Two-port geometry from a calibrated local response¶
Question and scope¶
The interior phase-dipole result
is a forward discrimination certificate for two acquired geometries. It does
not supply an independent observation from which to infer either geometry.
Moreover, the fixed critical (2,1,0) problem already has a unique target;
assuming that problem and returning its root would add no inverse information.
This result instead admits a continuous family of complete, generally nonstationary states on the joined support. Under a known law, input, clock and positive observation-gain interval, one supplied finite local increment gives a necessary outer interval for a donor angle. The receiver geometry remains a nuisance coordinate. The proof retains its nonzero port currents, all form and phase errors, and the finite response of the full law. It is a conditional theorem and computational enclosure, not a new frozen response assessment, independently calibrated experiment or physical inference.
A realizable transient state family¶
Use donor nodes 0,...,8, receiver nodes 9,...,17, their unit C9 cycles,
and unit contacts (0,9) and (1,10). Ports 0,1,9,10 have degree three;
the remaining nodes have degree two. Let \(L\) be the full Laplacian,
\(M\) its degree matrix, \(A=M^{-1}L\), and
\(\|z\|_M=(z^TMz)^{1/2}\). The total degree mass is forty.
Let \(P_M\) remove the single common degree-weighted mean.
Admit any closed parameter intervals within the compact radian rectangle
Define the donor short angle \(s\), receiver bulk angle \(d\), and contact offset \(\zeta\) by
Before subtracting their one full degree-weighted mean, take the lifts
Thus the donor short arc is \(s\), its eight long arcs are \(b\),
the receiver short arc is \(c\), and its eight long arcs are \(d\).
The increasing cycle orientations use closing-edge lift adjustments
\(4\pi\) and \(2\pi\), respectively. The contact gaps are
\(\zeta,-\zeta\), and the mixed cycle 0->9->10->1->0 has period zero.
All these nominal edges are strictly acute throughout (1). For example,
\(s\le4\pi-11<\pi/2\) follows from \(\pi<22/7\), while
\(s\ge4\pi-12>0\); the other bounds follow directly from (1)-(2).
No critical balance equation is imposed.
At the pre-probe time admit
The real common form and lifted phase means are arbitrary. Their removal is a proof coordinate choice, not a reset. Every node can carry a residual; no reflection or componentwise mean constraint is imposed. This family is supplied independently of the response. The previous capture and warmup certificates do not admit every point of this different transient family.
Complete law, event and observation¶
Retain unit held capacities, beta one, \(e=1023/1024\), \(w=1/1024\), and the fast structural clock \(\tau=et\). Define
The full rows after the supplied event are
where primes mean \(\tau\) derivatives. The graph, capacities and law remain fixed; no other forcing or event acts during the window. These are the same complete rows as the two-port compatibility model, without its equilibrium premise. Both common means are conserved.
At \(\tau=0\), supply
and read at \(0<h\le1\). The event is an external instantaneous phase input, not a finite pressure pulse or autonomously selected event. The primitive model increment is
Here \(\|q\|_M=2\), \(\|q\|_{M^{-1}}=1\), and the phase event preserves the weighted mean. Common form cancels in (7) and does not affect (5).
Each of the two scalar readings measures \(Gq^Tx\) plus a common constant offset and its own additive error of magnitude at most \(\delta\). One fixed gain \(G\in[G_-,G_+]\), with \(G_->0\), applies to both readings. The offset cancels; the two errors need not. If the recorded increment belongs to the supplied interval \(\mathcal M=[m_-,m_+]\), then
Interval division retains signed form. An unbounded offset drift or different unknown gains between readings is outside this model. The law, clock, input and calibration intervals are independent premises, not quantities fitted to this same increment.
Port locality and a finite full-law error bound¶
The degree operator is self-adjoint and nonnegative, with \(\|A\|_M\le2\). Thus \(e^{-\tau A}\) is a contraction. The complete sine map is globally Lipschitz with constant two in this norm: its derivative is a signed-cosine weighted Laplacian whose absolute quadratic form is bounded by that of \(A\).
