Geometry and gain from two successive phase inputs¶
Question and conditional claim¶
The calibrated local inverse constrains geometry after a positive sensor-gain interval is supplied. Its reserved software result also shows that widening that interval can substantially widen the necessary angle bound. That control is not an exact response collision.
Here two declared phase inputs and three scalar readings constrain an original source angle and one unknown held positive gain together. Both inputs act on one uninterrupted full-state trajectory. The second response retains the form and phase evolution caused by the first input, and the two recorded increments share the middle reading's error. The result is a conditional outer enclosure, not exact joint identification, state reconstruction, independent physical calibration or a new reserved experiment.
The ontology permits collective observations but retains their independently justified measurement premises. The strategy also distinguishes constitutive information from a fitted response. The complete sine law, clock, phase-input calibration, positive gain prior and held affine observation model below remain supplied premises.
F1: one full source, two events and three readings¶
Reuse the realizable affine source family
on donor nodes 0,...,8, receiver nodes 9,...,17, their unit C9 cycles,
and contacts (0,9) and (1,10). Ports 0,1,9,10 have degree three;
the other nodes have degree two. Let \(L\) be the full Laplacian,
\(M\) its degree matrix, \(A=M^{-1}L\), and \(P_M\) the
projection removing one common degree-weighted mean. All norms below
are \(\|z\|_M=(z^TMz)^{1/2}\), unless stated otherwise.
The nominal donor bulk and receiver short angles lie in supplied closed subintervals of
Their other angles are the donor short arc \(4\pi-8b\), receiver bulk arc \((2\pi-c)/8\), and contact gaps \(\pm(4\pi-8b-c)/2\). The shared source owner reconstructs all eighteen lifts \(\theta^0(b,c)\), subtracts their full weighted mean, and retains periods \((2,1,0)\). No equilibrium equation is imposed.
At the original pre-probe time admit
The common form and lifted phase means are arbitrary; every node may carry an error. The receiver coordinate and full residual vectors remain nuisance state. Source geometry requires the same strict acute guard \(\mu_0-Y>0\), where \(\mu_0\) is a lower bound for the nominal minimum acute margin over the declared rectangle. The algebraic response bounds below are global, but the offered inverse certificate retains this guard to interpret the original geometric statistic on its declared branch.
Set \(q=e_4-e_5\). Supply two cumulative phase amplitudes and equal elapsed windows satisfying
The actual events are
Thus the second jump is the difference of the two supplied cumulative amplitudes. It is not another jump of size \(a_2\), and neither event resets the state. Here \(\|q\|_M=2\), \(\|q\|_{M^{-1}}=1\), and \(\mathbf1^TMq=0\).
Read the scalar \(q^Tx\) at \(0,h,2h\). Form continuity makes the middle reading the same on either side of the second phase event. One unknown gain \(G\in[G_-,G_+]\), with \(0<G_-\le G_+\), and one arbitrary constant offset \(O\) apply to all three readings:
The supplied primitive recorded intervals are \(r_j\in[c_j-\tau_j,c_j+\tau_j]\), with exact midpoint \(c_j\) and nonnegative half-width \(\tau_j\). No statistical independence is assumed. The gain interval is prior information, not a calibration result manufactured from these readings. An unknown drifting gain, unbounded offset drift or uncertain input amplitude is outside the present theorem.
Form retains the source model's normalized structural units. Phases and input amplitudes are in radians, and \(h\) uses its fast structural clock \(\tau\). Gain has recorded units per local form-readout unit; \(O,\delta,\tau_j\) and the readings use recorded units. No laboratory unit or sensor identification is supplied by these declarations.
F2: complete dynamics and one retained history¶
Keep unit held capacities, beta one, \(e=1023/1024\), \(w=1/1024\), the fast structural clock \(\tau=et\), and
On both continuous windows the complete rows remain
There is no other input or support change. Both common means are conserved, including across (4). No phase coordinate or intermediate form is reconstructed from a reduced response or replaced by its initial value.
Define the continuous dynamical phase displacement
It is continuous at the second event. On the two respective windows, the actual phase is exactly
This common original reference is an algebraic description of the actual history, not a repeated preparation. In particular, the second window starts with the evolved \(x(h)\) and accumulated \(d(h)\).
