Gap-05-stability — Lambda radiative stability: the gate anchor ledger

The honest one-line: Gap-05-stability is the radiative-stability face of the cosmological-constant problem — no mechanism here, or anywhere, holds the vacuum energy ~122 orders of magnitude below the natural scale — but the gate has a rare, real win: it took the framework's own candidate cancellation, computed it to a clean FAIL, and proved why it had to fail (the vacuum-energy operator is the unit operator, blind to the grading the cancellation needs), then banked that negative as a theorem.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing Gap-05-stability touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape as the canonical SG-4 ledger.

This is the marquee open problem of fundamental physics. The gate's genuine deliverables are two computed negatives — they sharpen the wall and grade the door shut honestly; they do not close it. No grade here is raised above what the dossier supports.

1. Gate status header

• Gate-level Gap-05-stability roll-up: CERTIFIED-IRREDUCIBLE · RESOLVED +0 (board-canonical, ratified 2026-07-08; closure-of-record: the live gate dossier) · with a banked negative theorem (route honestly refuted); the earlier roll-up under the superseded least-closed-residual rule read OPEN. This is the radiative-stability mechanism face of $\Lambda$; the value face lives in the companion gate Gap-05-value, the catastrophe-half in Gap-05-catastrophe.
• Taxonomy reconciliation (2026-07-05): the gate-level grading is CERTIFIED-IRREDUCIBLE · RESOLVED +0 — the terminal the board carries (ratified 2026-07-08) — read as TERMINAL + RESIDUALS-SHOWN — the banked negative theorem (the two computed negatives plus the value/external-frontier handoff) is the reached terminal, with the open radiative-stability mechanism carried alongside as a shown residual family. The residual family listed in this ledger (A1–A4, R-uniqueness, S, R5, MO-1) remains listed and carried unchanged; the facts are unchanged. The earlier “OPEN” roll-up above reflects the superseded least-closed-residual rule, not different facts.
• The closed local/partial leg. There is no closed mechanism leg here — the radiative-stability mechanism pays no clean posit, because the only thing that would pin a per-tower vacuum-energy protector is the observed small $\Lambda$ itself, and reverse-engineering to that number is reverse-engineering from the measured value. What is rigorous and banked is a pair of computed negatives. The chamber route's structural no-go is the local certified fact $$O_{\rm chamber}(E_{\rm frozen}) = \mathbb{1} \quad\Rightarrow\quad [\,G,\,O_{\rm chamber}\,]=0\;\;\forall\,\text{gradings }G \quad\Rightarrow\quad \text{no chamber-label cancellation can act},$$ i.e. the vacuum-energy operator is the identity, so the graded sum equals the ungraded sum and the cancellation is impossible as a matter of operator structure.
• Status, split so it cannot be misread:
  • The radiative-stability mechanism (the gate's top object L05): OPEN — Weinberg-open. No symmetry-protected, no-re-tuning mechanism is known here or anywhere.
  • The chamber-cancellation no-go (certificates I2/I3): DERIVED (negative) / THEOREM_REFUTED — the framework's own cancellation lever is computed-FAIL and proved structurally refuted. This is a banked negative result, not a closure.
  • The relocation lane (BURDEN_FAIL × 3): AUDIT — decision-grade elimination — banked at decision grade, audited for consistent application only, NOT proven sound/complete.
  • The gravity-side Theorem-1 (three layers): PROVEN-positive (tree) / PROVEN-negative (quantum) / CONDITIONAL (all-orders) — a single theorem with its negative half intact; the Axiom alone is insufficient at the quantum level.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what Gap-05-stability stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 (anchor) / manifest meta a5b1e6f9d951. The branch is read-only, unmutated. These are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics — and crucially, $\Lambda$ has NO per-value first-principles hash, because the framework does not derive it.
• The 13D branch produces no $\Lambda$. The frozen geometry $$\mathfrak{B}_{\rm active} = [\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2]\oplus[F^+]\otimes[E],\qquad K_6=SU(3)/T^2,$$ carries no cosmological constant. The geometry's honest contribution here is negative and clarifying: it has one internal lever (chamber cancellation), which is computed-refuted, and it produces no $\Lambda$ to compare against (the no-reverse-engineering from the measured value guarantee).
• The Weinberg 1989 no-go is given / external / in-category — it enters the gate as a binding wall, not a Gap-05 output. Gap-05-stability does not clear Weinberg.

