Joint preparation resources for finite collective interaction¶
Prospective resource declaration¶
This analytical admission tests a different preparation of the same complete sine law. It does not modify the frozen sources, responses or verdicts in the outer acquisition, sequential observation, external interaction or relative-origin comparison. The original preparation remains valid under its original premises.
| Item | Fixed choice and scope |
|---|---|
| Question | Can wider original preparation allowances, a shorter recovery dwell and larger readout errors jointly retain the established finite response distinction and all marked identities? |
| Primitive constituents | Six independent class-one C9 preparations, each with nominal signed form m*(j-4), m=(2046/9)*(355/113)^2, and nominal phase zero; independent form and phase errors have exact separate zero sums |
| Source family | All twelve per-coordinate allowances lie in [0,1e-6]; retain all 96 relative error directions. These allowances are not Euclidean recovery radii |
| Complete isolated law | x_tau=(-L*x+gamma*S(theta))/4, theta_tau=gamma*L*x/4, gamma=1/(1023*pi), unit capacities, smooth full-circle sine, no continuous input or intervening event |
| Clock | tau=(1023/1024)*t in both rows; the quarter-factor isolated law equals the established half-factor law at half the elapsed tau |
| Formation and recovery | Quarter-factor formation time 200 and uninterrupted quarter-factor recovery dwell D=8*10^10; do not reset reached coordinates |
| Recovery proof | Class-one radius r=1/12, curvature cos(2*pi/9+sqrt(2)/12), shared modified-energy kernels, and exactly 2^-80 as the exponential upper bound; require kappa*D/2>=80 |
| Handoff radius | Require each actual constituent's form and target-phase Euclidean norms to be strictly below q=1e-8 at handoff |
| Outer preparation | Same five-child C9-by-C5 support, supplied child-common forms 4092*(c-2), outer acquisition duration 200, paired control with these common forms zero, and unchanged-state addition of all 45 outer edges |
| Observation word | Same marked order, twenty windows of length 1e-5, fifteen phase events with amplitude epsilon=1/10, and 25 raw mean-form readings on one continuing source |
| Observation accuracy | Every sequential raw reading error is bounded by eta_o in [0,1e-14]; these errors can be correlated |
| External constituent | Synchronize its preparation to finish at contact; supply zero common form and a rigid phase origin beta. The fixed signed contrast uses beta=pi/2; the capability comparison retains all origins |
| External support and law | Same 54-node C9-by-W6 support, 144 unit edges, full quarter-factor rows, both conserved means and all fine coordinates; no later input or support change |
| External accuracy and horizon | Two readings per contact, delay h=1e-3, each raw error bounded by eta_c in [0,1e-9]; retain four errors in the paired contrast |
| Decisions | Signed contrast above 1e-7; every outer-one capability below 1e-8; each outer-zero source has an origin with capability above 3e-7. The witness origin is not an operational alignment selector |
| Work and storage | Outer attachment and outer initial storage below 2e9; sequential absolute event work below 4 each and cumulative work below 40; new external attachment below 100 and joined present storage below 200. Account for six primitive preparations and earlier payments separately |
| Identity | All five host internal windings and nine outer windings survive the observation word; all six internal and nine old outer windings survive the subsequent finite contact. No indefinite outer-one maintenance is presumed |
| Assessment | One corner assessment with all twelve source allowances and both reading errors at their maxima; exact rational admission and existing outward dyadic128 arithmetic. Prove monotonic inclusion before using that corner for the whole box |
| Work budget and failure | No trajectory, source producer, parameter sweep, dwell adjustment, precision change or second candidate. If any premise or strict margin fails, report the specified box unavailable without generalizing to impossibility |
Commit this declaration before evaluating the new corner. Existing proof mechanisms may be rebuilt from their primitive premises; frozen producers and evaluated source archives are not replayed. The scope is conditional structural capability, with no laboratory-unit bridge or physical identification.
1. A coupled sufficient domain¶
Write x=EPI, and let \(\alpha=2\pi/9\), \(\gamma=1/(1023\pi)\),
\(V_n=n[1-\cos(2\pi/n)]\). On every fixed support in this history,
The support changes and phase commands have their actual complete storage jumps. They are supplied events, not continuous pressure flows or inferred occurrence laws. Common phase and form translations are isolated symmetries; relative child origins and common forms become consumed state after contact. The NFR ontology retains membership, constituent and boundary state separately from the observed winding label.
