap G7) that the Hadamard-renormalized stress tensor's state is not foliation/frame-independent on a genuinely radiating background (BMS-supertranslation vacuum ambiguity). G1's kernel is computed relative to a fixed state. This issue should fix one reasonable state choice (e.g., the standard asymptotic Bondi-frame vacuum, stated explicitly as an assumption) and compute$\Lambda_{\sup}$ relative to that choice — it should not attempt to simultaneously resolve G7's frame-independence question. If closing G1 turns out to depend sensitively on the state choice, that dependency is itself a valuable finding to report, not something to paper over.
Deliverable. Either:$\Lambda_{\sup}<\infty$ for the composed estimate — i.e., bound the Grönwall growth contribution from the coincidence-limit-derived source term, using the kernel's known causal-past ($J^-(x)$) support structure and whatever fall-off/decay properties of the retarded Green function are needed (state explicitly what's assumed about propagator fall-off near $\mathcal I^+$ , and whether it's established elsewhere in this repo's citation base or needs a fresh citation at exploratory tier); or$\Lambda(\tau)$ can genuinely diverge (or fails to have a uniform bound) for this source class — e.g. via an explicit construction where causal-past contributions accumulate unboundedly on the hyperboloidal foliation, or a resonance between the retarded propagator's fall-off and the hyperboloidal weight functions. A negative result here is a complete, valuable deliverable per this program's demotion-as-normal-maintenance convention, and would let the massless-sector Sketch claim in the linked Exploration be reassessed (not necessarily withdrawn — see the Exploration's own note that the M3 repair is a family of foliation-specific theorems; a G1 failure specific to hyperboloidal slicing wouldn't necessarily kill the general strategy).
(a) show
(b) a documented obstruction showing
Acceptance criteria. Self-checks per METHODOLOGY (dimensional analysis on any new bound introduced; consistency with the already-corrected §1.4 energy identity in the linked Exploration — do not re-derive it from scratch, cite it; order-of-magnitude sanity on$\Lambda_{\sup}$ against the existing κ-chain scales already discussed under the base paper's M1/M4 gaps, without attempting to resolve those separately-tracked issues). State explicitly which state-choice assumption (per the G7 scoping note above) the bound is computed relative to. Any new citations at exploratory tier unless independently verified per METHODOLOGY's strict tier — this session has twice caught fabricated author attributions on real, correctly-titled hyperboloidal/QFT-in-curved-spacetime papers; verify by direct primary-source fetch, not search-snippet recall.
Declared relations:
informs#167 (this is #167's own linked Exploration's top-ranked next target).informsfixed-point-existence's Gap M3 track generally (a documented outcome either way sharpens the "M3 repair is a family of foliation-specific theorems" structural finding). Does not touch Gap G7 (state-selection) — scoped around it per the dependency note above.