3. Object anchors (given-E / upstream)

The objects this gate acts on:

• The vacuum energy as a dimension-zero operator. The cosmological-constant operator is the unit/identity operator on the relevant space — grading-even, label-blind. Status: structural fact (load-bearing for the I3 no-go).
• The tower scales $\{M_{\rm Pl},\,m_t,\,v_{\rm EW},\,\Lambda_{\rm QCD}\}$ — the thresholds at which fresh additive vacuum-energy shifts turn on. A genuine mechanism must protect $\Lambda$ at all of them, simultaneously, with no per-scale re-tuning. Status: GIVEN / external physics.
• The measured $\Lambda$ value $\Lambda=(2.3\ {\rm meV})^4\approx1\times10^{-122}\,M_{\rm Pl}^4$ — Tier-1 row 5 of the irreducible ledger. Status: MEASURED-ANCHOR (handled in Gap-05-value) — not read, fitted, or compared anywhere in this gate.

4. Root and master-anchor traceability

Deep roots that are load-bearing for Gap-05-stability:

Causal order and shape are not the primary load-bearing roots for the radiative-stability obstruction.

Master anchors in play: finite invariant ledgers · no unpaid labels (no quantity reverse-engineered to $(2.3\ {\rm meV})^4$) · the frozen branch · given-$\Lambda$ (the measured anchor, in Gap-05-value) · the external Weinberg wall · open-residual discipline.

5. The Gap-05-stability anchor ledger (the universal table)

6. The arithmetic — the two computed negatives, in full

6.1 The I2 supertrace (the honest FAIL)

The $\oplus$-layer "chamber-cancellation" idea is the framework's deepest internal proposal for a vacuum-energy cancellation: the particle inventory splits into sign-graded "chambers," and a single signed sum — a supertrace — would vanish identically if the chambers' contributions paired up to cancel. Computed at successive coefficient orders $k$ over the 17-row physical inventory:

$$\frac{{\rm Str}_{\rm graded}}{{\rm Str}_{\rm ungraded}}\Big|_{k=0}=0.58,\qquad \frac{{\rm Str}_{\rm graded}}{{\rm Str}_{\rm ungraded}}\Big|_{k=1\ldots8}=1.000.$$

That is: no structural cancellation at any coefficient order. At $k=1$–$8$ the graded sum equals the ungraded sum exactly — the grading does nothing at all; at $k=0$ it shifts the sum only to $0.58$ of the ungraded value, nowhere near the zero a real cancellation would force. The structure-side witness number is recorded as

$$\mathrm{Str}\,\rho=\frac{-88.93\,\pm\,\text{band}}{R_Y^4}+c_{\rm loop},$$

the magnitude of the residual the failed cancellation leaves behind. It is never compared to $\Lambda_{\rm obs}$ — it is emphatically not a $\Lambda$ prediction.

6.2 The I3 no-go (why the chamber route had to fail)

The decisive piece is not the numerical FAIL but the structural reason for it. A sign-graded cancellation works only if the grading operator acts non-trivially on the object being summed — it must map "$+$chamber" contributions to "$-$chamber" contributions so they cancel in the signed sum. But the vacuum-energy operator is the identity:

$$O_{\rm vac}=\mathbb{1}\quad\Rightarrow\quad G\,O_{\rm vac}\,G^{-1}=O_{\rm vac}\;\;\forall G\quad\Rightarrow\quad {\rm Str}_{\rm graded}={\rm Str}_{\rm ungraded}.$$

The identity commutes with every chamber-label grading, so the graded and ungraded sums coincide. There is nothing for the grading to act on. The cancellation cannot happen — not because the numbers came out unlucky, but because the operator is structurally blind to the labels the cancellation needs. The $0.58$ residual at $k=0$ is the quantitative witness; the ratio $1.000$ at $k=1$–$8$ is the grading "doing literally nothing."

6.3 The gravity-side Theorem-1 — proved with its negative half intact

Theorem-1 (L1) — PROVEN, positive-but-limited (tree-level trace-drop). With $T_{\mu\nu}^{\rm vac}=-V g_{\mu\nu}$, the trace-free combination vanishes identically: $$T_{\mu\nu}^{\rm vac}-\tfrac14 g_{\mu\nu}T^{\rm vac}=-V g_{\mu\nu}+V g_{\mu\nu}=0$$ (verified for all $n$; exact at $n=4$). So a spatially-constant vacuum energy $V$ of any magnitude — including $O(M_{\rm Pl}^4)$ — does not source curvature at tree level. This is a theorem about tensor structure, not numbers.

