The Single-Metric Realization Problem for Dual-4DD: A No-Go Group, an Internal-Wedge Candidate, and a Transition-Surface Imprint

Qin, Han

doi:10.5281/zenodo.21393334

Abstract

The dark-sector installment of this series left two action-level open items: fixing the khronon–$C$ mixing coefficient $g$, and identifying the route by which the signed vacuum ledger $\Lambda_1+\Lambda_2=0$ enters the equations that matter sees. This paper recognizes both as facets of a single realization problem — how the two-sided dual-4DD structure is to enter a single-metric action — and measures precisely where that problem currently stands. The negative half is a theorem: in a healthy local bulk (single metric, time-orientable, positive-definite propagating sector, exact shift symmetry) a signed ledger cannot be realized as the difference of two independent canonical stress tensors, and $g$ is shown to be constructively underdetermined — the bulk does not generate it, the mixing operator does not drive it, symmetry fixes only its parity, and a bulk operator is symmetry-permitted but its coefficient is not pinned. The constructive half is a candidate: taking as geometric raw material the antipodality defect already published upstream (the two time axes are anti-aligned but not exactly $180^\circ$, the offset being first-order-linear in the frequency asymmetry), an internal top-form wedge furnishes a bounded-below, ghost-free existence witness that yields, on the non-trivial branch, a non-negative vacuum magnitude entering at $O(|\varepsilon|)$, whose zero is exactly the geometric content of the ledger axiom. The measurement is the paper's real result: observation fixes only the single combination $\mathcal N^2\lambda k=8$ — the flux normalization $\mathcal N$, the geometric tilt coefficient $k$, and the Wilson readout $\lambda$ are none of them independently derived, and their underdeterminacies are separately located as identifiable open problems. As a by-product, the HP2 hierarchy is established: intrinsic $w_{\rm int}\ge-1$ remains a hard bound within the positive-sign, ghost-free DBI baseline, while an effective phantom crossing is a provably underdetermined bulk-Wilson branch, adjudicated by a nested-hypothesis structure. All claims are tagged across four tiers — structural, posterior, motivated, open.

§1 Introduction: two open items, one realization problem

The dark-sector installment of this series unified, within a single action, dark matter (the khronon), dark energy and MOND (the $C$ field), and the cosmological constant (a 4-form sector); it closed by recording, honestly, two action-level open items. The first is the coefficient of the khronon–$C$ mixing operator $g\,\theta Q$ — the carrier of the apparent phantom crossing in the effective equation of state, whose size and sign were left unfixed. The second is the action-level route by which the signed vacuum ledger $\Lambda_1+\Lambda_2=0$, together with the observed value $\Lambda_{\rm obs}=2(\omega_2^2-\omega_1^2)/c^2$, reaches the equations matter obeys — open since the two-frame installment. This paper begins from a single recognition: these two items belong to the same realization class — how the two-sided dual-4DD structure enters a single-metric action. The strength of that statement must be qualified at once: it is the same class, not the same operator or the same coefficient. $g\theta Q$ is a derivative-order mixing and the signed ledger is a zero-derivative vacuum source; they share an obstruction to realization, but a no-go for one does not hand over the number for the other.

State the realization problem precisely. We seek a mathematical structure that lets a single-manifold, single-metric action satisfy simultaneously: (L) signed ledger — the vacuum sector's net contribution to the observed equations is proportional to $\omega_2^2-\omega_1^2$ (the difference, not the sum); (V) numerical boundary condition — the published relation $\Lambda_{\rm obs}=2(\omega_2^2-\omega_1^2)/c^2$ matches the Planck-derived value to within about $5\%$ (this $5\%$ is the mismatch between the SAE formula and the Planck-derived value, not Planck's observational uncertainty, which is far finer), the flux dictionary being revisable but subject to registration; (C) locality of $C$ — $C=\tau_1-\tau_2$ remains a local dynamical field on this spacetime, the field on which the dark-energy and MOND sectors are built; (G) ghost-freedom — both sectors share the metric, so any propagating field carrying wrong-sign kinetics is, through its gravitational coupling, a death sentence; (T) no closed causal loops — a single metric, with no bimetric structure or looped time topology; (S) exact shift — $C\to C+C_0$ is an exact symmetry of the action. Why this set is not decoration is shown by one load-bearing number: with flux normalization $f_i=\mathcal N M_P\omega_i/c$, the ratio of the vacuum-energy sum of two independent positive-sign 4-forms to the observed value is $\mathcal N^2(1+\varepsilon^2)/(8\varepsilon)$ — about $20$ under the original dictionary of the cosmological-constant installment ($\mathcal N=\sqrt2$), about $40$ for $\mathcal N=2$. Unless the ledger nearly cancels, the model is off by more than an order of magnitude at leading order.

The paper's results, stated at their true strength, are as follows. We prove that the signed dual-4DD vacuum ledger cannot be realized as the difference of two independent canonical stress tensors in a healthy single-metric bulk (the no-go group, §4); and that $g$ is provably constructively underdetermined within the locked baseline and operator classification — the bulk does not produce it, the mixing operator does not drive the background, symmetry fixes only its parity, and a bulk $\theta Q$ is symmetry-permitted but its coefficient is not pinned (§3). In the surviving gap that the no-go group's elimination points to, an internal top-form wedge supplies a bounded-below local existence witness: it naturally produces, on the non-trivial branch, a positive vacuum magnitude of order $O(|\varepsilon|)$, but its flux normalization $\mathcal N$, geometric tilt coefficient $k$, and Wilson readout $\lambda$ are constrained only by the single target relation $$\mathcal N^2\lambda k=8,$$ none of the three being independently derived (§5–§6). Accordingly, the minimal consistent architecture places the physical magnitude at a transition-surface imprint and the cross-cycle sign in an orientation ledger, while leaving the complete matching law explicitly to the next layer (§7–§8). The by-product is the nested HP2 hierarchy: intrinsic $w_{\rm int}\ge-1$ remains a hard bound, the effective crossing is a provably underdetermined bulk-Wilson branch, and the observational adjudication structure is given in §9.

What the paper does not do is listed with equal candor: it does not complete the transition-surface matching functional; it does not compute model-specific nucleation rates (only a thin-wall benchmark partition, with numbers in Appendix D); it does not touch the $a_0$ sector of the third installment (the HP1 adjudication discipline stands); it does not activate the effective-crossing branch; and it does not derive a numerical value for any one of $\mathcal N$, $\lambda$, $k$. Order of exposition: §2 sets the contract and the assumption box; §3 treats $\theta Q$; §4 proves the no-go group; §5 supplies the tilt raw material; §6 builds the wedge witness; §7 handles pairing and the transition surface; §8 synthesizes the architecture and registers revisions to earlier installments; §9 gives the nested falsifiable hypotheses; §10 closes and lists the keys to the next layer.

§2 Epistemic contract, baseline, and two ledgers

Tiers and accounting. This paper carries forward the four-tier tagging established in the zeroth installment — 〔structural〕 forced by the framework and falsifiable; 〔posterior〕 dependent on observation or parameters; 〔motivated〕 a well-motivated preference; 〔open〕 not yet adjudicated — and adds to every coefficient that appears a three-way accounting: prior-derived (given by framework structure), construction input (a modeling choice, explicitly flagged), posterior boundary (taken from observation). Any place where a "construction input" masquerades as "prior-derived" is a self-identified, checkable error of this paper.

Locked posterior inputs. Breathing periods $T_1=20$ Gyr, $T_2\approx19.5$ Gyr; frequency asymmetry $\varepsilon=(\omega_2-\omega_1)/(\omega_2+\omega_1)\approx0.0127$. Every "$O(\varepsilon)$" in this paper is an order in this parameter.

Baseline declaration. Throughout, the exact-shift baseline: a single metric; matter minimally coupled, $S_m[g_{\mu\nu}]$; $C$ entering the action only through derivatives ($Q$, $\nabla C$), its potential constant by the symmetry. The conformal machinery of the two-frame installment ($\mathscr A(C)$, $F(C)R$) migrates wholesale to a shift-breaking branch, which this paper neither uses nor refutes; that migration is registered as one of the revisions in §8.

Common assumption box. Every no-go proposition here shares one domain, stated once: four dimensions; Lorentzian signature; a single metric; a time-orientable bulk; a positive-definite propagating sector (ghost-free); a local action; exact shift symmetry for $C$. Wherever analyticity is used in addition (Theorem 3.3, the parity trichotomy of §6.4) it is flagged in place. The "escape routes outside the domain" listed after each proposition are the concrete ways of leaving this box.

Two ledgers. Two quantities run through the paper — one a definition, one a definition plus a candidate law — and must be kept apart; and two distinct indices must first be separated: the face index ($\pm$, the two faces of the dual pair within one cycle) and the cycle index ($n\to n+1$, adjacent time cycles). Collapsing them into a single $n+1$ was an error. Definition: $\rho_{\rm phys}\ge0$ is the physical vacuum energy entering the Einstein equation each cycle; the orientation ledger, written by face index, is $$\mathscr L_{n,+}=+\rho_n,\qquad \mathscr L_{n,-}=-\rho_n,\qquad \boxed{\mathscr L_{n,+}+\mathscr L_{n,-}=0},$$ the zero-sum of the two faces of the dual pair in cycle $n$ — this, and only this, is the identity of $\Lambda_1+\Lambda_2=0$; it concerns the two faces of one cycle and involves no adjacent cycle. The candidate matching law (not a definition; awaiting a complete boundary functional) governs the cycle index instead: $$\rho_{n+1}=\mathcal M(\rho_n,\Xi_{1,n},\Xi_{2,n}),$$ permitting the magnitude to drift cycle by cycle (the spiral). Thus "each side observes a positive cosmological constant" is a property of $\rho_{\rm phys}\ge0$; "dual-pair zero-sum" is a face-level ledger; "spiral drift" is the $n\to n+1$ evolution — the three live at different indices and no longer conflict (collapsing "the other-side label" and "the next-cycle label" into one $n+1$ would wrongly demand $\rho_{n+1}=\rho_n$). Throughout, the sign of a dual pair means the face-index $\pm$ (orientation label $\chi_{n,s}=s$, with no bare $\chi_n$ lacking a face index); the magnitude means $\rho_{\rm phys}$ — conflating the two is this paper's second self-forbidden error.

§3 The origin of θQ and the HP2 hierarchy

Fix the objects first. In this paper's baseline (§2) the aether is the breathing-foliation unit normal $A_\mu=-\partial_\mu\tau_b/\sqrt{-(\partial\tau_b)^2}$; write $$\theta\equiv\nabla_\mu A^\mu,\qquad Q\equiv A^\mu\partial_\mu C,$$ so that on FLRW $\theta=3H$, $Q=\dot C$. The operator $g\,\theta Q$ is the leading, shift-respecting, woven-type mixing between the khronon and $C$; the dark-sector installment identified it as the carrier by which the effective equation of state can present an apparent phantom crossing, and listed the coefficient $g$ as a free, $g\propto\varepsilon$–constrained Wilson branch. This section upgrades that EFT classification into three action-level theorems and one constructive-underdetermination proposition: where $g$ cannot come from, what constrains its functional form, and why the existing structure is insufficient to fix it.

