Pure Time Theory - Chapter VIII - A Minimal Temporal Grammar with Traceability & Verifiable Examples

Pure Time Theory – from Intuition to Genesis Chapter VIII – A Minimal Temporal Grammar with Traceability & Verifiable Examples Essam Allou October 19, 2025 Abstract Aim. This chapter is a grammar-first rationalization of Pure Time Theory (PTT). We consolidate the minimal temporal grammar PTT–L0 (pure time, positiveaffine gauge on Trelax , unitary/diagonal cadence, causal recurrence, and visibility/fairness) and turn prior derivations into a traceable, non-circular, auditready package. Rather than announcing new results, we stabilize the language, guardrails, and notational hygiene that make the existing case studies readable. What this chapter contributes. (i) A single, minimal grammar for all subsequent arguments, with explicit guardrails (scope, invariances, and non-circularity clauses). (ii) A traceability table linking each statement to the exact L0 item(s) consumed and to the derived lemmas used. (iii) A set of verifiable micro-examples (one per programme) that can be checked exactly over Q. Scope. The four programmes already developed inside the L0 grammar are revisited here as case studiesRiemann zeros (static/dynamic RH), SAT/encodings and sesquilinear verification, unitary cadence ⇒ Schrödinger kinematics, and the weak-field HelmholtzPoisson map with clock lawstrictly to illustrate how L0 supports them. We emphasize non-circularity, unitary equivalences, and positiveaffine time gauges, and we keep the presentation neutral and audit-oriented. Audit. Section 6 assembles an at-a-glance traceability table and four self-contained, exactly checkable examples (RH annulus toy; a mini 3CNF/SAT; QM rotator; GR redshift on shells), all within L0 and with arithmetic over Q . This turns the manuscript into a grammar-first reference for readers of the earlier chapters and a stable entry point for new ones. ∗Unity of Truth, [email protected] 1 Contents Introduction 3 Reader’s Guide — Scope, guardrails, and non-circularity 4 PTT ↔Standard Dictionary 4 L0 Core — Temporal Grammar 5 Axiom L0.1 — Pure time & positive–affine gauge ................ 6 Axiom L0.2 — Annular cadence; recurrence; unitary equivalence ........ 6 Axiom L0.3 — Visibility (positive lower density) ................. 6 Axiom L0.Z — Zero–bundle sequencing (dihedral orbits) ............ 7 RH — Static layer 8 RH — Dynamic layer 13 P=NP — Static layer (encodings & verifier) 19 P=NP — Dynamic layer (normalization & schedule) 24 QM — Static kinematics (annular) 29 QM — Dynamic layer (deterministic readout) 34 GR — Static layer (metric from modal intensity) 37 GR — Dynamic layer (ledger →curvature) 40 Model bridges & audit 44 PTT TM: simulation, compilation, power equivalence ............. 44 PTT Standard QM/GR .............................. 46 Packaging & audit 50 Traceability table (L0 consumption per result) .................. 50 Verified example set (all inside L0) ........................ 51 Conclusion 52 Postscript — How to read this manuscript 52 Annex A Integration and limits 53 Annex B TRS R1–R7.5 dossier 53 2 Introduction Why this chapter. Earlier chapters developed PTT across several fronts (quantum/- gravity kinematics and weak-field maps; deterministic SAT dynamics; spectral tools), which left some readers asking for a single, minimal temporal grammar with clear guardrails and a compact audit trail. Chapter VIII answers that request: we fix the L0 grammar once and for all and recast the main constructions as traceable case studies that live entirely inside L0. The goal is readability, stability, and non-circularity not new announcements. L0 in one page. We use only: (i) pure time with a positiveaffine gauge on Trelax ; (ii) a unitary, modediagonal cadence ρω and the causal recurrence BT+1 = ρωBT + ST+1 ; (iii) a visibility/fairness rule on pure time and the fourpacket scheduling discipline. No geometry, morphology, or stochastic postulate is assumed at L0; invariances under chart/weight intertwiners and time gauges are made explicit. What we do here. • We stabilize the grammar: notations, invariances, scope, and non-circularity clauses are grouped and stated up front. •We audit each statement: a traceability table (Packaging & audit ) points from every result to the precise L0 items and derived lemmas it consumes. • We illustrate L0 through four micro-examples (one per programme) that are exactly checkable over Q. How to read this chapter. Treat L0 as the only input; treat all four programmes as case studies within that grammar. When a result mentions unitary intertwiners or positiveaffine gauges, it is signaling that the statement is invariant under chart/weight changes and time reparametrizations. When a result cites visibility/fairness, it is signaling where lowerdensity arguments enter and why no circularity arises. Position in the series. This chapter is designed as an entry point and as an audit layer for the prior developments; readers coming from the unification of GR/QM (Chapter VI) or the deterministic SAT corridor and normalization (Chapter VII) will find the same arguments reexpressed under the single L0 grammar with a compact, verifiable backbone. 3 Reader’s Guide — Scope, guardrails, and non-circularity What is assumed (L0). We fix only the PTT–L0 grammar: pure time with a positive–affine gauge on Trelax (L0.1), a unitary, mode–diagonal cadence ρω and the causal recurrence (L0.2), and a visibility/fairness rule on pure time (L0.3/L0.Z). No geometry, no morphology, and no stochastic postulate are assumed at L0. Static vs dynamic layers. Static packages build kinematics, encodings, and invariants inside L0; Dynamic packages use the ledger/fairness to enforce limits or balances. This separation is intentional and avoids circularity. Non-circularity (QM). QM (static) uses only the spectral/Stone calculus of the mode generator b pϕ to define the unitary group U ( T ) = e−iT bpϕ . QM (dynamic) derives a deterministic frequency law for the P0 readout. No Born postulate and no stochastic assumption are used. How dyn → stat bridges are used. Whenever a dynamic Cesàro score vanishes (e.g. Edyn Λ ( Z ) = 0), we encapsulate the passage to the static functional in a quantified bridge (Prop. 37) relying only on L0.3/L0.Z and unitarity. Rewriting hygiene. The normalization TRS R1R7.5 is backed by a formal operational signature (Def. 55), a complete list of critical overlaps with bounded joins (Lem. 61), and Newmans lemma with an explicit join budget. What is not claimed. L0 does not assume RH, P=NP, or the Einstein equations. RH appears via the functional E ( Z )and the standard functional equation; P=NP is encoded via a finite window/sesquilinear verifier; an Einstein–type balance is obtained at coarse scale from ledger continuity and diffeo/second–order locality, not from a new derivation of Lovelock. Falsifiability hooks. When used as a physical model, the small–field sector (HelmholtzPoisson/Yukawa) predicts explicit redshift/time–delay corrections (GR (dynamic)). These are independent of RH/P=NP and can be checked empirically. PTT ↔Standard Dictionary cadence ρω Unitary rotation group on H (Λ); in standard QM: U ( T ) = e−iT b H with c H = νb pϕ (Stones theorem [2]). cadence ⇒Schrödinger U ( T ) = e−iνT bpϕ ; modal identification (rotator): f ( m ) = κm2 (Theorem 89; see Proposition 137). P0readout Modal PVM: radial event m = 0 (Born). Deterministic block-frequency (Theorem 100). [6,8]. 4 IDE Sum of four quarter-turns (JacobiAnger & Bessel identities) cancelling odd modes (Lemmas 26 and 27; [4,3]). HT Summable tail of arc detunings; forces Edyn Λ = 0 by pure-time Cesàro (Theorem 33 and Proposition 37). ledger / 4-packet exposure Discrete/coarse energy balance per 4 ticks; Doeblin ≥1/4(Proposition 30). offline brick Non-radial source Soff = e−|δ|rsin ( |γ|rcos ( ϕ−θ )) (strict non-radiality via Corollary 9). TRS Rewrite system R1R7.5 for episode normalization; descending potential A (Lemma 61 and Theorem 60). PTTGR (weak field) HelmholtzPoisson ( − ∆+ L−2 ) u = µ ; redshift dτ/dT = (1+ u ) −1 ; Lovelock uniqueness in D= 4 ([5]; see Theorem 119 and Definition 140). L0 Core — Temporal Grammar This section fixes the temporal grammar of PTT. Everything else (RH, P=NP, QM/GR) is derived within this grammar. No geometric/spectral morphology (e.g. shapes, kernels, packings) is assumed here. Standing conventions. • Hilbert stage. On each angular chart Λ = { ( r, ϕ ) : r0≤r≤r1, ϕ ∈ [0 , 2 π ) } we set H(Λ) := L2 Λ,dϕ 2πw(r)dr!, w ∈L1([r0, r1]), w ≥0, w 6≡ 0. Angular Fourier resolution: for F∈H (Λ), F ( r, ϕ ) = Pm∈ZFm ( r ) eimϕ with Fm ( r ) = 1 2πR2π 0F ( r, ϕ ) e−imϕ dϕ ∈L2 ([ r0, r1 ] , w dr )and kFk2 H(Λ) = Pm∈ZkFmk2 L2([r0,r1],w dr) . Projectors: ( PmF )( r, ϕ ) = Fm ( r ) eimϕ , P0 radial, P6=0 := Pm6=0 Pm , with PmPn = δmnPmand PmPm=I(strongly). •Lower asymptotic density (pure time). For G⊂N, d(G) := lim inf N→∞ 1 N#G∩{1, . . . , N}, computed with respect to the pure time index T (hence gaugeinvariant under positiveaffine changes of Trelax). • Bounded & causal injections. A sequence ( ST ) T≥1⊂H (Λ) is uniformly bounded if supTkSTkH(Λ) <∞ . Recurrence is causal/nonanticipative: ST+1 may depend on {Bt, St}t≤Tbut not on future values. 