The following conditions define the resource domain. Each must be admitted on the actual history: isolated acquisition and recovery, full-state outer acquisition and capture, inclusion in the observation class, sequential inference and work, then external response, work and finite identity. A successful last inequality cannot replace an unavailable earlier premise.
Isolated acquisition and class-one recovery¶
Use the fresh source estimates (20)--(23) of the own-law acquisition proof at deformation zero, half-factor time 100 and the admitted primitive errors \(\rho_x,\rho_\theta\). Their form and target-phase radii are denoted \(b_x,b_\theta\). The initial class-one cost bound is
Require the original source budget, initially zero winding, actual entry into the radius-\(r=1/12\) ball and its strict storage trap. No source error is multiplied by a forcing constant or silently replaced by a target. Inside this class-one ball every principal edge gap differs from \(\alpha\) by at most \(\sqrt2r\). Consequently
This is the actual nonlinear Hessian on the retained trap. The earlier two-class proof used a smaller common curvature to include class two; that frozen result is unchanged. Its existence does not require using its smaller curvature for this class-one-only preparation.
In the half-factor clock put \(A=L/2\), \(g_0=1/5\), \(\Lambda=2\), \(\eta=\gamma^2\), \(\epsilon_L=g_0/4\), \(\xi=A^{-1/2}(\theta-\Theta_1)\), \(v=\gamma A^{1/2}x\). On the relative leaf the modified energy is
With outward lower and upper bounds indicated by subscripts, define
The shared modified-energy proof gives \(\mathcal V'\le-\kappa\mathcal V\) and \(\mathcal V\ge\|v\|^2/4+a_-\|\xi\|^2\). A quarter-factor dwell D therefore admits the handoff radius q whenever
Indeed \(e^{-\kappa D/2}\le e^{-80}<2^{-80}\). Every bound concerns the uninterrupted reached state; q is a derived recovery enclosure, not a primitive accuracy requirement. All six separate sources must pass (2)--(5) before any joined-source conclusion is admitted.
The two return bounds in (5) retain more information than the common q enclosure. The preparation-aware initialization proof uses their separate form and phase allowances to bound constituent variances and carry a collective forecast from attachment. It preserves this source, dwell and decay budget; it neither infers a microstate nor replays the assessor.
Actual outer acquisition and observation-class inclusion¶
The exact same carrier, support and clock as the outer-acquisition owner retain their fresh nominal endpoint intervals and radii \(P_c,f_x,f_\theta\). For the actual five-child deviations its global heat argument gives
Require \(\alpha-2Z>0\), \(\alpha+2Z<\pi/2\) throughout the transit, and the nominal endpoint outer gaps enlarged by \(2Z\) strictly inside \((0,\pi/2)\). Rebuild the nominal storage excess \(U_{\rm nom}\) from the same carrier estimate. The actual excess obeys
Require \(U<\min\{b_9-V_9,b_5-V_5\}\), where \(b_n=1+(n-1)[1-\cos(3\pi/(2(n-1)))]\), and require the control bound \(40q^2<\min\{b_9-V_9,1\}\). These are the actual full-product barriers, not the primitive C5 capture verdict.
To enter the unchanged observation class set \(X=1/100\), \(\zeta=10^{-6}\), \(A_o=7/5\). Require
The factor two in (8) is essential: subtracting each child's actual mean phase error produces the observation class's zero-child-mean residual. Its coordinate magnitude is bounded by \(2Z\), while the mean corrections enlarge adjacent origin gaps by at most \(2Z\). Initial childwise means are not asserted conserved after the outer edges are added.
One observation word and the following contact¶
Keep \(T_o=1/5000\), \(h_o=1/100000\), \(\epsilon=1/10\), and put \(B_o=X+\gamma_+T_o\), \(D_o=4\gamma_+B_oT_o\). Use the complete background and backaction estimates (5), (13)--(15) of the sequential-observation proof. With the same quantities \(M_3,E_C,E_S\) and response factors \(f_C,f_S\), only the admitted reading errors change:
Rebuild the outward initial-angle width d from these errors and require \(5d<2\pi\), including the fixed interval-rounding allowance. Retain the positive cosine chart, finite-comparison domain and all three identity margins. Every event costs at most 3.6 in absolute storage and the cumulative absolute expenditure is at most 36. These are actual-state bounds and do not cancel inverse commands or spend previous loss. Inference remains conditional on an admitted source; arbitrary readings are not authenticated.