Theorem 3.1 (no bulk source). A two-DBI bulk containing only first derivatives does not generate a $\theta Q$ operator carrying $\nabla A$. Assumptions: the baseline two-clock action $-\sum_i\mathcal T_i\sqrt{1+(\partial\tau_i)^2}$; single metric; no interface term. Proof sketch: at quadratic order the sum/difference variables give $$\mathcal T_1(\partial\tau_1)^2+\mathcal T_2(\partial\tau_2)^2=(\mathcal T_1{+}\mathcal T_2)\Big[(\partial\tau_b)^2+\tfrac14(\partial C)^2\Big]+(\mathcal T_1{-}\mathcal T_2)\,\partial\tau_b\!\cdot\!\partial C,$$ the cross term being of the pure-first-derivative $Y\equiv\partial\tau_b\!\cdot\!\partial C$ type, diagonalizable at quadratic order by a field redefinition (note: diagonalization changes the physical definition of the aether, so the conclusion here is confined to "the quadratic kinetic matrix is diagonalizable," not "the mixing is physically void"). The full DBI $-\mathcal T_1\sqrt{U+Y}-\mathcal T_2\sqrt{U-Y}$ (with $U=1+X_b+\tfrac14X_C$) retains $Y^2$ and, for unequal tensions, the odd-in-$Y$ nonlinear mixing — but throughout only first derivatives; $\theta$ contains second derivatives of $\tau_b$ and has, structurally, nowhere to sit. The $3H$ appearing in the homogeneous field equation (via $\tfrac{d}{dt}(a^3P_Q)=0$) is the Hubble friction supplied by the covariant divergence, not an action-level $g\theta Q$ operator. Escape routes outside the domain: an interface/boundary term (on which the existence part of Proposition 3.4 rests); higher-derivative bulk operators beyond minimal DBI; matter couplings that break exact shift (outside the baseline). 〔structural〕

Theorem 3.2 (the $J\!\cdot\!\nabla C$ operator has no background source). On an exact FLRW background the aether–scalar mixing operator $J^\mu\nabla_\mu C$ (with $J^\nu\equiv A^\mu\nabla_\mu A^\nu$) vanishes pointwise; its integration-by-parts relation with $\theta Q$ supplies no independent background-level mixing source. Assumptions: exact FLRW; an action containing $J^\mu\nabla_\mu C$ (as in the Skordis–Złośnik–type vector–scalar mixing; note that theory is a complete vector–scalar–metric system, and this theorem addresses only that one mixing operator, not the remaining $Q$-dependent functions). Proof sketch: the identity $$\int\!\sqrt{-g}\;\theta Q=-\int\!\sqrt{-g}\Big[J^\nu\nabla_\nu C+A^\mu A^\nu\nabla_\mu\nabla_\nu C\Big]$$ binds $\theta Q$ to its companion normal Hessian; on FLRW $J^\mu=0$, $\theta Q=3H\dot C$, the Hessian $=\ddot C$, and $\int dt\,a^3(3H\dot C+\ddot C)=[a^3\dot C]_{\rm bdry}$ is a pure boundary term. Reading off, by parts, a lone "background coefficient of $\theta Q$" is operator-basis bookkeeping, not physics; the mixing operator does not by itself drive a background-level effective crossing. Escape routes: the perturbative level and inhomogeneous backgrounds — where a parameter-dependent $J\!\cdot\!\nabla C$ mixing survives, which is exactly the source of the §9 perturbation criterion; or an independently added bulk $\theta Q$ operator at an interface — whose coefficient is then governed by Theorem 3.3 and Proposition 3.4. 〔structural〕

Theorem 3.3 (parity and analyticity). If the action is invariant under face exchange $1\leftrightarrow2$, if $\varepsilon$ is the unique exchange-odd spurion, and if $g$ is analytic near $\varepsilon=0$, then $$g(\varepsilon)=g_1\varepsilon+g_3\varepsilon^3+\cdots.$$ Assumptions: the three above, each load-bearing. Proof sketch: under exchange $\tau_b\to\tau_b$, $C\to-C$, so $A,\theta$ are even, $Q$ odd, $\theta Q$ odd; invariance forces $g(-\varepsilon)=-g(\varepsilon)$; an analytic odd function contains only odd powers. Escape routes outside the domain: dropping analyticity leaves only "$g$ is odd" — $\operatorname{sgn}\varepsilon$, $\varepsilon|\varepsilon|$, $\varepsilon\log|\varepsilon|$ and other non-analytic odd structures are then permitted and the linear leading term is not forced ($|\varepsilon|$ is even and can never by itself be $g$); dropping "unique odd spurion" lets $g$ be written as another exchange-odd object times an even function — the $\chi W$ route in the §6.4 parity trichotomy is one instance of exactly this escape, the two being two faces of the same structural fact. 〔structural〕

Proposition 3.4 (symmetry-permission and constructive underdetermination). A bulk Wilson operator $g_{\rm df}(\varepsilon)\,\theta Q$, respecting exact shift and exchange covariance, is symmetry-permitted — $g_{\rm df}$ satisfies Theorem 3.3 — but within this paper's locked baseline and operator classification its coefficient is not fixed by structure, and the underdeterminacy is constructive. Assumptions: the baseline; some construction relating the operator to the vacuum sector, with its zero-derivative (vacuum) normalization already fixed. On type: what is asserted here is a four-dimensional bulk interaction $\int_M\sqrt{-g}\,g_{\rm df}(\varepsilon)\theta Q$ — it can persistently affect late-time $w_{\rm eff}(z)$, and is the object of the §9 interacting branch; its microscopic (interface or other) origin is itself open. This is a different type of object from the codimension-one temporal junction of §7–§8 (Proposition 4.1), which produces only a boundary matching or an instantaneous impulse and cannot serve as a bulk operator persistently driving the background; the two are not conflated. Proof sketch (permission + underdetermination): an exchange-odd scalar built from the expansion and normal derivative of the phase flows reduces, in the homogeneous co-directed limit, to the $\theta Q$ type with coefficient proportional to $\varepsilon$ — so the operator is not symmetry-forbidden. Record the dimensions once: under this paper's two-clock/DBI normalization, $[\theta]=M$ and $Q/c$ is dimensionless (so $Q$ is dimensionless at $c=1$), giving $[\theta Q]=M$ and Wilson coefficient $[g_{\rm df}]=M^3$ (consistent with the candidate scale $g_{\rm df}\sim M_P^2\bar\omega\,\varepsilon$). Its size is constructively underdetermined: let the single microscopic function generating it be $\mathscr F$ with dimensionless argument $X\equiv\theta Q/\mu_\Sigma$ ($[\mu_\Sigma]=M$); the zero-derivative gate fixes $\mathscr F(0)$, but the one-parameter family $\mathscr F_\beta(X)=1+\beta X$ ($\beta$ a dimensionless Wilson coefficient) passes the same gate for any $\beta$ while giving different $\mathscr F'(0)$ — $$\mathscr F(0)\ \text{fixed}\;\nRightarrow\;\mathscr F'(0)\ \text{fixed}.$$ In EFT the coefficients of operators at different derivative order are in principle independent; a vacuum normalization cannot lock a derivative coefficient. Escape route: a complete microscopic action that locks the zero-derivative order and the derivative order in a single expansion (supplying action, measure, coefficient dimension, and variation) — which §4's bookkeeping theorem (Proposition 4.3), together with the display above, shows the natural candidate (a vacuum factor used as a calibration gate) cannot deliver; or leaving the baseline. 〔structural (permission + underdetermination); complete microscopic construction: open〕

The HP2 hierarchy is thereby established. The first layer is untouched: the baseline's intrinsic equation of state satisfies $w_{\rm int}\ge-1$ — this follows from the non-ghost condition of positive-sign DBI ($\rho+p=2XP_X\ge0$), has nothing to do with anything in this section, and is a hard bound within the positive-sign, ghost-free DBI-$C$ baseline (this strength-qualified statement avoids being read as unconditional over every conceivable SAE completion). The second layer is upgraded: the bulk Wilson operator $g_{\rm df}\theta Q$ needed for an effective crossing, formerly an "uncomputed free branch" in the dark-sector installment, is turned by this section's theorems and proposition into a "branch provably underdetermined within the existing structure" — the minimal DBI bulk does not produce it (Theorem 3.1), the $J\!\cdot\!\nabla C$ operator does not drive a background crossing (Theorem 3.2), symmetry fixes only its parity (Theorem 3.3), and symmetry permits a bulk $\theta Q$ but does not lock its coefficient (Proposition 3.4). The framework neither predicts a crossing nor forbids one; the observational adjudication structure of this hierarchy — which data adjudicate the first layer, which combination the second — is given by the nested hypotheses of §9. 〔structural (classification); observational correspondence in §9〕

§4 The no-go group for a signed ledger

This section proves the paper's central negative result: within the healthy local bulk delimited by the §2 assumption box — four-dimensional, Lorentzian, single-metric, time-orientable, positive-definite propagating sector, local action, exact shift symmetry for $C$ — the signed dual-4DD vacuum ledger $\Lambda_1+\Lambda_2=0$ cannot be realized by two co-located sectors each coupling to the shared metric through a standard stress tensor. We split the result into seven propositions (the chain is Figure 1). Each is stated in one template: proposition, assumptions, proof sketch, escape route outside the domain. Listing the escape routes explicitly is both honesty and signposting: the constructive road taken in §6–§8 is selected from these escapes, not a way around the propositions.

Proposition 4.1 (codimension mismatch). A four-dimensional effective vacuum density and a three-dimensional junction tension are objects of different type; reading an effective "interface" term written $\int d^4x\sqrt{-g}\,\mathcal L_\Sigma$ directly as a codimension-one microscopic junction produces both a dimensional and a support mismatch. Assumptions: four-dimensional spacetime; the junction a codimension-one hypersurface carrying action through the induced volume element $\int d^3\xi\sqrt{|h|}$. Proof sketch: the former has density dimension $M^4$, the latter $M^3$, so the conversion must introduce a thickness scale $\ell_\Sigma$ — a new free microscopic scale. More to the point is the support: a codimension-one source enters the Einstein equations through the Israel junction condition (Israel 1966), its stress supported on $\delta_\Sigma$; a spacelike single transition surface gives the cosmological equations an instantaneous impulse, a timelike domain wall gives a local wall and generically breaks FRW homogeneity — neither is a homogeneous $-\rho\,g_{\mu\nu}$ Friedmann source. Escape routes: keep the four-dimensional effective-density reading (but its microscopic origin is then suspended); smear a wall network into a dense foliation; or lift the dimension so the four-dimensional universe is itself a brane. All three require structure beyond the assumption box. 〔structural〕

Proposition 4.2 (sign mismatch). In a single metric, the vacuum energies of two canonical 4-forms add rather than subtract; a spacetime orientation reversal cannot flip that sign. Assumptions: single metric; canonical kinetics $-\frac{1}{2\cdot4!}\sum_{ij}K_{ij}F_i^{\mu\nu\rho\sigma}F_{j\,\mu\nu\rho\sigma}$ with positive-definite kinetic matrix $K$; both fluxes co-located in the same bulk. Proof sketch: in four dimensions $F_i=f_i\,\mathrm{vol}_4$ and the vacuum energy is $\rho=\tfrac12 f^{\mathsf T}Kf\ge0$. Orientation reversal realizes at most $f_i\mapsto-f_i$, and $\rho(-f)=\rho(f)$. Adding a cross term does not help: as long as $K$ is positive-definite, an orthogonal diagonalization still gives a sum of positive squares. To obtain a $f_2^2-f_1^2$ difference, $K$ must carry at least one negative eigenvalue, or a physical orientation-odd spurion must be introduced. Escape routes: non-positive-definite kinetics (for the special status of the top form see the remark closing this section); a physical odd spurion; a bimetric structure (excluded by this series' anti-causal-loop discipline); or moving the sign out of the bulk to a boundary or non-local structure. 〔structural〕

Proposition 4.3 (bookkeeping theorem). A boundary-term coefficient appearing in the on-shell action of a 4-form cannot be read directly as a physical factor in the stress tensor. Assumptions: the 4-form field varied well-posedly in a fixed-flux ensemble, requiring a Bousso–Polchinski boundary term (Bousso & Polchinski 2000). Proof sketch: that boundary term's on-shell value is exactly minus twice the volume $F^2$ term; its job is to remove the surface mismatch between the on-shell action and the field equations (the difference between fixed-flux and fixed-potential ensembles), not to alter the physical vacuum energy. The latter must be read through $T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}$, whose value remains of the $+f^2/2$ type. Escape route: none. This is a methodological theorem: any derivation that takes an on-shell action value straight for an energy must redo it through the metric variation. 〔structural〕

Proposition 4.4 (causal-type mismatch). A Brown–Teitelboim membrane used for flux discharge and a khronon leaf satisfying $K_\Sigma=\theta$ cannot be the same hypersurface. Assumptions: Lorentzian signature; the discharge membrane a timelike worldvolume (spacelike normal, Brown & Teitelboim 1988); the khronon leaf $\tau_b=\mathrm{const}$ spacelike (timelike normal $A^\mu$). Proof sketch: $K_\Sigma=\theta$ requires $n^\mu=A^\mu$; but a single normal cannot be both spacelike and timelike. Escape routes: an explicit projection factor ($A\!\cdot\!n$, etc.) — at the cost of a new independent coefficient; or replacing the discharge object with a spacelike temporal matching surface (the temporal-junction path of §7, which is realized in essence by leaving the smoothness of the Lorentzian manifold) — at the cost that it is no longer a standard, nucleable dynamical membrane. 〔structural〕

Proposition 4.5 (the quadratic wall). Orientation reversal cannot flip any positive-definite quadratic vacuum energy; a signed ledger can enter only through orientation-odd objects. Assumptions: local field theory; energy a quadratic form in the fields with positive-definite kinetic structure. Proof sketch: by direct generalization of the argument of Proposition 4.2 to a general quadratic form. In the boson vacuum sector considered here — no fundamental fermions, no torsion, up to two derivatives — the directly relevant orientation-odd candidates are enumerable: $\int_M F_4$, $\int_\Sigma B_3$ (a WZ term), a linear $\star F$ (a pseudoscalar), and $d\tau\wedge B$–type $\varepsilon$-contractions — all linear or topological, none a quadratic energy capable of giving a homogeneous vacuum difference. (Note: higher characteristic densities such as $R\wedge R$ and Euler/Pontryagin classes, though also orientation-odd, are topological/boundary terms and likewise supply no homogeneous $-\rho\,g_{\mu\nu}$ vacuum source, so they do not change this conclusion; the list here is "directly relevant," not a claim of exhaustiveness over every four-dimensional theory.) Escape route: the channel list of Proposition 4.6. 〔structural〕