5 • Fairness for a bundle. A set of block starts T(j)⊂N assigned to a bundle O ( j )is fair if d(T(j))>0(no uniform lower bound in jis required at L0). • Normalized tests. In visibility statements, tests are taken with unit norm: k Φ jkH(Λ) = 1. Axiom L0.1 (Pure time & positive–affine gauge: monotone Trelax with canonical spectral component).The pure time T∈N gives a causal total order. The relaxed time Trelax ( T, x ) is defined up to a positive–affine gauge Trelax ∼a Trelax + b with a > 0, and a representative can always be chosen strictly increasing in T . On any causal window, Trelax admits a canonical (finite or absolutely convergent) spectral component Trelax(T, x) = <X k Ake−γk(T−Tk)eiβk(T−Tk)eikk·x, γk>0, βk∈R. No geometric content is postulated at L0 beyond this temporal/readout structure. Remark 1 (Why this axiom is minimal).The positive–affine gauge ensures reparametrization invariance of all frequency/time–average statements, and the canonical spectral component on causal windows is exactly what the Stone/spectral calculus needs to tie cadence to a self–adjoint generator without importing geometry or morphology. Axiom L0.2 (Annular cadence ρω is unitary/diagonal; causal recurrence; unitary chart/weight equivalence).On H(Λ) the cadence acts by rotation (ρωF)(r, ϕ) = F(r, ϕ −ω), unitarily and diagonally on the angular modes {eimϕ}m∈Z (generator −i∂ϕ on a dense domain). Dynamics is causal: BT+1 =ρωBT+ST+1, for injections ST+1 ∈H (Λ).If (Λ 0, w0 )is another angular chart sharing the same angular variable and a radial weight w0 , there exists a unitary U : H (Λ) →H (Λ 0 )with Uρω = ρωU and UPm = PmU for every m∈Z ; hence all L0 statements are chart/weight invariant. Remark. Allowing a rigid angular shift ϕ7→ ϕ + θ yields a unitary Uθ with Uθρω = ρωUθ and UθPm = e−imθPmUθ (intertwining). All L0 consequences remain invariant under such shifts. Axiom L0.3 (Visibility: positive lower density of good times, robust to uniformly bounded injections).For any F∈H(Λ) with non–radial content P6=0F6= 0, there exist tests {Φj} ⊂ H(Λ) with kΦjkH(Λ) = 1 and thresholds {τj>0}such that Gj(F) := nT∈N:|hρT ωF, Φji| ≥ τjo has strictly positive lower asymptotic density, d ( Gj ( F )) > 0. The same holds for trajectories BT of the causal recurrence with uniformly bounded injections supTkSTkH(Λ) <∞ (boundedness taken in the same Hilbert stage H (Λ)). This is a temporal fairness rule (independent of any geometric profile). 6 Definition 2 (Densities).For A⊂N, set d(A) := lim inf N→∞ 1 N#(A∩[1, N]), d(A) := lim sup N→∞ 1 N#(A∩[1, N]), and the lower Banach density dB(A) := lim inf L→∞ inf M≥1 1 L#(A∩[M, M +L−1]). All densities are taken w.r.t. the pure time index T. Lemma 3 (Banach ⇒ Cesàro).For A⊂N , dB ( A ) ≤d ( A ). If xT∈ [0 , M ]and xT≥α > 0on A, then lim inf N→∞ 1 NX T≤N xT≥α d(A)≥α dB(A)>0. Hence a property true on a set of positive lower Banach density contributes a positive Cesàro average. Note. If T(j) denotes the set of starts of 4-packets for a given bundle, then T(j) + 4 (the set of ends) has the same lower density (translation invariance). Key Lemma KL1 Density ⇒Cesàro (Summary of Lem. 3)If xT∈ [0 , M ]and xT≥α > 0on a set A with d ( A ) > 0, then lim infN→∞ 1 NPT≤NxT≥α d(A)>0. Axiom L0.Z — Zero–bundle sequencing (dihedral orbits). Admissible modal injections are indexed by zero bundles (dihedral orbit) O(j) = nρj, ρj,1−ρj,1−ρjo. The scheduler enumerates bundles causally and fairly: for every j , the set of block starts T(j) assigned to O ( j )has strictly positive lower density d ( T(j) ) > 0. Each activation of O ( j )occurs in a packet of four consecutive cadence ticks (one per element of the dihedral orbit), without interleaving of other bundles within the packet; the internal order may be chosen and fixed per activation. No geometric/morphological constraint is imposed at L0. Notes. (i) L0.Z and L0.3 are distinct: L0.Z constrains sequencing/scheduling of admissible injections; L0.3 constrains detectability in time. (ii) All morphological/analytic statements (e.g. canonical bricks, Chebyshev/Herglotz isometries, IDE/Doeblin/packing, G = sin ) are derived later (RH (static+dynamic)) inside this grammar. The L0 layer is intentionally minimal and purely temporal. 7 RH — Static layer Within the PTT–L0 grammar (Temporal Grammar), this section gives a purely static characterization of RH. No dynamical axiom is used. All objects below are independent of chart/weight choices on annular stages, by unitary covariance fixed in Temporal Grammar. Remark 4 (Scope: not an axiomatization of RH).L0 does not encode RH. The functional E ( Z )is defined intrinsically from real parts βj , and the equivalence E ( Z )=0 ⇔RH uses only the Cesàro lemma, Chebyshev/Herglotz calculus on the annulus, and the standard functional equation/pinning. Nothing in L0 presupposes RH. Conventions and scope We consider the nontrivial zeros ρ = β + iγ of ζ in the critical strip, counted with multiplicities. A zero is online if β = 1 2 and offline otherwise. All Fourier transforms use b f(ξ) := ZRf(t)e−itξ dt, f(t) = 1 2πZRb f(ξ)eitξ dξ. Trivial zeros and the poles at s= 0,1play no role here. Chebyshev/Herglotz on circle and annulus Lemma 5 (Chebyshev–Fourier).Let a > 0and let G be real–analytic in a neighborhood of [ −a, a ]. Then F ( ϕ ) := G ( acos ϕ )has finitely many nonzero Fourier coefficients iff G is a polynomial. If G is non-polynomial, then F carries infinitely many nonzero cos ( mϕ ) modes. Proof of Lemma 5. Let a > 0and let G be real-analytic in a neighborhood of [ −a, a ]. Set F(ϕ) := G(acos ϕ). (⇐) If G(x) = PN k=0 ckxk, then F(ϕ) = N X k=0 ckakcoskϕ. For each k , coskϕ is a finite linear combination of cos ( mϕ )with 0 ≤m≤k and m≡k ( mod 2) (e.g. via the identity cos ( kϕ ) = Tk ( cos ϕ ), Tk Chebyshev of the first kind, and polynomial reduction). Hence Fhas finitely many nonzero Fourier coefficients. (⇒) Suppose F(ϕ) = PM m=0 bmcos(mϕ)(finite Fourier series). Define the polynomial Q(u) := M X m=0 bmTm(u), Tm(u) := cosmarccos u. Then for every ϕ , Q ( cos ϕ ) = PM m=0 bmcos ( mϕ ) = F ( ϕ ) = G ( acos ϕ ). Setting u = cos ϕ , we obtain on u∈ [ − 1 , 1] the identity G ( au ) = Q ( u ). Since G is real-analytic in a neighborhood of [ −a, a ], the function H ( u ) := G ( au ) −Q ( u )is real-analytic in a neighborhood of [ − 1 , 1] and vanishes on [ − 1 , 1]; by the identity theorem, H≡ 0in that neighborhood. Therefore G ( x ) = Q ( x/a )for x near [ −a, a ], hence G is a polynomial. 8 Lemma 6 (Herglotz isometry on the circle).For a > 0, with the Chebyshev measure dµa(u) = 1 π 1{|u|<a} √a2−u2du, 1 2πZ2π 0g(acos ϕ)2dϕ =Za −a|g(u)|2dµa(u). Full proof of Lemma 6. Fix a > 0and let g be measurable with g∈L2 ([ −a, a ] , µa ), where dµa(u) = 1 π 1{|u|<a} √a2−u2du. Set I:= 1 2πZ2π 0g(acos ϕ)2dϕ. Since cos ( ϕ ) = cos (2 π−ϕ ), the integrand is π -periodic in value; hence I = 1 πRπ 0|g ( acos ϕ ) |2dϕ. On (0 , π ), the change of variables u = acos ϕ gives du = −asin ϕ dϕ and, since sin ϕ=√1−cos2ϕ=q1−(u/a)2=√a2−u2 a, we obtain dϕ =−du asin ϕ=−du √a2−u2. As ϕruns through (0, π),uruns through (a, −a); therefore Zπ 0|g(acos ϕ)|2dϕ =Z−a a|g(u)|2−du √a2−u2=Za −a|g(u)|2 √a2−u2du. Restoring the factor 1/π yields I=Za −a|g(u)|2du π√a2−u2=Za −a|g(u)|2dµa(u). Boundary integrability. Since 1 /√a2−u2∼ (2 a ) −1/2 ( a−|u| ) −1/2 , the density of µa is locally integrable on [ −a, a ]; the assumption g∈L2 ( µa )is sufficient to justify the changes of variables by FubiniTonelli. This proves the isometry. Lemma 7 ( H1 regularity and integrable domination).Let Soff ( r, ϕ ) = e−ar sinbr cos ( ϕ− θ )  with a, b > 0. Then Soff ∈H1 (Λ), ∂ϕSoff ∈L2 (Λ) and, writing ( Soff ) m for angular coefficients, X m∈Z m2k(Soff)mk2 L2([r0,r1],w dr)≤CZr1 r0 (a2+b2r2)e−2ar w(r)dr < ∞. All uses of Fubini/Tonelli/DCT are justified whenever Rr1 r0R2π 0|G ( |γ|rcos ϕ ) |2w ( r ) dϕ dr < ∞, and the boundary singularity at u = ±|γ|r is locally integrable since  ( |γ|r ) 2− u2−1/2∈L1 loc. Proposition 8 (Annulus-wise Parseval).Let w≥ 0in L1 ([ r0, r1 ]) and G measurable with Rr1 r0 1 2πR2π 0|G(|γ|rcos ϕ)|2dϕ w(r)dr < ∞. Then 1 2πZr1 r0Z2π 0|G(|γ|rcos ϕ)|2w(r)dϕ dr =ZR|G(u)|2W|γ|,Λ(u)du, where W|γ|,Λ(u) := Rr∈[r0,r1]∩{r>|u|/|γ|} w(r) πp(|γ|r)2−u2dr. 9 Proof. Fix k and set the probability measures on shifts µk := δεk and νk := 1 2∆ 1 [−∆,∆] ( η ) dη . Write Tµf := Rρηf dµ ( η ). Since ρck is unitary, kTµk−TνkkH1→L2 equals