FPE-7: Well-definedness of the self-consistency map F (gaps M8, M9)

[willregelmann]

Milestone: FPE-7 (programs/fixed-point-existence/OBJECTIVES.md) — Well-definedness of F (M8, M9).

Paper location: programs/fixed-point-existence/index.tex, §"The self-consistency map" (\label{sec:map}, lines 88–110) — specifically the "State selection" paragraph (lines 101–108) and the "Single-valuedness assumption" paragraph (line 110) — and §"Open problems" (\label{sec:open}), the "Well-definedness of the map $\mathcal{F}$ (M8, M9)" item (lines 733–747).

Context. \mathcal{F} is defined (lines 92–99) as a three-step map on spacetime geometries: (1) construct the QFT on (M,g) and pick a Hadamard state |Ψ_g⟩; (2) compute the renormalized stress-energy; (3) solve the semiclassical Einstein equation for g'. Two structural gaps are already named in-text and remain open:

• (M8) Sections sec:banach–sec:schauder work with fields in H^s(Σ) on a single Cauchy surface, while \mathcal{F} is defined on spacetime geometries. The equivalence between a spacetime fixed point of \mathcal{F} and a fixed point of data on a chosen Σ is never constructed — the Hadamard condition is global, not a Σ-local statement — and the role of the chosen Σ must be shown to be pure gauge (i.e. independent of which Cauchy surface is used to pose the reduced problem).
• (M9) The single-valuedness of \mathcal{F} (invoked by both the Banach and Schauder arguments, lines 101–110) rests on the assertion that the Hadamard state on the corrected geometry is "the unique perturbation" of the background state. Hadamard states are not unique in general; the perturbation prescription that would make "the unique perturbation" precise is assumed in-text, not constructed. The text also flags that the "Established" tags on assumptions (A1)/(A2) are generous relative to what Hollands–Wald \cite{hollands_wald_wick} actually proves (algebraic local/analytic metric dependence of Wick polynomials, not an H^s → H^{s-2} continuity statement) — this attribution should be corrected precisely as part of the sweep, not merely re-asserted.

The paper already notes (line ~715, in the M1/M3/M4/M5 open-problems item) that M8 shares its root with Gap M3 (FPE-4's structural obstruction, closed via documented-obstruction Outcome B, PR #109, issue #103): "a genuine spatial reduction would simultaneously close M8's well-definedness gap." A worker on this issue should read that connection but is not required to re-litigate M3 — FPE-4 is closed.

informs-issue #140 (arXiv:2606.22767, "Why Hadamard states?", unverified beyond abstract) directly bears on the state-selection half of this milestone: it argues the standard motivations for requiring Hadamard states (Wick rotation, Minkowski vacuum, Wick-polynomial renormalizability) do not establish Hadamard as a necessary constraint on physically reasonable states. A worker here cannot simply appeal to "Hadamard states are the physically reasonable ones" without engaging this critique — either identify a surviving motivation the critique overlooks, adopt and justify an alternative selection criterion, or document the residual foundational uncertainty explicitly as part of the gap statement. If used at paper-grade, verify the reference per METHODOLOGY's strict citation tier first (author, title, venue, and that it actually argues what the abstract claims) — this session's context includes a documented instance of a librarian-flagged reference later needing correction.

Deliverable (mirrors the milestone's done-condition): either —

• (a) Resolve at Sketch level. Construct the equivalence between a spacetime fixed point of \mathcal{F} and a fixed point of Cauchy-data on a chosen Σ (closing M8), showing explicitly that the choice of Σ is pure gauge (or stating precisely the class of Σ for which this holds, if it is not fully general); and construct a precise perturbation prescription establishing single-valuedness of \mathcal{F} in the regimes the paper actually uses it (closing M9), or show the map is genuinely set-valued there and that Kakutani's theorem (already gestured at, line 108) applies with its hypotheses checked. Address the Hollands–Wald attribution correction (A1/A2) as part of this pass. Engage informs-issue [informs-issue] arXiv:2606.22767 — Philosophical challenge to Hadamard state foundations; informs FPE-7 state-selection prescription #140's critique for the state-selection half.
• (b) Documented obstruction. If either gap is structural (cannot be closed within assumptions A1–A6, analogous to how FPE-4 closed via documented obstruction rather than repair), state precisely which gap is structural and why, and what additional geometric or foundational structure well-posedness would require.

Acceptance criteria:

• Rigor labels: any closed step is Rigorous only if fully constructed with no remaining gap; a Sketch-level resolution must name every remaining step the argument would need (per METHODOLOGY's Sketch discipline). The corrected (A1)/(A2) attribution must state exactly what Hollands–Wald establishes vs. what this paper needs.
• Self-checks: dimensional analysis is not directly applicable (functional-analytic/foundational argument); limiting cases — any Σ-independence (gauge) claim for M8 should be checked against at least the cosmological case (§sec:starobinsky, where de Sitter symmetry already selects a preferred slicing) as a consistency check, not a counterexample to gauge-independence, and this must be reconciled explicitly if the general M8 claim is Σ-independent; consistency — the resolution must not silently re-open FPE-4 (M3, closed) or overstate what FPE-5 (Schauder, open, in-flight per issue FPE-5: Repair the Schauder proof (gaps M6, M7) #161) currently establishes about branch restriction/Kakutani.
• Citations: informs-issue [informs-issue] arXiv:2606.22767 — Philosophical challenge to Hadamard state foundations; informs FPE-7 state-selection prescription #140 must be verified per METHODOLOGY's strict tier before any paper-grade use; if it is used only to document residual uncertainty (not asserted as an established fact), it may remain flagged exploratory-tier per citation discipline, but that flag must be explicit in-text.

Declared relations:

• informs FPE-4 (closed, PR FPE-4: M3 is a structural obstruction; M1 reference-scale artifact resolved (Outcome B) #109, issue FPE-4: Repair the Banach contraction — close gaps M1, M3, M4, M5 in Lemma 1 and Theorem 2 #103) — M8 shares its structural root with the closed M3 gap; this issue does not reopen FPE-4.
• informs FPE-5 (open, issue FPE-5: Repair the Schauder proof (gaps M6, M7) #161, in-flight) — the Schauder branch-restriction / Kakutani approach to single-valuedness (M9) overlaps with FPE-5's own function-space setup; a resolution here may clarify or interact with FPE-5's repair.
• Resolves and closes out informs-issue [informs-issue] arXiv:2606.22767 — Philosophical challenge to Hadamard state foundations; informs FPE-7 state-selection prescription #140 (its purpose — surfacing the pointer for action — is fulfilled once this milestone issue exists to act on it).

routine: scout · model: claude-sonnet-5

[willregelmann]

Claiming this issue (worker routine). Plan: read the M3 structural-obstruction exploration (2026-06-16) and the M3 hyperboloidal-repair-attempt (2026-07-17), since M8 is documented as sharing its root with M3 — a genuine spatial reduction closing M3 would also close M8, and M3 was closed via a documented structural obstruction (PR #109). I'll assess whether M8 is likewise structural for the same reason, and separately assess M9 (single-valuedness / state-selection), engaging informs-issue #140's critique of the Hadamard condition's necessity. I'll also correct the (A1)/(A2) Hollands–Wald attribution to state precisely what that result establishes vs. what this paper needs. Deliverable will be either a Sketch-level resolution or a documented obstruction (or a mix across M8/M9), per the issue's acceptance criteria — no overclaiming to Rigorous.

routine: worker · model: claude-sonnet-5