Proposition 4.6 (confinement theorem). In a bulk with positive-definite propagating sector, the ledger sign cannot be carried by the positivity or negativity of any propagating-mode Hamiltonian; its carrier must live in non-propagating objects, discrete/superselection sectors, boundary, or non-local constraints. Assumptions: the full §2 assumption box; both sectors sharing the metric. Proof sketch: by Proposition 4.5 the sign must enter through an orientation-odd object; and any propagating field carrying wrong-sign kinetics overall becomes, through its gravitational coupling to the shared metric, a ghost, violating the box's positive-propagation clause. Non-propagating top-forms (zero local degrees of freedom), discrete/superselection sectors, boundary data, and non-local constraints are not so bound. Two narrowing faces must be stated precisely. First, a UV completion may contain propagating modes: a healthy propagating field can perfectly well supply an exchange-odd spurion in its discrete vacuum/superselection sector — what is excluded is not "the theory contains propagating modes" but "the ledger sign is carried by the wrong sign of a propagating-mode Hamiltonian"; the sign cannot be carried by a propagating Hamiltonian, but its UV completion may contain propagating modes. Second, a healthy, exchange-even coupling (e.g. a positive-definite kinetic mixing) can still transmit perturbations between the two sectors — what is confined is only the sign bit, not all cross-sector communication. Escape route: the four carrier classes here are the channel list; §6 and §8 take the "non-propagating top-form" and "boundary" channels respectively. 〔structural〕

Proposition 4.7 (covering obstruction). If $C$ is taken as an odd section of a non-trivial time-orientation line bundle $L_t$, and exact shift requires adding a globally non-zero constant section to $C$, then the connected time-orientation double cover is incompatible with exact shift; a global khronon also trivializes the bundle. Assumptions: connected cover (equivalently, a non-time-orientable base); $C\in\Gamma(L_t)$; the shift an additive global constant. Proof sketch: on a non-trivial $\mathbb Z_2$ line bundle there is no globally parallel non-zero constant section — if there were, it would trivialize the bundle; and the globally timelike vector field required by the khronometric structure is time-orientability. Escape routes: a trivial cover (which organizes exchange parity cleanly but does not solve the ledger); or confining the non-triviality to the transition core — precisely where the causal structure itself dissolves — which is the boundary-matching reading of §7–§8, but that is a cross-boundary matching rule, not a bulk realization on a single smooth Lorentzian manifold. 〔structural〕

Remark (the special status of a wrong-sign top-form). In four dimensions the local propagating degrees of freedom of a 3-form gauge field are $\binom{D-2}{p}=\binom{2}{3}=0$; so assigning wrong-sign kinetics to one 4-form produces no perturbative ghost — this keeps the "non-positive-definite kinetics" escape of Proposition 4.2 from being immediately fatal as it usually would be. But the cost changes location: the flux energy of the wrong-sign sector is unbounded below, and any membrane allowed to change the flux opens a per-jump equal-step downward channel with no intrinsic stopper at fixed charge and tension. So the status of a wrong-sign top-form is: formally writable, healthy microscopic completion absent. This paper does not build on it, and cites its formal bookkeeping only in the comparison of §8. 〔structural〕

Net verdict. Combining Propositions 4.1–4.7: two independent, canonical, co-located 4-form sectors, each coupling to the shared metric through a standard stress tensor, cannot realize a signed difference ledger by time-orientation reversal. Every qualifier in that verdict is deliberate. It does not cover: two marked components of a single internal multiplet co-located in a near-antiparallel way (they do not each carry an independent standard stress tensor, but are components of a single functional) — the constructive part of §6 is built on exactly this un-sealed gap; nor does it cover boundary and non-local structures — the architectural synthesis of §8 is built there. The value of a no-go group is half in what it kills, half in the survivor it points to by elimination.

§5 Tilt: the antipodality defect and geometric underdetermination

Having evicted the signed ledger from the healthy bulk in §4, the constructive road of §6 needs one piece of geometric raw material: the two sectors' internal directions are near antiparallel, not exactly antiparallel. This section shows that this material already exists, at published first order, in the upstream Four-Forces installment VII, gives its inheritable strength, and then honestly measures the state of underdetermination of its coefficient in the cosmological case.

The vector solution and the defect angle from Four-Forces VII. That installment's Theorem 2.2 treats the closest-legal packing of six 4DD blocks (three antiparallel axis pairs) on $S^3\subset\mathbb R^4$: the symmetric reference configuration $\hat t_a^\pm=\pm e_a$ is strictly antiparallel; an intra-pair frequency asymmetry $\delta$ turns the two members' cone half-angles into $\tfrac\pi4(1\pm\delta)$, making the symmetric configuration violate the cross-axis no-overlap condition; the "remainder develops exactly" principle selects the legal-domain point nearest the reference, and the convex optimization gives uniquely $$\hat t_a^\pm\propto\pm e_a-\frac{\varepsilon_{\rm pack}}2\sum_{b\neq a}e_b,\qquad \varepsilon_{\rm pack}=\frac\pi2\delta.$$ This paper inherits the intra-pair offset angle from that vector solution (the strength must be stated precisely: the upstream theorem gives a first-order linear result $\theta=\tfrac{\pi}{3\sqrt2}\alpha+O(\alpha^2)$ and the display vectors linearized in the $S_3$-symmetric ansatz; so the arctangent below is the geometric angle of that set of first-order display vectors after normalization, not the exact angle of the full nonlinear packing solution): $$\varphi_{\rm pair}=\sqrt2\,\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)\qquad\big(\text{geometric angle of the normalized first-order representative}=2\arctan\tfrac{\varepsilon_{\rm pack}}{\sqrt2}\big).$$ That is: the two-sided time axes are antiparallel, but offset from exactly $180^\circ$ by an angle proportional to the frequency asymmetry, first-order-linear, its coefficient pinned by the (linearized) packing geometry. The same set of first-order vectors gives three distinct angles, to be named separately (Figure 2): the single-block off-axis angle $\theta_{\rm off}=\varepsilon_{\rm pack}/\sqrt2+O(\varepsilon_{\rm pack}^2)$; the intra-pair antipodality defect $\varphi_{\rm pair}=\sqrt2\,\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)$; and the cross-axis angular-distance increment $\Delta\theta_{\rm cross}=\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)$ (whose first-order value coincides with the no-overlap threshold shift). The quantity named "cross-axis tilt angle" in the Four-Forces VII text, $\sqrt2\,\varepsilon$, is geometrically the second of these (the defect). The existing numerical formulas ($c_1=\pi/(3\sqrt2)$ and its readout for $R_1$) use precisely its magnitude; this revision changes the geometric name, not the existing numerical readout; the formal erratum is carried by Four-Forces VII's own version upgrade, and this paper merely cites its first-order vector solution with this note. Here "unique" must be read in full as unique in the linearized closest-packing problem of the $S_3$-symmetric ansatz, not as an unqualified assertion that the entire nonlinear packing problem is unique. 〔first-order structure: structural (inherited upstream); erratum: registrational〕

Strict oddness of the directed angle (an independent step new to this paper). Beyond the first-order inheritance above, if one separately proves (by an exchange-uniqueness argument of the same type as Theorem 3.3: $\widetilde\varphi$ is the analytic odd function of the unique exchange-odd spurion $\varepsilon$) that the directed tilt is strictly odd, one may further write $\widetilde\varphi=k\varepsilon+O(\varepsilon^3)$ (odd powers, no $\varepsilon^2$). This is emphasized: it cannot be obtained automatically from the upstream "linearized solution"; it is an independent argument new to this paper, the cancellation of the even term depending on exchange symmetry, not on packing geometry.

Directed angle vs geometric defect angle. The parity accounting to follow requires the two quantities kept apart. The geometric defect angle $\varphi$ is the non-negative angle of the two internal directions relative to antiparallel, invariant under exchange; genuinely exchange-odd is a directed tilt $\widetilde\varphi$ with $\widetilde\varphi(-\varepsilon)=-\widetilde\varphi(\varepsilon)$ and $\varphi=|\widetilde\varphi|$. Three orders must be distinguished (resolving the apparent order gap between the §5 first-order response and the §6 matching expansion): $$\widetilde\varphi=O(\varepsilon)\ (\text{odd}),\qquad |\widetilde\varphi|=O(|\varepsilon|)\ (\text{even}),\qquad \widetilde\varphi^{\,2}=O(\varepsilon^2)\ (\text{even}).$$ Physical angles and energies run on the $|\widetilde\varphi|$ or $\widetilde\varphi^2$ branches (even); the exchange-odd parent runs on $\widetilde\varphi$ (odd).

The cosmological first-order response and the candidate table. Transported to the cosmological clock pair ($T_1,T_2$, $\varepsilon\approx0.0127$), the only universally statable thing is the form of the directed first-order response: with a linear displacement $a_\pm\varepsilon$ of the two intra-pair members along $m$ equivalent transverse directions, $$\widetilde\varphi=\sqrt m\,(a_++a_-)\,\varepsilon+O(\varepsilon^3);$$ in the natural subclass "threshold shift shared equally, cone half-angle $\beta_0(1\pm\varepsilon)$," $k\equiv\widetilde\varphi/\varepsilon=2\beta_0\sqrt m$. The coefficient then depends on the constraint geometry of the cosmological case, and that geometry is not a settled conclusion. The candidate table:

geometric input	$k$	note
phase-space $T^2$ equal-weight flow	$1$	$\tan\varphi_{T^2}=\varepsilon$ exactly; but depends on an equal-weight flat metric on the two clocks — a natural candidate, not a topological theorem
direct inheritance from Four-Forces VII	$\pi/\sqrt2\approx2.221$	requires six embedding assumptions: the same three-pair packing, $\varepsilon$ and $\delta$ on the same normalization, cone half-angle still $\pi/4$, the same constraints saturated, the same dual-pair constraints in surplus, the objective metric unchanged — none proven
time axis yielding to three space axes	$\pi\sqrt3/2\approx2.721$	the $m=3$, $\beta_0=\pi/4$ subclass
matching-condition point	$2$	from $\mathcal N^2\lambda k=8$ one gets only $\lambda k=2$ ($\mathcal N=2$); $k=2$ additionally requires $\lambda=1$, see §6 eq. (6.5)

Multiple natural, mutually compatible constraint geometries give different first-order coefficients; the matching value does not jump out of "one clock pair plus three-dimensional space." This is $D_{\rm geo}$: geometric underdetermination is this section's conclusion, not a regret — it locates precisely where the $k$ ledger is open (the adjudication of the constraint set), rather than vaguely declaring failure. Two further notes: first, "$2=n_{\rm dual}\times1$" (each face contributing one $T^2$ flow tilt) is motivated numerology, not a geometric derivation — it gives at most a readout multiplicity $\lambda=2$, $k=1$, of different identity from $k=2$, $\lambda=1$, the two degenerate only in the product; second, this section earns only the first-order response, the cubic response $k_3$ is not earned, and every cubic-order conclusion in §6 and Appendix D is therefore conditional on $k_3$. 〔$D_{\rm geo}$: structural honesty; numerology candidate: motivated〕

The exchange-parity classification. Under face exchange ($\varepsilon\to-\varepsilon$) all derived quantities of this framework's dark sector share one parity structure: the directed parent is exchange-odd of $O(\varepsilon)$ — $\Lambda_{\rm or}\propto\omega_2^2-\omega_1^2$, $a_{0,\rm or}\propto\omega_2-\omega_1$, the directed tilt $\widetilde\varphi$, and $g_{\rm df}$ (under the analyticity assumption of Theorem 3.3, leading term $O(\varepsilon)$); what is physically measured is its modulus, exchange-even. The sign itself is not an observable dynamical quantity but a label for "which side we are on" — a classification that both organizes the magnitude/sign accounting of §6 and grounds the parity of the face-level orientation-ledger reading of §8. 〔structural〕

§6 The wedge: a minimal EFT candidate and its underdetermination

Having sealed off "the difference of two independent stress tensors" in the previous section, this section gives one explicit construction in the surviving gap and measures honestly what it does and does not supply. It has two layers: Layer one is a bulk existence witness — a proof that a bounded-below, local wedge vacuum-energy functional can be written down; Layer two states how this paper's final architecture (§8) uses it — inheriting only its internal invariant as matching data on the transition surface. The two are not conflated: layer one proves "it can be written," layer two is the position actually adopted.