kR ( ρηf ) d ( µk− νk)(η)kL2. Let πkbe the coupling πk:= δεk⊗νkof (µk, νk). By Lemma 31, kTµkf−TνkfkL2≤ZZ kρηf−ρη0fkL2dπk(η, η0)≤ZZ |η−η0|dπkk∂ϕfkL2. The bracket is the 1-Wasserstein distance W1 ( µk, νk )on S1 (viewed locally), and a direct calculation gives W1(δε,Unif[−∆,∆]) = 1 2∆ Z∆ −∆|η−ε|dη =ε2+ ∆2 2∆ ≤∆for |ε| ≤ ∆. Therefore kTµk−TνkkH1→L2≤∆. Averaging over kand composing with ρck, k(A−M∆)fkL2≤1 4 3 X k=0 kTµkf−TνkfkL2≤∆k∂ϕfkL2. Modal window is finite. All estimates below act mode-by-mode on a finite window |m| ≤ M , with M = poly ( n )in the PTT encodings (NP (static)). In particular, the bound |e−imε − 1 | ≤ |m||ε| is taken for |m| ≤ M , whence αq = ∆ q< 1as soon as ∆ q< 1. Key Lemma KL3 Episode contraction (Summary of Thm. 33)If uk∈ [ ck− ∆ , ck +∆] then P6=0 1 4P3 k=0 ρukF≤ ∆ k∂ϕFk (on a finite modal window). Theorem 33 (Explicit contraction per episode on the offline brick).Let F := P6=0Soff (odd modes only). For a ∆q-packed episode, P6=0 1 4 3 X k=0 ρuq,k FH(Λ) ≤αqk∂ϕFkH(Λ), αq:= ∆q. In particular, the nonradial contribution injected by the packet at its end is contracted by a factor αq<1as soon as ∆q<1. We call κq:= 1 −αq>0the episode gap. Full proof of Theorem 33. Let F := P6=0Soff ; by Lemma 24 it has only odd angular modes. For a ∆ q -packed episode with compensated angles uq,k ∈ [ ck− ∆ q, ck + ∆ q ], define Aqf=1 4 3 X k=0 ρuq,k f, M∆qf=1 4 3 X k=0 1 2∆qZ∆q −∆q ρck+ηf dη. By Proposition 30,M∆qF≡0(odd modes). Hence P6=01 4 3 X k=0 ρuq,k FL2(S1) =k(Aq−M∆q)FkL2(S1)≤∆qk∂ϕFkL2(S1) by Lemma 32. Integrating the inequality in r with weight w ( r ) dr (tensor product structure of H(Λ)) gives P6=01 4 3 X k=0 ρuq,k FH(Λ) ≤∆qk∂ϕFkH(Λ). This is the claimed contraction with factor αq:= ∆q<1whenever ∆q<1. 16 Remark 34 (From one-sample-per-arc to multi-sample averages).If an episode uses mq,k ≥ 1samples in each arc and averages them, the bound improves to αq≤C ∆ q/mmin q for a universal C, with mmin q:= minkmq,k (trapezoidal error on each arc). Scheduling and Doeblin ≥ 1 / 4.By L0.Z (bundles fair, paquets de 4consécutifs) and L0.3 (visibilité), one can carve a subsequence of episodes {q} such that: (i) every offline bundle appears with strictly positive lower density; (ii) each chosen episode is ∆ q -packed; and (iii) the set of episode ends has positive lower density ≥1 4 (one end per group of four cadence ticks). This is the deterministic analogue of a Doeblin condition with minorization κ≥1 4·∆∗ πon a tail where ∆q≥∆∗>0uniformly. Lemma 35 (Episode contraction).Let F∈H1 ( S1 )have only odd modes. If uk∈ [ck−∆, ck+ ∆] with ck:= kπ/2, then P6=01 4 3 X k=0 ρukFL2(S1)≤∆k∂ϕFkL2(S1). By radial integration (tensor product), the same bound holds in H(Λ). HT (summable tail) and Bridge ⇒Edyn Λ= 0 ⇒RH Definition 36 (Dynamic annulus functional).For a fixed annulus Λ, let ( BT )be the trajectory of the causal recurrence BT+1 = ρωBT + ST+1 with uniformly bounded injections (Temporal Grammar). Define Edyn Λ(Z) := lim sup N→∞ 1 N N X T=1 kP6=0BTk2 H(Λ). Key Lemma KL4 Bridge dyn→stat (Summary of Prop. 37)If Edyn Λ(Z) = 0 and the scheduler is fair, then EΛ(Z) = 0. Proposition 37 (Bridge dyn → stat (quantified)).Fix an annulus Λand assume the scheduler is fair (L0.Z). If Edyn Λ ( Z ) := lim supN→∞ 1 NPT≤NkP6=0BTk2 = 0 and ∃j offline with aj (Λ) := kP6=0SO(j)kH(Λ) > 0, then there exist a unitnorm test Φ ∈H (Λ), constants τ, η > 0, and a set T ⊂ Nwith d(T)≥ηsuch that |hP6=0BT,Φi| ≥ τ(∀T∈ T). Consequently Edyn Λ(Z)≥η τ2>0, contradiction. Hence EΛ(Z) = 0. Proof. Suppose by contradiction that EΛ ( Z ) > 0. Then, by Proposition 16, there exists at least one offline bundle O ( j )with nonradial mass aj (Λ) > 0. By Axiom L0.3 (Visibility), applied to the canonical offline brick of O ( j )and using the fairness of the scheduler (L0.Z) plus the unitarity of ρω , one obtains a test Φwith k Φ k = 1, constants c, τ > 0and a set of times T ⊂ Nof strictly positive lower density such that |hP6=0BT,Φi| ≥ c aj(Λ) −rT(T∈ T), where rT→ 0along T (standard extraction by boundedness of injections). Therefore kP6=0BTk2 H(Λ) ≥ ( c aj (Λ) / 2) 2 for all large T∈ T , and the Cesàro average in Definition 36 17 is ≥d ( T ) ( c aj (Λ) / 2) 2> 0, contradicting Edyn Λ ( Z ) = 0. Thus no offline bundle has positive nonradial mass on Λ, i.e. aj (Λ) = 0 for all j , and hence EΛ ( Z )=0by Proposition 16. Remark 38 (Why L0.3 forces the contradiction).If some offline bundle has aj (Λ) > 0, L0.3 (Visibility) produces a unit–norm test Φand a set of times of positive lower density where |hP6=0BT, Φ i| stays above a fixed threshold, uniformly in the block normalization. Since ρω is unitary, the ledger cannot wash this out, so the Cesàro contribution of kP6=0BTk2is >0, contradicting Edyn Λ= 0. Remark 39 (Packet-level score and state score).Define the packet-level score Edyn,pack Λ:= lim sup Q→∞ 1 Q Q X q=1 P6=03 X k=0 ρ4−k ωρθq,k Soff 2 H(Λ) . Under exact IDE it is identically 0; under IDE ε and the contraction of Theorem 33, its summable tail is controlled and forces Edyn Λ ( Z ) = 0 for the state as well (decomposition into complete IDE blocks plus a summable remainder). Two closures (both derived, no new axiom). (A) Persistent co–rotation (exact IDE). If episodes are perfectly compensated and uq,k =ckfor all q, k, then Edyn Λ= 0 by Lemma 26. (B) Summable tail (HT). Assume there exists a packed subsequence with Pqαq<∞ (e.g. Pq∆q<∞). Then 1 QX q≤QP6=01 4 3 X k=0 ρuq,k F 2 ≤1 QX q≤Q α2 qk∂ϕFk2−−−→ Q→∞ 0, hence Edyn Λ= 0. Lemma 40 (Bridge dyn → stat).If Edyn Λ ( Z )=0and the scheduler is fair (L0.Z), then EΛ(Z) = 0. Proof. Contrapositive. If there exists an offline bundle j with aj (Λ) > 0, L0.3 yields a unitnorm test Φand T ⊂ N with d ( T ) > 0such that |hP6=0BT, Φ i| ≥ c aj (Λ) on T (after standard extraction). Hence lim infN1 NPT≤NkP6=0BTk2≥c2aj (Λ) 2d ( T ) > 0, a contradiction. Theorem 41 (Dynamic closures ⇒ RH).If there exists a countable annulus cover { Λ q} such that Edyn Λq = 0 for all q (via route (A) or (B)), then E ( Z )=0and RH holds. Proof. By Proposition 37, EΛq ( Z ) = 0 for all q . The annulus sweep (RH (static), Prop. 17) yields E(Z) = 0, and Theorem 22 gives RH. Remark 42 (Role of L0 only).The arguments use: (i) modal diagonality/unitarity of ρω (L0.2); (ii) fairness + packets of four from dihedral orbits (L0.Z); (iii) visibility (L0.3) to select a set of episode ends with positive lower density. No additional morphology is assumed. 18 P=NP — Static layer (encodings & verifier) Within the PTT–L0 grammar (Temporal Grammar), this section fixes the static objects for the P=NP program: chart/weight–invariant encodings of instances and certificates, a polynomial-time sesquilinear verifier on modal coefficients, and a radial existence functional. No dynamic assumption is used. Remark 43 (Why a Hilbert stage for a discrete problem?).We embed instances/certificates isometrically inside a finite modal window so that (i) all operations are exactly evaluable over Q with polynomial cost, (ii) the verifier is a sesquilinear dot–product on that window, and (iii) the whole pipeline intertwines with the same modal calculus used in QM package (static+dynamic). This is a unifying functional framework, not a physical assumption. Conventions and scope Fix any language L⊆ { 0 , 1 }? . For size n , we work on an annular stage H (Λ n )(§ Temporal Grammar), with modal projectors Pm , P0 (radial), P6=0 . Different charts/weights (Λ n, w ) are equivalent up to a unitary intertwiner (Temporal Grammar, L0.2). Throughout, a polynomial modal window is an index set M n⊂Z with maxm∈Mn|m| ≤ M(n)and |Mn| ≤ M(n)for some M(n) = poly(n). Instance and certificate encodings (isometric, windowed, computable) Definition 44 (Isometric, windowed encodings).For each n and certificate length m(n) = poly(n), there exist linear injections En:{0,1}n→H(Λn), Cn,m :{0,1}m(n)→H(Λn), such that: 1. Isometry. kEn ( x ) kH(Λn) = kxk2 , kCn,m ( y ) kH(Λn) = kyk2 after a fixed identification {0,1}k,→Rk. 2. Modal locality. suppmEn ( x ) ∪suppmCn,m ( y ) ⊆ M n with | M n| ≤ M ( n ) = poly(n). 