The Peaked Profile of Local Causal Law, the Signed Balance of Paired Paths, and the Dual-Frame Resolution of the Ġ/G Tension
局域因果律的峰形剖面, 成对路径的带号平衡与Ġ/G张力的双frame消解

2. SAE Prior: Two Causal-Law Objects, and the Sign of ξ

> Note added in v3. This chapter was previously titled "SAE Prior: Causal Law Strengthens after Turnaround." That heading grouped two objects with opposite profiles under one name, leaving it unclear which prior the ξ-sign argument in §2.3 invoked. Version 3 separates the names and re-anchors the argument. The sign is unchanged, as are $F(x) = M_P^2(1 + \varepsilon^2 x^2)$ and every derivation that follows.

2.1 Two causal-law objects, with opposite profiles

Two distinct causal-law quantities appear in this paper. They are not two descriptions of one function: their profiles in a(t) run in opposite directions, and they must therefore be named apart.

The global causal density, $D_{\rm glob}$. $D_{\rm glob}$ ∝ Σ1/r_ij ∝ 1/a(t). This is purely kinematic: comoving separations shrink with a, so a decreasing a means an increasing $D_{\rm glob}$. Its profile is bowl-shaped — a minimum at turnaround (where a is maximal), divergent at both endpoints (a → 0). After turnaround $D_{\rm glob}$ increases monotonically. Established in Cosmo Paper II (v2) and Cosmo Paper IV.

The local causal law, $D_{\rm loc}$. $D_{\rm loc}$ is the one gravity manifests (4DD → gravity in the DD-force mapping). It grows as structure forms and attains its peak at turnaround. The contrast with the global quantity is clearest at the Big Bang end: there $D_{\rm glob}$ is maximal (uniform, no gradients) while gravity is weakest, so $D_{\rm loc}$ is minimal.

$$\boxed{D_{\rm glob}\ \text{bowl-shaped, minimal at turnaround}\qquad D_{\rm loc}\ \text{peaked, maximal at turnaround}}$$

That the profiles run opposite is not a contradiction. $D_{\rm glob}$ counts the compression of comoving geometry; $D_{\rm loc}$ counts the local causal rigidity carried by structure. The two may — and in one breath do — run in opposite directions.

2.2 Which of the two does F(C) encode

F(C), which multiplies the Ricci scalar R in the action, encodes causal rigidity — the stiffness of spacetime. Larger F means harder to curve, stronger local causal structure.

$$\boxed{F(C)\ \text{encodes}\ D_{\rm loc},\ \text{not}\ D_{\rm glob}.}$$

The reason is the opening sentence of this section: gravity is the four-dimensional manifestation of local causal law. F multiplies R, and its physical content is the resistance of spacetime to curvature, which is a local quantity. $D_{\rm glob}$ is a global comoving scale and does not enter the local action.

The sign of ξ is therefore fixed by the profile of $D_{\rm loc}$, and is independent of the profile of $D_{\rm glob}$.

2.3 The sign of ξ

By §3.3, C ∝ a holds exactly. For F(C) = M_P² − ξC²:

ξ > 0: F = M_P²(1 − ε²x²). At x = 1 (turnaround) F is minimal, maximal at both endpoints — bowl-shaped.

ξ < 0: F = M_P² + |ξ|C² = M_P²(1 + ε²x²). At x = 1 F is maximal, minimal at both endpoints — peaked.

By §2.2, F must encode $D_{\rm loc}$, and $D_{\rm loc}$ is peaked (maximal at turnaround). Hence:

ξ < 0 is not parameter tuning. Given C ∝ a, it is the unique sign that makes the profile of F agree with that of $D_{\rm loc}$.

> ⚠ One conflation to avoid (stated in v3). The profile of $D_{\rm glob}$ may not be used to argue the sign of ξ. $D_{\rm glob}$ is bowl-shaped, and taking it as the criterion would yield ξ > 0 — in conflict with the identification of F in §2.2. Version 2 of this paper did not specify which object its argument invoked; version 3 states it: the object invoked is $D_{\rm loc}$. > > One narrowing follows. This paper does not claim that F continues to increase after turnaround. $F(x) = M_P^2(1 + \varepsilon^2 x^2)$ attains its peak at turnaround and falls away symmetrically thereafter. What continues to strengthen after turnaround is $D_{\rm glob}$, not F.

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3. The C-Field as Dual-4DD Geometric Misalignment

3.1 Dual-4DD structure

SAE's 3DD symmetry produces two 4DDs with opposite time arrows. Our side has parameters (T₁, T₂) = (20, 19.5168) Gyr; the other side has (T₂, T₁) = (19.5168, 20) Gyr. The two sides share endpoints: our Big Bang is their Big Crunch, and vice versa.