F3: a uniform finite error for both full-flow increments¶
The degree operator is self-adjoint, \(0\preceq A\preceq2I\), so its heat semigroup is contractive. The sine map is globally Lipschitz with constant two. On every nominal affine source, its interior rows cancel and only the four port rows may be nonzero. Consequently \(\|f(\theta^0)\|_M\le\sqrt{12}<F=4\). These facts and their exact graph realization are shared with the one-input proof. The present bound does not assume that any later state has port-supported forcing or reuse the earlier fourth-order locality cancellation.
Use \(g=1/3069>\gamma\), and define
Its denominator is positive for \(T\le1\). If \(Q_T\) is the actual supremum of \(\|P_Mx\|_M\) on \([0,T]\), Duhamel's formula, (7) and the global Lipschitz bound give
Hence throughout both windows
This includes the form created by the first input and the natural noncritical port currents. No acute trajectory, stability or source-reset premise enters (8)-(9).
The local nominal phase-input identity is
Let
Integrating the actual form row in each window gives
where
Indeed, \(|q^TAx|\le2Q\), while the difference between the actual phase in (7) and its nominal phase-input reference has norm at most \(Y+2gTQ\). Multiplying its sine-map error by \(\gamma\) and integrating for \(h\) proves (11). Both remainders concern the same trajectory. Bounding them separately by \(E\) is a conservative relaxation, not permission to choose separate intermediate states.
A sufficient whole-window acute margin is
It is optional additional evidence. Its failure neither invalidates (8)-(11) nor prevents the inverse when the original source guard passes. No event-work allowance, maintenance, winding retention after the inputs or eventual recovery is certified by this inverse.
F4: rank, shared observation errors and outer projections¶
Set \(\alpha_j=a_j/2\) and transform the two unknowns into
The leading response matrix is
It has rank two when \(a_1<a_2\). Equal cumulative amplitudes give a zero second jump and a rank-deficient leading matrix. This is an obstruction to the present inversion method, not a theorem that the full nonlinear two-time response contains no additional information. An unresolved positive factor is numerical unavailability, not an exact rank-one proof. Certify positivity of both scales and \(d=\sin(\alpha_2-\alpha_1)\) separately. Their positive product can have an outward lower endpoint of zero on a fixed absolute grid; that diagnostic interval does not invalidate the factor-based rank proof. Cancel the redundant scale factors before numerical inversion:
Evaluate the two divisions successively rather than materializing the products \(K_jd\). This also avoids losing a resolvable denominator through product rounding. The output retains the full determinant interval as a diagnostic. A nonzero determinant alone gives no useful resolution guarantee: conditioning must be compared with the full errors.
Form exact recorded midpoint differences
The arbitrary common offset cancels before outward coefficient arithmetic. Let \(w=(w_1,w_2)\) be either exact row of \(C^{-1}\). The corresponding transformed coordinate has center \(-w\cdot m\) and error radius at most
To see this, write each midpoint discrepancy from its true recorded reading together with the additive reading error as one bounded nuisance of size \(\tau_j+\delta\). The two increments use the incidence matrix
After applying \(-w\), their three coefficients are \((w_1,w_2-w_1,-w_2)\). The last three terms of (14) are precisely the support bound for that scalar combination of the three reading errors. The first term retains the two complete-flow remainders and the same unknown gain's upper bound.
For exact recorded points with per-reading error \(\delta\), the two increment errors obey
These describe their shared-middle-error hexagon, not two independently chosen reading pairs. Both inverse rows in (13) have opposite-sign entries. Thus, for equal reading-error allowances, (14)'s scalar noise radius equals the bound obtained from separate \(2\delta\) increment allowances. Correctly retaining the shared reading is not a claim of narrower marginal intervals in this particular sign pattern. The final coordinate rectangle still discards correlations between its two coordinates.
Outward arithmetic encloses the inverse rows, centers and radii rather than replacing them with point estimates. Intersect the resulting \((U,V)\) rectangle with the necessary prior coordinate ranges
The prior branch lies strictly in the first quadrant. Empty strict intersections exclude the combined premises; touching endpoints remain admitted. For a retained rectangle, project outward through
and intersect each projection with its corresponding prior. Do not normalize \((U,V)\) to a unit vector before checking the unknown gain. Unresolved argument or norm arithmetic cannot certify a result.
Every compatible full state and held gain yields values of \(b,G\) within these outer projections. Their Cartesian product need not be jointly realizable, and nonempty marginals do not prove that any full trajectory yields the supplied readings. An incompatible result excludes the joint assumptions without identifying which preparation, observation or model premise failed.