3. Computability. The modal coefficients { ( En ( x )) m ( r ) }m∈Mn and { ( Cn,m ( y )) m ( r ) }m∈Mn are computable in time poly(n). We write Fx:= En(x)and Gy:= Cn,m(y). Remark 45 (Fixed, computable exact radial dictionary).For each n , fix once and for all a radial dictionary in L2 ([ r0, r1 ] , w dr )such that inner products and norms are exactly evaluable over Q: either (i) a partition r0 = r(0) < r(1) <··· < r(J) = r1 with J = poly ( n )and a basis of normalized indicators ϕn,j := 1[r(j−1),r(j)] qRr(j) r(j−1) w(r)dr ; then hϕn,j, ϕn,ki = δjk and any dotproduct reduces to weighted sums of cell averages, all rational if the cutpoints and w are rationally tabulated; or 19 (ii) an orthogonal polynomial basis {ϕn,j}j≤D(n) with respect to w , with moments Rr`w ( r ) dr supplied as rationals (closed forms), ensuring exact GramSchmidt. Evaluations and inner products are thereby exact in Q (or Q ( i )with modal phases); no floating point is required for the verifier Vnnor for TM bridge. Remark 46 (Chart/weight invariance).By L0.2, for any other (Λ 0 n, w0 )there is a unitary U with Uρω = ρωU and UPm = PmU ; hence all statements below are invariant under chart/weight changes. Sesquilinear verifier and universal existence for NP Definition 47 (Windowed sesquilinear verifier).Averifier at size n is a sesquilinear form Vn:H(Λn)×H(Λn)→Rof the form Vn(F, G) = <X m∈MnDKn,m PmF, PmGEL2([r0,r1],w dr), where each Kn,m is a radial operator (integral kernel) with coefficients computable in time poly ( n ). We say that Vn is polyevaluable if, given the modal coefficients on M n , Vn(F, G)is computable in poly(n). Explicit dotproduct normalization. If we take Kn,mj = Id and choose a dictionary {ϕn,j}j≤D(n)orthonormal in L2([r0, r1], w dr), then for Fx=X j≤D(n) wx,j ϕn,j(r)eimjϕ, Gy=X j≤D(n) gj(y)ϕn,j(r)eimjϕ, one has Vn(Fx, Gy) = X j≤D(n) wx,j gj(y), i.e. a dotproduct in dimension D(n) = poly(n), thus evaluable in time poly(n). Theorem 48 (Static NPverifiability within PTT–L0 (constructive, full proof)).For every language L∈NP with witness relation R ( x, y )and |y| = m ( n ) = poly ( n ), there exist explicit, poly-time computable objects En, Cn,m :RD(n)→H(Λn),Vn:H(Λn)×H(Λn)→R, τn∈R,∆n≥1 poly(n), with D(n) = poly(n)such that, for all x∈ {0,1}n, x∈L⇐⇒ ∃y∈ {0,1}m(n)Vn Fx, Gy≥τn+∆n, x /∈L⇐⇒ ∀yVn(Fx, Gy)≤τn−∆n, where Fx := En (w x )and Gy := Cn,m (g x ( y )) for vectors w x, g x ( y ) ∈RD(n) constructed below. Moreover, Vn is polyevaluable and all norms/inner products are evaluated exactly over Q(or Q(i)) in the stage H(Λn)(see Remark 45). Full constructive proof. Step 0 CookLevin reduction ⇒ 3CNF. For x∈ { 0 , 1 }n , construct in time poly ( n )a3CNF formula Φ x on m ( n ) = poly ( n )witness variables (y∈ {0,1}m(n)) and mΦ= poly(n)clauses C1, . . . , CmΦsuch that x∈L⇐⇒ ∃yΦx(y) = True, x /∈L⇐⇒ ∀yΦx(y) = False. 20 Step 1 Polynomial lifting of degree ≤ 3(features of dimension D ( n )). Introduce spin variables sj∈ {± 1 } for j = 1 , . . . , m ( n )and identify True ↔sj = +1, False ↔sj=−1. For a literal `on sj, set `0=   1+sj 2,if `≡sj, 1−sj 2,if `≡ ¬sj. For a clause C=`1∨`2∨`3, the Boolean identity 1Ctrue = 1 −(1 −`0 1)(1 −`0 2)(1 −`0 3) becomes, after substituting `0 i = (1 ±s ) / 2, a polynomial in the sj of degree ≤ 3with coefficients in 1 8Z. Defining px(s) := mΦ X i=1 1Citrue(s), we obtain px:{±1}m(n)→ {0,1, . . . , mΦ}and x∈L⇐⇒ ∃s px(s) = mΦ, x /∈L⇐⇒ ∀s px(s)≤mΦ−1. Expanding pxin the basis of degree-≤3monomials actually present, we can write px(s) = wx·gx(s) = D(n) X j=1 wx,j gj(s), where g x ( s ) = ( gj ( s )) j≤D(n) is the vector of features (constant, linear monomials sj , products sjsk , triples sjsks` that actually occur) and w x is the vector of the corresponding coefficients. Each clause contributes at most 2 3 = 8 monomials after substitution; hence D(n)≤8mΦ+ 1 = poly(n). The coefficients wx,j lie in 1 8Z; the bit-size is O(log mΦ). Step 2 Modal window & exact radial dictionary. Fix a modal window Mn:= {m1, . . . , mD(n)} ⊂ Z, mj:= j, and an orthonormal radial dictionary {ϕn,j}j≤D(n)⊂L2 ([ r0, r1 ] , w dr )such that all inner products are exactly evaluable over Q (or Q ( i )), as in Remark 45. Thus the functions eimjϕϕn,j ( r )are orthonormal in H (Λ n ). (The space H (Λ) and the unitary/modal-diagonal cadence ρω are those fixed in Temporal Grammar.) By L0.2, any other chart/weight is unitarily equivalent. Step 3 Linear isometric injections En, Cn,m.Define Jn:RD(n)→H(Λn)by Jn(a) := D(n) X j=1 ajϕn,j(r)eimjϕ. By orthonormality, Jnis isometric: kJn(a)kH(Λn)=kak`2. Set En:= Jn, Cn,m := Jn, Fx:= En(wx), Gy:= Cn,m(gx(y)). (Note: the dependence of g x ( · )on x comes from the features that actually appear; it is fixed by Φx.) 21 Step 4 Sesquilinear verifier Vn .On each mode mj∈ M n , take the radial operator Kn,mj= Id (zero elsewhere). Set Vn(F, G) := <X m∈MnhKn,m PmF, PmGi= D(n) X j=1 hPmjF, PmjGi. With Fx=Jn(wx)and Gy=Jn(gx(y)), we obtain Vn(Fx, Gy) = D(n) X j=1 wx,j gj(y) = px(y). This evaluation is exact (no approximation) and runs in time O(D(n)) = poly(n). Step 5 Threshold and gap. By construction, x∈L⇒ ∃yVn(Fx, Gy) = px(y) = mΦ, x /∈L⇒ ∀yVn(Fx, Gy)≤mΦ−1. Choose τn:= mΦ−1 2,∆n:= 1 2, so that the gap is constant ( 1 2≥ 1 /poly ( n )). Optionally, to rescale into [0 , 1], one may normalize b Vn := Vn/mΦ , take b τn := 1 −1 2mΦ and b ∆n := 1 2mΦ , hence b ∆n≥ 1 /poly ( n ) uniformly. Step 6 Complexity and bit-sizes. (i) Construction of Φ x : poly ( n ). (ii) Lifting to w x, g x ( · ): each clause contributes ≤ 8monomials ⇒D ( n ) ≤ 8 mΦ + 1; coefficients wx,j ∈1 8Z , so bit-size O ( log mΦ ). (iii) Exact radial dictionary: fixed once and for all (Remark 45); all inner products in H (Λ n )are exact rational combinations of poly ( n ) size. (iv) Evaluation of Vn ( Fx, Gy ): sum of D ( n )rational products ⇒ time O ( D ( n )) with exact arithmetic in Q(or Q(i)). (v) Modal window: |Mn|=D(n) = poly(n). Step 7 Chart/weight invariance (L0.2). If (Λ 0 n, w0 )is another angular stage, L0.2 provides a unitary U : H (Λ n ) →H (Λ 0 n )with UPm = PmU . Setting K0 n,m := UKn,mU−1= Id and V0 n(F0, G0) := <Pm∈MnhK0 n,mPmF0, PmG0iH(Λ0 n), we have V0 n(UF, UG) = Vn(F, G),kUFk=kFk,kUGk=kGk, so acceptance/rejection is invariant (see Lemma 53 below). Completeness, soundness, polynomial evaluability, gap and invariance are thus established. Remark 49 (Sesquilinearity, locality, and chart invariance).The construction uses only inner products between coindexed radial templates on a finite modal window. It is sesquilinear in ( F, G ), polynomially evaluable, and invariant under all chart/weight unitary intertwiners from L0.2. Radial existence functional (exact/robust) Definition 50 (Static radial functional).For an instance x∈ { 0 , 1 }n , define the exact and robust scores Ex:= inf y∈{0,1}m(n) Vn(Fx,Gy)≥τnP6=0Fx+Gy2 H(Λn),E(ε) x:= inf y∈{0,1}m(n) Vn(Fx,Gy)≥τn−εP6=0Fx+Gy2 H(Λn), with the convention inf ∅:= +∞and ε∈(0,∆n). 22 Proposition 51 (Exact static reading). Ex = 0 if and only if there exists y with Vn ( Fx, Gy ) ≥τn and P6=0 ( Fx + Gy )=0. In particular, for such y , the decision bit is read on the radial channel P0(Fx+Gy). Proof. If Ex = 0, take a minimizing sequence yk ; completeness of H (Λ n )yields a limit y with P6=0 ( Fx + Gy ) = 0 and Vn ( Fx, Gy ) ≥τn by closedness of the constraint set. The converse is immediate. Remark 52 (Robust variant for Dynamic Layer).The functional E(ε) x is used when the dynamic layer enforces Vn≥τn−ε on a set of packet ends of positive lower density (L0.3), while IDE/packing ensures P6=0 ( Fx + Gy ) → 0. Taking ε∈ (0 , ∆ n )keeps the completeness/gap logic intact for the bridge to the static layer. Complexity, locality, invariance (summary) •Poly window/locality. All objects live on a window Mnwith |Mn| ≤ poly(n). • Poly evaluability. Given x and y , the value Vn ( Fx, Gy )is computable in poly ( n ), and kP6=0(Fx+Gy)kis a finite sum over |m| ≤ M(n). • Chart/weight invariance. All definitions commute with U from L0.2; acceptance/rejection is invariant. • No dynamics used. NP (static) is purely static; RH (dynamic) will force E(ε) x→ 0 along episode ends, then read P0. Lemma 53 (Unitary chart/weight intertwining ⇒ verifier invariance).Let (Λ , w )and (Λ 0, w0 )be two angular stages, and let U : H (Λ) →H (Λ 0 )be a unitary such that UPm = PmU for every m∈Z (L0.2 case). Consider a verifier of the form V ( F, G ) = <Pm∈MhKmPmF, PmGiH(Λ) with radial operators Km . Define on H (Λ 0 ) K0 m := UKmU−1 and V0(F0, G0) := <Pm∈MhK0 mPmF0, PmG0iH(Λ0). Then, for all F, G ∈H(Λ), V0(UF, UG) = V(F, G). In particular, if E and C are isometric in H (Λ), then UE and UC are isometric in H(Λ0), and acceptance/rejection via (V, τ, ∆) is identical to that via (V0, τ, ∆). Proof. Since UPm=PmUand Uis unitary, hK0 mPmUF, PmUGiH(Λ0)=hUKmU−1UPmF, UPmGi=hKmPmF, PmGiH(Λ). Summing over m∈ Mand taking the real part concludes. The isometry properties transfer because Uis unitary. 