Theorem-1 (L2) — PROVEN, NEGATIVE — the Axiom ALONE is insufficient at the quantum level. Under the Axiom, Bianchi + matter conservation force $\Lambda_{\rm grav}=\Lambda_0$, an integration constant fixed by a boundary datum. An additive matter-loop vacuum shift $\mathcal{L}_m\to\mathcal{L}_m+\delta V$ (with $\delta V\sim M^4$, constant) passes straight through: $$\Lambda_0\;\longrightarrow\;\Lambda_0+\delta V\qquad(\text{the map is the identity}).$$ Hence $\Lambda_{\rm grav}$ IS radiatively shifted at one loop already (Padilla–Saltas, arXiv:1409.3573). The earlier slogan "an integration constant has no beta function" is TRUE BUT IRRELEVANT — there is no running coupling, yet the boundary value that replaces it is shifted additively.

Theorem-1 (L3) — CONDITIONAL/PARTIAL. All-orders sequestering holds only for a strictly-stronger Kaloper–Padilla construction (arXiv:1606.04958) — rigid global scalars, a Gauss–Bonnet term, global flux/4-volume constraints — and only given an unproven smoothness/non-degeneracy assumption $S\;[\sigma(O(1)z)\sim O(1)\sigma(z)$, nonlinear$]$. The surviving residual $\Delta\Lambda$ is, in Kaloper–Padilla's own words, "radiatively stable, albeit with a value that is incalculable and should be set by measurement" — a verified 1:1 relocation of the value.

6.4 Specificity diagnostics — the negatives are real, not rigged

(i) The supertrace FAIL is specific, not trivial. If the grading did nothing because the inventory were empty or symmetric by construction, the ratio would be ill-defined or $1.000$ at every order including $k=0$. Instead the ratio is $0.58\neq1.000$ at $k=0$ — a genuine non-trivial shift — and exactly $1.000$ at $k=1$–$8$. That a non-trivial $k=0$ shift coexists with an exactly-flat $k\ge1$ tail is what makes the obstruction a real structural fact about the operator, not an artifact.

(ii) The harness has teeth (verified). The burden was checked to discriminate: an empty L0_NULL probe correctly returns FAIL, and a hypothetically-satisfied relocation-L1 correctly REOPENS (the terminal flips toward derived-reopens-toward-closure). So the three BURDEN_FAIL verdicts are a genuine elimination, not a rigged "everything fails."

(iii) Negative controls fail correctly. A granularity reduce-attack (a finite supertrace $\sim M_{\rm cutoff}^4$ at $M_{\rm cutoff}=1/R_0$) lands $\sim113$ orders off and predicts no value; an implicit-assumption audit finds no assumption-drop that predicts $(2.3\ {\rm meV})^4$. These deliberately-tripped controls confirm the route does not secretly back-solve the value.

7. Declared-structure splits — the two "L1/L2/L3" triples, kept apart

This gate carries two unrelated triples both labelled L1/L2/L3. The single phrase "L1/L2/L3" hides two entirely different claim-families, which must never be conflated:

1. The relocation candidates — relocation-L1 / L2 / L3. Three physical proposals for where a vacuum-energy cancellation could live: a chamber-pairing symmetry $G^+$; a non-perturbative wall $\Delta V_{\rm NP}$; chamber-projected boundary degrees of freedom on $S^1_Y$. All three return BURDEN_FAIL. Relocation-L1 is doubly-walled (Weinberg + I3); L2 fails on R2 (a wall pins a modulus, not the $122$-OOM zero-point — modulus stabilization $\neq$ vacuum-energy cancellation); L3 is parasitic on L1 (the boundary-Casimir term re-renormalizes per scale absent the bulk-tying symmetry, which is L1, refuted).
2. The Theorem-1 layers — Theorem-1 (L1) / (L2) / (L3). Three logical layers of a single gravity-side proof: a positive tree-level layer, a negative quantum-level layer, and a conditional all-orders layer.

So "L1/L2/L3" is a relocation-candidate label in §3.4 and a proof-layer label in §3.5 — same letters, different objects, separate statuses.

8. Open residuals — grouped by family

These distinct residuals make up the open gate; none is closed by the two banked negatives. None of these is the unsolvable object — the marquee unsolvable (a new L-construction passing all four predicates) is named and graded shut, not handed out.

Family I — the burden's authority (the keystone): - A1 (MO-3 / BLOCKER-A1) — soundness & completeness of R1–R4. OPEN. The burden is audited for consistent application only, not proven to exactly operationalize $122$-OOM radiative stability. Treating BURDEN_FAIL as a rigorous closure-by-elimination is forbidden until A1 closes. - A4 (BLOCKER-A4) — the teeth-test reopen-branch witness. OPEN. Whether the "hypothetically-satisfied L1 reopens" check runs a real constructed candidate through R1–R4 or merely toggles a satisfied flag is not determinable from the text. The most tractable hole (a bounded, fail-closed code inspection).