6.1 The surplus problem and the marked decomposition

First, why a general pair of positive-sign fluxes is unacceptable. With flux normalization $f_i=\mathcal N M_P\omega_i/c$ (the original value of this series' cosmological-constant installment being $\mathcal N=\sqrt2$), the ratio of the sum of two independent positive-sign 4-forms to the observed value is $$\frac{\rho_{\rm sum}}{\rho_\Lambda^{\rm pub}}=\frac{\mathcal N^2(1+\varepsilon^2)}{8\varepsilon}\;,\tag{6.1}$$ about $20$ at $\mathcal N=\sqrt2$, about $40$ at $\mathcal N=2$. This is the positive-flux surplus problem: unless the ledger nearly cancels somehow, the model is off by more than an order of magnitude at leading order. 〔structural, coefficient dictionary-dependent〕

The object of this section is therefore not two independent 4-forms but two marked components of a single rank-four internal top-form multiplet. For the exterior derivative and the vacuum energy to be mathematically clean, this paper locks the internal bundle to a fixed, trivial, flat-connection Euclidean bundle $E=M\times V_{\rm int}$ ($V_{\rm int}\cong\mathbb R^4$): then $B\in\Omega^3(M)\otimes V_{\rm int}$, $F=dB$ is legitimate (under a flat connection the covariant exterior derivative is the ordinary $d$), and the internal directions $e_1,e_2\in V_{\rm int}$ in the flux decomposition $\Phi_1=f_1e_1$, $\Phi_2=f_2e_2$ are global internal background sections, the marked projection an explicit construction input. This lock is deliberate: a general non-trivial internal bundle would require a connection $\mathcal D$, internal curvature, and the corresponding gauge covariance, substantially widening this paper's jurisdiction and introducing near the degeneracy point a transverse gauge dynamics not treated here; a trivial flat bundle avoids these, at the cost of recording "why the internal bundle is trivial and flat" as a construction input. This internal fiber is not newly invented: the block time axes of Four-Forces VII are objects in the internal configuration space (the stage of their $S^3$ packing), not spacetime directions; the two cosmological sectors each inherit one internal time axis, antiparallel at the symmetric point, and the frequency asymmetry offsets them from exact antiparallel by the defect mechanism of §5. Flag at once the construction's first open item: at the strict antiparallel point $e_2=-e_1$ the decomposition of the total flux $\Phi=(f_1-f_2)e_1$ degenerates, and the total field alone cannot reconstruct $f_1f_2$ and the wedge — the marked decomposition itself (two labeled components, a face index, or a rank-one projection) is a construction input, its microscopic origin listed in §10. 〔open〕

6.2 Conventions, action, and variation

Take signature $(-,+,+,+)$; dual scalars $\Phi_i=\star F_i$; and, under a metric variation at fixed covariant $F_i$, $\delta\Phi_i=\tfrac12\Phi_i\,g_{\mu\nu}\delta g^{\mu\nu}$. Layer one's action is $$S_F=+\int d^4x\,\sqrt{-g}\;\mathcal U_\lambda(\Phi_1,\Phi_2),\qquad \mathcal U_\lambda=\tfrac12\big|\Phi_1+\Phi_2\big|^2+\lambda\big|\Phi_1\wedge\Phi_2\big|,\quad\lambda\ge0,\tag{6.2}$$ the norm and wedge taken from the internal bundle's Euclidean structure. The positive sign is forced: the canonical single 4-form kinetic term $-\tfrac1{48}F^2=+\tfrac12\Phi^2$, and (6.2) must agree with it in the $\lambda=0$, single-component limit. $\mathcal U_\lambda$ is quadratically homogeneous in the fluxes; the Euler relation $\sum_i\Phi_i\,\partial\mathcal U/\partial\Phi_i=2\mathcal U$ together with the metric variation of $\delta\Phi_i$ gives $$T_{\mu\nu}=-\,\mathcal U_\lambda\,g_{\mu\nu},\qquad \rho=\mathcal U_\lambda\ge0.\tag{6.3}$$ This is a genuine $w=-1$ vacuum source, placed into the bulk density and obtained through the complete metric variation — provided $e_i$ are metric-independent internal boundary data. Three qualifications on (6.3) must be stated together. First, bounded below, not strictly positive-definite: along the equal-amplitude strictly-antiparallel direction $\Phi_2=-\Phi_1$ any common amplitude gives zero energy — a zero-energy flat flux direction, not a propagating ghost, but a microscopic-completion problem to be kept for §10. Second, what this paper calls "the witness's health" means only bounded-below and free of local propagating ghosts; it does not mean the microscopic completion, spectral stability, and membrane sector are closed, so throughout it is called "a bounded-below, ghost-free existence witness" with no stronger claim. Third, a hard cutoff at the degenerate core: as the system approaches the strict-antiparallel zero of (6.4), with $e_1$ and $e_2$ tending to collinear, $\mathcal U_\lambda$ itself contains no term protecting the positivity of the internal transverse (non-collinear) perturbation-mode kinetics; so this construction's range as a bulk existence witness is cut off above $\varphi\sim O(\varepsilon)>0$, and the behavior at the degenerate core ($\rho\to0$) must be given an infrared cutoff by higher-derivative terms or by the transition-surface completion mechanism (§10) — this cutoff being itself listed as a sub-item of $D_{\rm int}$, not claimed as solved. 〔structural (construction exists); health as a qualified definition; degenerate-core cutoff open〕

6.3 Three objects kept apart

The three quantities below carry different jobs in this paper; conflating any two would pass a conditional result off as unconditional, so they are separated here.

The wedge magnitude (and its odd parent) $\rho_{\rm wedge}=\lambda f_1f_2|\sin\widetilde\varphi|\ge0$; its exchange-odd parent is $\lambda f_1f_2\sin\widetilde\varphi$ (odd, $O(\varepsilon)$), the physical magnitude being its modulus (even, $O(|\varepsilon|)$). Its zero set is {$e_1\parallel e_2$, $e_1\parallel-e_2$, either flux zero}; it does not require equal amplitude.

The total physical vacuum energy $\rho_{\rm tot}=\mathcal U_\lambda$. For $f_i>0$, $\lambda\ge0$, within this existence witness, $$\rho_{\rm tot}=0\iff \Phi_2=-\Phi_1\iff\text{equal amplitude and strictly antiparallel}.\tag{6.4}$$ The job of (6.4) must be delimited precisely: it gives the geometric zero-point part of the ledger axiom — the witness functional's magnitude zero holds iff the two marked fluxes are equal-amplitude and the internal directions strictly antiparallel (in clock language, equal frequency and no defect angle); it is not the full identity of $\Lambda_1+\Lambda_2=0$. The signed two-face cancellation (dual-pair zero-sum, $\mathscr L_{n,+}+\mathscr L_{n,-}=0$) must be carried separately by the face-level matching law (§2, §8), not given by (6.4). 〔structural, within the witness〕

The orientation ledger $\mathscr L_{n,\pm}=\pm\rho_n$. The sign of the two faces of the dual pair is carried by the face index $\pm$, the zero-sum $\mathscr L_{n,+}+\mathscr L_{n,-}=0$ being an identity within one cycle (§2); it lives on the transition surface (§8), not in this section. The cross-cycle magnitude evolution $\rho_{n+1}=\mathcal M(\rho_n,\Xi_{1,n},\Xi_{2,n})$ is a candidate matching law, awaiting a complete boundary functional. Every harvest of this section is about magnitude, not sign.

6.4 Branch condition and harvest

Now state what layer one supplies — with the branch condition placed first. The current symmetry permits $\lambda=0$; if $\lambda=0$, the total energy starts at $O(\varepsilon^2)$. So every harvest below is confined to the minimal non-trivial wedge branch ($\lambda>0$, $k\neq0$), which the framework permits but does not yet force — exactly the other face of the coming $D_{\rm int}$, two faces of one coin that must be presented to the reader together.

On that branch: the physical magnitude starts at $O(|\varepsilon|)$ — the exchange-odd parent $\lambda f_1f_2\sin\widetilde\varphi$ is $O(\varepsilon)$, its modulus (the physical vacuum energy) exchange-even of $O(|\varepsilon|)$; it is non-negative, each side seeing a positive value under its own label; the total-energy zero (within the witness) is given by (6.4). 〔structural within the minimal non-trivial wedge branch〕

The parity cost of the linear order must be stated too. A smooth, exchange-invariant action can contain only even functions of the directed wedge $W$, starting at $W^2=O(\varepsilon^2)$; the linear order admits only two routes — take $|W|$ (the route taken here: at the cost of a cusp at the symmetric point, whose physical isolation condition is that $\varepsilon$ is cycle-by-cycle boundary data rather than a bulk dynamical quantity that can cross zero), or introduce another exchange-odd object $\chi$ and write $\chi W$ (smooth, but $\chi$ is then an orientation spurion migrated into the internal fiber). The shared lesson of the two routes is: the discrete orientation bit is not annihilated by the wedge, only given a healthier home. 〔structural〕

6.5 The matching invariant and two unclosed ledgers

Substituting the directed first-order response $\widetilde\varphi=k\varepsilon+O(\varepsilon^3)$ from §5, the wedge leading term is $\rho_{\rm wedge}=\lambda\mathcal N^2 k\,\varepsilon\,M_P^2\bar\omega^2/c^2+O(\varepsilon^3)$; against the observed relation $\rho_\Lambda^{\rm pub}=8\,\varepsilon\,M_P^2\bar\omega^2/c^2$, the matching condition is $$\boxed{\;\mathcal N^2\lambda k=8\;}\tag{6.5}$$ — one observational constraint pressing on three ledgers: flux normalization $\mathcal N$, geometric tilt coefficient $k$, wedge readout $\lambda$, none of the three independently derived. Writing the exchange-covariant expansion with $F=f_1+f_2$, $D=f_2-f_1$: $$\rho_{\rm wedge}^{\rm phys}=\frac{2}{\mathcal N^2}\,|FD|+O\!\Big(\frac{|D|^3}{F}\Big);\tag{6.6}$$ the rest of the paper is fixed on the $D>0$ observational branch, where the absolute value drops to $\rho_{\rm wedge}=\frac{2}{\mathcal N^2}FD+O(D^3/F)$. The cubic correction $-\frac{6+k^2}{3\mathcal N^2}\frac{D^3}{F}$ holds only under the minimal-angle ansatz ($\varphi=kD/F$ exact, no cubic response $k_3$), becoming $-\frac{6+k^2}{12}\frac{D^3}{F}$ at $\mathcal N=2$; $k_3$ is not earned in this paper, details in Appendix D.

The state of the three ledgers, honestly, one by one (the observational matching contours are Figure 3). $k$: geometrically underdetermined ($D_{\rm geo}$) — §5 already showed several natural cosmological constraint sets giving different first-order coefficients (candidate table: $1$, $\pi/\sqrt2$, $\pi\sqrt3/2$), the required $2$ (at $\mathcal N=2$) not jumping out of "one clock pair plus three-dimensional space." $\lambda$: constructively underdetermined ($D_{\rm int}$) — the one-parameter family $\mathcal U_\lambda$ satisfies, for any $\lambda\ge0$, single-metric, gauge invariance, exact shift, exchange-evenness, boundedness below, and ghost-freedom simultaneously: no symmetry or identity distinguishes $\lambda=0.9$, $1$, $2$ under the current inputs. The once-hoped Cauchy–Schwarz pinning does not hold either: that identity governs $|W|^2$ rather than the linear $|W|$ in the action; and one orientation does not uniquely fix a complex structure — the space of compatible complex structures is $SO(4)/U(2)\cong\mathbb{CP}^1$. $\mathcal N$: not derived — it is the dictionary between clock frequency and flux, its microscopic origin (§10) deciding whether (6.5) can be written $\lambda k=2$.

Under the three conditions of a minimal $\mathcal U_\lambda$ ansatz, fixed $\mathcal N$, and no independent operator at the same order, the next-order relative correction $\propto(4+k^2)\mathcal N^2\varepsilon/16$ depends only on $k$ and can in principle break the degeneracy of (6.5); it is indistinguishable within the roughly $5\%$ numerical deviation between the existing formula and observation (this $5\%$ being the matching error between the SAE formula and the Planck-derived cosmological constant, not Planck's own observational uncertainty, which is far finer). In a general EFT the discriminant additionally requires first proving there is no independent operator at the same order. The $SO(4)/U(2)\cong\mathbb{CP}^1$ used here (the space of compatible complex structures) is a standard fact, its proof in the Appendix C note.