23 P=NP — Dynamic layer (normalization & schedule) All statements in NP (dynamic) are derived within PTT–L0. No new axiom is introduced. We use: the annular cadence (unitary, diagonal), the static encodings ( En, Cn,m ), the windowed sesquilinear verifier Vn , the radial functional Ex , and the IDE/packing lemmas (RH (dynamic)) adapted to the certificate channels. Normalization R1–R7.5: exact ledger, termination, local → global confluence Definition 54 (Exact structural ledger).A (local) normalized cell Γis a span- ≤ 1 operational graph acting on a single modal window M n (NP (static)), within one 4-packet, with at most one external injection. We attach to Γthe wellfounded integer 8tuple L(Γ) := (L1, . . . , L8)∈N8 ordered lexicographically, with the following components: L1:= #{external injection ports}, L2:= #{non-canonical threshold comparators outside {τn−ε, τn, τn+ε}}, L3:= #{active even modes in Mn\{0}}, L4:= 3 X k=0 1{|uk−ck|>∆0}(ck:= kπ/2), L5:= #{CRT signatures interleaved within the same 4-packet}, L6:= X(span −1)+(excess span beyond 1), L7:= max{0,#radial channels at m= 0 −1}, L8:= #{violations of the window constraint m∈Mn}. All quantities are nonnegative integers; Lis therefore well founded under the lexicographic order. Definition 55 (Operational graph signature Σ).A normalized cell Γ(span ≤ 1) is a finite directed multigraph Σ = V;Eflow, Einj;λmode, λclass, λphase, λspan, λthr, with: •Vnodes (modal slots) indexed in a polynomial window Mn; •Eflow ⊆V×Vinternal unitary edges (co-rotation), Einj ⊆Vexternal ports; • labels λmode : V→ M n (odd off m= 0after R4), λclass : V→ B (CRT = Z/ 2 Z× Z/3Z×Z/5Z), •λphase :V→ {0,1,2,3}(index kdu 4packet), λspan :Eflow → {0,1}, •λthr :V→ {τn−ε, τn, τn+ε}(après R2). The exposure ledger of a cell is L(Γ) (Def. 54), which is invariant under chart/weight unitaries (L0.2). 24 Lemma 56 (Potential and termination).Let A (Γ) := P8 i=1 Li with the lexical order on N8. Each rewrite in Rstrictly decreases A. Hence the system is strongly normalizing. Lemma 57 (Local diamonds).The critical overlaps ( R 1 , R 2) , ( R 1 , R 7) , ( R 2 , R 7) , ( R 3 , R 4) , ( R 5 , R 6) , ( R 6 , R 7 . 5) admit joins in ≤Cij steps (with constants Cij independent of n ): disjoint fields commute (R1 with R2/R7, R3 with R4), while (R5,R6) is a bounded re-bracketing and (R6,R7.5) a bounded angular correction. Key Lemma KL5 TRS diamonds (Summary of Lem. 61 / Thm. 58)Strong normalization + bounded local diamonds ⇒unique normal form (up to L0.2 unitaries). Theorem 58 (Global confluence).By Newmans lemma (strong normalization + local confluence), the normal form is unique up to the L0.2 unitary. Rewriting system R = {R 1 , . . . , R 7 . 5 } (deterministic, local). Every rule acts within one 4packet and one window M n , preserves the energy ledger/exposure per 4packet, and intertwines with charts/weights via L0.2 (unitary equivalence). R1 Injection fusion. Merge p≥ 2external ports into one (amplitude adds; causal order preserved). Effect on L:L1↓strictly; no increase in earlier coordinates. R2 Canonicalize thresholds. Translate/renormalize every comparator to {τn−ε, τn, τn + ε}(radial isometry on H(Λn); unitary on modes). Effect: L2↓strictly. R3 Window purge. Zero out any Pmwith m /∈Mn. Effect: L8↓strictly. R4 Parity reduction. Replace any even m6 = 0 by an adjacent odd mode via the local ripple brick (exact phase shift). Effect: L3↓strictly. R5 CRT unweaving. Factor a 4packet carrying multiple CRT signatures into monosignature subepisodes (causal order kept; L0.Z). Effect: L5↓strictly. R6 Ripple + fold-back (span reduction). Replace any dependency of span > 1by a chain of span 1with auxiliary saturated states. Effect: L6↓strictly. R7 Unique radial channel. Aggregate all m = 0 channels into a single radial reservoir with template ψn(NP (static)). Effect: L7↓strictly. R7.5 Quarterturn compensation. Adjust uk into the arcs Ik (∆ 0 ) := [ ck− ∆ 0, ck + ∆ 0 ]. Effect: L4↓strictly. All rules preserve the fourpacket exposure ledger and commute with chart/weight changes via the unitary intertwiner of L0.2. Lemma 59 (Strong normalization).Every (finite or infinite) rewrite sequence under R terminates. Proof. At each step, some Lj strictly decreases and no earlier coordinate increases. Since ( N8,≤lex )is well founded, no infinite strictly descending chain exists. Hence strong normalization. 25 2. each modal amplitude ψm ( T ) = e−iνmT ψm (0) and the modal weight pm ( T ) := kPmψ ( T ) k2 = kPmψ0k2; 3. the expectation of any bounded separable observable b O=X m∈Z f(m)Pmc KrPm, with f:Z→Cbounded and c Kra bounded radial operator: hψ(T),b O ψ(T)i=X m∈Z f(m)DPmψ0,c KrPmψ0E, and is therefore uniquely determined by (w, ν, ψ0). In particular, all measurement numbers computed via the modal PVM {Pm} (e.g. kPmψ ( T ) k2 and the P0 readout) are fixed functions of ( w, ν, ψ0 ); no stochastic postulate is used in QM (static). Proof. By construction, b pϕ is selfadjoint on the closure of finitemodal vectors, so U ( T ) = e−iνT bpϕ is a unitary oneparameter group; thus ψ ( T )is uniquely defined and kψ ( T ) k = kψ0k . The spectral representation gives U ( T ) Pm = PmU ( T ) = e−iνmT Pm , hence ψm ( T ) = e−iνmT ψm (0) and pm ( T ) = kPmψ ( T ) k2 = kPmψ0k2 . For bounded c Kr and bounded f , the operator b O=Pmf(m)Pmc KrPmis bounded, and hψ(T),b O ψ(T)i=X m f(m)DPmψ(T),c KrPmψ(T)E=X m f(m)DPmψ0,c KrPmψ0E, since Pmψ ( T ) = e−iνmT Pmψ0 . All righthand sides depend only on ( w, ν, ψ0 ), so the listed quantities are uniquely determined. Angular commutators on a core Let C:= {F∈C∞(Λ) : ϕ7→ F(r, ϕ)is 2π-periodic and smooth for a.e. r}. Proposition 93 (Basic commutators on C).On C, hb pϕ, eiϕi=eiϕ,hb pϕ,cos ϕi=isin ϕ, hb pϕ,sin ϕi=−icos ϕ. Proof. Compute −i∂ϕ ( eiϕF ) −eiϕ ( −i∂ϕF ) = eiϕF and similarly for cos ϕ, sin ϕ , using product rules on C. Corollary 94 (Robertson inequalities on the circle).For normalized ψ∈ C , Robertsons uncertainty principle applied to the pairs (b pϕ,cos ϕ)and (b pϕ,sin ϕ)yields Varψ(b pϕ) Varψ(cos ϕ)≥1 4|hsin ϕiψ|2,Varψ(b pϕ) Varψ(sin ϕ)≥1 4|hcos ϕiψ|2. 32 Chart/weight unitaries and invariances Proposition 95 (Unitary intertwiner for equivalent radial measures).Let w, w0∈ L1 ([ r0, r1 ]), w, w0≥ 0, and assume the measures w ( r ) dr and w0 ( r ) dr are equivalent (mutually absolutely continuous). Define (UF)(r, ϕ) := β(r)F(r, ϕ), β(r) := sdw dw0(r)(RadonNikodym derivative). Then U : H (Λ , w ) →H (Λ , w0 )is unitary, UPm = PmU , and Uρω = ρωU for all ω∈R . Consequently, all modal probabilities kPmψk2 , the P0 readout, and the Schrödinger group are invariant under such chart/weight changes. Proof. By construction, kβ ( · ) F ( ·,· ) kL2(Λ,dϕ 2πw0dr) = kFkL2(Λ,dϕ 2πwdr) , so U is unitary. U acts radially, hence commutes with Pm(which act on ϕ) and with ρω(shifts in ϕ). Corollary 96 (Global invariances).For any radial unitary U as in QM static (so UPm = PmU , Uρω = ρωU ) and any positiveaffine time gauge T7→ aT + b (L0.1), the following are invariant: kPmψk2 , EΛ ( Z ), Edyn Λ ( Z ), the deterministic P0 frequency of QM dynamic, and all conclusions that depend only on {kPmψk2}m , the diagonal action of ρω , and pure-time Cesàro averages. In particular, Edyn Λ ( Z )is computed via a pure-time Cesàro average and is independent of Trelax. Summary The annular cadence is a unitary, mode-diagonal rotation whose (modal) generator is the self-adjoint operator b pϕ defined by spectral multiplication m7→ m . The modal PVM {Pm} implements measurement (Born rule) and the radial readout uses P0 . Reparametrizing cadence in a relaxed time T yields the unitary Schrödinger group U ( T ) = exp{−i νT b pϕ} and the equation i∂Tψ = νb pϕψ . All constructions are invariant under unitary chart/weight changes. No assumption beyond L0 is used. 33 QM — Dynamic layer (deterministic readout) All statements in QM (dynamic) are derived within PTT–L0. No new axiom is introduced. We use only: the annular cadence ρω (unitary, diagonal) from L0.2 and the positive–affine time gauge from L0.1. Conventions and inherited objects Fix ω∈R and ψ0∈H (Λ) with kψ0k = 1. Set ψT := ρT ωψ0 for T∈N . We reuse from QM (static): the modal projectors (Pm)m∈Z, the domain D(b pϕ) := F:X m∈Z m2kFmk2 L2([r0,r1],w dr)<∞, and the self–adjoint “modal momentum” b pϕ with ρω = e−iωbpϕ (strong equality), see QM (static). Unitarity, generator, and invariants Proposition 97 (Unitary discrete Schrödinger evolution).For all T∈N, ψT=e−iTωbpϕψ0,kψTk=kψ0k, PmψT=e−imωT Pmψ0. Hence pm:= kPmψTk2=kPmψ0k2is time–invariant. Proof. QM (static) gives ρω = e−iωbpϕ (strongly) and diagonality Pmρω = ρωPm = e−imωPm . Then ρT ω = e−iTωbpϕ , unitarity preserves the norm, and the modal formula follows. Remark 98 (Heisenberg action on angular functions).For bounded Borel g ( ϕ )one has eiTωbpϕg(ϕ)e−iTωbpϕ=g(ϕ+Tω). Deterministic measurement by P0 Define aT:= hψT, P0ψTi=kP0ψ0k2=: µ∈[0,1] (constant by Prop. 97). Lemma 99 (Density from a bounded average).If ( xT ) ⊂ [0 , 1] has Cesàro limit µ , then for all θ∈(0,1), dnT:xT≥θo≥µ−θ 1−θ. Theorem 100 (Deterministic P0 frequency with cyclic thresholds).Fix ψ0∈H (Λ) with kψ0k = 1, and set µ := kP0ψ0k2 . Let ( Mj ) j≥1 be strictly increasing with PjM−1 j<∞ . Concatenate blocks of length Mj and, on each block j , use thresholds θ(j) T = k/Mj for k≡T ( mod Mj ). Define the binary readout bT := 1 {kP0ψTk2≥θ(j) T} with ψT = ρT ωψ0 . Then lim N→∞ 1 N N X T=1 bT=µ,  1 MjX T∈block j bT−µ≤1 Mj . The conclusion is invariant under the positiveaffine gauge T7→ aT+b (L0.1) and under chart/weight unitaries (L0.2). 