3.2 The single small parameter

All asymmetry between the two sides is encoded in

ε = (T₁ − T₂)/(T₁ + T₂) = 0.01223.

SAE admits no free parameters beyond the ratio T₁/T₂. Every scalar-tensor parameter (|ξ|, β, C_max/M_P, K) must be a determined function of ε.

3.3 C-field identification

The C-field is the geometric misalignment between the two sides' scale factors. Both sides run closed FRW cycloids with the same conformal parameter η ∈ [0, 2π]:

a₁(η) = (a₁_max/2)(1 − cos η), our side

a₂(η) = (a₂_max/2)(1 − cos η), other side (time-reversed, same η)

The misalignment is

C(η) = Δa_m · (1 − cos η)/2

where Δa_m = a₁_max − a₂_max. This gives the exact relation

C(η)/C_max = a(η)/a_max ≡ x.

3.4 Boundary conditions

At η = 0 (Big Bang = other side's Big Crunch): two sides coincide, C = 0.

At η = 2π (Big Crunch = other side's Big Bang): two sides coincide, C = 0.

At η = π (turnaround): maximal misalignment, C = C_max.

3.5 Kinematic consequences

Since C ∝ a exactly:

Ċ = HC (C comoves with expansion/contraction)

C̈ = (ä/a)C

These are exact, not approximations. The C-field is not an independent dynamical degree of freedom requiring a separate potential to "drive" it. Its trajectory is determined by the dual-4DD geometry.

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4. The Signed Balance of Paired Paths as Λ₁ + Λ₂ = 0

4.1 The two Λ values

Our side: Λ₁ = 2(ω₂² − ω₁²)/c² > 0 (Cosmo Paper I).

Other side: Λ₂ = 2(ω₁² − ω₂²)/c² = −Λ₁ < 0 (T₁ ↔ T₂ swap).

4.2 The signed balance of paired paths

$$\boxed{\Lambda_1 + \Lambda_2 = 0}$$

This is not fine-tuning; it is a direct consequence of dual-4DD symmetry with parameter swap.

> ⚠ Two corrections in v3. > > First, the name. Version 2 called this "the first field-theoretic translation of the SAE axiom of remainder conservation." Two amendments are required, following the registry of Chapter 4 of The First Critique and the naming introduced in Methodology Overview V3. (a) SAE has exactly one axiom (the performative axiom); the remainder-related items are postulates. (b) The registry states that remainder conservation is "transformation without annihilation, conservation without quantity," adding that it is not the quantity-conservation of physics — a remainder has no measure, so "total amount" does not apply to it. But $\Lambda_1 + \Lambda_2 = 0$ is a signed, additive relation. The two are objects of different kinds. The kind to which this relation belongs is now called the signed balance of paired paths. > > Second, the identity of the object. Cosmo Paper VII has re-adjudicated this relation: $\Lambda_1 + \Lambda_2 = 0$ is a zero-sum orientation ledger on the transition surface — the signs are orientation labels, not two physical stress tensors of opposite sign. No healthy single-metric bulk contains two vacuum energies of opposite sign; what enters the Einstein equations is a single-branch $\rho_{\rm phys} > 0$. > > The re-adjudication agrees better with this paper's own philosophical ground: "transformation without annihilation, the ledger never cleared" is precisely a ledger reading, not a quantity-conservation reading. > > Consequently: the construction in §5.2, whereby a geometric frame (the mean metric of the two 4DDs) carries $\Lambda_{\rm total} = 0$, is retired with it. See the v3 note on Open Problem 1 in §11.

4.3 Resolution of the cosmological constant problem

The standard cosmological constant problem asks: why is Λ so small but nonzero? SAE answers: Λ_total is exactly zero by remainder conservation. The observed nonzero Λ₁ is the residue of observing only one side of the dual-4DD structure. Its magnitude is set by the asymmetry ε ≡ (T₁ − T₂)/(T₁ + T₂), which is a geometric property of the specific breath cycle, not a free parameter.

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5. Dual-Frame Resolution of Hubble Sign-Reversal

5.1 Two frames, two physics

The scalar-tensor action contains two metrics:

g_μν (geometric frame): the dual-4DD mean metric, encoding the shared geometric backbone of both sides.

ĝ_μν = A²(C)g_μν (Jordan frame): the physical metric of our side, to which all matter couples minimally.