At external contact let \(m_o=D_o/2\), \(e=Z+m_o+q\), \(h=1/1000\). Retain the larger initial form bound when necessary:
For each actual source its contact moment Q has the same source-derived rectangle as the relative-origin proof: sum \(\cos(P_c+[-e,e])\) for its real coordinate and \(-\sin(P_c+[-e,e])\) for its imaginary coordinate. The control uses \(P_c=\{0\}\). These enclosures retain actual recovery errors and the entire preceding observation history. They supply amplitude bounds \(A_{w,-}\le|Q_w|\le A_{w,+}\).
The exact donor mean obeys \(R_D'=-3R_D/2-\gamma\mathcal A/4\). The whole-law remainder (10) therefore gives
Require \(U_1<10^{-8}\), \(L_0>3\cdot10^{-7}\) and the fixed-origin signed contrast lower bound above \(10^{-7}\), retaining its separate \(2E+4\eta_c\) error. The first capability bound holds at every origin; the second has a source-dependent witness preceding the choice of reading errors. Neither supplies an alignment selector.
Work, present storage and finite identity¶
The outer attachment has exact nominal cost \(W_0=1507001760\) and \(W_0\le W_{\rm outer}\le W_0+20q^2\). Require both this upper bound and \(5V_9+W_0+40q^2\) below \(2\cdot10^9\). At the new contact put
Require \(W_+<100\) and both present-storage bounds \(6V_9+9V_5+U+36+4q^2+W_+<200\) and \(6V_9+36+44q^2+W_+<200\). Check every strict margin in (21) of the external-interaction owner using this q and \(D_o,D_j\). The six internal and nine old outer windings then persist through the whole finite contact, independently of the supplied donor origin. The historical account remains \(\sum_{s=1}^6B_s\), outer attachment, the fifteen absolute observation payments and the new contact payment. Present storage and dissipated storage cannot replace that account.
All these conditions are simultaneous. Acquisition radii, recovery V0, initial cost and transported errors increase with primitive allowances; phase and moment enclosures are nested as errors increase. The source-class and identity margins consequently decrease, observation widths increase with \(\eta_o\), and contact error increases with \(\eta_c\). Curvature, decay rate, dwell and q are fixed. Thus the declared maximal corner suffices for the full box without a parameter scan. This is an inclusion argument, not a monotonicity assertion about individual trajectories. It does not extend the single-corner verdict to different dwells, handoff radii or decay powers. Those require their own simultaneous checks.
2. Necessary restrictions that sufficient bounds cannot remove¶
Finite-time compression of an original phase family¶
Choose two admitted original sources with identical forms and opposite phase errors \(\pm r_0v\), where v is a mean-zero unit vector and \(0<r_0\le\rho_\theta\). Their initial phase separation is \(2r_0\). Eliminate their form difference in the quarter law, writing \(z=\Delta\theta\):
On C9, \(\|I-e^{-Lt/4}\|\le1\) and \(\|\Delta S\|\le4\|z\|\). Gronwall gives \(\|z(t)-z(0)\|\le2r_0(e^{\gamma^2t}-1)\), hence \(\|z(t)\|\ge2r_0(2-e^{\gamma^2t})\). If both reached phases lie within q of the same lifted target, then, when \(q<r_0\),
This global argument covers the nonacute formation transit. It does not prove that the earlier supplied dwell was necessary, and gives no positive restriction from this pair when \(q\ge r_0\). For the whole declared box one may take \(r_0=\rho_{\theta,\max}\). An exact rational lower witness can use \(\log(1+u)\ge2u/(2+u)\), \(u=1-q/r_0>0\), without introducing a new logarithm evaluator.
Reading error and the minimum contact horizon¶
The two contact readings permit a difference error of either sign and magnitude \(2\eta_c\). An upper capability threshold \(U_{\rm th}\) that must hold for every such error therefore necessarily requires \(2\eta_c<U_{\rm th}\). A witness response above a positive lower threshold \(L_{\rm th}\) for every error requires its true magnitude to exceed \(L_{\rm th}+2\eta_c\). Since the exact donor mean starts at zero, its damping equation and \(|\mathcal A|\le5\) give
For the declared capability these thresholds are \(10^{-8}\) and \(3\cdot10^{-7}\). These are necessary observation restrictions; they do not replace the sufficient nonlinear, work or identity checks. Replacing \(\gamma\) by \(\gamma_+\) in the signal margin gives an upper bound: a nonpositive upper margin proves an obstruction, whereas a positive one alone supplies no capability certificate.