Write \(f_0=f(\theta^0(b,c))\). The two equal and opposite bulk
sine currents cancel at every nonport node. Consequently \(f_0\)
is supported only on 0,1,9,10, and
Both readout nodes are at graph distance at least three from those ports. The diagonal entries of \(A\) do not shorten that distance, so
This removes only three Taylor moments, not all subsequent port influence. Using the contraction integral remainder,
Therefore the nominal noncritical background contribution to (7) is at most \(\gamma Fh^4/3\). It cannot be dropped by treating the family as a set of equilibria or by equating local adjacency with the complete finite response.
Use the rational upper bound \(g=1/3069>\gamma\), and define
The denominator is positive for \(h\le1\). Duhamel's formula and the global Lipschitz bound give, throughout \(0\le r\le h\),
Indeed, if \(Q_h\) is the actual supremum form norm, the initial sine norm is at most \(F+4a+2Y\), and phase motion contributes at most \(4g^2h^2Q_h\) to its form integral. Solving that scalar inequality proves (11)-(12). No acute assumption is needed for these global bounds.
At the nominal phase immediately after (6), the three affected interior edges have gaps \(b+a,b-2a,b+a\). Their local sine rows give exactly
The unknown receiver coordinate does not enter this leading term; its full finite influence is still covered by (9)-(12).
Let
Then every admitted full trajectory obeys
The five terms of \(E\) have separate sources. They bound free form relaxation, the port background, transport of the initial pulse current, the initial phase residual, and phase motion during the response. For the pulse term, its sine-map increment has norm at most \(4a\); \(\|(e^{-rA}-I)z\|_M\le2r\|z\|_M\) and integration give \(4gah^2\). The other terms follow from (9)-(12) and the same Duhamel formula. This calculation retains both rows of (5).
Branch retention and the conditional inverse¶
Let \(\mu_0\) be a lower bound for the nominal minimum acute edge margin over the declared parameter rectangle, including both contacts. Each edge difference has degree-dual norm at most one. The source guard \(\mu_0-Y>0\) admits the pre-probe acute branches and their geometric interpretation. A separate sufficient whole-window acute guard is
When it passes, the supplied event and subsequent window preserve the declared cycle branches and periods. It does not prove all-time recovery, acquisition or an allowance on the event's supplied work. The bound (15) and the inverse below remain valid when this additional guard is unresolved: their full-flow estimate is global. Whole-window acuteness is optional evidence, not a necessary inverse premise. The pre-probe source guard suffices for the actual geometric statistic defined below.
The shifted donor branch \(b-a/2\) lies strictly within \((0,\pi)\) under (1) and (6). Hence cosine is strictly decreasing, and its derivative has no zero on any admitted closed branch. Let \([b_-,b_+]\) be the declared donor interval, and form
If this intersection is empty, no state satisfying all the family, law, input, clock, gain and error premises produces the supplied interval. Touching endpoints must not be discarded as a strict exclusion. If \(\mathcal C=[C_-,C_+]\) is nonempty, every compatible nominal donor coordinate lies in
Outward interval arithmetic and monotone bisection can enlarge (17)-(18); unresolved numerical separation must retain the candidate or abstain. No equality with an interval midpoint or floating residual is a proof of exclusion. More precision refines an enclosure; it supplies no additional observation.
With arbitrary residuals, \(b\) is a nominal family coordinate and need not label the fine state uniquely. A unique actual geometric scalar at the pre-probe time is the mean of its eight oriented donor long arcs:
The two port degrees are three, so \(|v_1-v_0|\le\sqrt{2/3}\,Y\le Y\). Thus
Do not clip (20) back to the nominal prior: the actual fine-state mean can lie outside that prior while still satisfying (4). This inference is about the pre-probe geometry. The immediate interior phase jump leaves the two port phases and this statistic unchanged, but its value can change during the subsequent nonstationary response. Neither \(c\) nor the full residual vectors are reconstructed.
Meaning of the enclosure and remaining obligations¶
The implementation reuses the shared affine geometry and outward trigonometric arithmetic; it performs no equilibrium solve, trajectory integration or old producer replay. Its contract owns scalar admission, numerical budgets and availability. The tests exercise domains, the local algebra, bounds and inverse behavior. Synthetic response intervals used by those tests are software controls, not independent observations.