This subsection's closing sentence is deliberately plain: the matching relation is at present the simplest numerical target, but it is still a consilience — $\mathcal N$ is a dictionary and $\lambda$ a Wilson coefficient, two of the three ledgers not purely geometric — and so falls short of prior enforcement. The paper's result is not that it derived the $2$, but that it proved precisely that observation fixes only the single combination $\mathcal N^2\lambda k$, and located the underdeterminacy of each factor (geometric, constructive, dictionary) separately.

6.6 Layer two: the boundary-completion candidate

This paper's final architecture (§8) does not take (6.2) as the bulk's final action — that would put the two marked components back into the bulk of every breath. To forbid object drift, first fix an object dictionary. Define the clock-sourced transition-surface boundary internal vectors $$\Xi_{i,n}\equiv\mathcal N\,\frac{M_P\,\omega_{i,n}}{c}\,e_{i,n}\quad(i=1,2),$$ which are not independent stress tensors, and not the two 4-forms in cycle $n$'s bulk — they are boundary source data given on the transition surface by the two clocks and their internal axes. Accordingly, layer two inherits only the internal invariant of the layer-one functional, as matching data on the Crunch–Bang transition surface: $$\tfrac12 f_{n+1}^2=\mathcal W_\lambda\big(\Xi_{1,n},\,\Xi_{2,n}\big),\qquad \mathcal W_\lambda(\Xi_1,\Xi_2)\equiv\tfrac12|\Xi_1+\Xi_2|^2+\lambda|\Xi_1\wedge_{\rm int}\Xi_2|,\tag{6.7}$$ i.e. the wedge magnitude of the previous cycle's tilted internal pair (expressed through the boundary vectors $\Xi_{i,n}$) imprints the size of the single physical 4-form flux $f_{n+1}$ of the next cycle. Objects each in their place: $(\tau_i,\omega_i,e_i)$ are the two clocks and their internal axes; $\Xi_i$ is the clock-sourced boundary data on the transition surface; $f_n$ is cycle $n$'s single physical bulk flux; $\chi_{n,s}=s$ ($s\in\{+1,-1\}$) is the face orientation label within cycle $n$. The complete boundary functional $\mathcal B_\Sigma$ producing (6.7) — its constraint-surface map, symplectic preservation, and uniqueness — remains open (§10). Layer one proved "such a quantity can exist healthily," layer two states "where this paper places it"; the former is a theorem, the latter a candidate architecture, and the two are not conflated. 〔layer one: structural (construction exists); layer two: motivated architecture, component states tagged one by one in §8〕

§7 Pairing, the temporal junction, and the thin-wall benchmark

If the signed ledger has dynamics, its carrier is a membrane that changes the flux. This section handles three things: the jump algebra of pairing membranes (exact), what geometry forces the pairing mode (a conditional theorem), and a qualitative partition of bulk nucleation (a benchmark). Self-limitation: this section does not complete model-specific nucleation rates, determinants, and globalization; only a thin-wall benchmark partition.

Statement of this section's status (first). The jump algebra and thin-wall benchmark below diagnose a formal signed-bulk comparison branch that this paper rejects as its final foundation — the branch in which two fluxes are co-located and couple to gravity through their difference — not the single-positive-flux-plus-boundary-imprint architecture adopted in §8. It is nonetheless spelled out because it delimits precisely "what one would meet going the bulk-discharge route," thereby motivating the necessity of the boundary reading; but its numerical conclusions are not dynamical predictions of the boundary-completion candidate.

Jump algebra (exact identity). With $F=f_1+f_2$, $D=f_2-f_1$, the directed flux difference $\widetilde\rho_{\rm diff}\equiv\tfrac12(f_2^2-f_1^2)=\tfrac12FD$ (a quantity distinct from the orientation ledger $\mathscr L_{n,s}=s\rho_n$ of §2, hence a different name, and the section speaks of no "ledger" further) — this is an algebraic identity on the fluxes, dictionary-independent. Let a pairing membrane in the comparison branch send $(f_1,f_2)\mapsto(f_1+q_1,\,f_2+q_2)$:

pairing	$(q_1,q_2)$	$\Delta F$	$\Delta D$	$\Delta\widetilde\rho_{\rm diff}$	step
antisymmetric	$(q,-q)$	$0$	$-2q$	$-qF$ (exact)	$O(1)$, no $\varepsilon$ suppression
symmetric	$(q,+q)$	$+2q$	$0$	$+qD$ (exact)	$O(\varepsilon)$

The antisymmetric jump preserves the mean flux and consumes the difference directly: at $q_*=D/2$ one jump clears the current positive difference exactly — the natural candidate for turnaround/discharge. But the same algebra at once exposes its lack of a stopper: $F$ is unchanged under an antisymmetric jump, so at fixed $q$ each jump has constant $\Delta\widetilde\rho_{\rm diff}=-qF$, and the next jump past zero does not slow — runaway is not an engine, at most an engine part without a governor; a stopper must come from elsewhere (a finite charge reservoir, nucleation permitted only on a specific surface, or bubble-interior collapse cutoff). The symmetric jump instead preserves the difference and drifts the mean frequency: $\Delta(\omega_1+\omega_2)=2qc/(\mathcal N M_P)$ while $\omega_2-\omega_1$ is unchanged, so $\varepsilon$ drifts slightly per jump — the natural carrier of a spiral cross-cycle parameter drift. The internal geometry supports this division of labor: with the two internal directions near antiparallel, $|e_1+e_2|=2\sin(\varphi/2)\approx\varphi=O(\varepsilon)$ while $|e_1-e_2|=2\cos(\varphi/2)\approx2$; if the membrane charge is an internal vector $q\,(e_1\pm e_2)$, the symmetric charge is naturally $\varepsilon$-small and the antisymmetric one naturally $O(1)$ — "spiral small step, discharge large step" is not an imposed assignment but the modulus of the internal geometry. 〔jump algebra: structural; geometric support of the division: structural, conditional on §6's internal construction〕

Proposition 7.1 (a fixed point forces symmetric pairing). If the transition surface $W$ is the fixed-point set of the exchange involution $\sigma$, and the membrane's WZ action $q_1\!\int_W\!B_1+q_2\!\int_W\!B_2$ is invariant under exchange, then $q_1=q_2$. Assumptions: $\sigma$ a $t\mapsto-t$ reflection; $W=\{t=0\}$ fixed pointwise; the exchange realizing $B_1\leftrightarrow B_2$. Proof sketch: write $\Omega_M=dt\wedge\Omega_W$, normal $n\propto dt$; under reflection $\Omega_M\mapsto-\Omega_M$, $n\mapsto-n$, and the induced orientation $\Omega_W=\iota_n\Omega_M$ satisfies $\iota_{-n}(-\Omega_M)=\iota_n\Omega_M$ — the fixed surface's intrinsic orientation does not flip; so $\int_Wi^*B_1\mapsto\int_Wi^*B_2$ with no minus sign, and exchange invariance forces $q_1=q_2$. Escape routes outside the domain: a timelike bulk membrane — if one additionally adds the geometric condition "exchange reverses its worldvolume intrinsic orientation" (not guaranteed by "the two sides' arrows of time are opposite" alone, and to be written as an explicit assumption), then it forces antisymmetric $(q,-q)$; a mirror membrane pair at $\pm t_0$ — exchange symmetry guarantees only that each event has a mirror partner, a symmetric charge spectrum, and equal instanton actions, not co-nucleation in one history (which requires a $\sigma$-invariant state, a boundary-condition enforcement, or defining the two surfaces as one non-local composite junction). 〔structural, conditions explicit〕

Boundary pairing map (a P0-level statement of type). Proposition 7.1's WZ term is written with $B_1,B_2$, but in the final single-bulk-flux architecture (§8) there is only one physical flux $f_n$, and $B_1,B_2$ are not fields of that architecture; so one must give a typed boundary map, not carry the §7 jump algebra acting on $(f_1,f_2)$ into the final architecture. Let the pairing action on the transition surface be $$S_\Sigma^{\rm pair}=\int_\Sigma\big(q_1\,b_{1,\Sigma}+q_2\,b_{2,\Sigma}\big),$$ where $b_{i,\Sigma}$ are three-dimensional potentials on the transition surface associated with the two boundary internal directions $e_{i,n}$ (whether they are the pullback of some bulk 3-form or purely boundary degrees of freedom belongs to the open content of $\mathcal B_\Sigma$). This map must answer four questions to support a spiral: what $b_{i,\Sigma}$ is; whether it is the pullback of a bulk field; why a single-bulk-flux architecture still has two boundary charge channels (candidate answer: the two channels correspond to the two directions of the boundary internal pair $\Xi_{1,n},\Xi_{2,n}$, not two bulk fluxes); and how $q_1=q_2$ yields a drift of $\bar\omega$ through this map. Until this map is written in full, the most honest treatment is: the fixed-point exchange forcing equal charge ($q_1=q_2$, Proposition 7.1) is retained as a structural conclusion, while "equal charge causes a $\bar\omega$ spiral drift" is downgraded to a candidate awaiting a complete matching law. 〔equal charge: structural; equal charge → drift: candidate (map unclosed)〕

This gives the paper's most honest net picture, three rows kept apart: symmetric $(q,+q)$ pairing — the equal charge forced by the fixed-point exchange (Proposition 7.1), the boundary map to an $\omega_i$ drift (spiral) unclosed, hence a candidate; antisymmetric $(q,-q)$ discharge — living on a timelike bulk membrane or another non-fixed transition defect, residence undetermined; turnaround dynamics — not directly triggerable by a fixed surface at the cycle's end, open. A transition surface at a cycle's end at most imprints the next cycle's initial flux and lets the next cycle carry slightly different parameters; it is not the dynamical trigger of this cycle's turnaround. 〔three-row table: first row equal-charge structural, its drift a candidate; second and third rows open〕

Naming the temporal junction. The WZ integral $q\int_WB$ on a spacelike surface, as a directed three-dimensional boundary integral, remains real, and its variation with respect to $B$ does give a flux jump across time; the problematic piece is the tension term — a standard timelike membrane's $-\sigma\int\sqrt{-\det h}$ becomes imaginary on a spacelike induced metric, and spacelike branes in string theory must be defined otherwise, by a double Wick rotation or the like (Gutperle & Strominger 2002). So this paper does not invoke the name "S-brane nucleation," but calls this object a temporal junction: a real boundary-generating functional that gives the canonical momentum a jump across that time surface; it is not a standard causal membrane worldvolume, and does not automatically carry the usual probabilistic interpretation of a Coleman bounce or a membrane tension. This naming in fact favors the §8 architecture: a boundary formulation need not pretend it is a nucleable entity in the bulk. 〔methodological〕

Qualitative partition of the thin-wall benchmark (diagnosing only the rejected signed-bulk branch). Reiterating this section's opening status statement: the following benchmark diagnoses only the signed-bulk branch this paper rejects as its final foundation, and is not a dynamical prediction of the boundary-completion candidate; correspondingly, statements like "if bulk nucleation freezes then this cycle's $\Lambda$ is constant" are redundant in the final architecture — the final architecture simply has no such bulk membrane channel, and the constancy of $\Lambda$ within one breath follows directly from the single flux not being changed by a bulk membrane. If an antisymmetric membrane does exist in that comparison branch, its nucleation partitions in the tension–charge plane into three regions: a low-tension region with no exponential suppression or too many events per breath (over-active); a very narrow "about once per breath" engine band; and above it, frozen (thin-wall gravitational effects following Coleman & De Luccia 1980). Three qualitative conclusions can be stated without depending on details: first, that the membrane energy scale required by the engine band lands near the dark-energy scale is a dimensional necessity ($\sigma^{1/3}=O(\Delta V^{1/4})$, once the transition energy difference is taken at the observed dark-energy scale), not an independent coincidence of this framework; second, any membrane tension significantly above that scale freezes bulk nucleation extremely, and under the usual high-energy EFT intuition, freezing is a robust default (a conditional statement: absent a prior parameter measure, "generic" is not a mathematical theorem); third, even one local bubble per breath does not automatically equal a homogeneous turnaround of the whole universe — bubble expansion, percolation, and the time dependence of the nucleation rate are all globalization problems, open. The specific numbers ($\sigma$ thresholds, bandwidth, gravitational-blocking scale), together with all their assumptions, are in Appendix D; none of them is a structural constant. 〔qualitative trichotomy: structural (at formula level); freezing default: conditional; globalization: open〕

§8 The minimal consistent architecture and revisions to earlier installments

The architecture, three layers apart. This paper's minimal consistent architecture candidate is as follows (Figure 4), the epistemic status of each component tagged in place. Proven: two independent, canonical, co-located 4-form sectors, each coupling to the shared metric through a standard stress tensor, cannot realize a signed difference ledger (the §4 net verdict). Constructed: an internal top-form wedge can exist as a bounded-below local functional, giving on the non-trivial branch a positive vacuum magnitude of $O(|\varepsilon|)$ (§6 layer one). Open: how the transition-surface matching uniquely fixes the flux magnitude and the orientation ledger from the clocks' boundary data (§6 layer two, §10).