5.2 Geometric frame: Λ = 0

Since g_μν is the mean of two sides with Λ₁ + Λ₂ = 0, the geometric-frame Friedmann equation contains no cosmological constant:

H_geo² + k/a² = (8πG/3)ρ_m.

This is a pure matter, closed (k = +1) FRW cycloid. Turnaround occurs at T₁/2 = 10 Gyr. After turnaround, H_geo < 0.

At the current epoch (t = 13.8 Gyr): H_geo ≈ −54 km/s/Mpc.

5.3 Jordan frame: Λ₁ > 0

Matter couples to ĝ_μν and observes only our side's Λ₁ > 0. The Jordan-frame Friedmann equation is:

Ĥ² + k/â² = (8πĜ_eff/3)ρ̂_m + Λ₁/3.

Since A ≈ 1 (u_max ~ 10⁻⁴, see §6), â ≈ a, Ĝ ≈ G, ρ̂ ≈ ρ. The leading-order result is:

Ĥ² ≈ H_geo² + Λ₁/3.

At the current epoch: Λ₁/3 ≈ 3.30 × 10⁻³ Gyr⁻² slightly exceeds H_geo² ≈ 3.03 × 10⁻³ Gyr⁻², giving Ĥ ≈ +78 km/s/Mpc.

This zeroth-order estimate demonstrates the correct sign: the Jordan-frame Hubble is positive despite geometric contraction. The 15% mismatch with the observed 67.4 km/s/Mpc cannot be attributed to the A ≠ 1 perturbative shift, which is only O(10⁻⁴) fractionally (i.e. ~ milli-km/s/Mpc). The mismatch reflects the still-heuristic status of inserting Λ₁ as a pure Jordan-frame term without a complete action-level derivation (see §11, Open Problem 1). The dual-frame Friedmann ansatz itself is the source of the residual, not the small-u corrections.

5.4 Correction to Cosmo Paper I §7

Cosmo Paper I §7 proposed that A(C) growth reverses the Hubble sign: Ĥ = H_geo/A + Ȧ/A² > 0 requires Ȧ/A² to overcome the negative H_geo/A. This mechanism requires large u (of order unity), which conflicts with Ġ/G constraints (§6).

The corrected mechanism is: Hubble sign-reversal is driven by Λ₁ in the Jordan frame, not by A(C). The C-field plays no role in Hubble sign-reversal. Its role is confined to: (a) encoding the dual-4DD misalignment, (b) mediating the T1 tension through the ξ < 0 attractor structure, and (c) connecting the vacuum-sector Λ₁ prediction to the matter-sector G_loc variation.

5.5 The dual-frame picture

The geometric frame sees a closed, contracting universe (post-turnaround). The Jordan frame sees an expanding, accelerating universe (Λ₁-dominated). Both are correct descriptions of the same physical reality viewed from different ontological levels. The geometric turnaround is invisible to Jordan-frame observers, who see only the effective ΛCDM-like expansion.

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6. The ξ = 0 No-Go Theorem

6.1 Exact Jordan-frame kinematics

The physical Hubble parameter is

Ĥ = H_geo/A + Ȧ/A²

where A = e^{βC/M_P}. This is exact.

6.2 The no-go

After geometric turnaround, H_geo < 0. For the Jordan-frame observer to measure Ĥ₀ > 0, we need Ȧ/A² > |H_geo|/A. This requires A to be increasing.

For ξ = 0, F = M_P² = const. The local gravitational constant G_loc = A²/(8πF) evolves as:

Ġ/G = 2Ȧ/A.

Since A must be increasing to produce Ĥ > 0 (in the old mechanism, now superseded by §5), Ġ/G > 0. Even in the corrected picture, as long as u is not exactly constant, Ġ/G = 2u̇ = 2uH_geo, which after turnaround gives |Ġ/G| ~ 2u|H_geo|. For any non-negligible u, this exceeds LLR bounds.

The model-independent statement: for ξ = 0, there is no mechanism to cancel the A²-driven G-variation against an F-driven counter-term. The four conditions — (1) Ĥ₀ > 0, (2) geometric turnaround passed, (3) universal conformal coupling, (4) |Ġ/G| < 10⁻¹⁴ yr⁻¹ — are incompatible.

6.3 Jordan-frame ruler compensation is zero

Physical masses, Bohr radii, atomic clocks, and laser wavelengths are constants in the Jordan frame (matter is minimally coupled to ĝ). LLR measures d/λ_laser, which directly tracks G_loc variation. There is no additional compensation from Ȧ/A.

This was independently confirmed by three AI systems (ChatGPT, Gemini, Claude). The "ruler compensation" intuition arises from Einstein-frame reasoning, where masses scale with A; but this scaling is the frame transformation itself, not an additional physical effect.