A bounded candidate is a necessary outer constraint. It does not prove that any retained value, or even any complete state, realizes the supplied reading. Conversely, exclusion rejects the combined premises without identifying which premise failed. Overlap of outer response intervals is not a constructed observational collision. This differs from the finite orientation/law collision, which proves an actual equal finite reading under its own complete laws. The hidden-star inverse uses known incidence and multiple instantaneous rates; its rank conditions and coefficients do not transfer to this single finite increment.
The leading coefficient alone combines gain, clock scale and angle. If these calibration premises are removed, this theorem gives no joint identification. Equal leading coefficients would not by themselves prove equal finite full-law responses. A law-selection or physical interpretation would need independent calibration, admissible alternatives and a justified measurement bridge, following the research strategy.
The execution plan owns whether a later reserved inference experiment is admitted. Such an experiment must fix its unknown-state family, independent calibration, recorded observation, horizon and numerical budget before assessing its reserved response. This conditional inverse neither starts that campaign nor modifies any earlier frozen prediction, source archive or response.
Prospective reserved software inference protocol¶
The theorem above remains unchanged. This separate experiment evaluates three fixed complete-law responses and passes only their declared public observations to the inverse. The full source recipe, calibration, numerical budget, stopping criteria and producing source must be archived before any reserved response is generated. This is a software information-exclusion test with a separately specified forward calculation. It supplies neither external data nor physical calibration or cryptographic blindness.
Hidden sources and public preparation information¶
Use (1)-(7), the same eighteen-node support and full fast-clock law, and the following public inputs for every case:
| Input | Fixed value |
|---|---|
| Nominal donor prior | \([11/8,3/2]\) radian |
| Receiver nuisance prior | \([2/3,1]\) radian |
| Full degree-norm allowances | \(X=Y=2^{-40}\) |
| Supplied phase increment | \(a=2^{-10}\) radian |
| Fast elapsed time | \(h=2^{-16}\) |
| Error of each scalar response reading | \(\delta=2^{-60}\) in recorded units |
| Inverse refinement budget | Forty-eight bisections per boundary |
| Required actual-angle width | Strictly less than \(1/1024\) radian |
For case index \(k=0,1,2\), fix the hidden parameter pairs respectively as
These are three concrete transient sources, not critical targets or a claim that three evaluations cover every point of the theorem's family. For node index \(i=0,\ldots,17\), define raw residuals
Apply the full degree-mean projection to each vector, giving \(u^{(k)}=P_Mr^{(k)}\) and \(v^{(k)}=P_Ms^{(k)}\). The actual source is
All eighteen residual coordinates of each channel are nonzero. Before division by \(2^{52}\), the form mean is \((k+1)183/20\), and the three phase means are \(48/5,197/20,423/40\); none equals a corresponding raw integer. Orthogonal mean projection cannot increase the degree norm. The common conservative bound
admits both public allowances without discarding any node. Common means remain in the evolved state. Neither an equilibrium reset nor an old capture or warmup report supplies these sources.
Independent software calibration and recorded readings¶
One hidden held gain \(G=3/2\) and offset \(O=5/7\) apply to all calibration and response readings. Use two known scalar references \(z_-=-1,z_+=1\), with calibration error bound \(\delta_c=2^{-14}\), and the frozen realizations
The public calibration derives only
Its width is \(2^{-13}\), its lower endpoint exceeds one, and it contains the true held gain. The constant offset cancels. Neither the hidden gain nor offset is passed to the inverse, and none of the three reserved responses is used to determine (25). Exact reference values, error allowances and a held affine sensor are supplied software premises.
For case \(k\), set
Both magnitudes are at most \(\delta\). The recorded scalar readings are \(Gq^Tx^-_k+O+e_k^{\mathrm{before}}\) and \(Gq^Tx_k(h)+O+e_k^{\mathrm{after}}\). Retain both and their increment enclosure. The produced increment interval must have numerical width at most \(2^{-60}\), separately from the per-reading sensor error allowance of the same size. The inverse retains both reading errors again as unknown bounded errors; it receives no realized error correction.