The uniquely defined original geometric statistic is the pre-probe donor long-arc mean
Since both ports have degree three, \(|v_1-v_0|\le\sqrt{2/3}\,Y\le Y\). Expand the nominal angle projection by \([-Y/8,Y/8]\), using the original phase radius, and do not clip it back to the nominal prior. Neither instantaneous interior event changes the port phases, but their continuous evolution can change this statistic. The report concerns its original value, not its value at the middle or final reading.
A rational conditioning example without a response campaign¶
The following constants demonstrate an informative error regime; they are not a selected hidden source, recorded observation or frozen experiment:
Retain the full priors (1). Their original acute guard passes. The example does not assert that the phase-input states remain acute.
Elementary sine bounds and \(\gamma>1/3216\) give
so \(K_1>h/4824\) and \(K_2>h/2010\). If \(S_j\) is the sum of absolute entries of inverse row \(j\), then
The first bound uses \(\sin\alpha_2<3/8\) and \(\sin\alpha_1<1/8\); the second uses both cosines at most one. With \(\tau_j+\delta\le2\delta\), (14) bounds the Euclidean norm of the two coordinate radii by
The last inequality is an exact rational substitution into (8) and (11), so it evaluates no selected reading or trajectory.
For data produced by an admitted state, the true \(z\) belongs to the ideal rectangle and has norm \(G\ge1\). The rectangle has diameter at most \(2R\); every point has norm at least \(1-2R\). It remains in the first quadrant because \(\cos(3/2)>1/16\), \(\sin(11/8)>9/10\), and \(2R<1/1536\). The norm is one-Lipschitz, while the argument gradient has norm at most \(1/(1-2R)\). Consequently
Prior intersections cannot enlarge these projections. This establishes feasibility in exact arithmetic. A numerical report still has to retain outward coefficient and projection errors and pass its own availability checks. No synthetic chosen reading establishes independent calibration or a finite reserved-response verdict.
Implementation and evidence boundaries¶
The implementation reuses the original affine geometry, shared exact/represented scalar admission and outward trigonometric, argument and square-root arithmetic. It consumes three primitive recorded intervals, not a hidden source, incoming forward report or cached verdict. The contract owns the input and availability rules, and the tests exercise the independent algebra, scalar boundaries, rank and correlated-error behavior. These controls establish implementation behavior, not external observations.
The earlier two-time constitutive comparison also retains one state and one constant unknown through successive readings, but uses a different conservative doubled-C5 model and a constitutive coefficient. Its coefficients and verdicts do not transfer here. The hidden-state observation owner similarly distinguishes rank, necessary compatibility and joint existence; it does not supply this protocol's observation or gain model.
Known clock and law scales are essential to interpreting (17) as a gain bound. If either is left unknown, the leading response can combine it with gain; this theorem does not identify those scales separately. The receiver, full nodal residuals and common offset remain unobserved. Acquisition, event funding, subsequent maintenance and physical identification retain their own obligations.
The execution plan owns any later reserved two-input experiment. It would need one complete source trajectory, both declared events, a held sensor, three readings, information exclusion and frozen numerical/error/resolution budgets before evaluating its response. This conditional theorem starts no such campaign and changes none of the earlier calibrated theorem, prospective protocol, source archives or retained responses.
Prospective reserved software evaluation¶
This separately admitted protocol applies the preceding conditional theorem; it supplies no physical observation or independent sensor calibration. The preceding theorem and conditioning example remain unchanged. The source, observation and stopping conditions below are fixed before any of these reserved responses are evaluated. There is no evaluated result in this prospective section.
Full source family and hidden cases¶
Use the same eighteen-node support, degree metric, continuous phase lifts, complete law and fast structural clock as (1)--(6). The public inputs are
The gain interval is a supplied prior, not the output of a calibration experiment. Law, clock and both input amplitudes are held exactly as declared.
The three deterministic cases use the following primitives, withheld from the inverse worker:
| Case index \(k\) | Donor \(b\) | Receiver \(c\) | Held gain \(G\) |
|---|---|---|---|
| 0 | \(89/64\) | \(17/24\) | \(9/8\) |
| 1 | \(91/64\) | \(19/24\) | \(11/8\) |
| 2 | \(95/64\) | \(23/24\) | \(15/8\) |
For node \(i=0,\ldots,17\), define exact raw residuals
Use common means \(m_x=(k+2)/11\), \(m_\theta=-(k+2)/13\) and the full source (2). The nominal eighteen-phase vector is reconstructed from the affine source family, including its single full degree-weighted centering. No equilibrium, ideal target or acquired-source flag is substituted.