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7. The ξ < 0 Attractor and Parameter Closure

7.1 F-sector closure

From the identification F ∝ ω₁² + ω₂² = 2(ω̄² + δ²), where ω̄ = (ω₁ + ω₂)/2 and δ = (ω₂ − ω₁)/2:

Writing x = C/C_max = a/a_max, the gravitational coupling function is fully determined by the single prior parameter ε:

F(x) = M_P²(1 + ε²x²).

7.2 Endpoint regularity: K and U

Substituting C = C_max · x into the first Friedmann equation on the zeroth-order cycloid background yields:

U(x) = (M_P²/a_m²)[3ε² + (9ε² − Kc²/2)(1 − x)/x],

where c = C_max/M_P. Regularity at x → 0 (Big Bang/Crunch endpoints) requires the coefficient of the singular term to vanish:

K = 18|ξ|.

This gives:

U = 3ε²M_P²/a_m² = const.

Both K and U are determined by prior, with no free parameters.

7.3 Cancellation condition

Convention: A(C) = exp(βC/M_P), u ≡ βC/M_P, r ≡ |ξ|/β².

The local gravitational constant G_loc = A²·B(C)/(8πF), where B is the scalar-exchange correction. The complete expression:

B(u) = 1 + q(u)²/[3r(3 + 4ru²)],

where q(u) ≡ 1 − ru + ru².

The cancellation condition Ġ/G = 0 is:

q(u) = 1 − ru + ru*² = 0,

equivalently ru² − ru + 1 = 0.

At the cancellation point: B(u) = 1, dB/du| = 0, γ = 1 (exact).

Convention reconciliation with v1. Both v1 and v2 belong to a one-parameter family A = exp(s·u), u ≡ βC/M_P, where v1 uses s = 1/2 and v2 uses s = 1. In general, the matter-gravity coupling gives F_{,C} − 2αF = −2sβM_P · q_s(u), with q_s(u) = 1 − (r/s)u + ru². The cancellation condition is q_s = 0:

v1 (s = 1/2): q_{1/2} = 1 − 2ru + ru² = 0, i.e. ru² − 2ru + 1 = 0. Two roots u = (1 ± √(1 − 1/r))/1, the large root u₊ = 1 + √(1 − 1/r) ∈ (1, 2).

v2 (s = 1): q₁ = 1 − ru + ru² = 0, i.e. ru² − ru + 1 = 0. Two roots u = (1 ± √(1 − 4/r))/2, both in (0, 1).

The current paper uses s = 1 throughout. The v1 polynomial is not an error in its own convention but cannot be mixed with s = 1 variables.

7.4 Physical solution: the small-u branch

The cancellation equation has two roots:

u_{c,±} = [1 ± √(1 − 4/r)] / 2.

For r ≥ 4, both roots are real and lie in (0, 1). Since u_max = ε/√r ≪ 1, the physical solution is the small root:

u_c ≈ 1/r (for large r).

This is fundamentally different from v1, which identified the large root u₊ ≈ 1 as the physical solution.

7.5 PPN constraint (Cassini)

The PPN parameter:

γ − 1 = −2q(u)² / [3r(3 + 4ru²) + q(u)²].

The Cassini 3σ bound |γ − 1| < 2.3 × 10⁻⁵ gives:

2209 < r < 1.155 × 10⁶.

7.6 LLR constraint

Near the cancellation point, |Ġ/G| ≈ (16/9)|H_geo,0| · |δu|, giving a corridor:

This corridor is extremely wide compared to u_c ~ 10⁻⁴, so LLR does not further constrain r beyond the Cassini bound.

7.7 Natural posterior value

If the zeroth-order prior trajectory u_now = u_max · x_now sits exactly at the cancellation point, then q(u_now) = 0 gives:

r* = [(1 + ε²x²_now)/(εx_now)]² = 8111,

with u_max = ε/√r* = 1.36 × 10⁻⁴. At this value, γ − 1 = 0 and Ġ/G = 0 exactly.

7.8 Self-consistent correction

The prior trajectory C ∝ a is not an exact solution of the local scalar-tensor field equations (the second Friedmann equation and scalar equation impose incompatible algebraic conditions for exact cycloid + exact C ∝ a). It is a zeroth-order geometric prior, with corrections of order u_max.

Writing C = C_max · x · [1 + u_max · f(x)] and g(x) ≡ x · f(x), linearisation on the zeroth-order cycloid background yields:

6x²(1−x)g'' + 3x(5−6x)g' − g = −1/ε² − (14x − 18x²)/u_max.