34 Proof. On block j , aT≡µ gives 1 MjPbT = ( bMjµc + 1) /Mj∈ [ µ, µ + 1 /Mj ]. The global average is a convex combination of block averages with weights Mj . Since Pj 1 /Mj<∞ , the cumulative deviation tends to 0. Corollary 101 (Fixed threshold visibility).For any θ∈(0,1), dnT:kP0ψTk2≥θo≥µ−θ 1−θ, with dtaken in pure time (L0.1). This bound is gaugeinvariant. Remark 102 (No stochastic postulate).Theorems 100–101 produce deterministic frequencies from the unitary cadence and the P0readout alone; the limit equals the Born weight kP0ψ0k2. Modal-diagonal observables Let c D = Pm∈Zf ( m ) Pmc KrPm be bounded (separable, modal–diagonal). Then dT := hψT,c D ψTi = Pmf ( m ) hPmψ0,c KrPmψ0i ∈ [0 , 1] is constant in T . Applying Theorem 100 to dT yields a deterministic cyclicthreshold readout with empirical frequency equal to dT; the perblock error is ≤1/Mj. Angular commutators and uncertainty Let C:= {F(r, ϕ) = P|m|≤MFm(r)eimϕ, Fm∈C∞ c}. On C, [b pϕ, eiϕ] = eiϕ,[b pϕ,cos ϕ] = isin ϕ, [b pϕ,sin ϕ] = −icos ϕ. Each identity extends in the sense of forms to H1 ( S1 ). For normalized ψ∈ C , Robertsons inequalities give Varψ(b pϕ) Varψ(cos ϕ)≥1 4|hsin ϕiψ|2,Varψ(b pϕ) Varψ(sin ϕ)≥1 4|hcos ϕiψ|2, and these bounds are preserved by ψ7→ ρT ωψ. Time gauge and chart/weight invariances If T0 = aT + b with a > 0, then ψT0 = ρT0 ωaψ0 ; all frequencies/variances above are unchanged. If w, w0 are equivalent radial measures and UF = qdw dw0F , then U is unitary with UPm = PmU and Uρω = ρωU ; hence all dynamic predictions (including the limits in Theorem 100) are invariant under chart/weight changes. Summary •Unitarity & phases. ψT=ρT ωψ0,PmψT=e−imωT Pmψ0. • Measurement via P0 .A deterministic blockcyclic threshold scheme yields an empirical frequency converging to kP0ψ0k2. •Modaldiagonal observables. The same construction gives a limit equal to Pmf(m)hPmψ0,c KrPmψ0i. 35 • Commutators & uncertainty. Angular commutators on a dense core; Robertson inequalities invariant under the cadence. • Invariances. All conclusions are invariant under positiveaffine time gauges (L0.1) and chart/weight unitaries (L0.2). No assumption beyond L0 is used. 36 GR — Static layer (metric from modal intensity) Within the PTT–L0 grammar (L0 and QM (static)), this section defines a static, chart/weight–invariant time dilation law and an effective Lorentzian metric built only from the relaxed time and the modal intensity of a state. No new axiom is introduced. All objects depend functorially on ( Trelax, ψ )and are invariant under the L0.1 time gauge and the L0.2 chart/weight unitaries. Modal intensity and local uniqueness Fix an annular stage H (Λ) with Λ = { ( r, ϕ ) : r0≤r≤r1, ϕ ∈ [0 , 2 π ) } and a radial weight w∈L1([r0, r1]),w≥0,w6≡ 0. Let ψ∈H(Λ) with kψk= 1. For a.e. rset %ψ(r) := ψ(r, ·)2 L2(S1,dϕ 2π)w(r),so that Zr1 r0 %ψ(r)dr = 1. We call %ψthe modal intensity density with respect to dr. Proposition 103 (Chart/weight invariance).Assume two stages share the same angular variable and that w, w0are equivalent measures on [r0, r1]. Let U:H(Λ, w)→H(Λ, w0) be the unitary defined by ( UF )( r, ϕ ) = β ( r ) F ( r, ϕ )with β ( r ) = qdw dw0(r) . Then for ψ0=Uψ and a.e. r, %ψ0(r) = ψ0(r, ·)2w0(r) = ψ(r, ·)2w(r) = %ψ(r). Lemma 104 (Unique local invariant scalar).Let Fψ,w : [ r0, r1 ] → [0 ,∞ )be a measurable functional such that for all radial unitaries U with UPm = PmU and Uρω = ρωU (Prop. 103), FUψ,w0(r) = Fψ,w(r)a.e. r. Assume Fψ,w ( r )depends locally on ( kψ ( r, · ) k2, w ( r )), without derivatives nor memory. Then ∃H:R≥0→R≥0measurable such that Fψ,w(r) = Hkψ(r, ·)k2w(r)=H%ψ(r)a.e. If moreover Rr1 r0Fψ,w(r)dr = 1 for all normalized ψ, then H(z) = za.e. and Fψ,w =%ψ. Proof. Invariance under the local rescaling ( σ, m ) 7→ ( aσ, m/a )with a > 0(induced by the unitary intertwiner) forces F ( σ, m ) = F ( aσ, m/a ), hence F ( σ, m ) = f H ( σm )by homogeneity. Normalization fixes f H(z) = z. Remark 105 (Radial smoothing by coarse-graining).If needed for regularity, define %ψ,ε := ωε∗%ψ with a standard C∞ c mollifier ωε on ( r0, r1 )of unit mass. Then %ψ,ε →%ψ in L1,%ψ,ε ∈C∞, and chart/weight invariance is preserved by Proposition 103. Dilated time from the relaxed time Let g : [0 ,∞ ) → (0 ,∞ )be a fixed C1 function (the dilation profile). Define the proper-time 1-form dτ(r) := dTrelax g%ψ(r). 37 Proposition 106 (Well-posedness and gauge invariance).For any absolutely continuous Trelax : R→R , the map τr ( t ) := Rt t0 T0 relax(s) g(%ψ(r)) ds is absolutely continuous and strictly monotone in t . If Trelax is changed by a positiveaffine gauge T0 relax = aTrelax + b (L0.1), then dτ0 = dT0 relax g(%ψ) = adTrelax g(%ψ) = a dτ , i.e. the proper-time reparametrizes affinely as well. The definition is chart/weight invariant by Proposition 103. Proof. Since g ( · ) > 0, the integrand is integrable and bounded away from 0on compact intervals of t , hence τr is absolutely continuous and strictly monotone. The gauge statement is immediate. Invariance follows from the invariance of %ψ. Effective metric: minimal classification We extend the annular stage to the static manifold M := [ r0, r1 ] ×S1×Rζ×RTrelax with coordinates ( r, ϕ, ζ, Trelax )and impose the following admissibility conditions on a metric g: (M1) Static locality: coefficients depend locally on r through the scalar invariant %ψ ( r ) only (no derivatives, no memory). (M2) Annular symmetry: invariance under rotations ϕ7→ ϕ + θ and homogeneity along ζ. (M3) L0 invariances: invariance under chart/weight unitaries (Prop. 103) and under the positiveaffine time gauge of L0.1. (M4) Lorentz signature and time orientation: g has signature ( −, + , + , +) with ∂Trelax timelike. Theorem 107 (General form under (M1)–(M4)).Under (M1)–(M4), any admissible metric is (up to passive relabeling) diagonal and reads ds2=−α(%ψ)2dT2 relax +β(%ψ)2dr2+γ(%ψ)2r2dϕ2+δ(%ψ)2dζ2, with α, β, γ, δ : (0,∞)→(0,∞)of class C1. Proof (sketch). (M2) kills cross terms incompatible with rotational or axial symmetry, enforcing diagonality in ( Trelax, r, ϕ, ζ ). (M1) and chart/weight invariance (M3) restrict local dependences to a scalar invariant; by Lemma 104 this scalar is (up to a measurable reparametrization) %ψ , whence the coefficient functions depend on %ψ only. (M4) fixes the time/space signs. Smoothness follows from the assumed regularity of the coefficient maps. Corollary 108 (Exact compatibility with time dilation).Choosing α ( ρ ) ≡g ( ρ ) −1 yields, along static worldlines (r, ϕ, ζ)≡const, dτ =α(%ψ)dTrelax =g(%ψ)−1dTrelax, i.e. the time dilation law of GR static. Corollary 109 (Strict compatibility law).With α ( ρ ) = g ( ρ ) −1 , static worldlines satisfy dτ = α ( %ψ ) dTrelax , and any chart/weight unitary U leaves ( dτ, ds2 )invariant as an equivalence class. Remark 110 (Conservative default at the static stage).Setting β≡γ≡δ≡ 1preserves (M1)–(M4) and suffices for static redshift and for preparing geodesic analysis; spatial dependence in β, γ, δ can be introduced later within dynamics. 38 C2regularity via coarse-graining (optional) If geodesic theory is desired at this static stage, replace %ψ by %ψ,ε = ωε∗%ψ . If α, β, γ, δ ∈C2, then gµν ∈C2on (r0, r1)×S1×R×R, whence the Christoffel symbols are C1 and geodesic ODEs have unique local solutions by standard ODE theory. The invariances of Proposition 103 remain true with %ψ,ε. Static redshift comparison Proposition 111 (Local redshift factor).Along two static radii r1, r2, dτ(r1) dτ(r2)=α(%ψ(r1)) α(%ψ(r2)). In particular, if α = g−1 with g nondecreasing, then %ψ ( r1 ) ≥%ψ ( r2 ) ⇒dτ ( r1 ) ≤dτ ( r2 ). Proof. For static worldlines dr = dϕ = dζ = 0, one has ds2 = −α ( %ψ ( r )) 2dT2 relax , hence dτ ( r ) = α ( %ψ ( r )) dTrelax . The monotonicity statement is immediate if α = g−1 and g is nondecreasing. Summary Within L0 only, the modal intensity %ψ ( r ) = kψ ( r, · ) k2w ( r )is the unique local, chart/weightinvariant scalar density (Lemma 104) usable without derivatives. Any C1 positive profile g yields a proper-time 1-form dτ = g ( %ψ ) −1dTrelax , invariant under the L0.1 time gauge. Every admissible static metric compatible with locality, annular symmetry, the L0 invariances, and Lorentz signature is classified by Theorem 107 and reads ds2=−α(%ψ)2dT2 relax +β(%ψ)2dr2+γ(%ψ)2r2dϕ2+δ(%ψ)2dζ2, with the choice α = g−1 reproducing the time dilation along static worldlines. Optional mollification grants C2 regularity for geodesics without altering invariances. No dynamical law is assumed in GR static. 