Every centered form and phase residual is nonzero in each case. Their exact weighted sums vanish; their squared degree norms are at most \(2^{-80}\). These preparation checks use only rational primitives, not an evaluated response. For example, the centered raw coordinate ranges give the conservative squared bounds \(40\cdot76^2/2^{108}\) for form and \(40\cdot23^2/2^{108}\) for phase, both strictly below \(2^{-80}\). The hidden nominal and actual source angles are distinct:
The recorded source arrays and their checks must retain this difference.
Two full flows and one sensor history¶
At global time zero supply the phase jump \(q/4\), evolve the full law for \(h\), supply the second jump \(q/2\), and evolve for another \(h\). Form is continuous across both events. Use the existing full-state readout producer with two successive order-four calls. The second call's initial form and phase boxes must be exactly the respective eighteen-coordinate slices of the first call's complete validated endpoint. Its phase increment is \(a_2-a_1=1/2\), not \(a_2\).
Each call uses one fixed source-box Picard/Taylor step: the shared smooth-flow kernel, its sixteen-iteration Picard budget, dyadic-128 outward arithmetic, source-box coefficients through order four, and a fifth-order whole-tube remainder. Require strict Picard inclusion and the complete duration in both calls. The autonomous solver's local time starts at zero in each call; the retained protocol records their global starts as \(0\) and \(h\). This is a local clock origin for the same law, not a restart of the physical state.
The first enclosure contains every trajectory from its source box. At the second event, adding the fixed phase vector to that endpoint box therefore contains every actual post-event state. The second validated step encloses their continuations. Cartesian wrapping may discard correlation and widen bounds; it does not justify shrinking the carried state or resetting any coordinate. Retain all thirty-six initial, tube and endpoint rows of both calls, together with their derivative and inclusion evidence. The global smooth-law domain does not certify whole-window acuteness or identity.
Within case \(k\), one gain from the table and offset \(O=(2k+3)/11\) remain held for all three scalar readings. Their fixed additive errors are
Each lies within the public per-reading bound \(\delta\). Gains and offsets may differ between cases; neither may change within a case. The three true-readout enclosures are the first call's baseline, its endpoint readout, and the second call's endpoint readout. Applying the held affine sensor and each fixed error yields three recorded intervals. The middle reading is constructed once and shared by both increments; it equals the second call's pre-event form baseline before sensor errors. Do not draw or synthesize a separate second middle reading.
Require each recorded interval half-width to be at most \(\delta\). This numerical interval width is additional to the declared additive sensor-error bound. It encloses one software-generated scalar reading; it does not represent a second independent observation or sensor noise.
Public inverse packet and declared controls¶
A fresh inverse worker receives only the nine primitive inputs accepted by
infer_sine_two_pulse_geometry_gain: bulk_angle_bounds,
receiver_short_angle_bounds, form_radius, phase_radius,
phase_increments, probe_duration, recorded_reading_bounds,
readout_error_bound and readout_gain_bounds. Only the three recorded
intervals come from the producing path. The inverse receives no hidden
\(b,c,G,O\), nodal state, error realization, producer report, endpoint
box, actual angle or prior verdict. Retain the exact public JSON packets,
worker code and allowlist check. This checks information exclusion in the
declared software path; it is not cryptographic blindness or independent
execution authentication.
For each unchanged triple of recorded intervals, also evaluate:
- A false donor prior \([11/8,353/256]\), retaining the other primary
inputs; require
incompatible. - A false gain prior \([31/16,2]\), retaining the other primary inputs;
require
incompatible. - Equal cumulative amplitudes \((1/4,1/4)\), retaining the other primary
inputs; require
unavailablewith the rank-deficiency reason. This is an inversion-method control on the same packet, not a generated equal-input trajectory or a proof that the full nonlinear law has no information.
Finally compare both observed increments with the complete phase-blind alternative \(x'=-Ax,\ \theta'=\gamma Ax\), supplied with the same original source and phase events. Its form evolution ignores those events, and heat contraction preserves \(\|P_Mx\|_M\le X\) at all times. Since \(\|q\|_{M^{-1}}=1\) and \(\|A\|_M\le2\), each elapsed window, including the second, satisfies
Require each retained recorded-increment enclosure to be disjoint from this symmetric interval. This is a bound from the same original source, with no new heat trajectory or reset, and excludes only the declared alternative.