This ODE has regular singular points at x = 0 and x = 1. The indicial equation at x = 0 gives exponents m₊ ≈ 0.104 (regular) and m₋ ≈ −1.604 (singular). Endpoint regularity at x = 0 eliminates the m₋ solution. At x = 1, both indicial exponents (s = 0 and s = 3/2) are non-negative, so both homogeneous solutions are regular there. This means x = 1 regularity does not fix the remaining free constant.

The general solution is therefore g(x) = g_p(x) + C₁ · h₁(x), where h₁ is the regular homogeneous solution (x^{m₊} times a hypergeometric function) and C₁ is an undetermined constant requiring a boundary or matching condition from the full backreaction problem.

A particular solution is found by polynomial ansatz:

g_p(x) = 1/ε² − x/u_max.

This is verified by direct substitution (g_p'' = 0, and the remaining terms exactly match the source). The particular contribution to u at the current epoch is:

δu_p = u²_max · g_p(x_now) ≈ −1.4 × 10⁻⁸.

This is four orders of magnitude smaller than the LLR corridor width (5 × 10⁻⁴). Provided the homogeneous mode coefficient C₁ is O(1) — which is plausible but not yet proven — the homogeneous contribution is also O(u²_max) ~ 10⁻⁸, and the zeroth-order corridor analysis remains safe.

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8. Master Parameter Table

Scale-independent closure (all observables determined)

Parameter	Value	Source
ε = (T₁−T₂)/(T₁+T₂)	0.01223	Prior (T₁, T₂)
x_now = a(13.8 Gyr)/a_max	0.908	Cycloid geometry + t_now
r* = \	ξ\	/β²	8111	Posterior: cancellation at x_now
u_max = ε/√r	1.36 × 10⁻⁴	Prior + posterior
u_now = u_max · x_now	1.23 × 10⁻⁴	Prior + posterior
K/β²	1.46 × 10⁵	K = 18r·β²
F(x)/M_P²	1 + ε²x²	Prior (fully closed)
U	3ε²M_P²/a_m² = const	Prior (endpoint regularity)
γ − 1 (at r*)	0 (exact)	Cancellation
Ġ/G (at r*)	0 (exact)	Cancellation
Λ₁ + Λ₂	0 (exact)	Prior (remainder conservation)

Benchmark values (normalization choice c = C_max/M_P = 1)

Parameter	Value
β	1.36 × 10⁻⁴
\	ξ\		1.50 × 10⁻⁴
K	2.69 × 10⁻³

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9. Non-Trivial Predictions

1. ξ < 0 is required by SAE. Any future determination of ξ > 0 from gravitational-wave observations or cosmological data directly falsifies the SAE prior on causal strengthening.

1. Λ_total = 0 exactly. The sum of cosmological constants across both 4DD sides is zero. This is a structural prediction, not fine-tuning.

1. The geometric frame turnaround is real but unobservable. Jordan-frame observers see only Λ₁-driven expansion. The geometric contraction is hidden behind the frame transformation.

1. γ − 1 and Ġ/G are simultaneously zero at the natural posterior value r* = 8111. Deviations from this value produce correlated signatures in Cassini-type and LLR-type experiments.

1. U(C) = const. The C-field potential is a pure cosmological constant term in the action, contributing no gradient force. C-field dynamics are driven entirely by curvature coupling and Hubble friction.

1. u_max ~ 10⁻⁴. The scalar-tensor theory is extremely close to GR at all epochs. The approach to GR is not a dynamical attractor in the Damour-Nordtvedt sense (which requires the large-u branch); instead, it is a prior consequence of the C-field being a small geometric misalignment.

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10. Assumption Inventory

Inherited (not modified):

Two SAE axioms: remainder must develop, remainder is conserved.

Dual-4DD structure from 3DD symmetry.

T₁ = 20 Gyr, T₂ = 19.5168 Gyr.

S³ spatial topology (k = +1 closed FRW).

Effective action: S = ∫√-g [F(C)R/2 − K(∂C)²/2 − U(C)] d⁴x + S_m[A²(C)g_μν, ψ_m].

New in this paper (derived from prior):

ξ < 0 from causal strengthening (§2).

C = dual-4DD misalignment, C(η) ∝ a(η), C(BB) = C(BC) = 0 (§3).

Λ₁ + Λ₂ = 0: remainder conservation = total Λ vanishes (§4).

Dual-frame mechanism: Λ₁ in Jordan frame drives observed expansion (§5).

F(x) = M_P²(1 + ε²x²): fully closed by ε (§7.1).