Family II — cross-gate dependencies: - A2 (BLOCKER-A2) — is R4 ("non-perturbatively supplied") evaluable inside the corpus? OPEN — cascade. R4 may not be evaluable without the non-perturbative-QCD continuum engine that gates Gap-02 / UQF-11; if so, grading any relocation secretly cascades on Gap-02. - A3 (BLOCKER-A3) — an un-ruled-out boundary-only protection candidate. OPEN. Relocation-L3's full parasitism on L1 is asserted, not exhaustively shown; a fixed-point anomaly-inflow or $\mathbb{Z}_2$-localized boundary symmetry on $S^1_Y/\mathbb{Z}_2$ is un-ruled-out. Any such candidate is a new L05.D instance needing its own full burden run, NOT a revival of L3.

Family III — the gravity-side dissolution residuals: - R-uniqueness — is volume-mode freezing the UNIQUE MINIMAL trace-decoupling modification? OPEN. Until shown, the separating premise stays NON-ATOMIC and the all-orders sequestering claim is not load-bearing. - S — the smoothness assumption in the all-orders sequestering argument. OPEN / UNPROVEN. $S\,[\sigma(O(1)z)\sim O(1)\sigma(z)]$ is established in the action, not by an order-by-order BPHZ proof; a minimal-sequester no-go (gap G4) forces the non-minimal Gauss–Bonnet augmentation. - R5 — the residual finite-condensate engineering wall. OPEN. Finite electroweak/QCD condensate shifts (~$10^{44}$–$10^{55}$ above the observed scale) must be shown sequestering-clean, and the latent heat from the QCD/electroweak transitions must gravitate correctly.

Family IV — the named-and-graded-shut door (NOT a plugging task): - MO-1 = L05.D — a future L-construction ($\neq$ relocation-L1/L2/L3) supplying a vacuum-energy protector passing all four R1–R4. This is the open cosmological-constant problem (R1 + R2 together $\approx$ the Weinberg no-go). External-Clay-class. Producing one without a genuine new ingredient would be fabricated physics — the single worst outcome.

9. Anti-claims (what this page refuses to say)

• Gap-05-stability does not make $\Lambda$ radiatively stable / technically natural — frozen docs say REFUTED. No mechanism, here or anywhere, holds the vacuum energy $\sim122$ orders below the natural scale.
• No $\Lambda$ value is derived, predicted, or compared. The $122$-OOM figure is the size of the burden, explicitly not a result; the $-88.93/R_Y^4$ supertrace witness is the residual of the failed cancellation, never a $\Lambda$ prediction. Carrying $\Lambda$ as a measured anchor is not deriving it.
• BURDEN_FAIL is not a rigorous closure-by-elimination — it is a banked elimination at decision grade, audited for consistent application only (A1 open). The lane does not prove "L1, L2, L3 exhaust the relocation space."
• The chamber route cannot be repaired by re-grading toward a small $\Lambda$ — that is reverse-engineering (the $\kappa^3/\pi$ pattern); it RELOCATES and closes nothing. An independent re-run must reproduce the obstruction, never tune the grading toward zero.
• Report Theorem-1's negative half. $\Lambda_0\to\Lambda_0+\delta V$ is the identity map; the Axiom alone is insufficient at the quantum level. Reporting only the favorable tree-level L1 while dropping L2 is the exact honesty failure that misleads. Do not re-lean on the "integration constant has no beta function" slogan — it is TRUE BUT IRRELEVANT.
• Tree-level trace-drop $\neq$ quantum-level protection; modulus stabilization $\neq$ vacuum-energy cancellation; a wall $\neq$ a cancellation symmetry. These category distinctions are why a closed moduli gate (Gap-04) cannot be borrowed to close $\Lambda$.
• Dissolved $\neq$ solved. The catastrophe-half ("why not $M_{\rm Pl}^4$?") is dissolved-conditional; this gate does not solve the radiative stability of the value.
• The two "L1/L2/L3" triples are distinct — relocation candidates vs Theorem-1 layers — and must never be conflated.
• The frozen-branch hashes are audit anchors; they do not validate the physics, and $\Lambda$ has no per-value first-principles hash.

10. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. All physics; firewall the applications.