Joint stopping rule and retention¶
For every case require admitted source priors and full residual norms;
both complete thirty-six-coordinate horizons and strict Picard margins;
exact first-endpoint/second-source association and both declared events;
one held sensor with one middle reading; each recorded half-width within
budget; and the public-only input boundary. The primary inverse must be
available, rank certified and bounded_candidate; its nominal angle,
original actual angle and gain marginals must contain the respective
hidden primitives. Require both the actual-angle and gain widths to be
strictly below \(1/1024\). All three inverse controls and both
phase-blind increment exclusions must pass as declared. Optional
whole-window acuteness is not part of this stopping rule.
The source base is immutable revision
d86753763fcdd65eddf52c87dd3c5c74348129ce. Before the first response, retain
this prospective proof, the exact protocol, source inventory, producing
and inverse-worker code, numerical budgets and stopping predicates under
docs/assets/sine_formed_classes/two-pulse-inference-v1. The source archive
records the base and any actual overlays; unrecorded working-tree changes
cannot supply the evaluated code. Keep the original prior protocols,
archives and responses unchanged.
Retain the first attempt, including any unavailable response or export failure. Do not change the cases, priors, errors, horizon, arithmetic or thresholds to obtain a passing record. Save all hidden primitive arrays, both complete forward reports, the three readings, exact public packets, all inverse reports and per-condition outcomes, followed by a manifest. The saved result tests this finite software inference chain under its supplied premises. It does not identify a physical sensor, derive the preparation or support, prove joint realization of every retained angle/gain pair, account for event work, or certify subsequent maintenance.
Retained first evaluation and claim boundary¶
The saved response
has status certified_reserved_joint_inference. All sixty-six fixed
conditions passed in the first reserved attempt across the three cases.
Both complete-flow windows, their full-state handoff and phase-only events,
the one-middle-reading association and public-only inverse requests passed
without a retry, changed budget, source reset or earlier producer rerun.
The primary marginals contain every declared nominal angle, its actual original value \(B_{\mathrm{initial}}=b-5/2^{57}\), and the respective held gain \(9/8,11/8,15/8\). The following decimal endpoints are rounded outward for display; widths are approximations to the saved exact rational differences, rather than differences of those rounded display endpoints.
| Case | Original actual-angle enclosure (radians) | Angle width | Gain enclosure | Gain width |
|---|---|---|---|---|
| 1 | [1.390535815643, 1.390714000311] |
1.781846672682e-4 |
[1.124785196410, 1.125213380441] |
4.281840298225e-4 |
| 2 | [1.421806401222, 1.421943436179] |
1.370349557123e-4 |
[1.374785991809, 1.375212314611] |
4.263228007694e-4 |
| 3 | [1.484331260848, 1.484418628125] |
8.736727612697e-5 |
[1.874788260847, 1.875209614522] |
4.213536743246e-4 |
Every exact actual-angle and gain width is strictly below \(1/1024\).
The largest recorded-reading half-width is below \(3.09\times10^{-38}\),
well inside the separate \(2^{-60}\) numerical budget. Both false-prior
requests are incompatible in every case; each equal-amplitude request
abstains as unavailable for the declared leading-rank limitation. Both
recorded increments exclude the phase-blind interval (27), whose exact
radius here is \(2^{-58}=4\delta\). None of these controls substitutes
a second preparation or an independently generated middle reading.
The optional whole_window_acute_certified flag is false in all three
primary reports. The global full-law response bounds still support the
inverse, which refers to the original pre-probe geometric statistic.
The absent acute certificate is not a failed stopping condition and supplies
no post-input identity, recovery or maintenance claim.
The protocol,
source archive
and manifest
retain the complete source recipe, producing code, first responses and
fixed verdicts. The archived prospective proof has 26,821 bytes and SHA-256
2cae8b0d9257edf9ce7b78b9ca6d3feb0c11c331341ae517636a22da3ec5b285;
its content is preserved above, independently of routine checkout newline
conversion. The source archive SHA-256 is
5abff67d487287ad3d4949d970f5c6a448bd4e62dec8a212e817008ede031e38.
The retained evidence audit
checks the saved primitives and numerical evidence without regenerating
these reserved responses or calling the inverse again.
This is finite evidence that the declared software observation chain constrains original geometry and a held gain simultaneously, conditional on the known complete law, clock, exact inputs, source family and observation model. It is neither independent physical calibration nor exact joint identifiability: retained marginal pairs need not all be realizable, and receiver geometry and fine state remain unresolved. The result does not justify an unknown clock or law scale, autonomous support or preparation, event funding or physical constituent identification. Any continuation requires a separate admission in the sole execution plan.