Winding acquisition, duration and supplied contact work¶
For a fixed-support unforced interval of duration \(T_a\), Cauchy--Schwarz and (1) give, with \(\mathcal L=\int_0^{T_a}\|Lx\|^2/4\,dt\),
Initially each raw same-slot outer gap has magnitude at most \(2q\). At an outer-winding-one endpoint, principal gaps sum to \(2\pi\) but raw lifted gaps telescope to zero. Some raw gap in each slot is therefore below \(-\pi\), with the endpoint away from winding cuts. The corresponding two coordinates have squared total displacement at least \((\pi-2q)^2/2\). The nine slots are disjoint coordinate sets, so \(\|\Delta\theta\|^2\ge9(\pi-2q)^2/2\), for \(2q<\pi\).
If the endpoint retains the declared acute internal and outer cycles, Jensen's bounds give \(H_{\rm final}\ge5V_9+9V_5\); the initial disconnected source has storage at most \(5V_9+20q^2\). Thus
This necessary duration--work relation is weaker than the exact fixed-ramp cost \(W_0\), but demonstrates that the preparation cost is not entirely a pessimistic recovery estimate. It applies to the stated support, law, clock and endpoint identity, not arbitrary TNFR acquisition protocols.
3. Implementation and interpretation boundary¶
The private analytical owner is
_organized_joint_resources.py.
It must admit primitive allowances before rebuilding sources and reuse the
shared acquisition, recovery, heat-transport and observation kernels.
Incoming cached verdicts cannot certify the new preparation. Original
fixed-protocol consumers retain their original defaults and evidence.
The corner decision below concerns only the declared joint resource domain. A failed sufficient check would leave it unavailable rather than impossible. Neither outcome resolves indefinite outer-one maintenance, selects this law uniquely, supplies autonomous preparation or identifies a material property. Work and time here have structural units. Independent preparation, clock and observation bridges remain necessary for a fundamental-physics test.
4. Retained corner decision¶
The declaration and complete source were committed at 0ca8630d before
the single assessment. The attempt record
retains exact parameters, clock, support, arithmetic and the Git source
inventory. The first result
is joint_resource_box_admitted: all six source recoveries and every
coupled acquisition, observation, work, storage, response and finite-identity
condition pass. No trajectory, source producer, parameter sweep or second
candidate was evaluated. The monotonic inclusion argument in Section 1
therefore admits the full declared box, including all 96 relative source
directions at its positive corner.
The following decimals are outward summaries of retained exact rational bounds, not new numerical runs or replacements for the record:
| Quantity | Certified bound at the maximum corner |
|---|---|
Recovery exponent kappa*D/2 |
Above 101.08, exceeding the declared power 80 |
| Actual recovered form / target-phase Euclidean norms | Below 2.32e-13 / 4.57e-16, respectively; both below the handoff allowance 1e-8 |
| Sequential cycle-sum width | Below 1.477 radians, strictly below 2*pi |
| Outer-one response capability at every relative origin | Below 7.730e-9, against threshold 1e-8 |
| Outer-zero per-source witness capability | Above 3.866e-7, against threshold 3e-7 |
| Fixed-origin signed response contrast | Above 3.897e-7, against threshold 1e-7 |
| New contact work | Below 90.003 |
| Present joined storage, outer one / outer zero | Below 169.740 / 138.636, both below 200 |
| Six original preparation storage bounds combined | Below 1,087,375,716 |
| Initial-storage plus subsequent absolute-work account | Below 2,594,377,602; this is separate from present storage and the outer-attachment budget |
Relative to the earlier fixed preparation, this admits primitive component allowances 10,000 times wider, a supplied recovery dwell 250 times shorter, and reading errors 10 times larger in each of the two observation stages, simultaneously. The endpoint allowance is a recovery conclusion and transport premise; it is not the initial preparation or sensor precision. This establishes a new conditional capability region without rewriting any earlier evidence.
The necessary restrictions remain material. The exact supplied-ramp outer
attachment work is at least 1,507,001,760. The separate winding/time lower
bound exceeds 9,174,760 at acquisition duration 200; it is weaker than the
fixed-ramp bound and must not be added to it. For this original phase family,
the rational compression witness forces total preparation time above
6,839,825 quarter-clock units. It does not establish that the admitted dwell
8e10 is optimal. Contact noise and duration retain their separate necessary
restrictions from (15).
The result removes an unnecessarily severe inherited resource choice while retaining substantial supplied preparation and work. It does not establish experimental attainability, autonomous assembly, indefinite outer-one maintenance or physical identification. The next task remains solely in the execution plan.