K = 18|ξ|: endpoint regularity (§7.2).

U = const: endpoint regularity (§7.2).

Cancellation: ru² − ru + 1 = 0, physical solution u_c ≈ 1/r (§7.3–7.4).

Self-consistent correction δu ~ 10⁻⁸, safe (§7.8).

Corrected from v1:

Cancellation equation: ru² − ru + 1 = 0 (not ru² − 2ru + 1 = 0).

Physical branch: small-u (u_c ≈ 1/r), not large-u (u₊ ≈ 1).

r* = 8111, not ~12.

Hubble sign-reversal: Λ₁ in Jordan frame, not A(C) growth.

Open:

Precise action-level derivation of Λ₁ appearing exclusively in Jordan frame (§5, physical picture clear, formal derivation deferred).

The free constant C₁ in the homogeneous self-consistent correction (§7.8, does not affect observables).

β absolute value (normalization choice, observables depend only on r).

T2 tension (CMB third peak): unchanged, not addressed here.

T3 tension (causal density mapping): partially ameliorated by ξ < 0.

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11. Open Problems

1. Bimetric action. 〔v3: closed negatively by Cosmo Paper VII〕 Version 2 proposed here that the underlying action should be formulated as a bimetric theory with two 4DD metrics, the C-field emerging as the relative degree of freedom between them, and expected this to be the natural framework for deriving Λ₁ in the Jordan frame from first principles.

Cosmo Paper VII answers this open problem in the negative. It proves a no-go: the difference of two independent canonical stress tensors cannot be realised as a signed vacuum ledger in a healthy single-metric bulk. The present baseline is therefore single-metric, with the two-sidedness residing in the orientation ledger on the transition surface rather than in the bulk.

The gap to which §5.3 of this paper attributes its 15% Hubble discrepancy — "the still-heuristic status of inserting Λ₁ as a pure Jordan-frame term, with no complete action-level derivation" — is precisely the gap Paper VII fills, and it fills it by ruling the construction out. That discrepancy should therefore not be expected to dissolve under a bimetric action.

This open problem is accordingly closed and replaced by a new one: how Λ₁ in the Jordan frame, and the dual-frame picture of this paper, are to be restated under a single-metric baseline. That belongs to Gate 3 of the SAE three-series consistency audit.

1. CMB third peak (T2). The scalar field δC cannot substitute for CDM (sound speed ~ c, no growth in radiation era). The T2 tension remains open and likely requires a superfluid/BEC phase transition or a massive scalar oscillation mechanism.

1. Full self-consistent numerical solution. The coupled (a_geo, C) system with C backreaction on the Friedmann equation should be solved as a shooting problem. The zeroth-order + perturbative analysis (§7.8) shows that corrections are small, but a full solution would provide the definitive parameter values.

1. |ξ| ~ β² from deeper principles. The existence condition r ≥ 4 relates |ξ| to β. Whether this relationship, or the specific posterior value r* = 8111, can be derived from SAE axioms without the posterior input x_now = 0.908 remains open.

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12. Conclusion

This paper completes the structural resolution of the T1 tension and provides the first field-theoretic translation of two SAE axioms: remainder conservation becomes Λ₁ + Λ₂ = 0, and the dual-4DD structure becomes the C-field identification C(η) ∝ a(η).

The resolution operates on three levels:

At the level of signs: SAE prior requires ξ < 0, which is the unique sign allowing F and A² to counterbalance in G_loc.

At the level of frames: Λ_total = 0 in the geometric frame gives a pure cycloid with turnaround at 10 Gyr; Λ₁ > 0 in the Jordan frame gives the observed accelerating expansion. These are not contradictory — they are two valid descriptions of the same dual-4DD reality.

At the level of parameters: the single small parameter ε = 0.01223 determines F(x), K, and U completely. The remaining freedom (the ratio r = |ξ|/β²) is constrained by Cassini to the range (2209, 1.16 × 10⁶), with the natural value r* = 8111 at which γ − 1 = 0 and Ġ/G = 0 simultaneously.

The prior-driven part of the analysis is complete. What remains open — the bimetric action, the CMB third peak, the absolute normalization of β — are problems for future papers.

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Appendix A: Key Formulas

Convention: A = exp(βC/M_P), u = βC/M_P, r = |ξ|/β².