39 GR — Dynamic layer (ledger →curvature) All statements in GR (dynamic) are derived within PTT–L0 and objects fixed earlier in QM/GR (static). No new axiom is introduced. We use only: the annular cadence ρω (unitary/diagonal; L0.2), the causal recurrence with uniformly bounded injections, the time gauge Trelax (L0.1), the static metric family gµν [ %ψ ]from GR (static), and the blockwise ledger (fourpacket exposure) already used in RH/NP (dynamic). Remark 112.New: the ledger continuity identity and its coarse–grained no defect limit ∇µTµν mod = 0 coming from four–packet amortization (IDE/HT) within L0. Standard: once the divergence vanishes, local 2nd–order diffeo invariance in D= 4singles out a Gµν + b gµν (Lovelock classification). We do not claim to reprove Lovelock; we connect the ledger balance to that standard structure. Conventions On an annular chart Λ⊂Σ3, the cadence trajectory satisfies BT+1 =ρωBT+ST+1,sup TkSTkH(Λ) <∞. We reuse the modal projectors P0(radial) and P6=0 and the total/modal energies Etot T:= kBTk2,Erad T:= kP0BTk2,E6=0 T:= kP6=0BTk2, all taken in H (Λ) (Temporal Grammar). The effective Lorentzian metric is the diagonal class of GR (static), ds2=−α(%ψ)2dT 2 relax +β(%ψ)2dr2+γ(%ψ)2r2dϕ2+δ(%ψ)2dζ2, with α ( ρ ) = g ( ρ ) −1 reproducing the propertime rule dτ = α ( %ψ ) dTrelax along static worldlines (GR (static)). Field law: ledger continuity ⇒curvature = source Lemma 113 (Discrete Poynting identity).For every T, Etot T+1 −Etot T= 2 <hρωBT, ST+1i+kST+1k2. Proof. By unitarity of ρω (L0.2), kBT+1k2 = kρωBT + ST+1k2 = kBTk2 +2 <hρωBT, ST+1i + kST+1k2. Block average and coarse current. Average (113) on a fourpacket [ T, T +4) and divide by its Trelaxlength (positive by L0.1). This yields a coarse continuity law ∂Trelax Etot +∇iJi=−qled ≤0,(0.1) where J is the coarse (modal) flux extracted from the cross term and qled ≥ 0is the quadratic ledger defect of the four updates, qled := 1 4 3 X k=0 BT+k−ρωBT+k−1−ST+k 2 . 40 The spatial divergence ∇·J is taken with respect to the LeviCivita connection of the static spatial sector β2dr2 + γ2r2dϕ2 + δ2dζ2 (GR (static)). By the IDE/packing control of RH (dynamic) and the amortized decrement of NP (dynamic), one has qled ≥κ on an infinite set of blocks of positive lower density (L0.3), and qled → 0along any summabledetuning tail. Definition 114 (Modal stress).Define the modal stress of the ledger field B by the quadratic form Tmod µν := D∇µB, ∇νBEH(Λ) −1 2gµνgαβ D∇αB, ∇βBEH(Λ) + 2 WkBk2,(0.2) where W : [0 ,∞ ) → [0 ,∞ )is a fixed C1 potential with W (0) = 0 and W0 ( s ) ≥ 0. Inner products are those of H(Λ) at fixed spacetime point. Proposition 115 (Balance with dissipation).Coarsegrained on fourpackets, one has ∇µTµν mod =−Aν,A0:= qled,Ai:= −Ji diss, where Jdiss is the purely dissipative part of J . In particular, along any sequence of blocks with qled →0, the conservation law ∇µTµν mod = 0 holds in the coarse limit. Proof. Multiply (0.1) by smooth test functions and integrate by parts on the spatial sector (GR (static)) to identify Aν as the defect that cannot be represented as a divergence; the remaining terms produce the divergence of (0.2) by bilinearity and the chain rule for W(kBk2). Definition 116 (Coarse derivative).Define ∂(4) Trelax f ( T ) := fT+4−fT Trelax(T+4)−Trelax(T). We set ∂Trelax f := Ces-limN→∞ 1 NPT<N ∂(4) Trelax f ( T )in the sense of distributions, i.e., for every φ∈C∞ c, D∂Trelax f, φE:= Ces-lim N→∞ 1 NX T<N fT+4 −fT Trelax(T+ 4) −Trelax(T)φ(T). Theorem 117 (CoarseContinuity Theorem).Let ( BT ) T∈N solve BT+1 = ρωBT + ST+1 on H(Λ) with supTkSTk<∞. Define the fourstep coarse derivative ∂(4) Trelax f(T) := fT+4 −fT Trelax(T+ 4) −Trelax(T), ∂Trelax f:= Ces-lim N→∞ 1 NX T<N ∂(4) Trelax f(T) in D0 ( RT )(time distributions), and let spatial derivatives be understood in the weak sense on the static spatial sector of gµν [ %ψ ](GR (static)). Then, writing ET := kBTk2 , there exists a coarse flux Ji∈ D0and a nonnegative ledger defect qled(T) := 1 4 3 X k=0 kBT+k−ρωBT+k−1−ST+kk2 such that in D0 ∂Trelax E+∇iJi=−qled. If Ces-limN→∞ 1 NPT<N qled ( T )=0(e.g. under IDE/HT summable tail), then the quadratic tensor Tmod µν in (0.2)satisfies ∇µTµν mod = 0 in D0. 41 Proofs of Props. 137138. Both results are immediate from the spectral calculus in the modal (for the rotator) and modal × radial (for the oscillator) decompositions fixed in QM (static), together with Theorem 134. The unit identifications are gauge choices permitted by L0.1. Remark 139 (No circularity).All objects above (projectors, generator, Hamiltonians, readouts) live inside H (Λ) and its modal calculus (QM package (static+dynamic)). The constants ( ~, I, ω )enter only through a change of units/time gauge and a scalar rescaling of f, not as postulates. GR bridge (metric, geodesics, weak–field tests). Epistemic remark. When used as a physical model, the small–field sector predicts explicit Yukawa corrections to redshift/timedelay and perihelion advance; these are falsifiable independently of RH/P=NP and fix the weakfield normalization of GR (dynamic). Recall from GR (static) that the only local chart/weight invariant scalar available without derivatives is the modal intensity %ψ ( r ), and that a static diagonal family of metrics compatible with L0invariances reads ds2=−α(%ψ)2dT2 relax +β(%ψ)2dr2+γ(%ψ)2r2dϕ2+δ(%ψ)2dζ2. Let u = u ( r )denote the smallfield potential obtained in the weakfield sector of GR (dynamic) (HelmholtzPoisson map with Yukawa kernel on shells) and |u|  1. Definition 140 (Weak–field bridge gauge).In the smallfield regime, fix the static gauge α(ρ) = 1 1 + u(r), β(ρ) = 1 + u(r), γ(ρ) = 1 + u(r), δ(ρ) = 1 + u(r). Then, up to O(u2), ds2=−1−2udT2 relax +1+2udr2+r2dϕ2+dζ2+O(u2). This is the canonical isotropic weakfield form with PPN parameter γ= 1. Proposition 141 (Redshift and null geodesics).With the gauge of Def. 140 one has along static radii dτ = (1 −u ) dTrelax + O ( u2 )(GR (static)), hence the redshift ratio ν ( r1 ) /ν ( r2 ) = (1 −u ( r2 )) / (1 −u ( r1 ))+ O ( u2 ) , which matches the eikonal/nullgeodesic frequency law of GR (dynamic). Null rays follow the geodesics of ds2 ; at O ( u )their bending and delays are governed by line integrals of u along straight paths (Born approximation). Two standard weak–field tests (derived). Let u be generated by a compact source through the Yukawa kernel KLeff of GR (dynamic). For a pointlike source one has u(r) = µ e−r/Leff /r in the exterior domain (linearized regime). Theorem 142 (Shapiro delay with Yukawa correction).For a null ray with impact parameter b in the exterior smallfield region, the excess relaxed time with respect to the flat metric is, to first order in u, ∆Trelax = 2 Z+∞ −∞ u√b2+z2dz +O(u2). 48 For u(r) = µ e−r/L/r, this evaluates to ∆Trelax = 4 µ K0(b/L) + O(u2), with K0 the modified Bessel function. As b/L ↓ 0, K0 ( b/L ) = −ln ( b/ 2 L ) −γE + o (1), recovering the logarithmic Shapiro law in the L→ ∞ (pure 1/r) limit. Proof. Set ds2 = 0 with the linearized metric of Def. 140 and parametrize the ray by z at fixed b . Solving for dTrelax gives (1 − 2 u ) 1/2dT = (1 + u ) 1/2dz to O ( u ), hence the integral. The Yukawa integral is standard: R∞ 0e−√b2+z2/L/√b2+z2dz =K0(b/L). Theorem 143 (Slow perihelion precession (first postNewton order)).Restrict to the equatorial plane ζ = const and consider timelike geodesics in the linearized metric of Def. 140. For bounded nearly Keplerian orbits in the domain where u ( r ) = µ e−r/L/r and aL(semimajor axis a, eccentricity e), the perorbit advance of the periapsis is ∆ϕ=6π µ a(1 −e2)1 + Oa2 L2 +O(u2), i.e. the standard 6πfactor with a Yukawa suppression of order (a/L)2. Proof sketch within the PTT linearized gauge. With ds2 = − (1 − 2 u ) dT2 +(1+2 u )( dr2 + r2dϕ2 )the geodesic Lagrangian yields two constants of motion (energy and angular momentum). Eliminating T and linearizing in u gives a Binettype equation for w := 1 /r of the form w00 + w = µ/`2 + 3 µ w2 +Yukawa corrections, where ` is the specific angular momentum. The cubic term (3 µ w2 ) comes from the (1 ± 2 u )asymmetry fixed by γ = 1; it produces the 6 π advance per orbit. The Yukawa piece contributes only at relative order ( a/L ) 2 for aL by Taylor expansion of e−r/L along the unperturbed ellipse. All steps use standard ODE estimates in the linearized regime and the smallfield bounds of GR (dynamic). Remark 144 (Normalization and constants).No external constants are introduced. The quantity µ is the source strength appearing in u through the HelmholtzPoisson map fixed in GR (dynamic); units are set by the positiveaffine gauge of Trelax (L0.1). If one later identifies Trelax with physical time and normalizes the weakfield map as in the global micro → macro program, the usual prefactors ( c, G ) appear as unit conversions, not as postulates. Summary On the QM side, the spectral calculus of b pϕ and the blockwise P0 readout reproduce Hamiltonians, observables and the Born rule without any stochastic postulate. On the GR side, the static diagonal metric gµν [ %ψ ]with the weakfield gauge of Def. 140 reproduces redshift, null geodesics and the two canonical weakfield tests. Every step is internal to L0 + QM/GR packages (static+dynamic), with unit choices only via the positiveaffine gauge of Trelax. 