A separate full-state response certificate¶
The response producer evolves all thirty-six coordinates after the phase event using the shared complete sine rows. It consumes primitive source boxes, not the inverse formula, its leading coefficient or its response-error bound. Fix one step of duration \(h\), Taylor order four, at most sixteen Picard tube iterations, and the shared outward dyadic128 rational/trigonometric backend. There is no folded symmetry, equilibrium search, adaptive step, changed precision or response-dependent retry.
The shared direct source-box kernel retains a strict Picard tube \(Z\) for the entire step. Let \(J_j(B)\) be its interval extension of the normalized order-\(j\) solution derivative on a state box \(B\), evaluated by the shared formal jet recurrence. Every solution starting in the post-event source box \(B_0\) satisfies
The order-five derivative enclosure is evaluated on the entire certified tube. All lower coefficients are evaluated on the complete initial box, so no center substitution discards initial uncertainty. The complete sine field is globally smooth; strict Picard inclusion and its positive margin provide the whole-time enclosure independently of an acute-chart claim.
The increment in (27) omits the common order-zero coordinate before interval arithmetic. Applying the raw form readout \(q^T\) to that increment retains its correlation with the same initial state, instead of subtracting unrelated initial and endpoint boxes. All thirty-six endpoint coordinates, source boxes, tube, coefficients, derivative remainders, domain margins and achieved horizon remain in the report. A failed tube or remainder calculation is unavailable evidence, not an endpoint to substitute with a midpoint or a lower-order approximation.
Public inverse inputs and prospective controls¶
Construct each inverse request from an explicit allowlist: the two public angle priors, \(X,Y,a,h\), the produced recorded increment interval, \(\delta\), gain bounds and forty-eight refinements. No hidden angle, state coordinate, common mean, exact gain, offset, realized error, endpoint coordinate or producer diagnostic enters that request. The posterior truth audit is separate from the inverse call. This boundary must be checked in the actual evaluator, not inferred from a report label.
For each case make three requests with the same recorded increment:
- Calibrated inference: use \(\mathcal G\) from (25) and both original angle priors. Require a bounded candidate containing the hidden nominal \(b_k\) and the actual pre-probe long-arc mean (19), with actual-angle width strictly below \(1/1024\).
- Reduced calibration information: replace only the gain bounds by \([1,2]\). Require a bounded candidate containing the calibrated actual-angle interval and having width strictly above \(1/64\). This tests the effect of discarding gain information on a necessary outer bound. It does not construct two exact trajectories with equal readings or prove that every retained angle is realizable.
- False donor prior: use (25) and replace only the nominal donor prior by \([11/8,89/64]\). Require an incompatible result. This tests exclusion of the combined false-prior premises; it does not infer which premise would have failed for an unknown external source.
The calibration and noise budgets provide a prospective feasibility check without evaluating any reserved response. For these public inputs, \(K\ge K_-=ah/1206\), the shifted branch has sine greater than \(9/10\), and exact rational comparisons give \(E/K_-<2^{-15}\), \(Q<2^{-25}\). When the produced recorded interval has width at most \(2^{-60}\), its calibrated width divided by \(K_-\) is less than \(2^{-21}+2^{-12}\). Equations (17)-(20) then bound the ideal primary actual-angle width, including the slack of forty-eight resolved bisections at each boundary, by
The required strict signs and outward arithmetic availability must still be certified; an unresolved bisection cannot claim this numerical slack. This analytical budget check does not replace any of the reserved stopping criteria or manufacture the observation supplied to the inverse.
Also retain the complete phase-blind alternative
on the same support, source, clock and phase event. Its form flow is independent of phase. Contraction and \(\|A\|_M\le2\) give
Require each retained recorded increment interval to be disjoint from this closed symmetric interval. This includes its raw form background, held gain and both reading errors, with no ideal reset or extra numerical trajectory. Excluding this specified alternative does not select the sine law among every possible phase-dependent law.