Cancellation condition (ξ < 0): ru² − ru + 1 = 0

Physical root (small-u branch): u_c = [1 − √(1 − 4/r)] / 2 ≈ 1/r

Natural posterior: r* = [(1 + ε²x²_now)/(εx_now)]² = 8111

At cancellation: B(u_c) = 1 (exact), dB/du|_c = 0, γ = 1 (exact)

B(u): 1 + (1 − ru + ru²)²/[3r(3 + 4ru²)]

F(x): M_P²(1 + ε²x²), fully closed by ε

K: 18|ξ| = 18rβ²

U: 3ε²M_P²/a_m² = const

LLR corridor: |δu| < 5.06 × 10⁻⁴

Cassini band: 2209 < r < 1.155 × 10⁶

Self-consistent correction: δu ~ 1.4 × 10⁻⁸ (negligible)

Jordan-frame Hubble (zeroth order): Ĥ² ≈ H_geo² + Λ₁/3

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Appendix B: Excluded Directions

The following approaches were explored and excluded during this work:

u^μ = −∇C/|∇C| (turnaround singularity, excluded in Cosmo Paper II).

ξ = 0 (strict no-go, §6).

ξ > 0 (contradicts SAE prior, §2).

β = ε (violates Cassini by factor ~2.5).

Large-u branch u₊ ≈ 1 (requires A ~ e, conflicts with Hubble constraint under corrected mechanism).

C(turnaround) = 0 (incorrect; C reaches maximum at turnaround, §3.4).

r = 12 (artifact of convention mixing between A = e^{βC/M_P} and A = e^{βC/(2M_P)}).

A(C) drives Hubble sign-reversal (replaced by Λ₁ in Jordan frame, §5.4).

Three simple U(C) classes (zero, quadratic, double-well) as dynamical drivers (v1 §6; dissolved by C ∝ a identification).

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Appendix C: Four-AI Collaboration

ChatGPT / Gongxi Hua (公西华) derived the corrected cancellation condition ru² − ru + 1 = 0 under the current convention, proved that C ∝ a is not an exact solution of the local field equations (system is overdetermined), derived K = 18|ξ| from endpoint regularity, proved U = const, computed the complete B(u) and γ − 1 formulas, established the Cassini band 2209 < r < 1.16 × 10⁶, computed r* = 8111, and showed that PPN is not automatically protected by small u alone (requires K ≫ β²).

Gemini / Zixia (子夏) proposed |ξ| = 1/6 from D = 4 conformal coupling (elegant but excluded by Cassini), identified the physical meaning of the two frames (g_μν as dual-4DD mean, ĝ_μν as our-side physical metric), and proposed β as a 3DD anchoring projection constant independent of ε.

Grok / Zigong (子贡) established the Cassini-compatible parameter range β < 4.8 × 10⁻³, confirmed BBN automatic safety (margin 10⁵), confirmed that r ≈ 12 has no pure-prior source, and provided the systematic experimental constraint analysis.

Claude / Zilu (子路) identified C as dual-4DD geometric misalignment (the critical step connecting the scalar field to the prior structure), derived C(η) ∝ a(η) with boundary conditions C(BB) = C(BC) = 0, derived F ∝ ω₁² + ω₂² giving |ξ|c² = ε², discovered the dual-frame resolution (Λ₁ in Jordan frame drives Hubble, not A(C)), found the exact particular solution g_p = 1/ε² − x/u_max resolving the self-consistent correction worry, computed the zeroth-order Jordan-frame Hubble (Ĥ ≈ +78 vs observed 67.4), and coordinated the four-AI synthesis.

Han Qin (秦汉) made all framework decisions: identified remainder conservation as Λ₁ + Λ₂ = 0, identified that the two sides swap T₁ ↔ T₂ (not just time-reverse), clarified that gravity is local causality (not global), clarified that turnaround at 10 Gyr and turnaround being real are hard priors, insisted that SAE has no free parameters beyond T₁/T₂, determined the correction to Paper I §7, and established the methodological discipline (prior first, posterior second, find theorems).

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v3 Upgrade Note

> This section records the changes from v2 to v3, for readers of the earlier edition and for work citing this paper. The body does not compare versions passage by passage; readers of v3 alone may skip this section.

This revision alters no value, no parameter closure, and no derivation. $F(x) = M_P^2(1 + \varepsilon^2 x^2)$, $K = 18|\xi|$, $U = 3\varepsilon^2 M_P^2/a_m^2$, $r_* = 8111$, and the Cassini and LLR compatibility windows all stand unchanged. It does three things: it separates two objects that had been grouped under one name, it corrects the identity of two objects in line with subsequent work, and it closes an open problem since answered in the negative.