Atop these three layers, the architecture itself: the two clocks co-located in the bulk — $C=\tau_1-\tau_2$ remains local, the dark-energy and MOND sectors untouched; each cycle's bulk carries only one healthy positive-sign ledger flux $f_n$, bounded below with no runaway channel; that flux's magnitude imprinted by the wedge of the previous cycle's tilted internal pair on the Crunch–Bang transition surface (eq. 6.7); the sign of the two faces of the dual pair carried by the transition surface's face-index orientation ledger, $\mathscr L_{n,+}+\mathscr L_{n,-}=0$ being a zero-sum identity within one cycle. In this architecture the axiom $\Lambda_1+\Lambda_2=0$ is carried solely by the dual-face zero-sum ledger within one cycle ($\mathscr L_{n,+}+\mathscr L_{n,-}=0$, the two faces of one cycle equal-amplitude antiparallel, (6.4) giving its zero-point geometry); it is no longer a cancellation of two stress tensors in one bulk. The cycle-by-cycle inheritance/drift of the magnitude is a separate, independent cycle-to-cycle map candidate ($\rho_{n+1}=\mathcal M(\rho_n,\Xi_{1,n},\Xi_{2,n})$), not part of that identity; collapsing "the other-side label" and "the next-cycle label" into one index was an error, now strictly separated. The constancy of $\Lambda$ within one breath follows directly from the single flux having no bulk membrane channel to change it in the final architecture (not by invoking the nucleation freezing of the rejected signed-bulk branch). The equal charge of the symmetric pairing $(q,+q)$ is forced by the fixed-point exchange (Proposition 7.1), but its boundary map to an $\omega_i$ drift (spiral) is not yet closed (see the P0-level statement and $S_\Sigma^{\rm pair}$ above), so the spiral as the action-level carrier of a cross-cycle drift is downgraded to a candidate awaiting a complete matching law; the residence of the antisymmetric discharge and the trigger of the turnaround remain open per the §7 three-row table. 〔architecture: motivated; components: each per the three layers above; spiral carrier: candidate (map unclosed)〕

Revisions to earlier installments. This paper's results require explicit revision of several earlier statements in this series; registered one by one, in the discipline "corrected, and marked."

Zeroth installment §8 (the sign of the cosmological constant). That installment treated "the sign of the $\Lambda$ seen by the next cycle." This paper's magnitude/sign accounting and wedge mechanism give it a mechanism-level reinforcement: the physical magnitude $\rho_{\rm phys}\ge0$ makes each side always see a positive value under its own label (Proposition 4.5 of §4 supplies a theorem-level floor — an observable positive-definite quadratic vacuum energy is blind to orientation reversal, what flips being an orientation-odd object), and the sign is a label of the orientation ledger, not a dynamical quantity. Accordingly, any statement in that installment attributing "a sign flip" to "gravity appearing repulsive when viewed from reversed time" — if present, and in conflict with that installment's own "time reversal does not change the direction of acceleration" statement — should be deleted, replaced by this paper's $\rho_{\rm phys}\ge0$ modulus plus $\chi$ orientation-label framework; any selection-effect observation independent of that step may be kept as independent support. This registration is to be implemented in that installment's subsequent version, the specific edit list adjudicated verbatim against its final version.

Cosmological-constant installment (this series' first). The flux–frequency dictionary is generalized to $f_i=\mathcal N M_P\omega_i/c$, the working value $\mathcal N=2$ tagged as a construction input. Its epistemic status must be stated in two parallel clauses: no change at the observational level — the numerical relation $\Lambda_{\rm obs}=2(\omega_2^2-\omega_1^2)/c^2$ with Planck is kept as is; prediction status suspended at the realization level — the matching relation $\mathcal N^2\lambda k=8$ is locked by the observational target, and the three factors are none of them independently derived, so that agreement is at present a target relation plus an existing consilience; the no-parameter prediction status is restored only once the combination $\mathcal N^2\lambda k$ is independently closed by microscopic structure without using the $\Lambda$ observation (usually requiring all three fixed, or another independent relation exhausting the remaining freedom).

Two-frame installment (this series' fifth). Its conformal machinery ($\mathscr A(C)$, $F(C)R$) migrates wholesale to the shift-breaking branch, this paper's exact-shift baseline neither using nor refuting it; that installment's residual $15\%$ Hubble-constant deviation is diagnosed here as a symptom of the fact that "the action-level route of $\Lambda_1$ into the equations matter sees was never completed" — it neither confirms nor refutes any specific route.

Dark-sector installment (this series' sixth). Its informal reading "two co-located 4-forms with $\Lambda_1+\Lambda_2=0$" is superseded by this section's precise reading (a single per-cycle flux + transition-surface orientation ledger + zero-point geometry); no change at the observational level. Its EFT classification of $g$ (free, $\propto\varepsilon$) is upgraded by this paper to an action-level theorem group plus a constructive-underdetermination proof (§3).

Four-Forces installment VII (concept DOI 10.5281/zenodo.19433220). The terminology erratum (the §5 triangle disambiguation: the quantity named "cross-axis tilt angle" is geometrically the intra-pair antipodality defect) is carried by that installment's own version upgrade; this paper merely cites its Theorem 2.2 vector solution with a note (the vector solution and the definitions of the three angles are restated in Appendix C, so the reader need not consult the original to check the §5 disambiguation), and does not substitute an appendix for its version revision.

§9 Nested falsifiable hypotheses

The falsifiable structure of the effective-equation-of-state sector is stated as a three-layer nesting (the decision tree is Figure 5); what the background data directly reconstruct is the effective expansion history, not any single field's intrinsic $w_{\rm int}=p_C/\rho_C$, so the three layers' adjudicating data differ.

hypothesis	$g$	background prediction	adjudicator
minimal baseline	$0$	$w_{\rm eff}=w_{\rm int}\ge-1$, no crossing	background data alone: a robust background crossing falsifies this row
interacting branch	free; $g_{\rm df}$ odd in $\varepsilon$, leading term $O(\varepsilon)$ under a unique odd spurion and analyticity (Theorem 3.3)	$w_{\rm eff}$ may cross while $w_{\rm int}\ge-1$	joint background + growth + lensing + ISW; precondition: once the linear perturbation equations of this branch are completed and its unavoidable growth/lensing/ISW relations locked (not done in this paper), if observation sees only a background crossing and not the corresponding perturbation combination, this row is excluded
intrinsic phantom	not permitted by the positive-sign DBI baseline	true $p_C/\rho_C<-1$	requires a joint model discrimination; the expansion history alone cannot reconstruct intrinsic $w$

Three sentences close the table: background data adjudicate the minimal baseline; background plus perturbations adjudicate the interacting branch; "intrinsic phantom" is not a direct observable of the expansion history. This structure neither passes the free branch off as a prediction nor lets it be an unfalsifiable escape hatch — the branch's freedom is the underdeterminacy proved in §3, and its adjudication is the combination locked by the table above.

What this installment does not touch, and what is in principle adjudicable, each noted once. HP1 stands: the high-redshift evolution of the acceleration scale ($a_0\propto cH$, adjudicated at $z>3$) is unaffected by any result of this paper, its adjudication discipline holding as in this series' third installment. The matching degeneracy is in principle breakable: under the three conditions of a minimal wedge ansatz, fixed $\mathcal N$, and no independent operator at the same order, the vacuum energy's next-order relative correction $\propto(4+k^2)\mathcal N^2\varepsilon/16$ depends on $k$ and not only on the product $\mathcal N^2\lambda k$ — so the degeneracy of (6.5) can in principle be broken by a joint $\Lambda$-plus-period anchor at sub-percent precision; indistinguishable within the roughly $5\%$ numerical deviation between the existing formula and observation (this $5\%$ a matching error, not Planck's observational uncertainty), noted as such. The closed-space requirement stands: the framework still needs a globally closed three-sphere; the current curvature measurement is highly close to flat, placing a lower bound on the closure radius rather than directly excluding topological closure — this requirement and pressure restated as in this series' zeroth installment.

§10 Conclusion and the keys to the next layer

This paper's answer to the dual-4DD realization problem closes, at its true strength, into three results. The negative result is a theorem: a signed vacuum ledger cannot live in the difference of two independent canonical stress tensors in a healthy single-metric bulk; and the khronon–$C$ mixing coefficient is constructively underdetermined within the locked baseline and operator classification. The constructive candidate: the internal tilt (the first-order antipodality defect given by the upstream published packing vector solution) plus the wedge readout gives a bounded-below, ghost-free existence witness, whose magnitude starts on the non-trivial branch at $O(|\varepsilon|)$ and is always non-negative; the total witness functional's zero is the two marked components equal-amplitude strictly antiparallel ($\varepsilon=0$ making the magnitude vanish), while the axiom $\Lambda_1+\Lambda_2=0$ is carried solely by the dual-pair two-face zero-sum within one cycle ($\mathscr L_{n,+}+\mathscr L_{n,-}=0$); the cycle-by-cycle inheritance/drift of the magnitude is an independent cycle map candidate $\rho_{n+1}=\mathcal M(\rho_n,\Xi_{1,n},\Xi_{2,n})$, not part of that identity; the minimal consistent architecture places the magnitude at a transition-surface imprint and the sign in a face-level orientation ledger. The underdeterminacy measurement is the paper's real result: we did not derive the factor $2$ — we proved precisely that observation fixes only the single combination $\mathcal N^2\lambda k$, and located the underdeterminacy of each factor — dictionary ($\mathcal N$), geometric ($k$), constructive ($\lambda$) — as an identifiable open problem. A framework's honesty lies not in how much it claims to have derived, but in whether it can draw a checkable boundary between "given by structure" and "still dependent on normalization, response, and readout."

The keys to the next layer, listed without pretending they are already turned: the uniquification of $k$ (adjudicating the cosmological clock pair's packing constraint set); the pinning of $\lambda$ (including "what structure forces the non-trivial wedge branch"); the microscopic origin of $\mathcal N$ (which decides whether the central relation is $\lambda k=2$ or the general form); the cubic tilt response $k_3$ (the premise of every cubic-order conclusion); the structure of the marked decomposition and the internal kinetic matrix (why the cross-term coefficient is one); the smoothing or spurion completion of the $|W|$ cusp; the linear perturbation stability of the internal axes and the wedge sector; the complete matching functional $\mathcal B_\Sigma$ (including the derivation of the cycle map candidate $\rho_{n+1}=\mathcal M(\rho_n,\Xi_{1,n},\Xi_{2,n})$, the closure of the boundary pairing map $S_\Sigma^{\rm pair}$, the constraint-surface map, symplectic preservation, and uniqueness); the nucleation rate and the globalization of the turnaround; whether the symmetric transition surface and the antisymmetric discharge require two different defects; and the version upgrade of Four-Forces installment VII. These are not the paper's oversights but the boundary it draws.