1. A1 (soundness/completeness of R1–R4). Either (i) prove $R1\wedge R2\wedge R3\wedge R4$ exactly operationalizes $122$-OOM radiative stability, or (ii) supply a counterexample — a construction passing all four yet provably radiatively unstable (refutes soundness), or a known-stable mechanism failing one predicate (refutes completeness). A negative is a valid close. Falsifier: an independently checkable counterexample. Closing A1 upgrades the lane from decision-grade to rigorous closure-by-elimination — the keystone for the harness's authority.
2. A2 (R4 evaluability). State how R4 is evaluated, plus a well-defined non-perturbative evaluation of $\Delta V_{\rm NP}$ from the Gap-02 / UQF-11 continuum measure. Falsifier / valid close: a proof that R4 is not evaluable inside the corpus without the continuum engine — which makes the cascade-tie explicit and exports R4 cleanly to UQF-11. Do not label a perturbative proxy "non-perturbative."
3. A3 (boundary-only candidate). Run the four-predicate burden on any independent boundary-only protection candidate (a fixed-point anomaly-inflow; a $\mathbb{Z}_2$-localized symmetry on $S^1_Y/\mathbb{Z}_2$). If it passes, it is a new L05.D instance (the one genuine reopen door — do not grandfather it in as "L3"). If no candidate exists / every candidate fails, A3 closes.
4. A4 (reopen-branch witness). Inspect the harness code: confirm the reopen branch accepts a real constructed input evaluated through R1–R4 (genuine) versus a stipulated flag (weak). Valid close either way: if it only toggles a flag, re-classify the reopen branch as a stipulation and scope the teeth claim to the fail side. The most tractable hole — a bounded, fail-closed code inspection.
5. R-uniqueness. Prove (or refute) that volume-mode freezing is the unique minimal trace-decoupling change. If proven, the premise sharpens to a named conditional theorem; if refuted, sequestering is one of many such constructions — itself a clean terminal disclosure.
6. S (smoothness). Prove assumption $S$ (or supply an order-by-order BPHZ argument for the augmented construction without $S$). Falsifier: a counterexample where $S$ fails and a vacuum bubble is not absorbable — which bounds the sequestering route precisely. Do not re-introduce the misattributed arXiv citation removed pending re-verification.
7. R5 (condensate shifts). Demonstrate that finite phase-transition condensate shifts are absorbed by the trace-free/sequestering structure without re-tuning, and that the released latent heat gravitates correctly across both the QCD and electroweak transitions. A bounded computation; partially cascades on A2.

The single unreachable object, named and graded shut (NOT a plugging task): MO-1 = L05.D. A future L-construction passing all four R1–R4 is the open cosmological-constant problem — external-Clay-class. We name the door and grade it shut; we refuse to fabricate one.

11. Completion tests for this page

Required presence (all met): gate roll-up CERTIFIED-IRREDUCIBLE · RESOLVED +0 · banked-negative-theorem label · the local certified leg $O_{\rm chamber}=\mathbb{1}$ and its DERIVED-(negative) label · $\Lambda$ not derived · frozen hashes (AUDIT ONLY) · the I2 supertrace ratios ($0.58$ at $k{=}0$, $1.000$ at $k{=}1$–$8$) · the $-88.93/R_Y^4$ structure-side witness · the I3 unit-operator no-go · Theorem-1 (L1) tree-drop · Theorem-1 (L2) PROVEN-negative $\Lambda_0\to\Lambda_0+\delta V$ · Theorem-1 (L3) conditional/assumption-$S$ · the burden R1–R4 with teeth-test · all three relocation BURDEN_FAIL · the two-triples split · every open residual (A1–A4, R-uniqueness, S, R5) as its own row · MO-1 named-and-graded-shut · the specificity diagnostics · the no-reverse-engineering from the measured value, dissolved-$\neq$-solved, report-the-negative-half, and hashes-don't-validate anti-claims.

Required absence (all held): no claim that Gap-05-stability is closed · $\Lambda$ is protected / made technically natural · a $\Lambda$ value is derived/predicted/compared · BURDEN_FAIL is a rigorous elimination · the relocation space is proven exhausted · the chamber sum can be re-graded toward small $\Lambda$ · only Theorem-1's favorable half is reported · the no-beta-function slogan is re-leaned on · the two "L1/L2/L3" triples are conflated · tree-level trace-drop extends quantum-mechanically · a closed moduli gate closes $\Lambda$ · hashes validate physics.

This gate follows the same eleven-part shape and universal table as the canonical SG-4 ledger.

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why no quantity may be reverse-engineered to the observed $\Lambda$) · Layer 4 — carrier-forcing & the given-E wall · the Gap-02 anchor ledger (the shared non-perturbative-QCD cascade behind R4 / A2) · the full Gap-05-stability dossier.