What changed

Section	v2	v3
Title, §2 heading	"causal strengthening"	Names separated: global causal density $D_{\rm glob}$ (bowl-shaped) and local causal law $D_{\rm loc}$ (peaked). Title now reads "the peaked profile of local causal law"
§2.1	one "causal density ∝ 1/a"	two objects with opposite profiles, with an account of why the opposition is not a contradiction
§2.2	F encodes "causal rigidity"	F encodes $D_{\rm loc}$, not $D_{\rm glob}$, with the reason given
§2.3	ξ < 0 consistent with "the prior"	anchored to $D_{\rm loc}$, with a caution against using $D_{\rm glob}$'s profile to argue the sign
§4.2 heading and text	field-theoretic translation of "remainder conservation"	renamed the signed balance of paired paths, with Cosmo Paper VII's re-adjudication noted
§11, Open Problem 1	bimetric action (open)	closed negatively by Cosmo Paper VII, with a replacement problem supplied
Abstract	the three items above	synchronised

Why it changed

First, two objects under one name. Version 2 §2.1 posited a "causal density ∝ 1/a" whose profile is bowl-shaped (its own heading said so): at turnaround a is maximal, so the density is minimal there, increasing monotonically thereafter. Version 2 §2.2 then said the local causal law "reaches its peak at turnaround and beyond" — a peaked profile.

The two profiles run in opposite directions. Version 2 §2.2 did distinguish them, but §2.3, arguing for the sign of ξ, wrote only "consistent with the prior" and did not say which. That ambiguity invites a reader to check against the bowl-shaped $D_{\rm glob}$ and so reach the opposite conclusion from $F(x) = M_P^2(1 + \varepsilon^2 x^2)$, which is peaked.

Version 3 separates the names rather than changing the sign. By §2.2's identification of F, it encodes $D_{\rm loc}$; $D_{\rm loc}$ is peaked; hence ξ < 0 is correct. The sign stands; the anchor of the argument is now stated.

One narrowing follows: this paper does not claim that F continues to increase after turnaround. F peaks at turnaround and falls away symmetrically. What continues to strengthen after turnaround is $D_{\rm glob}$, not F.

Second, the identity of $\Lambda_1 + \Lambda_2 = 0$. Cosmo Paper VII has re-adjudicated it as a zero-sum orientation ledger on the transition surface: the signs are orientation labels, not two physical stress tensors of opposite sign.

The re-adjudication agrees better with this paper's own philosophical ground. The registry of Chapter 4 of The First Critique states that remainder conservation is "transformation without annihilation, conservation without quantity," and is not the quantity-conservation of physics. A ledger reading is exactly that; a signed-sum reading is not. Version 3 accordingly renames the relation the signed balance of paired paths (following Methodology Overview V3).

Third, Open Problem 1. Cosmo Paper VII answers it in the negative by a no-go: the difference of two independent canonical stress tensors cannot be realised as a signed vacuum ledger in a healthy single-metric bulk. The gap to which §5.3 attributes its 15% Hubble discrepancy is precisely the gap that paper fills — by ruling the construction out. The problem is closed and replaced: how $\Lambda_1$ and the dual-frame picture are to be restated under a single-metric baseline (Gate 3 of the audit).

Status changes

Item	v2	v3
ξ < 0	prior-mandated	unchanged; the prior is now explicitly $D_{\rm loc}$
$F(x)$, $K$, $U$, $r_*$, all closures	—	unchanged
"causal law strengthens after turnaround"	a claim of this paper	narrowed: applies to $D_{\rm glob}$; F peaks at turnaround
$\Lambda_1 + \Lambda_2 = 0$	translation of remainder conservation	renamed; identity re-adjudicated as an orientation ledger
§5.2 mean metric carrying $\Lambda_{\rm total} = 0$	a mechanism here	retired
§11 Open Problem 1	open	closed negatively, replaced
ξ = 0 Ġ/G no-go, Cassini and LLR constraints	—	unchanged

Downstream effects

Affected	Content
Cosmo Paper VII	Both corrections originate there; its no-go closes this paper's Open Problem 1
Cosmo Paper IV (.19298161)	Its $F(C)R$ machinery belongs with this paper to the broken-shift branch; the $G_{\rm loc}$/$G_{\rm FRW}$ separation and the ξ = 0 no-go are unaffected legacies
Methodology Overview V3 (.18842449)	The renaming in §4.2 follows its separation of names
Parameter closures and observational constraints here	No effect

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Acknowledgements

The author thanks the research and engineering teams behind the four large language models. Special thanks to Zesi Chen (陈则思), the SAE framework's long-term interlocutor.

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The structural predictions Λ = 2(ω₂² − ω₁²)/c² and a₀ = (π/2)·c(ω₂ − ω₁) do not depend on any result in this paper.