Appendix A · Claim–tier master table

claim	tier
the two open items belong to one realization class	structural architectural classification (not theorem-level; supported by the no-go group in retrospect)
Theorem 3.1 (no bulk source)	structural
Theorem 3.2 ($J\!\cdot\!\nabla C$ operator has no independent background source on exact FLRW)	structural (domain-limited)
Theorem 3.3 ($g$ odd; linear leading term under analyticity)	structural (three assumptions each load-bearing)
Proposition 3.4 (bulk Wilson permission + constructive underdetermination; $[g_{\rm df}]=M^3$)	structural; microscopic origin open
HP2 first layer $w_{\rm int}\ge-1$	structural (a hard bound within the positive-sign, ghost-free DBI baseline)
HP2 second layer = a provably underdetermined bulk Wilson branch	structural (classification); branch value free
Propositions 4.1–4.7	structural (each with assumption box)
wrong-sign top-form: no perturbative ghost, unbounded below	structural
net verdict (narrowed)	structural
defect angle $\varphi_{\rm pair}=\sqrt2\,\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)$ (first order)	structural (inheriting the Four-Forces VII linearized vector solution)
triangle disambiguation and term attribution	registrational
first-order response form $\widetilde\varphi=\sqrt m(a_++a_-)\varepsilon+O(\varepsilon^3)$	structural (form); coefficient open
$k$ candidate table; required value not automatic	open ($D_{\rm geo}$)
"$2=n_{\rm dual}\times1$"	motivated (numerology)
exchange-parity classification (odd parent / even modulus / sign is a label)	structural
wedge witness exists; $\rho=\mathcal U\ge0$	structural (construction exists; called "a bounded-below, ghost-free witness")
zero-energy flat direction; degenerate core needs an IR cutoff	structural (qualified) + open (a $D_{\rm int}$ sub-item)
$\rho_{\rm tot}$ zero (equal-amplitude strictly antiparallel, within the witness, eq. 6.4)	structural
$\Lambda_1+\Lambda_2=0$ = same-cycle dual-face zero-sum $\mathscr L_{n,+}+\mathscr L_{n,-}=0$	inherited axiom / ledger definition; its action-level boundary realization: open
internal bundle locked to trivial flat $E=M\times V_{\rm int}$	construction input
object dictionary $\Xi_{i,n}$ (boundary internal vector, not bulk flux)	definition
magnitude $O(	\varepsilon	)$ start within the non-trivial branch	conditional structural ($\lambda=0$ permitted, branch not forced)
parity trichotomy; orientation bit persists	structural
$\mathcal N^2\lambda k=8$	posterior matching relation (not a derivation)
the three ledgers $k$, $\lambda$, $\mathcal N$ unclosed	open
next-order discriminant (depends on $k$ under three conditions)	structural (domain-limited); currently indistinguishable
jump algebra ($-qF$, $+qD$)	structural (exact identity)
Proposition 7.1 (fixed-point symmetric pairing)	structural (conditions explicit)
three-row table (spiral candidate / discharge residence undetermined / turnaround open)	structural + open
temporal junction naming	methodological
thin-wall trichotomy; meV a dimensional necessity; freezing default	structural (formula) / conditional (default)
cycle map candidate $\rho_{n+1}=\mathcal M(\cdots)$ / equal-charge → spiral map	motivated (candidate, map unclosed)
cosmological-constant installment realization-level prediction status suspended	registrational
nested hypothesis structure	structural (classification)
HP1 untouched	structural (an established lock, not touched in this installment)

Appendix B · Proposition-dependency and assumption audit

proposition	assumptions	conclusion	mode of failure
quadratic wall (4.2/4.5)	single metric; positive-definite kinetics; local quadratic form	cannot give a signed difference	drop positive-definite kinetics (top-form special case: no perturbative ghost but unbounded below); odd spurion; move to boundary; non-local
covering obstruction (4.7)	$C$ an odd section of $L_t$; exact additive shift	connected cover incompatible	reduce to a monodromy shift; or confine non-triviality to the transition core (the boundary-matching reading)
bookkeeping theorem (4.3)	well-posed variation of a fixed-flux ensemble	on-shell coefficient not a physical factor	none; a methodological theorem
causal type (4.4)	discharge membrane timelike; khronon leaf spacelike	cannot be the same surface	an explicit projection factor (a new coefficient); or a temporal junction (a non-standard nucleable membrane)
constructive underdetermination (3.4)	baseline; operator classification; vacuum normalization fixed	$g_{\rm df}$ not fixed ($[g_{\rm df}]=M^3$)	a complete microscopic action locking the zero-derivative and derivative orders in a single expansion (an interface being only one possible source); or leaving the baseline
$J\!\cdot\!\nabla C$ operator no background source (3.2)	exact FLRW	that operator has no independent background $\theta Q$	the perturbative level / an inhomogeneous background keeps the $J\!\cdot\!\nabla C$ mixing
wedge witness (6.2–6.3)	marked components; $\lambda\ge0$; $e_i$ metric-independent; quadratic homogeneity	a bounded-below $w=-1$ source	the origin of the marked decomposition; the cusp variation; the positivity of the axis kinetics when dynamized; perturbation stability
fixed-point pairing (7.1)	a reflection fixed surface; $B_1\leftrightarrow B_2$	$q_1=q_2$	a timelike membrane + a worldvolume-orientation-reversal condition → antisymmetric; a mirror pair → co-nucleation must be enforced by a state / boundary condition

Appendix C · A brief note on Four-Forces installment VII

The vector solution and the three angles (a self-contained restatement, for independent checking). Four-Forces VII (concept DOI 10.5281/zenodo.19433220) Theorem 2.2's convex-optimization symmetric reference configuration is $\hat t_a^\pm=\pm e_a$ (three antiparallel axis pairs, $a=1,2,3$; strictly antiparallel); an intra-pair frequency asymmetry $\delta$ makes the cone half-angles $\beta^\pm=\tfrac\pi4(1\pm\delta)$ and makes the symmetric configuration violate cross-axis no-overlap; in the linearized closest-packing problem of the $S_3$-symmetric ansatz, the nearest legal representative vector is unique (the display vectors accurate only to the stated perturbative order, not the full nonlinear packing solution), and is $$\hat t_a^\pm\propto\pm e_a-\frac{\varepsilon_{\rm pack}}2\sum_{b\neq a}e_b,\qquad \varepsilon_{\rm pack}=\frac\pi2\delta.$$ From this first-order vector solution, three geometrically distinct angles to be named separately (all accurate to $O(\varepsilon_{\rm pack}^2)$): (i) the single-block off-axis angle $\theta_{\rm off}=\varepsilon_{\rm pack}/\sqrt2+O(\varepsilon_{\rm pack}^2)$ ($\hat t_a^+$ to axis $e_a$); (ii) the intra-pair antipodality defect $\varphi_{\rm pair}=\sqrt2\,\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)$ ($\hat t_a^+$ to $-\hat t_a^-$; the geometric angle of the normalized first-order representative $=2\arctan(\varepsilon_{\rm pack}/\sqrt2)$; the two members each yield to $\theta_{\rm off}$ in opposite transverse directions, hence twice); (iii) the cross-axis angular-distance increment $\Delta\theta_{\rm cross}=\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)$ (its first-order value equal to the no-overlap threshold shift, the saturation criterion). The first-order ratio $\theta_{\rm off}:\varphi_{\rm pair}:\Delta\theta_{\rm cross}=1:2:\sqrt2$ (first order), not to be mixed. The quantity named "cross-axis tilt angle" in that text, $\sqrt2\varepsilon$, is geometrically the second (defect), not the third; the existing numerical readout ($c_1=\pi/(3\sqrt2)$ and $R_1$) uses precisely that quantity's ($\sqrt2\varepsilon$'s) magnitude — misusing the off-axis angle $\varepsilon/\sqrt2$ would give a $\pi/6$-type different number inconsistent with the existing readout — so this disambiguation changes the name, not the number. The formal erratum is carried by that installment's version upgrade.

On $SO(4)/U(2)\cong\mathbb{CP}^1$ (used in §6.5). The space of complex structures $J$ on $\mathbb R^4$ compatible with a fixed Euclidean metric ($J^2=-1$, $J$ metric-preserving): $SO(4)$ acts transitively on such $J$, with stabilizer $U(2)$, so the quotient is $SO(4)/U(2)$; the latter is diffeomorphic to $\mathbb{CP}^1\cong S^2$. The point is that it is non-trivial (dimension 2), so "given an orientation" (a $\mathbb Z_2$ bit) is far from enough to determine a $J$ uniquely — which is the geometric content of the Cauchy–Schwarz pinning's failure in §6.5.

Appendix D · Numerical anchor table (each number with its assumptions)

quantity	value	assumptions
$\varepsilon$	$0.0127$	locked parameters ($T_1,T_2$)
$\bar\omega$	$5.4\times10^{-61}\,M_P$	$T_1=20$ Gyr
surplus ratio	$\approx20$ ($\mathcal N=\sqrt2$); $\approx40$ ($\mathcal N=2$)	$\mathcal N^2(1+\varepsilon^2)/(8\varepsilon)$; dictionary-dependent
clearing charge $q_*$	$\mathcal N\varepsilon\bar\omega M_P/c$ ($\approx1.4\times10^{-62}M_P^2$ at $\mathcal N=2$)	benchmark: one jump clears the current ledger
engine-band tension	$\sigma^{1/3}\approx1.1$ meV	thin wall, flat bounce, one-loop-type prefactor, $V_4\simeq2\pi^2H^{-4}$, $q=q_$; not a structural constant*
bandwidth	$\sim0.4\%$ (tension)	$B\propto\sigma^4$ sensitivity; prefactor uncertainty widens it
gravitational blocking	$\sigma_c\approx2\times10^{-61}M_P^3$	thin-wall Israel expanding-true-bubble branch; other branches not excluded
meV identity	$\sigma_B^{1/3}=(2B/27\pi^2)^{1/12}\Delta V^{1/4}$	a dimensional necessity, not an independent coincidence
cubic correction	$-\tfrac{6+k^2}{3\mathcal N^2}\tfrac{D^3}F$ ($\mathcal N=2$: $-\tfrac{6+k^2}{12}\tfrac{D^3}F$)	a double condition: minimal-angle ansatz (no $k_3$)
next-order discriminant	$(4+k^2)\mathcal N^2\varepsilon/16$; $+2\varepsilon\approx+2.5\%$ at $(k,\lambda,\mathcal N)=(2,1,2)$	three conditions: minimal ansatz, fixed $\mathcal N$, no independent operator at the same order; indistinguishable within the $5\%$ band
direct Four-Forces VII inheritance total deviation	$1.139$ (+13.9%)	$(k,\lambda,\mathcal N)=(\pi/\sqrt2,1,2)$; six embedding assumptions unproven

Figure list (separate files)

Figure 1 (§4) The implementation problem and the no-go chain: the constraint set (L)–(S) at top, the seven propositions as nodes, each tagged with an assumption-box clause and an escape-route destination. Figure 2 (§5) The defect-angle geometry: left, the strict-antiparallel reference configuration; right, the tilted solution; the three angles labeled (off-axis $\varepsilon/\sqrt2$, defect $\sqrt2\varepsilon$, cross-axis increment $\varepsilon$). Figure 3 (§6) Observational matching contours: two slices ($\mathcal N=\sqrt2$ and $\mathcal N=2$) of the single constraint surface; the four candidate $k$ points marked only on the $\mathcal N=2$ slice. Figure 4 (§8) The architecture: a bold divider separating the bulk existence witness from the adopted boundary-completion candidate; two clocks and a single flux in the bulk; on the junction, the orientation ledger $\chi$, the wedge imprint eq. (6.7), and the symmetric pairing; the antisymmetric discharge dashed on the rejected signed-bulk branch and marked "residence undetermined." Figure 5 (§9) The nested-hypothesis decision tree: three hypotheses each to its adjudicating data; the interacting branch's perturbation precondition marked as a gate node.

References

External literature

Israel, W. (1966). Singular Hypersurfaces and Thin Shells in General Relativity. Il Nuovo Cimento B, 44(1), 1–14. DOI: 10.1007/BF02710419.
Coleman, S., & De Luccia, F. (1980). Gravitational Effects on and of Vacuum Decay. Physical Review D, 21(12), 3305–3315. DOI: 10.1103/PhysRevD.21.3305.
Brown, J. D., & Teitelboim, C. (1988). Neutralization of the Cosmological Constant by Membrane Creation. Nuclear Physics B, 297(4), 787–836. DOI: 10.1016/0550-3213(88)90559-7.
Bousso, R., & Polchinski, J. (2000). Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant. JHEP, 2000(06), 006. DOI: 10.1088/1126-6708/2000/06/006.
Gutperle, M., & Strominger, A. (2002). Spacelike Branes. JHEP, 2002(04), 018. DOI: 10.1088/1126-6708/2002/04/018.
Skordis, C., & Złośnik, T. (2021). New Relativistic Theory for Modified Newtonian Dynamics. Physical Review Letters, 127(16), 161302. DOI: 10.1103/PhysRevLett.127.161302.
Planck Collaboration (2020). Planck 2018 Results VI: Cosmological Parameters. Astronomy & Astrophysics, 641, A6. DOI: 10.1051/0004-6361/201833910.

SAE series (by this author on Zenodo)

Qin, H. (2026). SAE Cosmology 0: One Complete Cosmic Breath. Zenodo. DOI: 10.5281/zenodo.21319569.
Qin, H. (2026). SAE Cosmology I: The Cosmological Constant. Zenodo. DOI: 10.5281/zenodo.19245267.
Qin, H. (2026). SAE Cosmology III: From Λ to a₀. Zenodo. DOI: 10.5281/zenodo.19281983.
Qin, H. (2026). SAE Cosmology V: Two Frames and Λ₁+Λ₂=0. Zenodo. DOI: 10.5281/zenodo.19329771.
Qin, H. (2026). SAE Cosmology VI: The Aether-Scalar-Tensor Dark Sector. Zenodo. DOI: 10.5281/zenodo.21267555.
Qin, H. (2026). SAE Four Forces VII: The Three-Layer Correction Structure (S³ Packing Theorem). Zenodo. DOI: 10.5281/zenodo.19433220.

(The Zenodo records are the authoritative versions of the SAE series.)