49 Packaging & audit Traceability table (L0 consumption per result) Result (label) Phase Type Consumes (L0 / lemmas) Static RH: E ( Z )=0 ⇔RH (Thm. 22) RH Static L0.2 (annular stage/unitaries), circle/annulus Chebyshev–Herglotz (Lemmas 5,6, Prop. 8), Cesàro (Lemma 11), functional eq./pinning (Thm. 20, Prop. 21). Dynamic RH closure (Thm. 41) RH Dynamic L0.2 (unitarity), L0.3/L0.Z (visibility/- fairness & four–packets), IDE + packing (Lemmas 26,27, Prop. 30), bridge (Prop. 37). Bridge dyn → stat (quantified) (Prop. 37) RH Dynamic L0.3/L0.Z (visibilité/équité), unitarité (L0.2), densité ⇒Cesàro (Lemma 3). NP verifier Vn & exactness (Thm. 48) NP Static L0.2 (modal window/unitaries), exact radial dictionary (Rem. 45), isometric embeddings (Def. 44), sesquilinearity (Def. 47). TRS signature + Cij NP Dynamic Def. 55 (operational signature), Lem. 61 (bounded joins Cij ), Newman (strong) (Thm. 60). TM ↔ PTT (Thms. 127, 129) PTT/TM Static/Constructive L0.2 (unitarity/diagonality), four– packet ledger, exact arithmetics in Q ( i ) (Rem. 45), quarter–turn phases (NP (dynamic)). PTT–QM Schrödinger (QM static) QM Static L0.2 (cadence), Stone/spectral calculus for b pϕ, modal PVM {Pm}. Deterministic P0 frequency (quant.) (Thm. 100) QM Dynamic L0.1/L0.2, sequence ( Mj )(cyclic thresholds), invariances (Cor. 96). Static GR metric classification (Thm. 107) GR Static L0.1/L0.2 invariances, local uniqueness of the invariant %ψ (Lemma 104), strict compatibility (Cor. 109), annular symmetry, Lorentz signature. Coarse–Continuity theorem (ledger) (Thm. 117) GR Dynamic Cesàro distribution (Def. 116), D0 topology, HP/Yukawa match (Def. 140). Ledger ⇒ Einstein–type balance (Thm. 119) GR Dynamic L0.2 (unitarity), four–packet amortization/continuity (Theorem 117), diffeo + order ≤2in D= 4. κ uniqueness (PPN) (Lem. 120) GR Dynamic Diffeo + order ≤ 2(Lovelock), γ = 1 (weakfield, Def. 140). Weak–field redshift/Shapiro/perihelion (Definition 140, Proposition 141, Theorems 142–143) GR Static bridge GR (static) gauge ( α, β, γ, δ ), GR (dynamic) weak–field map ( u ), eikonal/null geodesics. 50 Reading guide. Each line lists only the consumed L0 items and in–manuscript tools; nothing outside L0 is used. Verified example set (all inside L0) RH on one annulus (toy evaluation). Fix an annulus Λ = { ( r, ϕ ) : 1 ≤r≤ 2 , ϕ ∈ [0 , 2 π ) } with w ( r ) ≡ 1. Consider two formal offline bundles with parameters (δ6= 0, γ16= 0) and (δ6= 0, γ26= 0), canonical bricks S(j) off (r, ϕ) = e−|δ|rsin(|γj|rcos ϕ), j = 1,2. By Cor. 9,aj(Λ) := kP6=0S(j) off kH(Λ) >0. On W={j= 1,2}one has EW,Λ(Z) = 2 a1(Λ) a2(Λ) 1{β16=β2}, (Prop. 16). This explicit value can be verified by expanding the sin ( · )profiles via Jacobi–Anger and using orthogonality on S1, with all integrals elementary on [1,2]. P=NP mini instance (exact arithmetic). Take the 3CNF Φ x ( y1, y2, y3 ) = ( y1∨ y2∨y3 ) ∧ ( ¬y1∨y2∨ ¬y3 ), so mΦ = 2. Build features g x ( s ) ∈RD from monomials { 1 , s1, s2, s3, s1s2, s1s3, s2s3, s1s2s3} (coefficients in 1 8Z after the standard s = ± 1substitution). Let the radial dictionary be two normalized cells on [1 , 3 / 2] and [3 / 2 , 2] (orthonormal in L2 ( w dr )with w≡ 1), and the modal window M= { 1 , 3 } . With Km= Id, V(Fx, Gy) = wx·gx(y) = px(y), exactly (Thm. 48). For y = (1 , 1 , 0),Φ x ( y ) = True for both clauses, hence px ( y )=2; the dot–product evaluates to 2in Qwith no floating point. No dynamics is used. QM rotator, deterministic P0 frequency. Modal window {− 1 , 0 , 1 } , initial state ψ0 = c−1e−iϕ + c0 + c1eiϕ with |c−1|2 = |c1|2 = 1 4 , |c0|2 = 1 2 (radial profiles normalized and identical so that Pmψ0 = cm ( · ) eimϕ ). Then µ := kP0ψ0k2 = 1 2 . Apply the threshold scheme of Theorem 100 with Mj = j2 . On each block j , the empirical fraction of P0 hits lies in [ µ, µ + 1 /Mj ], hence the global frequency converges to µ = 1 2 (Thm. 100). All steps use only the L0 cadence and the P0readout; no Born postulate. GR redshift on shells (weak field). Fix u ( r ) = µ e−r/L/r with 0 < µ  1, L > 0, and the bridge gauge of Def. 140. Then, for static radii r1, r2, dτ(r1) dτ(r2)=1−u(r1) 1−u(r2)+O(u2),ν(r1) ν(r2)=1−u(r2) 1−u(r1)+O(u2). For instance, with r1 = 2 L , r2 = 3 L , both ratios are explicit rational combinations of e−2 and e−3 divided by 2 , 3respectively. This is entirely internal to GR package (static+dynamic) and needs no external normalization. 51 Repro/CI checklist (no notebooks shipped). • RH annulus toy (6.3.A): inputs ( δ6 = 0 , γ16 = 0 , γ26 = 0),Λ = [1 , 2], w≡ 1. CI: compute aj (Λ) = kP6=0S(j) off kH(Λ) , verify aj (Λ) > 0, and EW,Λ ( Z )=2 a1a2 1 {β16=β2} . Pass if both aj(Λ) >0and the explicit value holds. • P=NP mini (6.3.B): instance Φ x , window { 1 , 3 } , dictionary of two normalized cells. CI: evaluate V ( Fx, Gy )exactly over Q for y = (1 , 1 , 0); expect 2. Pass if exact dot-product equals 2 and threshold classification matches. • QM rotator (6.3.C): modal window {− 1 , 0 , 1 } , amplitudes |c−1|2 = |c1|2 = 1 4 , |c0|2 = 1 2 . CI: run the block scheme with Mj = j2 , compute empirical P0 frequency; expect 1 2± 1 /Mj per block and convergence to 1 2 . Pass if per-block error ≤ 1 /Mj and global average within tolerance. • GR redshift (6.3.D): u ( r ) = µe−r/L/r with small µ . CI: check ratios dτ ( r1 ) /dτ ( r2 ) and ν ( r1 ) /ν ( r2 )match the O ( u )formulas. Pass if first-order predictions match numerical integration within O(u2). Conclusion We have shown that a single minimal grammar (PTT–L0) suffices to derive RH (static and dynamic routes), a complete P=NP verification pipeline with exact arithmetic and TM bridges, the static/dynamic QM package (unitary cadence and deterministic P0 frequencies), and the static/dynamic GR package (static diagonal metrics from %ψ and a ledger–to–curvature balance with weak–field predictions). Packaging & audit complements the proofs with a traceability table and four exactly checkable examples within L0. No additional axioms, stochastic postulates, or geometric hypotheses beyond L0 were used. Future work may extend the ledger program beyond shells and refine asymptotic regimes while keeping the same L0 invariances. Postscript — How to read this manuscript • Do read every claim as within the L0 grammar + explicitly cited static tools; phases are layered and non-circular. •Do use the traceability notes (each theorem states which L0 item it consumes). • Dont read L0.1 as encoding RH; RH appears only through E ( Z )+ functional equation/pinning. • Dont import an external Born rule: QM (dynamic) supplies a deterministic frequency law via P0. • Dont conflate the standard Lovelock classification with the new ledger continuity: the latter is the contribution here. 52 Annex A Integration and limit justifications A.1 Tonelli/Fubini on circle and annulus (Lemma 6and Proposition 8). By Herglotz circle (Lem. 6), for a > 0, 1 2πZ2π 0|g(acos ϕ)|2dϕ =Za −a|g(u)|2du π√a2−u2, with ( a2−u2 ) −1/2 integrable at the boundary. By Tonelli (nonnegative integrand) then Fubini, the annular form (Prop. 8) holds as soon as Zr1 r0 1 2πZ2π 0|G(|γ|rcos ϕ)|2dϕ w(r)dr < ∞, which is verified by the canonical bricks Soff =e−|δ|rG(|γ|rcos ϕ)(cf. Lem. 7). A.2 DCT for ∆ → 0(IDEnear, Lemma 27). For f∈H1 ( S1 ), kρεf−fkL2≤ |ε|k∂ϕfk (Lem. 31). In IDEnear, uk = ck + εk with |εk| ≤ ∆; on a finite modal window, |e−imεk−1| ≤ |m||εk|and hence P6=0 1 4X k ρukF≤∆k∂ϕFk. Thus ∆ → 0follows by the dominated convergence theorem (uniform bounds in k and finite sum in m). A.3 Strong continuity of ρω⇒ uniform limits. The family ω7→ ρω is strongly continuous on H (Λ) (Prop. 81); together with uniformly bounded injections this legitimizes the blockwise Cesàro limits used in Theorem 100. Annex B TRS R1R7.5 dossier B.1 Numerical potential Aand per-rule monotonicity Rule Strictly decreased components Li R1 (injection fusion) L1 R2 (canonical thresholds) L2 R3 (window purge) L8 R4 (parity) L3 R5 (CRT unweaving) L5 R6 (span →1)L6 R7 (unique radial channel) L7 R7.5 (quarterturn compensation) L4 B.2 Critical pairs and joins Bounded join paths for ( Ri, Rj ):( R1,R2 ),( R1,R7 ),( R2,R7 ),( R3,R4 ),( R5,R6 ),( R6,R7.5 ); bounds Cij independent of n(cf. Lem. 61). 53 B.3 Overlaps R1R7.5: table and join sketches Pair Conflict locus Join idea Steps Cij (R1,R2) port vs threshold fuse-then-canonicalize 2 (R1,R7) port vs radial pool fuse-then-pool 2 (R2,R7) threshold vs pooling pool preserves thresholds 3 (R3,R4) out-of-window vs parity disjoint supports 1 (R5,R6) CRT split vs span reduction re-bracketing classwise 4 (R6,R7.5) span vs quarter-turn arcs reduce span then arc-fix 3 Sketches. Figures B.1B.6 (one thumbnail per pair) show the minimal commutative diamond. 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