Stopping rule and retained evidence¶
Predict that all three fixed cases pass source and calibration admission, the complete-horizon numerical certificate and numerical-width limit, public-only input separation, both primary truth-containment checks, the strict primary width threshold, the reduced-calibration containment and width control, the false-prior exclusion, and the phase-blind exclusion. Record every criterion separately; success is their conjunction across all cases. Preserve the first verdict even if an arithmetic, numerical, export or discrimination condition fails. A correction needs separately retained evidence and cannot rewrite the evaluated prediction.
The frozen protocol owns the exact source arrays, hidden and public records, calibration readings, fixed laws and units, event, backend and budgets. Archive it with the complete producing source and prospective proof before the first reserved calculation. Retain the resulting full response reports, three inverse requests and outputs per case, posterior audits, stopping verdicts and a manifest of source/configuration hashes. Hashes provide integrity checks, not provenance authentication. Keep all earlier frozen experiments and their original producers unchanged. The execution plan alone owns the active status and later result closure.
Retained result of the reserved software inference experiment¶
The first frozen evaluation returned certified_reserved_inference with
all forty-three stopping conditions passing. No response was retried and
no source, calibration, horizon, numerical budget or threshold was changed.
The saved record
retains the three complete response certificates, public packets, inverse
outputs and posterior audits. Its
protocol,
source archive
and manifest
retain the preparation and producing implementation. The archived prospective
proof has 23,649 bytes, with SHA-256
488c51e9adc0ecbbf2007cef4c113073e08c61222607cc7a72c8c7a5c630c50f.
Its content is preserved above; routine Git checkout conversions may change
the working copy's line-ending bytes without changing that archived artifact.
For every fixed source, (22) gives \(v_1-v_0=7/2^{52}\). Its exact pre-probe long-arc mean is therefore
The primary nominal interval contains \(b_k\), and the primary actual-angle interval contains \(B_k\), in all three cases. The displayed interval endpoints below are rounded outward; widths are approximate displays of the saved exact rational differences. They do not replace the exact tests.
| Case and hidden nominal \(b_k\) | Primary actual-angle interval, radian | Primary width | Width with gain bounds \([1,2]\) |
|---|---|---|---|
| 1: \(45/32\) | \([1.4062191227,1.4062800901]\) | \(6.0967316150\times10^{-5}\) | \(0.07278097884\) |
| 2: \(23/16\) | \([1.4374705738,1.4375287176]\) | \(5.8143756231\times10^{-5}\) | \(0.09610233068\) |
| 3: \(47/32\) | \([1.4687219897,1.4687773795]\) | \(5.5389766204\times10^{-5}\) | \(0.07734653861\) |
Each primary actual-angle width is strictly below \(1/1024\). Each
broad-gain actual interval contains the corresponding primary interval and
has width strictly above \(1/64\). All three fixed false-prior requests
return incompatible. Each retained recorded increment also lies outside
the complete phase-blind alternative's closed raw-response interval.
These controls used the same recorded response as their respective primary
inference; they did not trigger another full-flow evaluation.
The calibration produced the positive interval \([196597/131072,196613/131072]\) from its two separate reference readings. Each full thirty-six-coordinate certificate reached the fixed endpoint in its one fourth-order step. All source and reading errors were admitted, and each recorded increment's numerical width met its separate \(2^{-60}\) limit. Fresh worker processes consumed only the serialized public inference packets; hidden source and sensor realizations remained in the separate source generation and posterior audit. The retained packet hashes and frozen worker code make that information boundary reviewable.
This is finite software evidence for conditional partial inference on the three stated transient sources. The broad-gain widths quantify this outer inverse's loss of resolution; they are not exact observational collisions. Excluding the supplied false prior and phase-blind law does not establish unique geometry under arbitrary models or a uniquely selected constitutive law. Every interval still has the theorem's necessary-constraint meaning; the receiver state and complete nodal state are not reconstructed. Source preparation, complete law, structural clock and held affine observation model remain supplied. No physical calibration, external measurement bridge, source acquisition or post-probe maintenance result follows from this record. Content hashes support consistency and source recovery, not independent timing, execution authentication or cryptographic blindness.