Acknowledgments

The framework direction, the core arguments, and the final adjudication of all claims are the responsibility of Han Qin (秦汉). In the course of writing, the author made use of several artificial-intelligence systems for multiple rounds of adversarial derivation-checking, conceptual organization, geometric-consistency checking, and constraint-based critique, with additional systems participating in divergent discussion and external evaluation; these systems' contributions are confined to auxiliary derivation, checking, and criticism, and do not constitute authorship of the paper's scientific claims. Several of this paper's negative results (the no-go proposition group and the constructive-underdetermination proof) were in fact established through exactly this adversarial-checking process; any errors in their statement are the author's.

Author responsibility statement

All claims of this paper, their tier tags (structural / posterior / motivated / open), the three-way accounting discipline (prior-derived / construction input / posterior boundary), and every open problem and possible error flagged, are the responsibility of the author. The constructive part of the paper (the wedge witness and the architectural synthesis) is a mechanism candidate, its coefficients flagged honestly at the underdetermined state of $\mathcal N^2\lambda k=8$, and entirely possibly to be corrected or falsified by later work; the author presents it in full, in the spirit of "not seeking to be without fault, but seeking to be of use," so as to give later research a scrutinizable, falsifiable target. Any oversight or error is the author's.

几何输入	$k$	备注
相空间 $T^2$ 等权流	$1$	$\tan\varphi_{T^2}=\varepsilon$ 精确;但依赖两钟等权平直度量,是自然候选而非拓扑定理
四力篇 VII 直接继承	$\pi/\sqrt2\approx2.221$	须六条嵌入假设:同一三对 packing、$\varepsilon$ 与 $\delta$ 同归一、锥半角仍 $\pi/4$、同组约束饱和、dual-pair 约束同富余、objective 度量不变——均未证
时间轴对三空间轴让步	$\pi\sqrt3/2\approx2.721$	$m=3$、$\beta_0=\pi/4$ 子类
匹配条件点	$2$	由 $\mathcal N^2\lambda k=8$ 仅得 $\lambda k=2$($\mathcal N=2$);$k=2$ 尚须另取 $\lambda=1$,见 §6 式 (6.5)

假说	$g$	背景预测	裁判
最小 baseline	$0$	$w_{\rm eff}=w_{\rm int}\ge-1$,不穿越	背景数据即可:稳健的背景穿越证伪本行
相互作用分支	自由;$g_{\rm df}$ 对 $\varepsilon$ 为奇函数,在唯一奇 spurion 且解析时首项 $O(\varepsilon)$(定理 3.3)	$w_{\rm eff}$ 可穿越而 $w_{\rm int}\ge-1$	背景 + 增长 + 透镜 + ISW 联判;前置条件:一旦完成该分支的线性扰动方程并锁定其不可避免的增长、透镜与 ISW 关系(本文未做此推导),若观测只见背景穿越而不见相应扰动组合,则排除本行
内禀 phantom	正号 DBI baseline 不容许	真 $p_C/\rho_C<-1$	须模型联合判别;单凭膨胀史不能重构内禀 $w$

主张	层级
两开放项属同一实现类	结构性架构分类(非定理级,由 no-go 群回溯支持)
定理 3.1(bulk 无源)	结构性
定理 3.2(精确 FLRW 上 $J^\mu\nabla_\mu C$ 算符无独立背景源)	结构性(域限定)
定理 3.3($g$ 奇性;解析下线性首项)	结构性(三假设逐条为要)
命题 3.4(bulk Wilson 允许性 + 构造性欠定;$[g_{\rm df}]=M^3$)	结构性;微观来源开放
HP2 第一层 $w_{\rm int}\ge-1$	结构性(正号非鬼 DBI baseline 内的硬边界)
HP2 第二层 = 被证明欠定的 bulk Wilson 分支	结构性(分类);分支取值自由
命题 4.1–4.7	结构性(各带假设盒)
错号 top-form:无微扰鬼、无下界	结构性
净判(收窄版)	结构性
亏损角 $\varphi_{\rm pair}=\sqrt2\,\varepsilon_{\rm pack}+O(\varepsilon_{\rm pack}^2)$(一阶)	结构性(承四力篇 VII 线性化向量解)
三角辨析与术语归属	登记性
一阶响应形式 $\widetilde\varphi=\sqrt m(a_++a_-)\varepsilon+O(\varepsilon^3)$	结构性(形式);系数开放
$k$ 候选表;匹配所需值不自动	开放($D_{\rm geo}$)
"$2=n_{\rm dual}\times1$"	动机性(数字学)
ε-宇称分类(奇母量/偶模/符号即标签)	结构性
wedge 见证存在;$\rho=\mathcal U\ge0$	结构性(构造存在;称"有下界、无局域传播鬼的见证")
零能平坦方向;退化内核须红外切断	结构性(限定)+ 开放($D_{\rm int}$ 子项)
$\rho_{\rm tot}$ 零点(等幅严格反平行,witness 内,式 6.4)	结构性
$\Lambda_1+\Lambda_2=0$ = 同循环 dual-face 零和 $\mathscr L_{n,+}+\mathscr L_{n,-}=0$	继承公理/账本定义;其作用量级边界实现:开放
内部丛锁定为平凡平直 $E=M\times V_{\rm int}$	构造输入
对象字典 $\Xi_{i,n}$(边界内部向量,非 bulk 通量)	定义
非零分支内量值 $O(\lvert\varepsilon\rvert)$ 起始	条件性结构($\lambda=0$ 被容许,分支未强制)
宇称三岔口;取向比特存续	结构性
$\mathcal N^2\lambda k=8$	后验匹配关系(非推导)
$k$、$\lambda$、$\mathcal N$ 三账未闭	开放
次阶判别式(三条件下依赖 $k$)	结构性(域限定);当前不可分辨
跳变代数($-qF$、$+qD$)	结构性(精确恒等式)
命题 7.1(不动点对称配对)	结构性(条件明标)
三行表(螺旋候选/放电住所未定/回转开放)	结构性 + 开放
temporal junction 正名	方法论
薄壁三分区;meV 为量纲必然;冻结默认	结构性(公式)/条件性(默认)
架构综合	动机性(构件分标)
cycle 匹配候选 $\rho_{n+1}=\mathcal M(\cdots)$ / 等荷→螺旋映射	动机性(候选,映射未闭)
宇宙学常数篇实现层预测身份暂停	登记性
嵌套假说结构	结构性(分类)
HP1 不动	结构性(既定锁定,本篇不触碰)

命题	假设	结论	失效方式
二次墙(4.2/4.5)	单度规;正定 kinetic;局域二次型	不能给符号差	放弃正定 kinetic(top-form 特例:无微扰鬼但无下界);奇 spurion;移边界;非局域
覆盖障碍(4.7)	$C$ 为 $L_t$ 奇截面;精确加性平移	连通覆盖不相容	降为 monodromy 平移;或非平凡性限于过渡核心(边界匹配读法)
簿记定理(4.3)	固定通量系综的良定变分	on-shell 系数非物理因子	无;方法论定理
因果型(4.4)	放电膜类时;khronon 叶类空	不能同面	显式投影因子(新增系数);或 temporal junction(非标准核化膜)
构造性欠定(3.4)	baseline;算符分类;真空归一已定	$g_{\rm df}$ 未被固定($[g_{\rm df}]=M^3$)	同一展开锁定零导数阶与导数阶的完整微观作用量(界面只是可能来源之一);或离开 baseline
$J\!\cdot\!\nabla C$ 算符无背景源(3.2)	精确 FLRW	该算符无独立背景 $\theta Q$	扰动层/非均匀背景保留 $J\!\cdot\!\nabla C$ 混合
wedge 见证(6.2–6.3)	标记分量;$\lambda\ge0$;$e_i$ 度规无关;二次齐次	有下界的 $w=-1$ 源	标记分解来源;cusp 变分;轴动力学化时的动能正定;扰动稳定性
不动点配对(7.1)	反射不动面;$B_1\leftrightarrow B_2$	$q_1=q_2$	类时膜 + 世界体取向反转条件 → 反对称;镜像对 → 须态/边界条件强制共同核化

量	值	假设
$\varepsilon$	$0.0127$	锁参($T_1,T_2$)
$\bar\omega$	$5.4\times10^{-61}\,M_P$	$T_1=20$ Gyr
超额比	$\approx20$($\mathcal N=\sqrt2$);$\approx40$($\mathcal N=2$)	$\mathcal N^2(1+\varepsilon^2)/(8\varepsilon)$;字典依赖
清零荷 $q_*$	$\mathcal N\varepsilon\bar\omega M_P/c$($\mathcal N=2$ 时 $\approx1.4\times10^{-62}M_P^2$)	benchmark:一跳清零当前账本
引擎带张力	$\sigma^{1/3}\approx1.1$ meV	薄壁、平直反弹、一圈型 prefactor、$V_4\simeq2\pi^2H^{-4}$、$q=q_$;非结构常数*
带宽	$\sim0.4\%$(张力)	$B\propto\sigma^4$ 灵敏度;prefactor 不确定性会放宽
引力阻断	$\sigma_c\approx2\times10^{-61}M_P^3$	薄壁 Israel 膨胀真泡分支;不排除其他分支
meV 身份	$\sigma_B^{1/3}=(2B/27\pi^2)^{1/12}\Delta V^{1/4}$	量纲必然,非独立巧合
三次修正	$-\tfrac{6+k^2}{3\mathcal N^2}\tfrac{D^3}F$($\mathcal N=2$:$-\tfrac{6+k^2}{12}\tfrac{D^3}F$)	双重条件:最小角度 ansatz(无 $k_3$)
次阶判别	$(4+k^2)\mathcal N^2\varepsilon/16$;$(k,\lambda,\mathcal N)=(2,1,2)$ 时 $+2\varepsilon\approx+2.5\%$	三条件:最小 ansatz、$\mathcal N$ 固定、无同阶独立算符;当前 5% 窗内不可分辨
四力篇 VII 直搬总偏差	$1.139$(+13.9%)	$(k,\lambda,\mathcal N)=(\pi/\sqrt2,1,2)$;六条嵌入假设未证

配对	$(q_1,q_2)$	$\Delta F$	$\Delta D$	$\Delta\widetilde\rho_{\rm diff}$	步长
反对称	$(q,-q)$	$0$	$-2q$	$-qF$(精确)	$O(1)$,无 $\varepsilon$ 压低
对称	$(q,+q)$	$+2q$	$0$	$+qD$(精确)	$O(\varepsilon)$

The Single-Metric Realization Problem for Dual-4DD: A No-Go Group, an Internal-Wedge Candidate, and a Transition-Surface ImprintDual-4DD 单度规实现问题——带符号账本的 no-go、内部楔积候选与过渡面烙印

§1 Introduction: two open items, one realization problem

§2 Epistemic contract, baseline, and two ledgers

§3 The origin of θQ and the HP2 hierarchy

§4 The no-go group for a signed ledger

§5 Tilt: the antipodality defect and geometric underdetermination

§6 The wedge: a minimal EFT candidate and its underdetermination

6.1 The surplus problem and the marked decomposition

6.2 Conventions, action, and variation

6.3 Three objects kept apart

6.4 Branch condition and harvest

6.5 The matching invariant and two unclosed ledgers

6.6 Layer two: the boundary-completion candidate

§7 Pairing, the temporal junction, and the thin-wall benchmark

§8 The minimal consistent architecture and revisions to earlier installments

§9 Nested falsifiable hypotheses

§10 Conclusion and the keys to the next layer

Appendix A · Claim–tier master table

Appendix B · Proposition-dependency and assumption audit

Appendix C · A brief note on Four-Forces installment VII

Appendix D · Numerical anchor table (each number with its assumptions)

Figure list (separate files)

References

Acknowledgments

Author responsibility statement

§1 引言:两个开放项,一个实现问题

§2 认识论契约、baseline 与两本账

§3 θQ 的起源与 HP2 的分层

§4 带符号账本的 no-go 群

§5 倾角:反平行亏损角与几何欠定

§6 wedge:最小 EFT 候选与欠定性

6.1 超额问题与标记分解

6.2 约定、作用量与变分

6.3 三个对象的分立

6.4 分支条件与收获

6.5 匹配不变量与两本未闭账

6.6 层二:边界完成候选

§7 配对、temporal junction 与薄壁 benchmark

§8 最小一致架构与对前篇表述的修订

§9 嵌套可证伪假说

§10 结论与下一层钥匙

附录 A · 主张—层级总表

附录 B · 命题依赖与假设审计表

附录 C · 四力系列篇 VII 术语简注

附录 D · 数值锚表(每数带假设)

图件清单(独立文件)

参考文献

致谢

作者责任声明

The Single-Metric Realization Problem for Dual-4DD: A No-Go Group, an Internal-Wedge Candidate, and a Transition-Surface Imprint
Dual-4DD 单度规实现问题——带符号账本的 no-go、内部楔积候选与过渡面烙印